Spike solutions for nonlinear Schrödinger equations in 2D with vanishing potentials

Abstract

We consider $\varepsilon$-perturbed nonlinear Schrödinger equations of the form

$$\begin{aligned} - \varepsilon ^2\Delta u + V(x)u = Q(x)f(u) \quad \text {in } \mathbb {R}^2, \end{aligned}$$

where V and Q behave like $(1+|x|)^{-\alpha }$ with $\alpha \in (0,2)$ and $(1+|x|)^{-\beta }$ with $\beta \in (\alpha , + \infty )$, respectively. When f has subcritical exponential growth—by means of a weighted Trudinger–Moser-type inequality and the mountain pass theorem in weighted Sobolev spaces—we prove the existence of nontrivial mountain pass solutions, for any $\varepsilon >0$, and in the semi-classical limit, these solutions concentrate at a global minimum point of $\mathcal A=V/Q$. Our existence result holds also when f has critical growth, for any $\varepsilon >0$.

1 Introduction and main results

We deal with $\varepsilon$-perturbed nonlinear Schrödinger equations of the form

$$\begin{aligned} - \varepsilon ^2\Delta u + V(x)u = Q(x)f(u) \quad \text { in } \mathbb R^2 \end{aligned}$$

($\mathcal P_{\varepsilon }$)

where $\varepsilon >0$, and we mainly consider nonlinear terms $f=f(s)$ with subcritical exponential growth. We are interested in the case when the potential V and the weight Q are positive, bounded from above, and decay to zero at infinity. More precisely, we assume that for some $\alpha$ and $\beta$ in the range

$$\begin{aligned} \alpha \in (0,2) \quad \text { and } \quad \beta \in (\alpha , + \infty ) \end{aligned}$$

(1.1)

the following decay conditions hold

• (V) $V \in \mathcal C^{1} (\mathbb {R}^2)$, $V(x) \sim |x|^{- \alpha }$ as $|x| \rightarrow + \infty$, and

  $$\begin{aligned}\frac{a}{1 + |x|^{\alpha }} \le V(x) \le A, \quad \text {where } a, \, A>0;\end{aligned}$$

• (Q) $Q \in \mathcal C^{1} (\mathbb {R}^2)$, $Q(x) \sim |x|^{- \beta }$ as $|x| \rightarrow + \infty$, and

  $$\begin{aligned}0 < Q(x) \le \frac{b}{1 + |x|^{\beta }}, \quad \text {where } b>0.\end{aligned}$$

Inspired by [6], and complementing the result obtained in [21], we study the existence of mountain pass solutions for ($\mathcal P_{\varepsilon }$), for any fixed $\varepsilon >0$, and their concentration behavior in the semi-classical limit, i.e., when $\varepsilon \rightarrow 0$.

In the case of subcritical polynomial nonlinearities in higher dimensions, with more general vanishing potentials V and more general weights Q, concentration properties of solutions are extensively studied by means of suitable adaptations of the penalization method of del Pino and Felmer [17]. See, e.g., [10,11,12, 27] and the references therein.

In the case of exponential nonlinearities in 2D, up to our knowledge, we have only the result of Fei and Yin [22], and a further generalization [15]. In [22] and [15], via penalization method, the authors construct solutions to ($\mathcal P_{\varepsilon }$), with $\varepsilon >0$ sufficiently small, concentrating around a single point when

• V can vanish at infinity, and Q does not have to be bounded,

• $f(s)=|s|^{p-2} s\mathrm{e}^{\alpha _0 s^2}$ with $p>2$ and $\alpha _0>0$.

Adapting the method of proof introduced by Ambrosetti, Felli and Malchiodi in [6], we analyze the concentration properties of mountain pass solutions to ($\mathcal P_{\varepsilon }$) for a more restrictive class of weights and nonlinearities (indeed, we consider subcritical exponential nonlinearities), but what we gain is a more precise information about the limiting profile in the semi-classical limit, see Theorem 1.3. Moreover, differently from [22] and [15], our existence result holds without any restrictions on $\varepsilon >0$ in both the subcritical and critical case, see Theorems 1.2 and 1.4.

We point out that, in the case of exponential nonlinearities in 2D, few results about the study of the semi-classical limit are available in the literature even for potentials V which are positive and bounded away from zero. We refer the reader to [4, 5, 9, 13, 20, 25, 36, 38].

1.1 Variational framework

We apply the minimax variational method in a suitable weighted space. Let $p \ge 1$, and let us consider a weight function $w: \, \mathbb {R}\rightarrow [0,+\infty )$; we denote by $L^p_w(\mathbb {R}^2)$ the weighted $L^p$-space consisting of all measurable functions $u: \, \mathbb {R}^2 \rightarrow \mathbb {R}$ satisfying

$$\begin{aligned}\int _{\mathbb {R}^2} w(x) |u|^p \, \mathrm{d}x < + \infty .\end{aligned}$$

Assuming (V) with $\alpha \in (0,2]$, we introduce the weighted Sobolev space

$$\begin{aligned}H^1_V(\mathbb {R}^2):= \left\{ \, u \in L^2_V (\mathbb {R}^2) \, : \, |\nabla u| \in L^2(\mathbb {R}^2) \, \right\} ,\end{aligned}$$

and for any fixed $\varepsilon >0$, we denote by $\mathcal H_{\varepsilon }$ the Hilbert space $H^{1}_{V}(\mathbb {R}^{2})$ with inner product

$$\begin{aligned} \langle u,v\rangle _\varepsilon := \int _{\mathbb {R}^2} \left[ \varepsilon ^2\nabla u \cdot \nabla v + V(x)u v \right] \, \mathrm{d}x, \end{aligned}$$

and norm

$$\begin{aligned} \Vert u\Vert _\varepsilon ^2 : = \langle u,u\rangle _\varepsilon = \varepsilon ^2\Vert \nabla u\Vert _2^2 + \int _{\mathbb {R}^2} V(x) u^2 \, \mathrm{d}x. \end{aligned}$$

The embedding results obtained in [32] (see also Theorem 2.5) ensure that if the weight Q is of the form $(1+|x|^{\beta })^{-1}$, then

$$\begin{aligned} \mathcal H_{\varepsilon } \hookrightarrow L_{Q}^{p}(\mathbb {R}^{2}) \quad \forall p \in [2, + \infty ) \end{aligned}$$

provided (Q) holds with $\beta \in [\alpha , + \infty )$ (see [32, Example 20.6]). The above embeddings are compact if we restrict the range of $\beta$ and we assume $\beta > \alpha$.

It turns out that functions belonging to $\mathcal H_{\varepsilon }$ satisfy a weighted exponential integrability condition and the maximal growth allowed for a variational study of ($\mathcal P_{\varepsilon }$) is determined by the following weighted Trudinger–Moser-type inequality in $\mathcal H_{\varepsilon }$.

Theorem 1.1

Suppose that (V) and (Q) hold with $\alpha \in (0,2)$ and $\beta \in [\alpha , + \infty )$, respectively. For any $\gamma >0$ and any $u \in H^{1}_{V}(\mathbb {R}^{2})$, we have

$$\begin{aligned} \int _{\mathbb {R}^2} Q(x) (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x < + \infty . \end{aligned}$$

(1.2)

Moreover, for any fixed $\varepsilon >0$,

$$\begin{aligned} \sup _{u \in \mathcal H_\varepsilon , \, \Vert u\Vert _\varepsilon \le \varepsilon } \, \int _{\mathbb {R}^2} Q(x) (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x {\left\{ \begin{array}{ll}< + \infty &{} \text {if } \, 0<\gamma <4 \pi , \\ = + \infty &{} \text {if } \, \gamma > 4 \pi . \end{array}\right. } \end{aligned}$$

(1.3)

The above theorem is proved in [21, Theorem 4.1] with $\alpha \in (0,2)$ and $\beta \in [2, + \infty )$, and hence, it improves [21, Theorem 4.1] with respect to the range of the parameter $\beta$.

In view of Theorem 1.1, we say that a nonlinear term f has subcritical exponential growth at infinity if

$$\begin{aligned} \lim _{|s| \rightarrow + \infty } \frac{|f(s)|}{\mathrm{e}^{\gamma s^2}}=0\quad \text {for any } \gamma >0, \end{aligned}$$

($f_0$)

while f has critical exponential growth at infinity if

$$\begin{aligned} \text {there exists } \gamma _0>0 \text { such that } \lim _{|s| \rightarrow + \infty } \frac{|f(s)|}{\mathrm{e}^{\gamma s^2}}= {\left\{ \begin{array}{ll} 0 &{} \text {if } \gamma > \gamma _0,\\ + \infty &{} \text {if } \gamma < \gamma _0. \end{array}\right. } \end{aligned}$$

($f_0'$)

We recall that the above notions of subcritical and critical growth were first introduced in [2, 16] to study Dirichlet problems on bounded domains of $\mathbb {R}^{2}$. Clearly, in the case of bounded domains the notion of critical growth is governed by the classical Trudinger–Moser inequality [28].

1.2 Subcritical case

The following classical assumptions on the nonlinear term $f=f(s)$ are settled to guarantee a good mountain pass geometry:

$$\begin{aligned} \lim _{s\rightarrow 0}\frac{f(s)}{s}=0, \end{aligned}$$

($f_1$)

$$\begin{aligned} \exists \mu >2 \text { such that } 0 < \mu F(s):= \mu \int _0^s f(t) \, \mathrm{d}t \le s f(s) \quad \forall s \in \mathbb R \setminus \{0\}. \end{aligned}$$

($f_2$)

Theorem 1.2

[Existence] Assume (V) and (Q) with $\alpha$ and $\beta$ in the range (1.1). Let $f: \, \mathbb {R}\rightarrow \mathbb {R}$ be a continuous function, with subcritical exponential growth ($f_0$), satisfying ($f_1$) and ($f_2$). Then equation ($\mathcal P_{\varepsilon }$) has a positive mountain pass solution $u_\varepsilon \in \mathcal H_{\varepsilon }$, and $u_{\varepsilon } \in L^{2}(\mathbb {R}^{2})$, for any $\varepsilon >0$. Moreover, there exists $C>0$ such that for any $\varepsilon >0$ sufficiently small we have

$$\begin{aligned} \int _{\mathbb {R}^2} u_{\varepsilon }^2 \, \mathrm{d}x \le C \varepsilon ^2. \end{aligned}$$

(1.4)

The solution $u_\varepsilon$ found in Theorem 1.2 belongs to $\mathcal H_{\varepsilon }$, hence a priori it is not obvious that indeed $u_{\varepsilon } \in H^1(\mathbb {R}^2)$, and $u_{\varepsilon }$ is a weak solution of ($\mathcal P_{\varepsilon }$) in the sense that

$$\begin{aligned} \langle u_\varepsilon ,v\rangle _\varepsilon - \int _{\mathbb {R}^2} Q(x) f(u_\varepsilon ) v \, \mathrm{d}x =0 \quad \forall v \in \mathcal H_{\varepsilon }. \end{aligned}$$

If in addition we assume that $f \in \mathcal C^{0,\sigma }(\mathbb {R})$ with $\sigma \in (0,1)$, then, by local elliptic regularity theory, $u_{\varepsilon } \in \mathcal C^{2}(\mathbb {R}^{2})$ is a classical solution to ($\mathcal P_{\varepsilon }$). In the semi-classical limit, i.e., when $\varepsilon \rightarrow 0$, we show that $u_{\varepsilon }$ are spike solutions concentrating at a global minimum point of the auxiliary potential $\mathcal A:= V/Q$. Clearly, (V) and (Q) with $\alpha$ and $\beta$ in the range (1.1) ensure that $\mathcal A \in \mathcal C(\mathbb {R}^{2})$ is positive, $\lim \nolimits _{|x|\rightarrow + \infty } \mathcal A(x)=+ \infty$, and $\mathcal A$ has a positive global minimum.

Theorem 1.3

[Concentration] Assume (V) and (Q) with $\alpha$ and $\beta$ in the range (1.1), and assume that $f\in \mathcal C^{0, \sigma }(\mathbb {R})$ with $\sigma \in (0,1)$ satisfies ($f_0$), ($f_1$) and ($f_2$). The mountain pass solutions $u_{\varepsilon }$ found in Theorem 1.2 concentrate at a global minimum point $x^{*}$ of $\mathcal A$. More precisely, for any $\varepsilon >0$ sufficiently small, there exists a maximum point $x_{\varepsilon }$ of $u_\varepsilon$ such that $x_{\varepsilon }\rightarrow x^{*}$ as ${\varepsilon }\rightarrow 0$, and for any such $x_{\varepsilon }$,

$$\begin{aligned} u_{\varepsilon }(x)=U^{*}\left( \frac{x-x_{\varepsilon }}{\varepsilon } \right) +\omega _{\varepsilon }(x) \quad \text {as } \varepsilon \rightarrow 0, \end{aligned}$$

where the error term $\omega _{\varepsilon }\rightarrow 0$ in $\mathcal C^2_\mathrm{loc}(\mathbb {R}^2)$ and in $L^\infty (\mathbb {R}^2)$, as ${\varepsilon }\rightarrow 0$, and $U^{*}$ is a ground-state solution of the limiting problem

$$\begin{aligned} -\Delta U +V(x^{*})U=Q(x^{*})f(U)\quad \text {in}\quad \mathbb {R}^2. \end{aligned}$$

Moreover, the maximum point $x_{\varepsilon }$ is basically unique, in the sense that if $y_{\varepsilon }$ is a maximum point of $u_{\varepsilon }$ with $y_{\varepsilon } \ne x_{\varepsilon }$ then $|x_{\varepsilon }-y_{\varepsilon }| \le C \varepsilon$. If in addition f is increasing on $(0,+ \infty )$, then $x_{\varepsilon }$ is indeed the unique local (and hence global) maximum point of $u_{\varepsilon }$.

1.3 Critical case

In order to apply the mountain pass theorem in the critical case, we require the following additional assumptions:

$$\begin{aligned} \exists s_0, \, M_0 >0 \text { such that } 0 < F(s) \le M_0 |f(s)| \quad \forall |s| \ge s_0, \end{aligned}$$

($f_3$)

$$\begin{aligned} \lim _{|s| \rightarrow + \infty } \frac{s f(s)}{\mathrm{e}^{\gamma _0 s^2}} = + \infty , \quad \text {with } \gamma _{0}>0 \text { given by }f_0'. \end{aligned}$$

($f_4$)

Note that, condition ($f_3$) is used to prove the convergence of the nonlinear part of the energy functional along Palais–Smale sequences. This convergence ensures that the critical point obtained by means of the mountain pass theorem is in fact nontrivial. The growth condition ($f_4$) is needed to guarantee that Palais–Smale sequences at the mountain pass level are convergent up to subsequences.

The following existence result is already proved in [21, Theorem 2.1, Proposition 2.2] with $\alpha \in (0,2)$ and $\beta \in [2,+ \infty )$, our improvement concerns the range of the parameter $\beta$.

Theorem 1.4

[Existence] Assume (V) and (Q) with $\alpha$ and $\beta$ in the range (1.1). Let $f: \, \mathbb {R}\rightarrow \mathbb {R}$ be a continuous function with critical exponential growth ($f_0'$) for some $\gamma _{0}>0$. If f satisfies ($f_1$), ($f_2$), ($f_3$) and ($f_4$), then equation ($\mathcal P_{\varepsilon }$) has a nontrivial mountain pass solution $u_\varepsilon \in \mathcal H_{\varepsilon }$, and $u_{\varepsilon } \in L^{2}(\mathbb {R}^{2})$, for any $\varepsilon >0$.

For any fixed $\varepsilon >0$, the growth condition ($f_4$) can be replaced by the weaker condition

$$\begin{aligned} \liminf _{|s| \rightarrow + \infty } \frac{s f(s)}{\mathrm{e}^{\gamma _0 s^2}}> \mathcal M_\varepsilon = \mathcal M_\varepsilon (V,Q):= \inf _{r >0} \frac{4\varepsilon ^2 e^{\frac{1}{2\varepsilon ^2} \, r^2 \, V_{\text {max},r}}}{\gamma _0 r^2 \, Q_{\text {min},r}}, \end{aligned}$$

($f_4'$)

where

$$\begin{aligned} V_{\text {max},r}:= \max _{|x| \le r} V(x)>0 \quad \text { and } \quad Q_{\text {min},r}:= \min _{|x| \le r} Q(x) >0. \end{aligned}$$

Remark 1.5

Clearly, if $\alpha$ and $\beta$ are in the range (1.1) and we set $V(x) = a (1 + |x|^\alpha )^{-1}$ and $Q(x)=b (1 +|x|^\beta )^{-1}$, for some $a>0$ and $b>0$, then $\mathcal M_\varepsilon \in (0, + \infty )$ and $\lim \nolimits _{\varepsilon \rightarrow 0}\mathcal M_\varepsilon =+\infty$.

1.4 Organization of the paper

In Sect. 2, we prove the weighted Trudinger–Moser inequality stated in Theorem 1.1, and we pay particular attention to deduce some exponential integral uniform estimates for $\mathcal H_{\varepsilon }$-functions (see Lemma 2.6) which will be crucial to show the $L^2$-integrability of weak solutions to ($\mathcal P_{\varepsilon }$), and to study the concentration behavior of mountain pass solutions to ($\mathcal P_{\varepsilon }$).

Sections 3–5 are devoted to the case of nonlinearities with subcritical exponential growth. More precisely in Sect. 3, we prove the first part of Theorem 1.2, i.e., the existence result in the subcritical case. In Sect. 4, we obtain the $L^2$-integrability of weak solutions to ($\mathcal P_{\varepsilon }$) (see Proposition 4.4), and the $L^2$-uniform bound for weak solutions described by Proposition 4.3, which holds for mountain pass solutions to ($\mathcal P_{\varepsilon }$) in view of Corollary 5.4. Hence, the results of Sect. 4 complete the proof of Theorem 1.2.

Also in Sect. 4, we pay special attention to integral uniform estimates for weak solutions to ($\mathcal P_{\varepsilon }$), and in particular, Lemma 4.2 will be crucial to analyze the concentration behavior of mountain pass solutions to ($\mathcal P_{\varepsilon }$). This analysis will be developed in Sect. 5.

Finally, Sect. 6 is devoted to the proof of Theorem 1.4, i.e., the existence result in the critical case.

1.5 Open problems

Both the case $\alpha =2$ with $\beta \in [2,+ \infty )$ and the case $\gamma = 4 \pi$ are left open by Theorem 1.1. Does a weighted Trudinger–Moser inequality hold in these cases?

The study of the concentration behavior of the mountain pass solutions found in Theorem 1.4, in the critical exponential case, is still open. Following our argument, it is not clear how to extend the validity of Lemma 4.2 to the critical case.

1.6 Notations

We denote by $B(x,R) \subset \mathbb {R}^2$ the closed ball of radius $R>0$ centered at $x \in \mathbb {R}^2$ and, to simplify notations, we set $B_R:=B(0,R)$ and $B^{\text {c}}_{R}:= \mathbb {R}^2 \setminus B_R$.

2 A weighted Trudinger–Moser-type inequality and integral uniform estimates for $\mathcal H_{\varepsilon }$-functions

This section is mainly devoted to the proof of the weighted Trudinger–Moser inequality expressed by Theorem 1.1. As a by-product of the proof of Theorem 1.1, we will deduce again the embedding results obtained in [32], see Theorem 2.5. Finally, having in mind the study of the concentration behavior of solutions to ($\mathcal P_{\varepsilon }$), we complete this section with some integral estimates for functions belonging to $\mathcal H_{\varepsilon }$ which are uniform with respect to $\varepsilon >0$, see Lemma 2.6.

As already mentioned, Theorem 1.1 improves [21, Theorem 1.1] with respect to the range of the parameter $\beta$ which determines the decay of the weight Q at infinity. More precisely, the arguments of the proof of [21, Theorem 1.1] enable to deduce the validity of (1.3)

$$\begin{aligned}\text {for any } \alpha \text { and } \beta \text { in the range } \alpha \in (0,2) \text { and } \beta \in [2,+ \infty ),\end{aligned}$$

while here we will show that (1.3) actually holds

$$\begin{aligned}\text {for any } \alpha \text { and } \beta \text { in the range } \alpha \in (0,2) \text { and } \beta \in [\alpha ,+ \infty ).\end{aligned}$$

This improvement can be reached by following closely the method adopted in [21], and for this reason, we will only give the details of the proof of some estimates with explicit dependence on $\varepsilon$ that will be crucial to obtain the concentration result for equation ($\mathcal P_{\varepsilon }$).

The main difference with respect to the proof of [21, Theorem 1.1] can be summarized as follows. The method adopted in [21] combines ideas of Kufner and Opic [32] with a local version of the classical Trudinger–Moser inequality pointed out by Yang and Zhu in [37] (see also [21, Lemma 4.3]). Instead of applying that local Trudinger–Moser inequality, we will use locally the following well known global result.

Theorem 2.1

([1, 18, 30, 31, 33]) There exist constants $C_{\gamma }>0$ such that

$$\begin{aligned} \sup _{u \in H^1(\mathbb {R}^2), \, \Vert \nabla u \Vert _2 \le 1} \, \frac{1}{\Vert u\Vert _2^2}\int _{\mathbb {R}^2} (\mathrm{e}^{\gamma u^2} -1)\, \mathrm{d}x {\left\{ \begin{array}{ll} \le C_{\gamma } &{} \text {if } 0<\gamma < 4 \pi , \\ = + \infty &{} \text {if } \gamma \ge 4 \pi . \end{array}\right. } \end{aligned}$$

First, we derive an integral estimate in the exterior of a large ball which is uniform with respect $\varepsilon$, provided $\varepsilon >0$ is sufficiently small.

Proposition 2.2

Suppose that (V) and (Q) hold with $\alpha \in (0,2)$ and $\beta \in [\alpha ,+ \infty )$, respectively, and let $0< \gamma < 4 \pi$.

• (I) There exists $\tilde{n}= \tilde{n} (\varepsilon , \gamma , a, \alpha ) >1$ such that for any $n \ge \tilde{n}$ we have

  $$\begin{aligned} \int _{B^{\text {c}}_{3n}} Q(x) (\mathrm{e}^{\gamma u^2} -1 ) \, \mathrm{d}x \le C_{\tilde{n}, a, \alpha , b, \beta } \int _{B^{\text {c}}_{n}} V(x) u^2 \, \mathrm{d}x \quad \forall u \in \mathcal H_\varepsilon \text { with } \Vert u\Vert _\varepsilon \le \varepsilon . \end{aligned}$$

• (II) If in addition $\beta > \alpha$, then for any $\eta >0$ there exists $\tilde{n}= \tilde{n} (\varepsilon ,\eta , \gamma , a, \alpha ) >1$ such that for any $n \ge \tilde{n}$ we have

  $$\begin{aligned} \int _{B^{\text {c}}_{3n}} Q(x) ( \mathrm{e}^{\gamma u^2} -1 ) \, \mathrm{d}x \le \eta \int _{B^{\text {c}}_{n}} V(x) u^2 \, \mathrm{d}x \quad \forall u \in \mathcal H_\varepsilon \text { with } \Vert u\Vert _\varepsilon \le \varepsilon . \end{aligned}$$

In both cases, if $\varepsilon \in (0,1]$ then $\tilde{n}$ can be chosen independently of $\varepsilon$.

Proof

For any fixed $n \ge \tilde{n}$, with $\tilde{n}>>1$ to be chosen during the proof, we consider the covering of $B^{\text {c}}_{n}$ consisting of annuli of the form

$$\begin{aligned}A^\sigma _n:= \left\{ x \in B^{\text {c}}_{n} \, \big | \, |x|< \sigma \right\} = \left\{ x \in \mathbb {R}^2 \, \big | \, n< |x| < \sigma \right\} , \quad \text {with } \sigma >n.\end{aligned}$$

As in [21, Section 4.2], we apply the Besicovitch covering lemma (see [21, Lemma 4.5] and [21, (4.8)]) which gives the existence of a sequence of points $\{x_k\}_k \in A_{\tilde{n}}^\sigma$ and the existence of a universal constant $\theta >0$ such that

$$\begin{aligned}\displaystyle A_{\tilde{n}}^\sigma \subseteq \bigcup _k \, U_k^{1/2}, \quad \text {where } U_k^{1/2}:= B\left( \, x_k, \frac{1}{2} \, \frac{|x_k|}{3} \,\right) ,\end{aligned}$$

and

$$\begin{aligned} \sum _k \chi _{U_k} (x) \le \theta \text { for any }x \in \mathbb {R}^2, \end{aligned}$$

(2.1)

where $\chi _{U_k}$ is the characteristic function of $\displaystyle U_k:= B(x_k, |x_k|/3)$.

Let $u \in \mathcal H_\varepsilon$ be such that $\Vert u\Vert _\varepsilon \le \varepsilon$. We begin with the estimate of the weighted exponential integral of u on $A_{3n}^\sigma$ with $n \ge \tilde{n}$ and $\sigma >3n$. Clearly,

$$\begin{aligned} A_{3n}^\sigma \subset A_{\tilde{n}}^\sigma \subseteq \bigcup _k \, U_k^{1/2}, \end{aligned}$$

and if we consider the set of indices $K_{n, \sigma }:= \{k \in \mathbb N \, \big | \, U_k^{1/2} \cap B^{\text {c}}_{3n} \ne \varnothing \}$ then we have

$$\begin{aligned} A_{3n}^\sigma \subseteq \bigcup _{k \in K_{n, \sigma }} \, U_k^{1/2}. \end{aligned}$$

Therefore, we get

$$\begin{aligned} \int _{A_{3n}^\sigma } Q(x) (\mathrm{e}^{\gamma u^2} -1 ) \, \mathrm{d}x \le \sum _{k \in K_{n, \sigma }} \, \int _{U_k^{1/2}} Q(x) (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x. \end{aligned}$$

(2.2)

In order to estimate the single terms of the series on the right-hand side, few remarks about the choice of the balls $U_k^{1/2}$ and $U_k$ are needed. Since

$$\begin{aligned} 2|x_k|/3 \le |y| \le 4 |x_k|/3 \quad \forall y \in U_k, \end{aligned}$$

we have the existence of $C_{\alpha }$, $C_{\beta }>0$ such that

$$\begin{aligned} V(y) \ge \frac{a}{1 + |y|^\alpha } \ge \frac{a}{1+ C_\alpha |x_k|^\alpha } \quad \forall y \in U_k, \end{aligned}$$

(2.3)

and

$$\begin{aligned} Q(y) \le \frac{b}{1+ |y|^\beta } \le \frac{b}{1 + C_\beta |x_k|^\beta } \quad \forall y \in U_k. \end{aligned}$$

Furthermore, it is easy to see that if $U_k \cap B^{\text {c}}_{3n} \ne \varnothing$ then $U_k \subset B^{\text {c}}_{n}$, and this yields

$$\begin{aligned} \bigcup _{k \in K_{n, \sigma }} U_k^{1/2} \subseteq \bigcup _{k \in K_{n,\sigma }} U_k \subseteq B^{\text {c}}_{n} \subseteq B^{\text {c}}_{\tilde{n}}. \end{aligned}$$

(2.4)

Let us fix $k \in K_{n, \sigma }$. We have

$$\begin{aligned} \int _{U_k^{1/2}} Q(x) (\mathrm{e}^{\gamma u^2} -1 ) \, \mathrm{d}x \le \frac{b}{1 + C_\beta |x_k|^\beta } \int _{U_k^{1/2}} (\mathrm{e}^{ \gamma u^2} -1) \, \mathrm{d}x, \end{aligned}$$

(2.5)

and we will apply Theorem 2.1 to estimate the unweighted integral on $U_{k}^{1/2}$. To this aim, we introduce a cutoff function $\varphi _k \in \mathcal C^\infty _0(U_k)$ satisfying

$$\begin{aligned} 0 \le \varphi _k \le 1 \text { in } U_k, \quad \varphi _k \equiv 1 \text { in } U_k^{1/2}, \quad \text { and } \quad |\nabla \varphi | \le C /|x_k| \, \text { in } U_k, \end{aligned}$$

for some universal constant $C>0$. Then $\varphi _k u \in H_0^1(U_k) \subset H^{1}(\mathbb {R}^{2})$ and we may estimate

$$\begin{aligned} {\begin{matrix} \int _{U_k} |\nabla (\varphi _k u)|^2 \, \mathrm{d}x &{} \le (1+ \delta ) \int _{U_k} |\nabla u|^2 \, \mathrm{d}x + \left( 1+ \frac{1}{\delta }\right) \, \frac{C^2}{|x_k|^2} \, \int _{U_k} u^2 \, \mathrm{d}x \\ &{} \le (1+ \delta ) \int _{U_k} |\nabla u|^2 \, \mathrm{d}x + \left( 1+ \frac{1}{\delta }\right) \, \frac{C^2}{a} \, \frac{1 + C_\alpha |x_k|^\alpha }{|x_k|^2} \, \int _{U_k} V(x) u^2 \, \mathrm{d}x, \end{matrix}} \end{aligned}$$

where we also used (2.3). The above estimate holds for any $\delta >0$; by assumption $\gamma \in (0,4 \pi )$, and we can choose $\delta \in (0,1)$ so that

$$\begin{aligned} \gamma /(1 - \delta ) < 4 \pi . \end{aligned}$$

(2.6)

Recalling that $k \in K_{n, \sigma }$, in view of (2.4), we have that $x_k \in B^{\text {c}}_{\tilde{n}}$. Since $\alpha \in (0,2)$, there exists $\tilde{n} = \tilde{n} (\varepsilon , \delta , a, \alpha )$ sufficiently large such that

$$\begin{aligned}\left( 1+ \frac{1}{\delta }\right) \, \frac{C^2}{a} \, \frac{1 + C_\alpha |x_k|^\alpha }{|x_k|^2} \le \frac{1 + \delta }{\varepsilon ^2} \quad \forall k \in K_{n, \sigma }, \, n \ge \tilde{n},\end{aligned}$$

and we can take $\tilde{n} = \tilde{n} (1, \delta , a, \alpha )$ for any $\varepsilon \in (0,1]$. In this way, we get

$$\begin{aligned} \int _{U_k} |\nabla (\varphi _k u)|^2 \, \mathrm{d}x \le \frac{(1+ \delta )}{\varepsilon ^2} \int _{U_k}(\varepsilon ^{2}|\nabla u|^2 + V(x) u^2) \, \mathrm{d}x \le (1+ \delta ). \end{aligned}$$

If we set $v_k:= \sqrt{1-\delta } \, \varphi _k u \in H^1(\mathbb {R}^{2})$, then $\Vert \nabla v_k \Vert _2^2 \le 1- \delta ^2 \le 1$. In view of (2.6), we can apply Theorem 2.1 to $v_k$ obtaining

$$\begin{aligned} \int _{U_k^{1/2}} (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x = \int _{U_k^{1/2}} (\mathrm{e}^{\gamma (\varphi _k u)^2} -1) \, \mathrm{d}x \le \int _{\mathbb {R}^2} (\mathrm{e}^{[\gamma /(1-\delta )] v_k^2} -1) \, \mathrm{d}x \le C \int _{\mathbb {R}^2} | v_k|^2 \, \mathrm{d}x, \end{aligned}$$

for a positive constant $C=C(\gamma ,\delta )$. Then we deduce

$$\begin{aligned} \int _{U_k^{1/2}} (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x \le C (1- \delta )\int _{U_k}u^2 \, \mathrm{d}x \le C (1- \delta )\frac{ 1+C_\alpha |x_k|^\alpha }{a} \int _{U_k}V(x) u^2 \, \mathrm{d}x. \end{aligned}$$

(2.7)

Combining (2.2), (2.5) and (2.7), we get

$$\begin{aligned} {\begin{matrix} \int _{A_{3n}^\sigma } Q(x) ( \mathrm{e}^{\gamma u^2} -1 ) \, \mathrm{d}x &{} \le \frac{b}{a} C (1-\delta ) \sum _{k \in K_{n, \sigma }} \frac{1+C_\alpha |x_k|^\alpha }{1 + C_\beta |x_k|^\beta } \, \int _{U_k} V(x) u^2 \, \mathrm{d}x \\ &{} \le \frac{b}{a} C (1-\delta ) \sum _{k \in K_{n, \sigma }} \frac{1+C_\alpha |x_k|^\alpha }{1 + C_\beta |x_k|^\beta } \, \int _{B^{\text {c}}_{n}} V(x) u^2 \chi _{U_k} (x) \, \mathrm{d}x \end{matrix}} \end{aligned}$$

where the last inequality follows from (2.4). Using again (2.4), we have

$$\begin{aligned} \frac{1+C_\alpha |x_k|^\alpha }{1 + C_\beta |x_k|^\beta } \le \mathcal B_n:= \sup _{x \in B^{\text {c}}_{n}} \frac{1+C_\alpha |x|^\alpha }{1 + C_\beta |x|^\beta } \quad \forall k \in K_{n, \sigma }. \end{aligned}$$

Hence

$$\begin{aligned} \int _{A_{3n}^\sigma } Q(x) ( \mathrm{e}^{\gamma u^2} -1 ) \, \mathrm{d}x \le \frac{b}{a} C (1-\delta ) \mathcal B_n \sum _{k \in K_{n, \sigma }} \int _{B^{\text {c}}_{n}}V(x) u^2 \chi _{U_k}(x) \, \mathrm{d}x, \end{aligned}$$

and, in view of (2.1),

$$\begin{aligned} \int _{A_{3n}^\sigma } Q(x) ( \mathrm{e}^{\gamma u^2} -1 ) \, \mathrm{d}x \le \frac{b}{a} C (1-\delta ) \theta \mathcal B_n \int _{B^{\text {c}}_{n}} V(x) u^2 \, \mathrm{d}x. \end{aligned}$$

Letting $\sigma \rightarrow + \infty$, we deduce the existence of $\tilde{n}=\tilde{n}(\varepsilon , \delta , a, \alpha )>>1$ such that for any $n \ge \tilde{n}$ we have

$$\begin{aligned} \int _{B^{\text {c}}_{3n}} Q(x) (\mathrm{e}^{\gamma u^2} -1 ) \, \mathrm{d}x \le b C \theta \mathcal B_n \int _{B^{\text {c}}_{n}} V(x) u^2 \, \mathrm{d}x. \end{aligned}$$

(2.8)

Note that

$$\begin{aligned}\lim _{n \rightarrow + \infty } \mathcal B_n = \lim _{n \rightarrow + \infty } \frac{1+C_\alpha n^\alpha }{1 + C_\beta n^\beta } = {\left\{ \begin{array}{ll} 0 &{} \text { if } \beta >\alpha , \\ C_\alpha / C_\beta &{} \text { if } \beta =\alpha , \end{array}\right. } \end{aligned}$$

and in conclusion (2.8) yields both (I) and (II). $\square$

In view of Proposition 2.2(I), the sharp inequality (1.3) follows by a simple adaptation of the arguments in [21]. More precisely, arguing as in [21], we can prove

Proposition 2.3

Suppose that (V) and (Q) hold with $\alpha \in (0,2)$ and $\beta \in [\alpha ,+ \infty )$, respectively.

• (I) Let $0< \gamma < 4 \pi$. There exists $\tilde{R}=\tilde{R}(\varepsilon , \gamma , a, \alpha )>0$ such that for any $R \ge \tilde{R}$ we have

  $$\begin{aligned}\int _{B_{R}} Q(x) (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x \le C R^{2} \quad \forall u \in \mathcal H_{\varepsilon } \text { with } \Vert u \Vert _{\varepsilon } \le \varepsilon ,\end{aligned}$$

  where $C>0$ is a universal constant. In particular, if $\varepsilon \in (0,1]$ then $\tilde{R}$ can be chosen independently of $\varepsilon$.

• (II) For any $\gamma > 4 \pi$ we have

  $$\begin{aligned}\sup _{u \in \mathcal H_\varepsilon , \, \Vert u\Vert _\varepsilon \le \varepsilon } \, \int _{\mathbb {R}^2} Q(x) (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x = + \infty .\end{aligned}$$

If $\varepsilon =1$, $\alpha \in (0,2)$ and $\beta \in [2, + \infty )$, Proposition 2.3(I) is nothing but the result obtained in [21, Section 4.1]. Since the argument of the proof in [21] is independent of $\beta$, it can be easily adapted to cover the case $\beta \in [\alpha , + \infty )$ with obvious modifications taking into account the presence of the parameter $\varepsilon >0$.

For the proof of the sharpness statement expressed by Proposition 2.3(II), we refer to the beginning of [21, Section 4]. We only point out the key point, that is: since V and Q are bounded away from zero by positive constants on compact subsets of $\mathbb {R}^2$, and $H^{1}{(\mathbb {R}^{2})} \hookrightarrow \mathcal H_{\varepsilon }$, the sharpness statement is a direct consequence of the sharpness of the Trudinger–Moser inequality in $H^{1}(\mathbb {R}^{2})$ due to Ruf [35] (see also [3] and [14, Remark 6.1]).

Since the space $\mathcal C^{\infty }_{0}(\mathbb {R}^{2})$ is dense in $\mathcal H_{\varepsilon }$ (see [21, Section 3]), the validity of (1.2) can be derived from (1.3) using standard density arguments (see for instance [21, Section 4.3]).

With these remarks, the proof of Theorem 1.1 is completed.

Remark 2.4

For any fixed $\gamma \in (0,4\pi )$, the family of suprema $\{S_{\gamma ,\varepsilon }\}_{\varepsilon \in (0,1]}$, with $S_{\gamma ,\varepsilon }$ defined by

$$\begin{aligned}S_{\gamma ,\varepsilon }:=\sup _{u \in \mathcal H_\varepsilon , \, \Vert u\Vert _\varepsilon \le \varepsilon } \, \int _{\mathbb {R}^2} Q(x) (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x,\end{aligned}$$

is bounded. In other words, for any $\gamma \in (0,4 \pi )$ there exists $C_{\gamma }>0$ such that

$$\begin{aligned}S_{\gamma , \varepsilon } \le C_{\gamma } \quad \text { for all } \varepsilon \in (0,1].\end{aligned}$$

This is a consequence of the fact that both $\tilde{n}$ in Proposition 2.2(I) and $\tilde{R}$ in Proposition 2.3(I) can be chosen independently of $\varepsilon \in (0,1]$.

As a by-product of the proof of Proposition 2.2, we re-obtain the embedding results of Kufner and Opic [32]. More precisely,

Theorem 2.5

[32] If (V) and (Q) hold with $\alpha \in (0,2)$ and $\beta \in [\alpha , + \infty )$, respectively, then

$$\begin{aligned} \mathcal H_{\varepsilon } \hookrightarrow L^p_Q(\mathbb {R}^2) \quad \forall p \in [2, + \infty ), \end{aligned}$$

(2.9)

and there exists $C_{\varepsilon , p}>0$ such that

$$\begin{aligned}\int _{\mathbb {R}^{2}} Q(x) |u|^{p} \, \mathrm{d}x \le C_{\varepsilon , p} \Vert u \Vert _{\varepsilon }^{2} \quad \forall u \in \mathcal H_{\varepsilon }.\end{aligned}$$

Moreover, if $\beta > \alpha$, then the above embeddings are compact.

Proof

Since the spaces $\mathcal H_{\varepsilon }:= \left( H_{V}^{1}(\mathbb {R}^{2}), \Vert \cdot \Vert _{\varepsilon }\right)$ are isomorphic for all $\varepsilon >0$, without loss of generality we take $\varepsilon =1$. On the one hand $H^{1}_{V}(\mathbb {R}^{2}) \hookrightarrow H^{1}_\mathrm{loc}(\mathbb {R}^{2})$. On the other hand, for any $p\ge 2$ there exists $C_p>0$ such that $|t|^p\le C_p(e^{t^2}-1)$ for any $t\in \mathbb {R}$, and hence (2.8) yields

$$\begin{aligned} \lim _{n\rightarrow \infty } \, \sup _{u\in H^1_V(\mathbb {R}^{2}), \, \Vert u\Vert \le 1} \, \int _{B^{\text {c}}_{n}} Q(x)|u|^p \, \mathrm{d}x = \left\{ \begin{array}{ccl} 0&{}\text {if}&{}\beta >\alpha \\ <+\infty &{}\text {if}&{}\beta =\alpha . \end{array}\right. \end{aligned}$$

Therefore, we can apply [32, Theorems 17.6 and 17.10] to deduce the continuity of the embeddings (2.9), and their compactness when $\beta > \alpha$. $\square$

The inequality expressed by Proposition 2.2(II) can be seen as a counterpart of [6, Proposition 7] for the 2D case. Since the integrals on the two side of the inequality are taken on different domains, following [6, Proposition 11], we show that the dependence on the same domain can be restored, and at once, we restate the result to obtain a more suitable tool for the study of concentration of solutions to ($\mathcal P_{\varepsilon }$).

Lemma 2.6

Suppose that (V) and (Q) hold with $\alpha \in (0,2)$ and $\beta \in (\alpha ,+ \infty )$, respectively, i.e., with $\alpha$ and $\beta$ in the range (1.1). Let $M>0$, and let us consider any function $u \in \mathcal H_\varepsilon$ satisfying

$$\begin{aligned} \Vert u\Vert _\varepsilon ^2\le M\varepsilon ^2\quad \text { for some } \varepsilon >0. \end{aligned}$$

(2.10)

Then for any $\gamma \in (0,\overline{\gamma })$, with $\overline{\gamma }=\overline{\gamma }(M)>0$, and for any $\eta >0$ there exists $\overline{R}>1$ such that

$$\begin{aligned} \int _{B^{\text {c}}_{R}} Q(x) (\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x \le \eta \, \int _{B^{\text {c}}_{R}} (\varepsilon ^2|\nabla u|^2 + V(x) u^2) \, \mathrm{d}x\quad \forall R \ge \overline{R}. \end{aligned}$$

In particular, if $\varepsilon \in (0,1]$ then $\overline{R}$ can be chosen independently of $\varepsilon$.

Proof

Let $R>1$. We consider a smooth non-decreasing function $\psi _{R,\varepsilon }:\mathbb {R}^+ \rightarrow [0,1]$ satisfying

$$\begin{aligned} \psi _{R,\varepsilon }(r)=0\,\,\text{ if }\,\,0\le & {} r \le R-\varepsilon R^{\alpha /2},\quad \psi _{R,\varepsilon }(r)=1\,\,\text{ if }\,\,r \ge R,\quad \text{ and }\quad \\ |\psi _{R,\varepsilon }' (r)|\le & {} 2/(\varepsilon R^{\alpha /2})\,\, \forall r\in \mathbb {R}^+.\end{aligned}$$

In polar coordinates $(r, \theta ) \in [0, + \infty ) \times \mathbb S^1$, we define

$$\begin{aligned} u_{R,\varepsilon }(r, \theta ):= {\left\{ \begin{array}{ll} 0 &{} \text {if } 0 \le r \le R - \varepsilon R^{\alpha /2} \\ \psi _{R,\varepsilon }(r) u(2R-r, \theta ) &{} \text {if } R - \varepsilon R^{\alpha /2} \le r \le R \\ u(r, \theta ) &{} \text {if } r \ge R. \end{array}\right. } \end{aligned}$$

On the annulus

$$\begin{aligned} A_{R,\varepsilon }:= \{\, x \in \mathbb {R}^2 \, | \, R- \varepsilon R^{\alpha /2} \le |x| \le R \,\}, \end{aligned}$$

arguing as in [6, Proposition 11], we get the estimate

$$\begin{aligned} \int _{A_{R,\varepsilon }} (\, \varepsilon ^2|\nabla u_{R,\varepsilon }|^2 + V(x) u_{R,\varepsilon }^2 \,) \, \mathrm{d}x \le C \int _{B^{\text {c}}_{R}} (\, \varepsilon ^2 |\nabla u|^2 + V(x) u^2 \,) \, \mathrm{d}x, \end{aligned}$$

where $C>0$ does not depend neither on $\varepsilon >0$ nor on $R>1$. Since $u_{R,\varepsilon } \equiv 0$ when $|x| \le R-\varepsilon R^{\alpha /2}$ and $u_{R,\varepsilon } \equiv u$ when $|x| \ge R$, we deduce

$$\begin{aligned} \Vert u_{R,\varepsilon }\Vert _\varepsilon ^2&= \int _{B^{\text {c}}_{R - \varepsilon R^{\alpha /2}}} (\, \varepsilon ^2 |\nabla u_{R,\varepsilon }|^2 + V(x) u_{R,\varepsilon }^2 \,) \, \mathrm{d}x \le (1+C) \int _{B^{\text {c}}_{R}} (\, \varepsilon ^2|\nabla u|^2 + V(x) u^2 \,) \, \mathrm{d}x \\&\le (1+C)M\varepsilon ^2, \end{aligned}$$

where we also used (2.10).

Next, the idea is to apply Proposition 2.2(II) to the auxiliary function

$$\begin{aligned}w_{R,\varepsilon }:= u_{R,\varepsilon }/\sqrt{(1+C)M}.\end{aligned}$$

For any $\gamma >0$, we can estimate

$$\begin{aligned} {\begin{matrix} \int _{B^{\text {c}}_{R}} Q(x)(\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x &{} = \int _{B^{\text {c}}_{R}} Q(x)(\mathrm{e}^{\gamma u_{R,\varepsilon }^2} -1) \, \mathrm{d}x \le \int _{B^{\text {c}}_{R- \varepsilon R^{\alpha /2}}} Q(x)(\mathrm{e}^{\gamma \, u_{R,\varepsilon }^2} -1) \, \mathrm{d}x \\ &{} = \int _{B^{\text {c}}_{R- \varepsilon R^{\alpha /2}}} Q(x)(\mathrm{e}^{\gamma (1+C)M\, w_{R,\varepsilon }^2} -1) \, \mathrm{d}x. \end{matrix}} \end{aligned}$$

If $\eta >0$ is arbitrarily fixed and $\tilde{n}=\tilde{n} (\varepsilon , \eta M, \gamma , a, \alpha )$ is given by Proposition 2.2(II), then, by choosing $\overline{R}>(\varepsilon \alpha /2)^{2/(2-\alpha )}$ such that $\overline{R} - \varepsilon \overline{R}^{\alpha /2} \ge 3 \tilde{n}$, we have for any $R \ge \overline{R}$

$$\begin{aligned} {\begin{matrix} \int _{B^{\text {c}}_{R}} Q(x)(\mathrm{e}^{\gamma u^2} -1) \, \mathrm{d}x &{} \le \int _{B^{\text {c}}_{R- \varepsilon R^{\alpha /2}}} Q(x)(\mathrm{e}^{\gamma (1+C)M\, w_{R,\varepsilon }^2} -1) \, \mathrm{d}x \\ &{} \le (M\eta ) \int _{\mathbb {R}^{2}} V(x) w_{R,\varepsilon }^2\, \mathrm{d}x \le \eta \int _{B^{\text {c}}_{R}} (\, \varepsilon ^2|\nabla u|^2 + V(x) u^2 \,) \, \mathrm{d}x, \end{matrix}} \end{aligned}$$

provided $0< \gamma < \overline{\gamma }$ with

$$\begin{aligned} \overline{\gamma }:=4 \pi /[(1+C)M]. \end{aligned}$$

Clearly, Proposition 2.2(II) also guarantees that $\tilde{n}$, and hence $\overline{R}$, can be chosen independently of $\varepsilon \in (0,1]$. $\square$

The useful aspect of Lemma 2.6 is that neither the exponent $\overline{\gamma }>0$ nor the radius $\overline{R}>1$ depend on $\varepsilon \in (0,1]$.

3 Subcritical case—existence

In this section, we prove the existence of a mountain pass solution to ($\mathcal P_{\varepsilon }$), for any $\varepsilon >0$, under the assumptions:

• (V) and (Q) hold with $\alpha$ and $\beta$ in the range (1.1),

• $f: \mathbb {R}\rightarrow \mathbb {R}$ is a continuous function, with subcritical exponential growth ($f_0$), satisfying ($f_1$) and ($f_2$).

In other words, we prove the existence statement of Theorem 1.2, while the $L^{2}$-integrability of mountain pass solutions will be established in Proposition 4.4.

The functional associated with a variational approach to ($\mathcal P_{\varepsilon }$) is defined as

$$\begin{aligned}I_{\varepsilon }(u):= \frac{1}{2} \, \Vert u\Vert ^2_{\varepsilon } - \int _{\mathbb {R}^2} Q(x) F(u) \, \mathrm{d}x \quad \forall u \in \mathcal H_{\varepsilon }.\end{aligned}$$

Moreover, $I_{\varepsilon } \in \mathcal C^1(\mathcal H_{\varepsilon }, \mathbb {R})$ with

$$\begin{aligned}I_{\varepsilon }'[u](v):= \langle u,v\rangle _{\varepsilon } - \int _{\mathbb {R}^2} Q(x) f(u) v \, \mathrm{d}x \quad \forall u, v \in \mathcal H_{\varepsilon },\end{aligned}$$

and weak solutions to ($\mathcal P_{\varepsilon }$) are critical points of $I_{\varepsilon }$.

The above properties can be deduced using standard arguments: it is enough to combine the continuous embeddings (2.9) with the Trudinger–Moser bound (1.2), and the following estimates. The growth assumptions ($f_0$) and ($f_1$) on nonlinear term f yield, for any $\gamma >0$, $q \ge 0$, and $\delta >0$,

$$\begin{aligned} |f(s)| \le \delta |s| + C(\gamma , q, \delta ) |s|^{q} (\mathrm{e}^{\gamma s^2} -1) \quad \forall s \in \mathbb {R}. \end{aligned}$$

(3.1)

Therefore, using the Ambrosetti–Rabinowitz condition ($f_2$), we get

$$\begin{aligned} |F(s)| \le \delta s^2 + C(\gamma , q, \delta ) |s|^{q+1} (\mathrm{e}^{\gamma s^2} -1) \quad \forall s \in \mathbb {R}. \end{aligned}$$

(3.2)

The functional $I_{\varepsilon }$ has a mountain pass geometry on $\mathcal H_{\varepsilon }$. More precisely,

Lemma 3.1

There exist $\tau =\tau ({\varepsilon })$ and $\varrho =\varrho ({\varepsilon })>0$ such that $I_{\varepsilon }(u) \ge \tau$ for any $u \in \mathcal H_{\varepsilon }$ with $\Vert u\Vert _{\varepsilon } = \varrho$, and there exists $e_{\varepsilon } \in H_{\varepsilon }$ with $\Vert e_{\varepsilon }\Vert _{\varepsilon } > \varrho$ such that $I_{\varepsilon }(e_{\varepsilon })<0$.

Proof

The proof is standard and can be carried out as in [21, Lemma 5.1]. $\square$

The functional $I_{\varepsilon }$ satisfies the Palais–Smale compactness condition at any level $c \in \mathbb {R}$. We recall that $\{u_{n}\}_{n} \subset \mathcal H_{\varepsilon }$ is a Palais–Smale sequence for $I_{\varepsilon }$ at the level $c \in \mathbb {R}$ (for short, $(PS)_{c}$-sequence) if $I_{\varepsilon }(u_{n}) \rightarrow c$ and $I_{\varepsilon }'[u_{n}]\rightarrow 0$ as $n \rightarrow + \infty$.

Lemma 3.2

Let $\{u_{n}\}_{n} \subset \mathcal H_{\varepsilon }$ be a $(PS)_{c}$-sequence for $I_{\varepsilon }$ with $c \in \mathbb {R}$.

• (I) The sequence $\{u_{n}\}_{n}$ is bounded in $\mathcal H_{\varepsilon }$, and

  $$\begin{aligned} \sup _{n} \, \int _{\mathbb {R}^{2}} Q(x)f(u_{n}) u_{n} \, \mathrm{d}x< + \infty . \end{aligned}$$

  (3.3)

• (II) Up to a subsequence, $u_{n} \rightarrow u_{\varepsilon }$ in $\mathcal H_{\varepsilon }$, and $u_{\varepsilon } \in \mathcal H_{\varepsilon }$ is a weak solution to ($\mathcal P_{\varepsilon }$).

Proof

The proof of (I) follows by standard arguments as in [21, Lemma 5.3], and we omit it. In order to prove (II), we point out that the boundedness of $\{u_{n}\}_{n}$ in $\mathcal H_{\varepsilon }$ implies the weak convergence $u_{n} \rightharpoonup u_{\varepsilon }$ in $\mathcal H_{\varepsilon }$. Here and below, up to the end of the proof, convergence has to be understood up to a subsequence.

We can rewrite

$$\begin{aligned}\Vert u_{n}-u_{\varepsilon } \Vert _{\varepsilon }^{2}= \left( I'_{\varepsilon }[u_{n}] - I'_{\varepsilon }(u_{\varepsilon }) \right) (u_{n} - u_{\varepsilon }) + \int _{\mathbb {R}^{2}} Q(x) \left( f(u_{n}) - f(u_{\varepsilon })\right) (u_{n}-u_{\varepsilon }) \, \mathrm{d}x.\end{aligned}$$

Clearly, $I'_{\varepsilon }[u_{n}](u_{n}-u_{\varepsilon }) \rightarrow 0$ as $n \rightarrow + \infty$. Moreover, in view of (3.3), we can apply [16, Lemma 2.1] to conclude that

$$\begin{aligned}Q(x)f(u_{n}) \rightarrow Q(x)f(u_{\varepsilon }) \quad \text { in } L^{1}_\mathrm{loc}(\mathbb {R}^{2}),\end{aligned}$$

and hence $u_{\varepsilon }$ is a weak solution to ($\mathcal P_{\varepsilon }$). Therefore, $I'_{\varepsilon }[u_{\varepsilon }](u_{n}-u_{\varepsilon })= 0$ for any n. The proof is complete if we show that

$$\begin{aligned} \lim _{n \rightarrow + \infty } E_{n} =0, \quad \text {where } E_{n}:=\int _{\mathbb {R}^{2}} Q(x) \left( f(u_{n}) - f(u_{\varepsilon })\right) (u_{n}-u_{\varepsilon }) \, \mathrm{d}x. \end{aligned}$$

(3.4)

To simplify notations, we will write

$$\begin{aligned}\Vert v \Vert _{L^{2}_{Q}}^{2}:= \int _{\mathbb {R}^{2}} Q(x) v^{2} \, \mathrm{d}x \quad \forall v \in L^{2}_{Q}(\mathbb {R}^{2}).\end{aligned}$$

Applying (3.1) and Hölder’s inequality, we can estimate

$$\begin{aligned} |E_{n}| \le \,&C_{1} \int _{\mathbb {R}^{2}} Q(x) \left( |u_{n}| + |u_{\varepsilon }|\right) |u_{n}-u_{\varepsilon }| \, \mathrm{d}x \\&+C_{2} \int _{\mathbb {R}^{2}} Q(x) \left[ \, (\mathrm{e}^{\gamma u_{n}^{2}} -1) + (\mathrm{e}^{\gamma u_{\varepsilon }^{2}} -1) \, \right] |u_{n}-u_{\varepsilon }| \, \mathrm{d}x \\ \le \,&C_{1} \left( \, \Vert u_{n} \Vert _{L^{2}_{Q}} + \Vert u_{\varepsilon } \Vert _{L^{2}_{Q}} \,\right) \Vert u_{n} - u_{\varepsilon } \Vert _{L^{2}_{Q}} \\&+ C_{2} \biggl [\, \left( \, \int _{\mathbb {R}^{2}} Q(x)( \mathrm{e}^{2\gamma u_{n}^{2}} -1 ) \, \mathrm{d}x \,\right) ^{\frac{1}{2}} + \left( \, \int _{\mathbb {R}^{2}} Q(x)( \mathrm{e}^{2\gamma u_{\varepsilon }^{2}} -1 ) \, \mathrm{d}x \,\right) ^{\frac{1}{2}} \,\biggr ] \Vert u_{n} - u_{\varepsilon } \Vert _{L^{2}_{Q}}. \end{aligned}$$

Using the continuity of the embedding $\mathcal H_{\varepsilon } \hookrightarrow L^{2}_{Q}(\mathbb {R}^{2})$ (see Theorem 2.5), the Trudinger–Moser estimate (1.2), and the boundedness of $\{u_{n}\}_{n}$ in $\mathcal H_{\varepsilon }$, we get

$$\begin{aligned} |E_{n}| \le \tilde{C}_{1} \Vert u_{n} - u_{\varepsilon } \Vert _{L^{2}_{Q}} + C_{2} \left( \, \int _{\mathbb {R}^{2}} Q(x)\left( \mathrm{e}^{2\gamma u_{n}^{2}} -1 \right) \, \mathrm{d}x \,\right) ^{\frac{1}{2}} \Vert u_{n} - u_{\varepsilon } \Vert _{L^{2}_{Q}} + \tilde{C}_{2} \Vert u_{n} - u_{\varepsilon } \Vert _{L^{2}_{Q}}. \end{aligned}$$

Since the embedding $\mathcal H_{\varepsilon } \hookrightarrow L^{2}_{Q}(\mathbb {R}^{2})$ is also compact (see again Theorem 2.5), $u_{n} \rightarrow u_{\varepsilon }$ in $L^{2}_{Q}(\mathbb {R}^{2})$. Therefore, the proof of (3.4) is complete if we obtain the existence of $\gamma >0$ such that

$$\begin{aligned}\sup _{n} \, \int _{\mathbb {R}^{2}} Q(x)\left( \mathrm{e}^{2 \gamma u_{n}^{2}} -1 \right) \, \mathrm{d}x < + \infty ,\end{aligned}$$

and this is a direct consequence of the Trudinger–Moser inequality (1.3). In fact, since there exists $C_{\varepsilon }>0$ such that $\Vert u_{n} \Vert _{\varepsilon }^{2} \le C_{\varepsilon }$ for any n, it is enough to choose $\gamma >0$ sufficiently small so that $2 \gamma C_{\varepsilon }/\varepsilon ^2< 4 \pi$.$\square$

Completion of the proof of the existence statement of Theorem1.2 We introduce the mountain pass level

$$\begin{aligned} c_{\varepsilon }:= \inf _{ \gamma \in \Gamma _{\varepsilon }} \, \sup _{t \in [0,1]} \, I_{\varepsilon }(\gamma (t)), \quad \text {where } \Gamma _{\varepsilon }:= \{\, \gamma \in \mathcal C([0,1], \mathcal H_{\varepsilon }) \, | \, \gamma (0)=0 \text { and } I_{\varepsilon }(\, \gamma (1)\,) <0 \,\}. \end{aligned}$$

(3.5)

In view of Lemmas 3.1 and 3.2, the classical mountain pass theorem ensures that $c_\varepsilon >0$ is a critical level for $I_\varepsilon$. In other words, $I_\varepsilon$ has a critical point $u_{\varepsilon } \in \mathcal H_{\varepsilon }$ satisfying $I_{\varepsilon }(u_\varepsilon )= c_\varepsilon$. Since $c_\varepsilon >0$, $u_\varepsilon$ is a nontrivial weak solution for ($\mathcal P_{\varepsilon }$).$\square$

4 Integral uniform estimates and $L^{2}$-integrability of weak solutions

In this section, we establish preliminary integral estimates for weak solutions to ($\mathcal P_{\varepsilon }$) under the assumptions:

• (V) and (Q) hold with $\alpha$ and $\beta$ in the range (1.1),

• $f: \mathbb {R}\rightarrow \mathbb {R}$ is a continuous function with subcritical exponential growth ($f_0$) satisfying ($f_1$) and ($f_2$).

These estimates are uniform with respect to $\varepsilon$, and this aspect is crucial to study the concentration behavior of mountain pass solutions. As a by-product, at the end of this section, we will show that any weak solution $u_\varepsilon \in \mathcal H_\varepsilon$ for ($\mathcal P_{\varepsilon }$) satisfies $u_\varepsilon \in L^2(\mathbb {R}^2)$, and hence $u_\varepsilon \in H^{1}(\mathbb {R}^{2})$. We will follow the arguments introduced in [6, Lemma 17 and Lemma 18], see also [26, Section 3] and [21, Section 6].

Lemma 4.1

Let $M>0$, and $\varepsilon _0 \in (0,1]$. For any $\varepsilon \in (0, \varepsilon _0)$, let $u_\varepsilon \in \mathcal H_\varepsilon$ be a weak solution to ($\mathcal P_{\varepsilon }$) satisfying

$$\begin{aligned} \Vert u_{\varepsilon }\Vert _\varepsilon ^2\le M\varepsilon ^2. \end{aligned}$$

(4.1)

Then there exists $R_{0}>0$ such that, for any $n \in \mathbb N$ with $R_{n,\varepsilon }:=\varepsilon n^{2/(2- \alpha )} \ge R_{0}$, we have

$$\begin{aligned} \int _{B^{\text {c}}_{R_{n+1,\varepsilon }}} (\,\varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x \le \frac{3}{4} \int _{B^{\text {c}}_{R_{n,\varepsilon }}} (\, \varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x. \end{aligned}$$

Proof

Let $\chi _{n,\varepsilon } \, : \, \mathbb {R}^2 \rightarrow [0,1]$ be a piecewise affine function such that

$$\begin{aligned}\chi _{n,\varepsilon }\equiv 0\,\,\text{ in }\,\,B_{R_{n,\varepsilon }} \quad \text {and} \quad \chi _{n,\varepsilon }\equiv 1\,\,\text{ in }\,\,B^c_{R_{n+1,\varepsilon }}.\end{aligned}$$

Arguing as in [6, Lemma 17], we get the estimate

$$\begin{aligned} \varepsilon ^2|\nabla \chi _{n,\varepsilon }(x)|^2 \le V(x)\quad \forall x\in \mathbb {R}^2. \end{aligned}$$

By construction $\chi _{n,\varepsilon } u_\varepsilon \in H^1_V(\mathbb {R}^2)$,

$$\begin{aligned} \int _{B^{\text {c}}_{R_{n+1,\varepsilon }}} (\,\varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x \le \int _{B^{\text {c}}_{R_{n,\varepsilon }}} \chi _{n,\varepsilon }(\,\varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x,\end{aligned}$$

and we can compute

$$\begin{aligned} \langle u_\varepsilon , \chi _{n,\varepsilon } u_\varepsilon \rangle _\varepsilon = \int _{B^{\text {c}}_{R_{{n,\varepsilon }}}} \chi _{n,\varepsilon }(\, \varepsilon ^2|\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, dx + \varepsilon ^2\int _{B^{\text {c}}_{R_{n,\varepsilon }}} u_\varepsilon \nabla u_\varepsilon \cdot \nabla \chi _{n,\varepsilon } \, \mathrm{d}x.\end{aligned}$$

Moreover, taking $\chi _{n,\varepsilon } u_\varepsilon \in \mathcal H_\varepsilon$ as test function, we obtain

$$\begin{aligned} \langle u_\varepsilon , \chi _{n,\varepsilon } u_\varepsilon \rangle _\varepsilon - \int _{\mathbb {R}^2} Q(x) f(u_\varepsilon ) \chi _{n,\varepsilon } u_\varepsilon \, \mathrm{d}x =0.\end{aligned}$$

Therefore, we can estimate

$$\begin{aligned} {\begin{matrix} \int _{B^{\text {c}}_{R_{{n,\varepsilon }}}}\!\!&{} \chi _{n,\varepsilon }(\, \varepsilon ^2|\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x = \int _{B^{\text {c}}_{R_{{n,\varepsilon }}}} Q(x)f(u_\varepsilon ) \chi _n u_\varepsilon \, \mathrm{d}x - \varepsilon ^2\int _{B^{\text {c}}_{R_{n,\varepsilon }}} u_\varepsilon \nabla u_\varepsilon \cdot \nabla \chi _{n,\varepsilon } \, \mathrm{d}x \\ &{} \le \int _{B^{\text {c}}_{R_{{n,\varepsilon }}}} Q(x)f(u_\varepsilon ) u_\varepsilon \, \mathrm{d}x + \frac{\varepsilon ^2}{2} \left( \, \int _{B^{\text {c}}_{R_{n,\varepsilon }}} |\nabla u_\varepsilon |^2 \, \mathrm{d}x + \int _{B^{\text {c}}_{R_{n,\varepsilon }}} |\nabla \chi _{n,\varepsilon }|^2 u_\varepsilon ^2 \, \mathrm{d}x \, \right) \\ &{} \le \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x)f(u_\varepsilon ) u_\varepsilon \, \mathrm{d}x + \frac{1}{2} \, \int _{B^{\text {c}}_{R_{n,\varepsilon }}} (\,\varepsilon ^2|\nabla u_\varepsilon |^2 +V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x. \end{matrix}} \end{aligned}$$

The proof is complete if we establish the existence of $R_{0}>0$ such that, for any $n \in \mathbb N$ with $R_{n,\varepsilon } \ge R_{0}$, the following inequality holds

$$\begin{aligned} \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x) f(u_\varepsilon ) u_\varepsilon \, \mathrm{d}x \le \frac{1}{4} \, \int _{B^{\text {c}}_{R_{n,\varepsilon }}} (\,\varepsilon ^2|\nabla u_\varepsilon |^2 +V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x. \end{aligned}$$

(4.2)

Applying (3.1) and Hölder’s inequality, for any fixed $\gamma >0$ we have

$$\begin{aligned} {\begin{matrix} \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x) f(u_\varepsilon ) u_\varepsilon \, \mathrm{d}x \le &{} \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x) u_\varepsilon ^2 \, \mathrm{d}x \\ &{} + C(\gamma ) \, \left( \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x) u_\varepsilon ^2 \, \mathrm{d}x \right) ^{\frac{1}{2}} \, \left( \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x) (\mathrm{e}^{2 \gamma u_\varepsilon ^2} -1) \, \mathrm{d}x \right) ^{\frac{1}{2}}. \end{matrix}} \end{aligned}$$

To estimate the exponential integral, we choose $\gamma \in (0,\overline{\gamma }/2)$ and $R_{0} \ge \overline{R}$, where $\overline{\gamma }$ and $\overline{R}$ are given by Lemma 2.6 with $\eta =1$. With this choice of $\gamma$ and $R_{0}$, for any $n \in \mathbb N$ satisfying $R_{n,\varepsilon } \ge R_{0}$, we have

$$\begin{aligned} \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x) (\mathrm{e}^{2 \gamma u_\varepsilon ^2} -1) \, \mathrm{d}x \le \int _{B^{\text {c}}_{R_{n,\varepsilon }}} (\, \varepsilon ^2|\nabla u_\varepsilon |^2 +V(x)u_\varepsilon ^2 \,) \, \mathrm{d}x. \end{aligned}$$

(4.3)

To estimate the weighted $L^{2}$-norm, we point out that, for any $n \in \mathbb N$ with $R_{n,\varepsilon }\ge R_{0}$, we have

$$\begin{aligned} \sup _{x \in B^{\text {c}}_{R_{n,\varepsilon }}} \frac{Q(x)}{V(x)} \le \sup _{x \in B^{\text {c}}_{R_{0}}} \frac{Q(x)}{V(x)} \le \sup _{x \in B^{\text {c}}_{R_{0}}} \frac{b}{a} \, \frac{1 + |x|^\alpha }{1 + |x|^\beta } \le \frac{b}{a} \, \frac{1 + |R_{0}|^\alpha }{1+|R_{0}|^\beta } =: \mathcal B(R_{0}), \end{aligned}$$

where we used assumptions (V) and (Q) with $\alpha$ and $\beta$ in the range (1.1). Hence,

$$\begin{aligned} \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x) u_\varepsilon ^2 \, \mathrm{d}x \le \mathcal B(R_{0}) \, \int _{B^{\text {c}}_{R_{n,\varepsilon }}} V(x) u_\varepsilon ^2\, \mathrm{d}x, \end{aligned}$$

(4.4)

and it is important to point out that

$$\begin{aligned} \lim _{R_{0} \rightarrow + \infty } \mathcal B(R_{0})= \lim _{R_{0} \rightarrow + \infty } \frac{b}{a} \, R_{0}^{\alpha - \beta } =0. \end{aligned}$$

Combining (4.3) and (4.4), we obtain

$$\begin{aligned} \int _{B^{\text {c}}_{R_{n,\varepsilon }}} Q(x) f(u_\varepsilon ) u_\varepsilon \, \mathrm{d}x \le \left[ \, [\mathcal B(R_{0})]^{\frac{1}{2}} + C(\gamma ) \, \right] \, [\mathcal B(R_{0}) ]^{ \frac{1}{2}} \, \int _{B^{\text {c}}_{R_{n,\varepsilon }}} (\, \varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x. \end{aligned}$$

Since $\gamma \in (0,\overline{\gamma }/2)$ is fixed, we can choose $R_{0} \ge \overline{R}$ sufficiently large so that

$$\begin{aligned} \left[ \, [\mathcal B(R_{0})]^{\frac{1}{2}} + C(\gamma ) \, \right] \, [\mathcal B(R_{0}) ]^{ \frac{1}{2}}\le \frac{1}{4}, \end{aligned}$$

and this completes the proof of (4.2).$\square$

Lemma 4.2

Let $M>0$, and $\varepsilon _0 \in (0,1]$. For any $\varepsilon \in (0, \varepsilon _0)$, let $u_\varepsilon \in \mathcal H_\varepsilon$ be a weak solution to ($\mathcal P_{\varepsilon }$) satisfying (4.1). Then there exists $R_{0} >0$ and a constant $C=C(M)>0$ such that

$$\begin{aligned} \int _{B^{\text {c}}_{\varrho }} (\,\varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x \le C \varepsilon ^{2} \exp \left\{ -\left| \log (3/4) \right| \, \varepsilon ^{-(2-\alpha )/2}\left[ \varrho ^{(2-\alpha )/2}-R^{(2-\alpha )/2}\right] \right\} , \end{aligned}$$

for any $R\ge R_{0}$ and $\varrho > 2 R$.

Proof

Let $R_{0}$ and $\{R_{n,\varepsilon }\}_n$ be as in Lemma 4.1. If $R\ge R_{0}$ and $\varrho > 2 R$, then we have the existence of two positive integers $\overline{n} > \tilde{n}$ such that

$$\begin{aligned}R_{\tilde{n}, \varepsilon } \le R \le R_{\tilde{n} +1, \varepsilon } \quad \text { and } \quad R_{\overline{n} -1, \varepsilon } \le \varrho \le R_{\overline{n}, \varepsilon }.\end{aligned}$$

By definition of $R_{n,\varepsilon }$, it is easy to see that

$$\begin{aligned} \overline{n} - \tilde{n} \ge \varepsilon ^{-(2- \alpha )/2}\left( \varrho ^{(2- \alpha )/2} - R^{(2- \alpha )/2} \right)> (R/\varepsilon )^{(2- \alpha )/2} \left( 2^{(2- \alpha )/2} -1\right) >2 \end{aligned}$$

provided $R_{0}$ is sufficiently large, and it can still be chosen independently of $\varepsilon \in (0,1]$. Therefore, $R_{\overline{n} -1, \varepsilon } \ge R_{\tilde{n} +1, \varepsilon } \ge R\ge R_0$, and in view of (4.1), we can apply Lemma 4.1. In this way, we can estimate

$$\begin{aligned} {\begin{matrix} \int _{B^{\text {c}}_{\varrho }} (\,&{} \varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x \le \int _{B^{\text {c}}_{R_{\overline{n} -1, \varepsilon }}} (\, \varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x \\ &{}\le ( 3/4)^{\overline{n} - \tilde{n} -2} \int _{B^{\text {c}}_{R}} (\, \varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x \\ &{}\le (4/3)^2 \, \exp \left\{ - \left| \log (3/4) \right| \varepsilon ^{-(2-\alpha )/2} (\varrho ^{(2-\alpha )/2} - R^{(2- \alpha )/2})\right\} \, \int _{B^{\text {c}}_{R}} (\, \varepsilon ^2 |\nabla u_\varepsilon |^2 + V(x) u_\varepsilon ^2 \,) \, \mathrm{d}x, \end{matrix}} \end{aligned}$$

and the desired result follows again from (4.1).$\square$

We will show that the mountain pass solutions found in Theorem 1.2 satisfy (4.1) for any $\varepsilon >0$ sufficiently small, see Corollary 5.4. Therefore, the $L^2$-uniform bound (1.4) for mountain pass solutions to ($\mathcal P_{\varepsilon }$) is a consequence of the following result.

Proposition 4.3

Let $M>0$, and $\varepsilon _0 \in (0,1]$. For any $\varepsilon \in (0, \varepsilon _0)$, let $u_{\varepsilon } \in \mathcal H_{\varepsilon }$ be a weak solution to ($\mathcal P_{\varepsilon }$) satisfying (4.1). Then there exists $C>0$ such that

$$\begin{aligned} \int _{\mathbb {R}^{2}} u_{\varepsilon }^{2} \, \mathrm{d}x \le C\varepsilon ^2 \quad \forall \varepsilon \in (0, \varepsilon _{0}). \end{aligned}$$

(4.5)

Proof

First, we recall that $L^{2}_{V}(\mathbb {R}^{2}) \hookrightarrow L^{2}_\mathrm{loc}(\mathbb {R}^{2})$, and for any $\Omega \subset \mathbb {R}^{2}$ bounded there exists $C(\Omega )>0$ such that

$$\begin{aligned} \int _{\Omega } u^{2} \, \mathrm{d}x \le C(\Omega ) \Vert u \Vert _{\varepsilon }^{2} \quad \forall u \in H^{1}_{V}(\mathbb {R}^{2}), \, \forall \varepsilon >0. \end{aligned}$$

(4.6)

Combining (4.1) with (4.6), for any $\Omega \subset \mathbb {R}^{2}$ bounded we have

$$\begin{aligned} \int _{\Omega } u_{\varepsilon }^{2} \, \mathrm{d}x \le C(\Omega ) M \varepsilon ^{2} \quad \forall \varepsilon \in (0, \varepsilon _{0}). \end{aligned}$$

(4.7)

If we obtain a suitable uniform estimate of the $L^2$-norm of $u_\varepsilon$ on $B^{\text {c}}_{2}$, then the proof is complete, and to this aim, the idea is to apply Lemma 4.2. If $r \ge 2$ and $y \in B_{2r}^{c}$, we have

$$\begin{aligned}\sup _{x \in B(y,r)} \frac{1+ |x|^\alpha }{a |y|^\alpha } \le \frac{1 +(r +|y|)^\alpha }{a |y|^\alpha } \le \frac{1+ \left( \frac{3}{2} \, |y|\right) ^\alpha }{a|y|^\alpha } \le \sup _{y \in B^{\text {c}}_{4}} \frac{1+ \left( \frac{3}{2} \, |y|\right) ^\alpha }{a|y|^\alpha } =: C(\alpha ) < + \infty ,\end{aligned}$$

and hence

$$\begin{aligned}\int _{B(y,r)} u_\varepsilon ^2 \, \mathrm{d}x \le \int _{B(y,r)} \frac{1 + |x|^\alpha }{a} \, V(x) u_\varepsilon ^2 \, \mathrm{d}x \le C(\alpha ) \, |y|^\alpha \int _{B^{\text {c}}_{|y|/2}} V(x) u_\varepsilon ^2 \, \mathrm{d}x,\end{aligned}$$

where we also used (V) and the inclusion $B(y,r) \subseteq B^{\text {c}}_{|y|/2}$.

If $R_0>1$ is given by Lemma 4.2, and we choose $r > 2 R_0$, then applying Lemma 4.2 we get

$$\begin{aligned} \int _{B(y,r)} u_\varepsilon ^2 \, \mathrm{d}x \le \tilde{C}(\alpha ) \, |y|^\alpha \,\varepsilon ^2 \, \exp \left\{ -\overline{C}(\alpha ) \left| \log (3/4) \right| \,\varepsilon ^{-(2-\alpha )/2}\, |y|^{(2-\alpha )/2}\right\} . \end{aligned}$$

(4.8)

Let $m \in \mathbb N$ and $y_{i} \in B_{2}^{c}$ with $i \in \{1, \dots , m\}$ be such that

$$\begin{aligned}B_5 \setminus B_2 \subset \bigcup _{i=1}^m B(y_i, 1)\end{aligned}$$

and let $y_{i,k}:= 2^k y_i$. If $K_0$ is a positive integer such that $2^{K_0} > 2 R_0$ then, using (4.8) with $r=2^k$ and $y=y_{i,k}$, we obtain

$$\begin{aligned}\int _{B(y_{i,k}, 2^k)} u_\varepsilon ^2 \, \mathrm{d}x\le \tilde{C}(\alpha )\,\varepsilon ^2 \, |y_{i,k}|^\alpha \, \exp \left\{ -\overline{C}(\alpha ) \left| \log (3/4) \right| \,\varepsilon ^{-(2-\alpha )/2}\, |y_{i,k}| ^{(2-\alpha )/2}\right\} \quad \forall k \ge K_0,\end{aligned}$$

and

$$\begin{aligned} {\begin{matrix} \int _{B^{\text {c}}_{2}} u_\varepsilon ^2 \, \mathrm{d}x &{} \le \sum _{k=0}^{+ \infty } \int _{2^k B_5 \setminus B_2} u_\varepsilon ^2 \, \mathrm{d}x \le \sum _{i=1}^{m} \sum _{k=0}^{+ \infty } \int _{B(y_{i,k}, 2^k)} u_\varepsilon ^2 \, \mathrm{d}x \le \sum _{i=1}^{m} \sum _{k=0}^{K_0-1} \int _{B(y_{i,k},2^k)} u_\varepsilon ^2 \, \mathrm{d}x \\ + &{} \tilde{C}(\alpha )\,\varepsilon ^2\, \sum _{i=1}^{m} \sum _{k=K_0}^{+ \infty } \left\{ |y_{i,k}|^\alpha \, \exp \left[ -\overline{C}(\alpha ) \left| \log (3/4) \right| \,\varepsilon ^{-(2-\alpha )/2}\, |y_{i,k}|^{(2-\alpha )/2}\right] \right\} . \end{matrix}} \end{aligned}$$

(4.9)

Since $\alpha \in (0,2)$, the series on the right-hand side is finite and uniformly bounded for any $\varepsilon \in (0, \varepsilon _{0})$. This combined with (4.7) enables us to conclude that (4.5) holds.$\square$

We conclude this section showing that any weak solution $u_{\varepsilon } \in \mathcal H_{\varepsilon }$ for ($\mathcal P_{\varepsilon }$) is a bound state solution, i.e., it has finite $L^2$-norm.

Proposition 4.4

Any weak solution $u_{\varepsilon } \in \mathcal H_{\varepsilon }$ to ($\mathcal P_{\varepsilon }$) satisfies $u_{\varepsilon } \in L^2(\mathbb {R}^2)$, and hence $u_{\varepsilon } \in H^1(\mathbb {R}^2)$.

Proof

Let $\varepsilon >0$ be fixed, and let $u_{\varepsilon }$ be any weak solution to ($\mathcal P_{\varepsilon }$). Clearly, $u_{\varepsilon }$ satisfies (4.1) with $M=M(\varepsilon ):= \Vert u_{\varepsilon } \Vert _{\varepsilon }^{2}/\varepsilon ^{2}$, and Lemma 4.1 and Lemma 4.2 still hold for some $R_0>0$ depending on $\varepsilon$. Therefore, we can argue as in the proof of Proposition 4.3 to obtain (4.8), and hence (4.9). Since $\alpha \in (0,2)$, the series on the right-hand side of (4.9) is finite, and this combined with (4.6) enables us to conclude that $u_{\varepsilon } \in L^{2}(\mathbb {R}^{2})$.$\square$

In view of Proposition 4.3 (see also Corollary 5.4) and Proposition 4.4, the proof of Theorem 1.2 is complete.

5 Proof of Theorem 1.3

In this section, we consider the mountain pass solutions $u_{\varepsilon }$ found in Theorem 1.2, and we assume:

• (V) and (Q) hold with $\alpha$ and $\beta$ in the range (1.1),

• $f\in \mathcal C^{0, \sigma }(\mathbb {R})$ with $\sigma \in (0,1)$ satisfies ($f_0$), ($f_1$), and ($f_2$).

Since $u_{\varepsilon }$ is a weak solution to ($\mathcal P_{\varepsilon }$), $u_{\varepsilon }$ is also a classical solution and $u_{\varepsilon } \in \mathcal C^{2}(\mathbb {R}^2)$. This can be shown through interior $L^{2}$-estimates, the classical Sobolev embeddings, and Schauder estimates (see, e.g., [34, Proposition 2.15]).

Since $u_{\varepsilon } \in H^{1}(\mathbb {R}^{2})$, by standard arguments (see, e.g., [34, Remark 2.34]),

$$\begin{aligned}\lim _{|x|\rightarrow + \infty } u_{\varepsilon }(x)=0,\end{aligned}$$

and hence, by the strong maximum principle, $u_{\varepsilon }>0$ in $\mathbb {R}^{2}$.

The proof of Theorem 1.3 will be carried out by means of several steps.

5.1 The limiting problem

First, we recall some known results about the limiting problem associated with ($\mathcal P_{\varepsilon }$). If $\xi \in \mathbb {R}^{2}$ is appropriately chosen, then the limiting equation for ($\mathcal P_{\varepsilon }$) is of the form

$$\begin{aligned} -\Delta U+V(\xi )U=Q(\xi )f(U) \quad \text {in}\,\mathbb {R}^2. \end{aligned}$$

(5.1)

Let $L_\xi :H^1(\mathbb {R}^2)\rightarrow \mathbb {R}$ be the related energy functional, i.e.,

$$\begin{aligned} L_\xi (U)=\frac{1}{2}\int _{\mathbb {R}^2}|\nabla U|^2\, \mathrm{d}x+\frac{V(\xi )}{2}\int _{\mathbb {R}^2}U^2\, \mathrm{d}x-Q(\xi )\int _{\mathbb {R}^2}F(U)\, \mathrm{d}x. \end{aligned}$$

(5.2)

Clearly, $L_\xi$ is of class $C^1$ and its critical points are solutions to (5.1). The existence of a least energy solution for equation (5.1) was established in [7], and we denote by $m_{\xi }$ the corresponding least energy level, i.e.,

$$\begin{aligned}m_{\xi }:= \inf \left\{ \, L_{\xi }(u) \, : \, u \in H^{1}(\mathbb {R}^{2}) \setminus \{0\} \text { is a solution to } (5.1) \,\right\} \end{aligned}$$

Next, we recall the following characterization of the least energy level.

Theorem 5.1

[24] Let $\xi \in \mathbb {R}^{2}$. The least energy level $m_{\xi }$ of the functional $L_{\xi }$ coincides with the mountain pass level, i.e.,

$$\begin{aligned} m_\xi =\inf _{\gamma \in \Gamma _\xi } \, \max _{ t\in [0,1]}L_\xi (\gamma (t)), \end{aligned}$$

where $\Gamma _\xi =\left\{ \gamma \in C([0,1], H^1(\mathbb {R}^2)):\,\gamma (0)=0\,\,\text {and}\,\,L_\xi (\gamma (1))<0\right\}$. Moreover, for any least energy solution $U \in H^{1}(\mathbb {R}^{2})$ of (5.1), there exists a path $\gamma \in \Gamma _\xi$ such that

$$\begin{aligned} U\in \gamma ([0,1])\quad \text {and}\quad \max _{t\in [0,1]}L_\xi (\gamma (t))=L_\xi (U)=m_\xi . \end{aligned}$$

(5.3)

Remark 5.2

Exploiting the properties of dilations, if we set

$$\begin{aligned}U_{\lambda }(x):= U(\lambda x), \quad x \in \mathbb {R}^{2}, \, \lambda >0,\end{aligned}$$

then $L_\xi (U)=\tilde{L}_\xi ( U_{\lambda })$ where $\tilde{L}_{\xi }$ is the auxiliary functional

$$\begin{aligned} \tilde{L}_\xi (U):= \frac{1}{2}\int _{\mathbb {R}^2}|\nabla U|^2\, \mathrm{d}x+\frac{\mathcal A(\xi )}{2}\int _{\mathbb {R}^2}U^2\, \mathrm{d}x-\int _{\mathbb {R}^2}F(U)\, \mathrm{d}x, \end{aligned}$$

defined by means of $\mathcal A:=V/Q$. It is worth pointing out that the mountain pass level $m_\xi$ of $L_{\xi }$ coincides with the mountain pass level of the auxiliary functional $\tilde{L}_{\xi }$. Hence, using the definition of $\tilde{L}_\xi$, we deduce

$$\begin{aligned} \text {if } \mathcal A(\xi _1)< \mathcal A(\xi _2), \text { for some } \xi _1, \, \xi _2\in \mathbb {R}^2, \quad \text { then } \quad m_{\xi _1}< m_{\xi _2}. \end{aligned}$$

(5.4)

We also observe that, in view of (V) and (Q), since $\beta >\alpha$, the function $\mathcal A$ has a minimum point $\xi _0\in \mathbb {R}^2$. So $m_{\xi _0}\le m_{\xi }$ for all $\xi \in \mathbb {R}^2$.

5.2 Uniform estimate of the mountain pass level

Let $c_{\varepsilon }$ be the mountain pass critical level associated with ($\mathcal P_{\varepsilon }$) defined in (3.5).

Proposition 5.3

For any $\xi \in \mathbb {R}^{2}$, there exists $\varepsilon _0=\varepsilon _0(\xi ) > 0$ such that

$$\begin{aligned} \varepsilon ^{-2} c_\varepsilon = \varepsilon ^{-2} I_\varepsilon (u_\varepsilon )\le m_\xi +o_\xi (1), \quad \forall \varepsilon \in (0,\varepsilon _0) \end{aligned}$$

where $\lim \nolimits _{\varepsilon \rightarrow 0} o_\xi (1)=0$. In particular, there exists $C> 0$ such that $c_\varepsilon \le C\varepsilon ^2$ for all $\varepsilon \in (0,\varepsilon _0)$.

Proof

Let $\xi \in \mathbb {R}^2$. We consider a least energy solution $U\in H^1(\mathbb {R}^2)$ for (5.1) and a path $\gamma \in \Gamma _{\xi }$ satisfying (5.3). Let $\varphi \in C^\infty _0 \left( B(\xi ,2)\right)$ be a cutoff function such that

$$\begin{aligned}0 \le \varphi \le 1 \, \text { in } \, B(\xi , 2) \quad \text { and } \quad \varphi \equiv 1 \, \text { in } \, B(\xi , 1),\end{aligned}$$

For any $\varepsilon >0$ and $t \in [0,1]$, we define

$$\begin{aligned} \varphi _{\varepsilon ,t}(x)=\varphi (x)\; \gamma (t)\left( \frac{x-\xi }{\varepsilon }\right) ,\quad x\in \mathbb {R}^2. \end{aligned}$$

Since $\gamma (t)\in H^1(\mathbb {R}^2)$, we get $\varphi _{\varepsilon ,t}\in H^1(\mathbb {R}^2)\subset \mathcal H_\varepsilon$, and we can compute

$$\begin{aligned} {\begin{matrix} \int _{\mathbb {R}^2}| \nabla \varphi _{\varepsilon ,t}|^2\, \mathrm{d}x=&{}\int _{\mathbb {R}^2}| \nabla \varphi (\varepsilon y+\xi )|^2[\gamma (t)(y)]^{2}\, dy+2\int _{\mathbb {R}^2} \nabla \varphi (\varepsilon y+\xi )\cdot (\nabla \gamma (t)(y))\, dy\\ &{}+\int _{\mathbb {R}^2}\varphi ^2(\varepsilon y+\xi )|\nabla \gamma (t)(y)|^2\, dy\\ =&{}\int _{\mathbb {R}^2}|\nabla \gamma (t)(y)|^2\, dy+o_\xi (1), \end{matrix}} \end{aligned}$$

$$\begin{aligned} \frac{1}{\varepsilon ^2}\int _{\mathbb {R}^2}V(x)\varphi _{\varepsilon ,t}^2\, \mathrm{d}x =\int _{\mathbb {R}^2}V(\varepsilon y+\xi )\varphi (\varepsilon y+\xi )^2[\gamma (t)(y)]^2\, dy = {V(\xi )}\int _{\mathbb {R}^2}[\gamma (t)(y)]^2\, dy+o_\xi (1), \end{aligned}$$

and

$$\begin{aligned} \frac{1}{\varepsilon ^2}\int _{\mathbb {R}^2}Q(x)F(\varphi _{\varepsilon ,t})\, \mathrm{d}x =&\int _{\mathbb {R}^2}Q(\varepsilon y+\xi )F\left( \varphi (\varepsilon y+\xi )\gamma (t)(y)\right) \, dy \\ =&{Q(\xi )}\int _{\mathbb {R}^2}F \left( \gamma (t)(y)\right) \, dy+o_\xi (1), \end{aligned}$$

where $\lim \nolimits _{\varepsilon \rightarrow 0}o_\xi (1)=0$ uniformly in $t\in [0,1]$. Then

$$\begin{aligned} {\begin{matrix} \varepsilon ^{-2}I_\varepsilon (\varphi _{\varepsilon ,t})&{}= \frac{1}{2}\int _{\mathbb {R}^2}| \nabla \varphi _{\varepsilon ,t}|^2\, \mathrm{d}x+\frac{1}{2\varepsilon ^2}\int _{\mathbb {R}^2}V(x)\varphi _{\varepsilon ,t}^2\, \mathrm{d}x-\frac{1}{\varepsilon ^2}\int _{\mathbb {R}^2}Q(x)F(\varphi _{\varepsilon ,t})\, \mathrm{d}x\\ &{}=L_\xi (\gamma (t))+o_\xi (1)\le m_\xi +o_\xi (1). \end{matrix}} \end{aligned}$$

By assumption $L_\xi (\gamma (1))<0$, hence $I_\varepsilon (\varphi _{\varepsilon ,1})<0$, for sufficiently small values of $\varepsilon >0$. Therefore, $\varphi _{\varepsilon ,t}\in \Gamma _\varepsilon$, for any $0< \varepsilon<<1$, and we get the estimate

$$\begin{aligned} {\begin{matrix} \varepsilon ^{-2} c_\varepsilon&\le \varepsilon ^{-2}\max _{t\in [0,1]}I_\varepsilon (\varphi _{\varepsilon ,t})\le m_\xi +o_\xi (1), \end{matrix}} \end{aligned}$$

as desired. $\square$

As a consequence of the above proposition, we have

Corollary 5.4

There exist $\varepsilon _0>0$ and $M>0$ satisfying

$$\begin{aligned} \Vert u_\varepsilon \Vert _\varepsilon ^2\le M\varepsilon ^2\quad \forall \varepsilon \in (0,\varepsilon _0). \end{aligned}$$

Proof

In fact, since $u_\varepsilon$ is a mountain pass solution for ($\mathcal P_{\varepsilon }$) and f satisfies ($f_2$), we have

$$\begin{aligned} c_\varepsilon =I_\varepsilon (u_\varepsilon )-\frac{1}{\mu }I^\prime _\varepsilon [u_\varepsilon ](u_\varepsilon )\ge \left( \frac{\mu -2}{2\mu }\right) \Vert u_\varepsilon \Vert _\varepsilon ^2 \end{aligned}$$

and the result follows from Proposition 5.3. $\square$

5.3 Pointwise uniform exponential decay of mountain pass solutions

Our next result shows that, at least for sufficiently small values of $\varepsilon >0$, the mountain pass solutions $u_\varepsilon$ have a pointwise uniform exponential decay at infinity determined by $\alpha \in (0,2)$.

Theorem 5.5

[Uniform decay estimate] There exists $\varepsilon _0>0$ such that $u_{\varepsilon } \in L^{\infty }(\mathbb {R}^2)$ for any $\varepsilon \in (0,\varepsilon _0)$. Moreover, there exist $C>0$, $d>1$ and $R>0$ such that for any $\varepsilon \in (0,\varepsilon _0)$ and any $|x|\ge 2R+2$ it holds

$$\begin{aligned} |u_\varepsilon (x)|\le C\varepsilon ^{1-d}|x|^{\alpha /2}\exp \left\{ -\frac{1}{2}\left| \log \frac{3}{4} \right| \, (2\varepsilon )^{-(2-\alpha )/2}\left[ |x|^{(2-\alpha )/2}-(2R)^{(2-\alpha )/2}\right] \right\} . \end{aligned}$$

(5.5)

Proof

We already know that $u_{\varepsilon } \in L^\infty _\mathrm{loc}(\mathbb {R}^2)$, and clearly this property combined with (5.5) yields $u \in L^\infty (\mathbb {R}^2)$. Therefore, the rest of the proof will be devoted to the pointwise uniform bound (5.5). To simplify notations, let

$$\begin{aligned} g_\varepsilon (x)=Q(x)f(u_\varepsilon (x)), \quad x\in \mathbb {R}^2. \end{aligned}$$

From standard interior $L^2$-estimates (see, e.g., [23, Theorem 9.11]), for any $\varepsilon >0$ and any $y \in \mathbb {R}^2$, we have the local inequality

$$\begin{aligned} \Vert u_\varepsilon \Vert _{H^2( B(y,1))}\le C\varepsilon ^{-d}\left[ \Vert u_\varepsilon \Vert _{L^2( B(y,2))}+\Vert g_\varepsilon \Vert _{L^2 (B(y,2))}\right] ,\\ \end{aligned}$$

where $d>1$ is a universal constant, and $C=C(A)>0$ with A given by (V) – in other words, the constant C depends only on the norm of V in $L^\infty (\mathbb {R}^2)$. Recalling that $H_\mathrm{loc}^2(\mathbb {R}^2)\hookrightarrow L_\mathrm{loc}^\infty (\mathbb {R}^2)$, we get

$$\begin{aligned} \Vert u_\varepsilon \Vert _{L^\infty ( B(y,1))}\le C' \Vert u_\varepsilon \Vert _{H^2( B(y,1))}\le { C''}\varepsilon ^{-d}\left[ \Vert u_\varepsilon \Vert _{L^2( B(y,2))}+\Vert g_\varepsilon \Vert _{L^2 (B(y,2))}\right] . \end{aligned}$$

(5.6)

The idea is to estimate the right-hand side of (5.6) in terms of the $\Vert \cdot \Vert _{\varepsilon }$-norm of $u_{\varepsilon }$ in the exterior of a large ball. With this in mind, we consider $|y|>2R + 2$, with $R>1$, so that $B(y,2)\subset B^c_{|y|-2}\subset B^c_{2 R}$.

First, recalling that (V) implies $a\le 5V(x)|y|^\alpha$ for any $x\in B(y,2)$ and $|y|>2$, we have

$$\begin{aligned} \int _{B(y,2)}u_\varepsilon ^2(x)\, \mathrm{d}x\le \frac{5}{a} |y|^{\alpha }\int _{B(y,2)}V(x)u_\varepsilon ^2(x)\, \mathrm{d}x \le \frac{5}{a}|y|^{\alpha }\int _{B^c_{|y|-2}}\left( \varepsilon ^2 |\nabla u_\varepsilon |^2+V(x)u_\varepsilon ^2\right) \, \mathrm{d}x. \end{aligned}$$

Next, we will estimate the second term on the right-hand side of (5.6). To this aim, it is useful to point out that Corollary 5.4 gives the validity of the main assumption (2.10) of Lemma 2.6; indeed, there exists $\varepsilon _0 \in (0,1]$ and $M>0$ such that

$$\begin{aligned} \Vert u_{\varepsilon } \Vert _{\varepsilon }^2 \le M \varepsilon ^2 \quad \forall \varepsilon \in (0, \varepsilon _0). \end{aligned}$$

(5.7)

From (3.1), for any $\gamma >0$, we get

$$\begin{aligned} g^2_{\varepsilon }(x) \le C_1 Q(x)u_{\varepsilon }^2 + C_2(\gamma ) Q(x) \left( \mathrm{e}^{\gamma u_{\varepsilon }^2} -1 \right) , \quad x \in \mathbb {R}^2, \end{aligned}$$

and hence

$$\begin{aligned} {\begin{matrix} \int _{B(y,2)}g_\varepsilon ^2(x)\, \mathrm{d}x&{}\le C_1 \int _{B^c_{|y|-2}}Q(x)u_\varepsilon ^2\, \mathrm{d}x+ C_2(\gamma )\int _{B^c_{|y|-2}}Q(x)\left( \mathrm{e}^{\gamma u_\varepsilon ^2}-1\right) \, \mathrm{d}x\\ &{}\le 2 \frac{b}{a} C_1 \int _{B^c_{|y|-2}}V(x)u_\varepsilon ^2\, \mathrm{d}x+ C_2(\gamma )\int _{B^c_{|y|-2}}Q(x)\left( \mathrm{e}^{\gamma u_\varepsilon ^2}-1\right) \, \mathrm{d}x, \end{matrix}} \end{aligned}$$

(5.8)

where in the last inequality we used (V) and (Q) to deduce that $Q/V\le 2 b/a$ on $B_1^c$. To estimate the second term on the right-hand side of (5.8), we apply Lemma 2.6. This can be done, in view of (5.7), if we choose $\gamma \in (0, \overline{\gamma })$ and $R\ge \overline{R}$, with $\overline{\gamma }$ and $\overline{R}$ given by Lemma 2.6; since $\varepsilon _0 \in (0,1]$, the choices of both $\gamma$ and $R_0$ are independent of $\varepsilon \in (0, \varepsilon _0)$. In this way, we get

$$\begin{aligned} \int _{B(y,2)}g_\varepsilon ^2(x)\, \mathrm{d}x \le C_3 \int _{B^c_{|y|-2}}\left( \varepsilon ^2 |\nabla u_\varepsilon |^2+V(x)u_\varepsilon ^2\right) \, \mathrm{d}x, \end{aligned}$$

with $C_3>0$ independent of $\varepsilon \in (0,\varepsilon _0)$ and y, provided $|y|>2R+2$.

What we proved so far shows the existence of $R_0>1$ such that, for any $\varepsilon \in (0, \varepsilon _0)$ and any $y \in \mathbb {R}^2$ with $|y|> 2 R_0 +2$, we have

$$\begin{aligned}\Vert u_{\varepsilon }\Vert _{L^\infty ( B(y,1))} \le C'' \varepsilon ^{-d}|y|^{\alpha /2} \left[ \int _{B^c_{|y|-2}}\left( \varepsilon ^2 |\nabla u_\varepsilon |^2+V(x)u_\varepsilon ^2\right) \, \mathrm{d}x\right] ^{1/2}.\end{aligned}$$

Finally, in view of (5.7), we can apply Lemma 4.2. Again, this can be done if we choose $R \ge R_0$, with $R_0$ given by Lemma 4.2, and this choice is independent of $\varepsilon \in (0, \varepsilon _0)$. Since $|y|-2\ge \frac{|y|}{2}$, we easily obtain (5.5) and the proof is complete.$\square$

Remark 5.6

Any weak solution $u_\varepsilon$ to ($\mathcal P_{\varepsilon }$) satisfies $u_{\varepsilon } \in L^\infty (\mathbb R^2)$, for any $\varepsilon >0$. In fact, let $\varepsilon >0$ be fixed, and let $u_{\varepsilon }$ be any weak solution to ($\mathcal P_{\varepsilon }$). Clearly, $u_{\varepsilon }$ satisfies (2.10), or equivalently (4.1), with $M=M(\varepsilon ):= \Vert u_{\varepsilon } \Vert _{\varepsilon }^{2}/\varepsilon ^{2}.$ Therefore, we can argue as in the proof of Theorem 5.5 to conclude the validity of (5.5) with C and R depending on $\varepsilon$.

5.4 Concentration behavior of mountain pass solutions

The validity of the pointwise uniform exponential decay estimate expressed by Theorem 5.5 will play a crucial role, and from now on we will implicitly assume $\varepsilon >0$ to be sufficiently small.

Let $x_\varepsilon \in \mathbb {R}^2$ be any local maximum point of $u_\varepsilon$. As in [6, Lemmas 24 and 25], we show that $\{x_\varepsilon \}$ is bounded in $\mathbb {R}^2$ and $\Vert u_\varepsilon \Vert _\infty \ge c>0$. More precisely, we have the following result.

Lemma 5.7

There exist $c>0$ and $C>0$ such that for any local maximum point $x_{\varepsilon } \in \mathbb {R}^2$ of $u_\varepsilon$, it holds

• (I) $u_\varepsilon (x_\varepsilon )\ge c$,

• (II) $|x_\varepsilon |\le C$.

Proof

By contradiction, we assume that $u_\varepsilon (x_\varepsilon )\rightarrow 0$, for a sequence of local maximum points $x_{\varepsilon } \in \mathbb {R}^2$. Since $x_\varepsilon$ is a maximum point of $u_\varepsilon$, we have $\Delta u_\varepsilon (x_\varepsilon )\le 0$, and hence

$$\begin{aligned} V(x_\varepsilon )u_\varepsilon (x_\varepsilon )\le -\Delta u_\varepsilon (x_\varepsilon )+V(x_\varepsilon )u_\varepsilon (x_\varepsilon )=Q(x_\varepsilon )f(u_\varepsilon (x_\varepsilon )). \end{aligned}$$

Recalling that $u_\varepsilon >0$ and $\min \nolimits _{x\in \mathbb {R}^2}\mathcal A(x)>0$, where $\mathcal A=V/Q$, we deduce that

$$\begin{aligned} 0< \min _{x\in \mathbb {R}^2}\mathcal A(x)\le \frac{V(x_\varepsilon )}{Q(x_\varepsilon )}\le \frac{f(u_\varepsilon (x_\varepsilon ))}{u_\varepsilon (x_\varepsilon )}, \end{aligned}$$

which is in contradiction with condition ($f_1$). Thus, (I) holds. To prove (II), it is enough to point out that if $|x_\varepsilon |\rightarrow +\infty$ then Theorem 5.5 yields $u_\varepsilon (x_\varepsilon )\rightarrow 0$, which contradicts (I).$\square$

In view of Lemma 5.7, given $\{x_{\varepsilon }\}$, where $x_\varepsilon \in \mathbb {R}^2$ is a local maximum point of $u_\varepsilon$, we have $x_\varepsilon \rightarrow x^*$ in $\mathbb {R}^{2}$ as $\varepsilon \rightarrow 0$, up to a subsequence. If we introduce the auxiliary function

$$\begin{aligned}\psi _\varepsilon (x)=u_\varepsilon (\varepsilon x+x_\varepsilon ), \quad x \in \mathbb {R}^{2},\end{aligned}$$

then $x=0$ is a local maximum point of $\psi _\varepsilon$. Moreover, $\psi _{\varepsilon }$ solves

$$\begin{aligned} -\Delta \psi _\varepsilon (x)+V(\varepsilon x+x_\varepsilon )\psi _\varepsilon (x)= Q(\varepsilon x+x_\varepsilon )f(\psi _\varepsilon (x)), \quad x \in \mathbb {R}^2, \end{aligned}$$

(5.9)

and we can compute

$$\begin{aligned}\int _{\mathbb {R}^{2}} \left( |\nabla \psi _{\varepsilon }|^{2} + V(\varepsilon x + x_{\varepsilon }) \psi _{\varepsilon }^{2} \right) \, \mathrm{d}x = \varepsilon ^{-2} \Vert u_{\varepsilon } \Vert _{\varepsilon }^{2}.\end{aligned}$$

Using again condition (V) and the boundedness of $\{x_{\varepsilon }\}$ expressed by Lemma 5.7(II), we have the existence of a constant $\tilde{a}>0$ such that

$$\begin{aligned} \int _{\mathbb {R}^2}\left( |\nabla \psi _\varepsilon |^2+\frac{\tilde{a}}{1+|x|^\alpha }\psi _\varepsilon ^2(x)\right) \, \mathrm{d}x\le \varepsilon ^{-2} \Vert u_{\varepsilon } \Vert _{\varepsilon }^{2} \le M, \end{aligned}$$

where we also applied Corollary 5.4. Therefore, by Schauder estimates, $\{\psi _\varepsilon \}$ is bounded in $\mathcal {C}^2_\mathrm{loc}(\mathbb {R}^2)$, and hence $\psi _\varepsilon \rightarrow U^{*}$ in $\mathcal {C}^2_\mathrm{loc}(\mathbb {R}^2)$ as $\varepsilon \rightarrow 0$. Passing to the limit in (5.9), we deduce that $U^{*}$ is a classical solution to the limiting problem (5.1) with $\xi =x^{*}$, i.e.,

$$\begin{aligned} -\Delta U^{*} +V(x^{*})U^{*}=Q(x^{*})f(U^{*})\quad \text {in}\quad \mathbb {R}^2. \end{aligned}$$

(5.10)

Arguing as in Theorem 5.5, we deduce that $\psi _\varepsilon \rightarrow U^*$ in $L^\infty (\mathbb {R}^2)$ as $\varepsilon \rightarrow 0$. The positivity of $\psi _{\varepsilon }$ yields $U^*\ge 0$, and since $x=0$ is a maximum point of $\psi _{\varepsilon }$ then $x=0$ is also a maximum point of $U^{*}$. Moreover, $\psi _{\varepsilon }(0)=u_{\varepsilon }(x_{\varepsilon })$ and, as a consequence of Lemma 5.7(I), $U^{*}(0)\ne 0$ which yields $U^{*} \not \equiv 0$. It follows from the maximum principle that $U^*>0$. Because $\psi _\varepsilon \rightarrow U^{*}$ in $\mathcal {C}^2_\mathrm{loc}(\mathbb {R}^2)$, for any $R>0$, we have

$$\begin{aligned} \int _{B_{R}}\left( |\nabla U^{*}|^2+V(x^{*})(U^{*})^2\right) \, \mathrm{d}x= & {} \lim _{\varepsilon \rightarrow 0}\int _{B_{R}}\left( |\nabla \psi _\varepsilon |^2+V(\varepsilon x+x_\varepsilon )\psi _\varepsilon ^2\right) \, \mathrm{d}x\\\le & {} \limsup _{\varepsilon \rightarrow 0}\varepsilon ^{-2}\Vert u_\varepsilon \Vert _\varepsilon ^2\le M, \end{aligned}$$

and hence $U^{*}\in H^1(\mathbb {R}^2)$.

Next, we consider the energy functional $L_{x^{*}}: \, H^1(\mathbb {R}^2)\rightarrow \mathbb {R}$ defined in (5.2) with $\xi = x^{*}$.

Lemma 5.8

The following inequality holds

$$\begin{aligned} L_{x^{*}}(U^{*})\le \liminf _{\varepsilon \rightarrow 0}\varepsilon ^{-2}I_\varepsilon (u_\varepsilon )=\liminf _{\varepsilon \rightarrow 0}\varepsilon ^{-2}c_\varepsilon . \end{aligned}$$

Proof

We have

$$\begin{aligned} \varepsilon ^{-2} c_\varepsilon = \varepsilon ^{-2}I_\varepsilon (u_\varepsilon )=\frac{1}{2}\int _{\mathbb {R}^2}\left( |\nabla \psi _\varepsilon |^2+V(\varepsilon x+x_\varepsilon )\psi _\varepsilon ^2\right) \, \mathrm{d}x-\int _{\mathbb {R}^2}Q(\varepsilon x+x_\varepsilon )F(\psi _\varepsilon )\, \mathrm{d}x, \end{aligned}$$

and we define

$$\begin{aligned} h_\varepsilon (x)=\frac{1}{2}\left( |\nabla \psi _\varepsilon |^2+V(\varepsilon x+x_\varepsilon )\psi _\varepsilon ^2\right) -Q(\varepsilon x+x_\varepsilon )F(\psi _\varepsilon ), \end{aligned}$$

(5.11)

so that

$$\begin{aligned} \varepsilon ^{-2} c_\varepsilon = \int _{\mathbb {R}^{2}} h_{\varepsilon }(x) \, \mathrm{d}x. \end{aligned}$$

(5.12)

For any $R>0$,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{B_{R}}h_\varepsilon (x)\, \mathrm{d}x=\frac{1}{2}\int _{B_{R}}\left( |\nabla U^{*}|^2+V(x^{*})(U^{*})^2\right) \, \mathrm{d}x-\int _{B_{R}}Q(x^{*})F(U^{*})\, \mathrm{d}x. \end{aligned}$$

Since $U^{*}\in H^1(\mathbb {R}^2)$, for any $\nu >0$ there exists $R_{\nu }>0$ such that

$$\begin{aligned} \int _{B^c_{R}}\left( |\nabla U^{*}|^2+V(x^{*})(U^{*})^2\right) \, \mathrm{d}x<\nu , \text { for any } R \ge R_{\nu }, \end{aligned}$$

and hence

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{B_{R}}h_\varepsilon (x)\, \mathrm{d}x\ge & {} \frac{1}{2}\int _{\mathbb {R}^2}\left( |\nabla U^{*}|^2+V(x^{*})(U^{*})^2\right) \, \mathrm{d}x-\int _{\mathbb {R}^2}Q(x^{*})F(U^{*})\, \mathrm{d}x-\frac{\nu }{2}\nonumber \\= & {} L_{x^{*}}(U^{*})-\frac{\nu }{2}. \end{aligned}$$

(5.13)

In view of (5.12), the proof is complete if we show that, for some $R \ge R_{\nu }$,

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0}\int _{B^c_{R}}h_\varepsilon (x)\, \mathrm{d}x\ge \frac{-\nu }{2}. \end{aligned}$$

To this aim, we consider $\eta \in \mathcal {C}^\infty (\mathbb {R}^2,[0,1])$ satisfying

$$\begin{aligned} \eta \equiv 0\,\,\text{ in }\,\,B_{R-1},\quad \eta \equiv 1\,\,\text{ in }\,\,B^c_{R},\quad \text{ and }\quad |\nabla \eta (x)|\le 2\,\, \forall x\in \mathbb {R}^2. \end{aligned}$$

Using $\eta \psi _\varepsilon$ as a test function in equation (5.9), by ($f_2$) we obtain

$$\begin{aligned} {\begin{matrix} 0=\,&{}\int _{B^c_{R}}\left( |\nabla \psi _\varepsilon |^2+V(\varepsilon x+x_\varepsilon )\psi _\varepsilon ^2\right) \, \mathrm{d}x- \int _{B^c_{R}}Q(\varepsilon x+x_\varepsilon )f(\psi _\varepsilon )\psi _\varepsilon \, \mathrm{d}x\\ &{}+\int _{B_R\backslash B_{R-1}}\left[ |\nabla \psi _\varepsilon |^2\eta +\psi _\varepsilon \nabla \eta \nabla \psi _\varepsilon +V(\varepsilon x+x_\varepsilon )\eta \psi _\varepsilon ^2- Q(\varepsilon x+x_\varepsilon )f(\psi _\varepsilon )\eta \psi _\varepsilon \right] \, \mathrm{d}x\\ =\,&{}2\int _{B^c_{R}}h_\varepsilon \, \mathrm{d}x-\int _{B^c_{R}}Q(\varepsilon x+x_\varepsilon )\left[ f(\psi _\varepsilon )\psi _\varepsilon -2F(\psi _\varepsilon )\right] \, \mathrm{d}x +E_\varepsilon \le 2\int _{B^c_{R}}h_\varepsilon \, \mathrm{d}x+E_\varepsilon \end{matrix}} \end{aligned}$$

where

$$\begin{aligned} E_\varepsilon :=\int _{B_R\backslash B_{R-1}}\left[ |\nabla \psi _\varepsilon |^2\eta +\psi _\varepsilon \nabla \eta \nabla \psi _\varepsilon +V(\varepsilon x+x_\varepsilon )\eta \psi _\varepsilon ^2- Q(\varepsilon x+x_\varepsilon )f(\psi _\varepsilon )\eta \psi _\varepsilon \right] \, \mathrm{d}x. \end{aligned}$$

The convergence $\psi _\varepsilon \rightarrow U^{*}$ in $B_R\backslash B_{R-1}$, as $\varepsilon \rightarrow 0$, implies $\limsup \nolimits _{\varepsilon \rightarrow 0} |E_\varepsilon |\le \nu$ for $R \ge R_{\nu }$ sufficiently large, and hence

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0}\int _{B^c_{R}}h_\varepsilon (x)\, \mathrm{d}x\ge \liminf _{\varepsilon \rightarrow 0}\frac{-E_\varepsilon }{2}\ge \frac{-\nu }{2}. \end{aligned}$$

$\square$

Lemma 5.9

The limit point $x^{*}$ is a global minimum point of the auxiliary potential $\mathcal A=V/Q$ and $U^{*}$ is a ground-state solution of the limiting problem (5.10).

Proof

Let $\xi \in \mathbb {R}^{2}$ and let $m_\xi$ be the least energy level of the functional $L_\xi$ introduced in (5.2). By contradiction, we assume the existence of $\xi ^{*}\in \mathbb {R}^2$ such that $\mathcal A(\xi ^{*})<\mathcal A(x^{*})$. Recalling (5.4), then we have $m_{\xi ^{*}}<m_{x^{*}}$. On the other hand, combining Proposition 5.3 and Lemma 5.8, we get

$$\begin{aligned} m_{x^{*}}\le L_{x^{*}}(U^{*})\le \liminf _{\varepsilon \rightarrow 0}\varepsilon ^{-2} I_\varepsilon (u_\varepsilon )\le m_\xi ,\quad \forall \xi \in \mathbb {R}^2. \end{aligned}$$

Therefore, $x^{*}$ must be a global minimum point of $\mathcal A$. Moreover, $m_{x^{*}}= L_{x^{*}}(U^{*})$, and $U^{*}$ is a ground-state solution of (5.10). $\square$

Lemma 5.10

If $x_\varepsilon$ and $y_\varepsilon$ are maximum points of $u_\varepsilon$ then $|x_{\varepsilon } - y_{\varepsilon }| \le C \varepsilon$. If in addition f is increasing on $(0,+ \infty )$, then $u_{\varepsilon }$ has a unique maximum point in $\mathbb {R}^{2}$.

Proof

Let $x_\varepsilon$ and $y_\varepsilon$ be maximum points of $u_\varepsilon$. In view of Lemma 5.7(II), without loss of generality, we can assume that $x_\varepsilon \rightarrow x^{*}$ and $y_\varepsilon \rightarrow y^{*}$, in $\mathbb {R}^{2}$. We proved that the rescaled functions $\psi _\varepsilon (x):=u_\varepsilon (\varepsilon x+x_\varepsilon )$ and $\tilde{\psi }_\varepsilon (x):=u_\varepsilon (\varepsilon x+y_\varepsilon )$ satisfy $\psi _{\varepsilon } \rightarrow U^{*}$ and $\tilde{\psi }_{\varepsilon } \rightarrow \tilde{U}^{*}$ in $C^{2}_\mathrm{loc}(\mathbb {R}^{2})$, where $U^{*}$ and $\tilde{U}^{*}$ are positive ground-state solutions of the limiting problem (5.1) with $\xi =x^{*}$ and $\xi =y^{*}$, respectively. We also pointed out that $x=0$ is a maximum point of both $U^*$ and $\tilde{U}^*$. Let $z_{\varepsilon }:=\varepsilon ^{-1}(y_\varepsilon -x_\varepsilon )$. Clearly, we have the following alternative:

• (i) either $\{z_{\varepsilon }\}$ is bounded, and up to a subsequence, $z_{\varepsilon } \rightarrow z^{*}$ in $\mathbb {R}^2$;

• (ii) or $\{z_{\varepsilon }\}$ is unbounded, and up to a subsequence, $|z_{\varepsilon }|\rightarrow \infty$.

The first part of the statement of the lemma is equivalent to show that (ii) cannot occur. Assume by contradiction that $|z_{\varepsilon }|\rightarrow \infty$. For any $R>0$, the balls $B_{R}$ and $B\left( z_{\varepsilon },R \, \right)$ are disjoint, provided $\varepsilon >0$ is sufficiently small. Let $h_\varepsilon$ be the function defined by (5.11) in terms of $\psi _{\varepsilon }$. In particular (5.13) holds, i.e., for any $\nu >0$ there exits $R=R(\nu )>0$ such that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _{B_{R}}h_\varepsilon (x)\,\mathrm{d}x\ge L_{x^{*}}(U^{*})-\frac{\nu }{2}, \end{aligned}$$

and arguing as in the proof of Lemma 5.8 (after a suitable change of variable), we get also

$$\begin{aligned}\lim _{\varepsilon \rightarrow 0}\int _{B(z_{\varepsilon },R )}h_\varepsilon (x)\,\mathrm{d}x\ge L_{y^{*}}(\tilde{U}^{*})-\frac{\nu }{2}, \quad \text {along with} \quad \liminf _{\varepsilon \rightarrow 0}\int _{(\, B(z_{\varepsilon },R)\cup B_{R} \,)^c}h_\varepsilon (x)\,\mathrm{d}x\ge -\nu . \end{aligned}$$

Therefore,

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \int _{\mathbb {R}^2}h_\varepsilon (x)\,\mathrm{d}x\ge L_{x^{*}}(U^{*})+L_{y^{*}}(\tilde{U}^{*})-2\nu , \end{aligned}$$

and combining (5.12) with the fact that $\nu >0$ is arbitrary, we get

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \varepsilon ^{-2} c_\varepsilon = \liminf _{\varepsilon \rightarrow 0} \int _{\mathbb {R}^2}h_\varepsilon (x)\,\mathrm{d}x\ge L_{x^{*}}(U^{*})+L_{y^{*}}(\tilde{U}^{*}). \end{aligned}$$

Proposition 5.3 yields $\liminf _{\varepsilon \rightarrow 0} \varepsilon ^{-2} c_\varepsilon \le m_\xi$, for all $\xi \in \mathbb {R}^2$. Since $x^{*}$ and $y^{*}$ are global minimum points of $\mathcal A$, we have $\mathcal A(x^{*})=\mathcal A(y^{*})$ and $m_{x^{*}}=m_{y^{*}}$ (see Remark 5.2). Hence

$$\begin{aligned} L_{x^{*}}(U^{*})+L_{y^{*}}(\tilde{U}^{*})\le \frac{1}{2}\left( m_{x^{*}}+m_{y^{*}}\right) =\frac{1}{2}\left( L_{x^{*}}(U^{*})+L_{y^{*}}(\tilde{U}^{*})\right) , \end{aligned}$$

which is impossible, and this rules out (ii).

From now on, we furthermore assume that f is increasing on $(0,+ \infty )$, and we aim to show that $x_{\varepsilon }=y_{\varepsilon }$. To this aim, it is enough to prove that (i) cannot occur unless $x_{\varepsilon }=y_{\varepsilon }$.

Since $U^{*}$ is a ground-state solution to the limiting problem (5.1) with $\xi =x^{*}$, it is well known that, for some $x_0\in \mathbb R^2$, $U^{*}(\cdot +x_0)$ is radially symmetric and strictly decreasing (see [8] and [7]). Since $x=0$ is a maximum point of $U^*$, we deduce that $x_0=0$. Using the symmetry, positivity and monotonicity properties of $U^{*}$ together with the monotonicity of f, we can easily show that $\Delta U^{*}(0)<0$. In particular, $U^{*}_{rr}(0)<0$, with $r=|x|$, and in view of the convergence $\psi _\varepsilon \rightarrow U^*$ in $C^2_\mathrm{loc}(\mathbb {R}^2)$, we can apply [29, Lemma 4.2] to get the existence of $R>0$ such that, for any small $\varepsilon >0$, $x=0$ is the unique critical point of $\psi _\varepsilon$ in $B_{R}$.

If (i) occurs, since $z_{\varepsilon }$ is a maximum point of $\psi _\varepsilon$, then $z^{*}$ is a maximum point of $U^{*}$, and hence $z^{*}=0$. Thus, $z_{\varepsilon }\in B_{R}$ for any small $\varepsilon >0$, and the choice of R implies $z_{\varepsilon }=0$, equivalently $x_{\varepsilon }=y_{\varepsilon }$. Therefore, for small $\varepsilon >0$, $u_\varepsilon$ has a unique local (hence global) maximum point in $\mathbb {R}^2$. $\square$

Remark 5.11

The monotonicity assumption on f in Lemma 5.10 is required to guarantee that $U^{*}_{rr}(0)<0$. The same conclusion can be reached, by means of a classical fixed point argument, if f is locally Lipschitz continuous.

6 Critical case—existence

This section is devoted to the proof of Theorem 1.4. More precisely, for any fixed $\varepsilon >0$, we prove the existence of a mountain pass solution to ($\mathcal P_{\varepsilon }$) under the assumptions:

• (V) and (Q) hold with $\alpha$ and $\beta$ in the range (1.1),

• $f: \mathbb {R}\rightarrow \mathbb {R}$ is a continuous function, with critical exponential growth ($f_0'$), satisfying ($f_1$), ($f_2$), ($f_3$) and ($f_4'$).

The proof follows the same lines of [21, Section 5], with obvious modifications due to the presence of the parameter $\varepsilon >0$, and it is based on the classical mountain pass theorem. Therefore, we will only sketch the main steps needed to ensure that the mountain pass theorem can actually be applied to obtain a nontrivial mountain pass solution. For the proof of the fact that mountain pass solutions belong to $L^2(\mathbb {R}^2)$, we refer the reader to [21, Proposition 2.2].

Let

$$\begin{aligned}I_{\varepsilon }(u):= \frac{1}{2} \, \Vert u\Vert ^2_{\varepsilon } - \int _{\mathbb {R}^2} Q(x) F(u) \, \mathrm{d}x \quad \forall u \in \mathcal H_{\varepsilon },\end{aligned}$$

and

$$\begin{aligned} c_{\varepsilon }:= \inf _{ \gamma \in \Gamma _{\varepsilon }} \, \sup _{t \in [0,1]} \, I_{\varepsilon }(\gamma (t)), \quad \text {where } \Gamma _{\varepsilon }:= \{\, \gamma \in \mathcal C([0,1], \mathcal H_{\varepsilon }) \, | \, \gamma (0)=0 \text { and } I_{\varepsilon }(\, \gamma (1)\,) <0 \,\}. \end{aligned}$$

• Step 1$I_{\varepsilon } \in \mathcal C^1 (\mathcal H_{\varepsilon }, \mathbb {R})$, $I_{\varepsilon }$ has a mountain pass geometry on $\mathcal H_{\varepsilon }$, and hence the mountain pass level $c_{\varepsilon }$ is well defined and positive. In view of ($f_0'$) and ($f_1$), we have the validity of (3.1) and (3.2) for any $\gamma > \gamma _0$, $q \ge 0$ and $\delta >0$ (here, $\gamma _0$ is given by ($f_0'$) and determines the critical growth of f at infinity). This is enough to deduce, as in Sect. 6, that $I_{\varepsilon }$ is a $\mathcal C^1$-functional on $\mathcal H_\varepsilon$. Moreover, the conclusion of Lemma 3.1 still holds (see [21, Lemma 5.1]).

• Step 2 We have

  $$\begin{aligned} c_\varepsilon <2\pi \varepsilon ^2/\gamma _0, \end{aligned}$$

  (6.1)

  where $\gamma _0$ is given by condition ($f_0'$). In view of ($f_4'$), the proof of [21, Lemma 5.2] can be repeated using the same classical Moser’s sequence of functions and taking into account the presence of the parameter $\varepsilon >0$. Arguing in this way, we deduce (6.1).

• Step 3 Let $c \in \mathbb {R}$, and let $\{u_n\}_n \subset \mathcal H_{\varepsilon }$ be any $(PS)_c$-sequence for $I_{\varepsilon }$. Then $\{u_n\}_n$ is bounded in $\mathcal H_{\varepsilon }$, and

  $$\begin{aligned} \sup _{n} \, \int _{\mathbb {R}^{2}} Q(x)f(u_{n}) u_{n} \, \mathrm{d}x< + \infty . \end{aligned}$$

  (6.2)

  Moreover, if $u_n \rightharpoonup u_{\varepsilon }$ in $\mathcal H_{\varepsilon }$ then

  $$\begin{aligned} \lim _{n \rightarrow + \infty }\int _{\mathbb {R}^2}Q(x)F(u_n) \mathrm{d}x= \int _{\mathbb {R}^2}Q(x)F(u_\varepsilon ) \mathrm{d}x. \end{aligned}$$

  (6.3)

  In view of ($f_2$), the proof of the boundedness of $\{u_n\}_n$ and the proof of (6.2) are standard and can be carried out as in [21, Lemma 5.3]. Moreover, using ($f_3$) and the compactness of the embedding $\mathcal H_{\varepsilon } \hookrightarrow L^2_Q(\mathbb {R}^2)$ expressed by Theorem 2.5, (6.3) can be deduced as in [21, Lemma 5.4].

• Step 4$I_{\varepsilon }$ satisfies the $(PS)_{c_{\varepsilon }}$-condition at the mountain pass level $c_{\varepsilon }$. We follow closely the arguments introduced in [16]. Let $\{u_n\}_n$ be a $(PS)_{c_{\varepsilon }}$-sequence for $I_{\varepsilon }$, i.e.,

  $$\begin{aligned} \lim _{n \rightarrow + \infty } I_{\varepsilon }(u_n)=c_{\varepsilon }, \end{aligned}$$

  (6.4)

  and

  $$\begin{aligned} \biggl |\, \langle u_n, v \rangle _{\varepsilon } - \int _{\mathbb {R}^2} Q(x) f(u_n) v \, \mathrm{d}x \,\biggr | \le \delta _n \Vert v\Vert _{\varepsilon } \quad \forall v \in \mathcal H_{\varepsilon }, \text { where } \lim _{n \rightarrow + \infty } \delta _n=0. \end{aligned}$$

  (6.5)

  In view of the boundedness of $\{u_n\}_n$ in $\mathcal H_{\varepsilon }$, up to a subsequence, $u_n \rightharpoonup u_{\varepsilon }$ in $\mathcal H_{\varepsilon }$.

First, we prove that $u_{\varepsilon } \not \equiv 0$. If not then $u_{\varepsilon } \equiv 0$, and combining (6.4) with (6.3), we get

$$\begin{aligned}\lim _{n\rightarrow \infty }\Vert u_n\Vert _\varepsilon ^2=\lim _{n\rightarrow \infty }2\biggl [c_\varepsilon +\int _{\mathbb {R}^2}Q(x)F(u_n) \mathrm{d}x\biggr ]=2c_\varepsilon >0.\end{aligned}$$

On the other hand, using the compact embeddings expressed by Theorem 2.5 and arguing as in [21, Section 5.3], we deduce that $u_n \rightarrow 0$ strongly in $\mathcal H_{\varepsilon }$, and we reach a contradiction.

Clearly, $I_{\varepsilon }(u_{\varepsilon }) \le c_{\varepsilon }$. Our aim is to show the strong convergence $u_n \rightarrow u_{\varepsilon }$ in $\mathcal H_{\varepsilon }$, and this is equivalent to $I_{\varepsilon }(u_{\varepsilon })=c_{\varepsilon }$. We will prove that if

$$\begin{aligned} I_{\varepsilon }(u_{\varepsilon })< c_{\varepsilon } \end{aligned}$$

(6.6)

then

$$\begin{aligned} \lim _{n \rightarrow + \infty } \int _{\mathbb {R}^2} Q(x)f(u_n)(u_n - u_{\varepsilon }) \, \mathrm{d}x=0, \end{aligned}$$

(6.7)

and hence (6.6) cannot happen. Indeed, (6.5) and (6.7) yield $\langle u_n, u_n - u_{\varepsilon } \rangle _{\varepsilon }\rightarrow 0$, and we deduce that $u_n \rightarrow u_{\varepsilon }$ strongly in $\mathcal H_{\varepsilon }$ which contradicts (6.6).

To prove (6.7), let $\gamma > \gamma _0$, so that (3.1) holds with $q \ge 0$ and $\delta >0$. Using (3.1) with $q=0$, using the continuity of the embedding $\mathcal H_{\varepsilon } \hookrightarrow L^2_Q(\mathbb {R}^2)$ expressed by Theorem 2.5, and the boundedness of $\{u_n\}_n$ in $\mathcal H_{\varepsilon }$, for any $\delta >0$ we have

$$\begin{aligned} \biggl | \int _{\mathbb {R}^2} Q(x)&f(u_n)(u_n -u_{\varepsilon }) \, \mathrm{d}x \biggr | \le C_1 \delta + C_2(\gamma , \delta ) \int _{\mathbb {R}^2} Q(x) |u_n - u_{\varepsilon }| (\mathrm{e}^{\gamma u_n^2} -1 ) \, \mathrm{d}x \\&\le C_1 \delta + C_2(\gamma , \delta ) \biggl (\int _{\mathbb {R}^2} Q(x) |u_n-u_{\varepsilon }|^{p'} \, \mathrm{d}x\biggr )^\frac{1}{p'} \biggl (\int _{\mathbb {R}^2} Q(x) (\mathrm{e}^{\gamma p u_n^2} -1 ) \, \mathrm{d}x\biggr )^\frac{1}{p}, \end{aligned}$$

where in the last estimate we applied Hölder’s inequality with $p>1$ and $\frac{1}{p} + \frac{1}{p'}=1$. Note that, Theorem 2.5 yields the compactness of the embedding $\mathcal H_{\varepsilon } \hookrightarrow L^{p'}_Q(\mathbb {R}^2)$ for any $p' \ge 2$. Therefore, if we assume the existence of $\gamma > \gamma _0$ and $p \in (1,2]$, such that

$$\begin{aligned} \sup _n E_n(\gamma , p) < \infty , \quad \text {where} \quad E_n(\gamma , p):= \int _{\mathbb {R}^2} Q(x) (\mathrm{e}^{\gamma p u_n^2} -1 ) \, \mathrm{d}x \end{aligned}$$

(6.8)

then

$$\begin{aligned}\biggl | \int _{\mathbb {R}^2} Q(x) f(u_n)(u_n -u_{\varepsilon }) \, \mathrm{d}x \biggr | \le C_1 \delta + C_2(\gamma , \delta ) \biggl (\sup _n E_n(\gamma , p)\biggr )^\frac{1}{p} \biggl (\int _{\mathbb {R}^2} Q(x) |u_n-u_{\varepsilon }|^{p'} \, \mathrm{d}x\biggr )^\frac{1}{p'},\end{aligned}$$

and the proof of (6.7) is complete.

We obtain (6.8) as a consequence of the following Lions-type concentration-compactness lemma in $\mathcal H_{\varepsilon }$.

Lemma 6.1

Suppose that (V) and (Q) hold with $\alpha \in (0,2)$ and $\beta \in [\alpha , + \infty )$, respectively, and let $\varepsilon >0$. If $\{v_n\}_n \subset \mathcal H_{\varepsilon }$ satisfies $\Vert v_n\Vert _{\varepsilon }=\varepsilon$ for any $n \ge 1$, and $v_n \rightharpoonup v_{\varepsilon }$ in $\mathcal H_{\varepsilon }$ with $\Vert v_{\varepsilon }\Vert _{\varepsilon }< \varepsilon$ then

$$\begin{aligned} \sup _n \int _{\mathbb {R}^2} Q(x) (\mathrm{e}^{qv_n^2} -1) \, \mathrm{d}x < + \infty \quad \forall q \in \biggl (0, \frac{\varepsilon ^2}{\varepsilon ^2 - \Vert v_{\varepsilon }\Vert _{\varepsilon }^2} \, 4 \pi \biggr ). \end{aligned}$$

(6.9)

Proof

In view of the subcritical Trudinger–Moser inequality expressed by Theorem 1.1, and recalling that $\mathcal H_{\varepsilon }$ is a Hilbert space, the proof of (6.9) follows the same lines of [19, Lemma 2.6]. $\square$

We apply the above lemma to

$$\begin{aligned}v_{\varepsilon }:= \varepsilon u_{\varepsilon } \biggl [2 \left( c_{\varepsilon } + \int _{\mathbb {R}^2} Q(x) F(u_{\varepsilon }) \, \mathrm{d}x \right) \biggr ]^{- \frac{1}{2}} \quad \text { and } \quad v_n:= \frac{\varepsilon u_n}{\, \Vert u_n\Vert _{\varepsilon }}.\end{aligned}$$

Indeed, $\Vert v_n\Vert _{\varepsilon }=\varepsilon$ and $v_n \rightharpoonup v_{\varepsilon }$ in $\mathcal H_{\varepsilon }$. Moreover, using (6.6), it is easy to see that $\Vert v_{\varepsilon }\Vert _{\varepsilon }< \varepsilon$, and hence (6.9) holds. Note that, since $v_{\varepsilon } \not \equiv 0$, (6.9) improves the Trudinger–Moser inequality (1.3) along the sequence $\{v_{n}\}_n$, and we can rewrite

$$\begin{aligned}\frac{\varepsilon ^2}{\varepsilon ^2 - \Vert v_{\varepsilon }\Vert _{\varepsilon }^2} \, 4 \pi = \frac{2 \pi }{c_{\varepsilon } - I_{\varepsilon }(u_{\varepsilon })} \lim _{n \rightarrow + \infty } \Vert u_n\Vert _{\varepsilon }^2.\end{aligned}$$

Clearly

$$\begin{aligned}\sup _n E_n(\gamma , p) = \sup _n \int _{\mathbb {R}^2} Q(x) \left( \mathrm{e}^{\gamma p \frac{\Vert u_{n}\Vert _{\varepsilon }^2}{\varepsilon ^2} v_n^2} -1 \right) \, \mathrm{d}x < + \infty ,\end{aligned}$$

provided the following inequality holds

$$\begin{aligned} \gamma p \Vert u_{n}\Vert _{\varepsilon }^2 < \varepsilon ^2 \frac{2 \pi }{c_{\varepsilon } - I_{\varepsilon }(u_{\varepsilon })} \lim _{n \rightarrow + \infty } \Vert u_n\Vert _{\varepsilon }^2, \end{aligned}$$

(6.10)

at least for any n sufficiently large. This is actually the case in view of the fact that $c_{\varepsilon }< 2 \pi \varepsilon ^2/\gamma _0$ and $I_{\varepsilon }(u_{\varepsilon }) \ge 0$, equivalently

$$\begin{aligned}\varepsilon ^2 \frac{2 \pi }{c_{\varepsilon } - I_{\varepsilon }(u_{\varepsilon })} > \gamma _0,\end{aligned}$$

and hence (6.10) holds for any n large if we choose $\gamma > \gamma _0$ sufficiently close to $\gamma _0$ and $p \in (1,2]$ sufficiently close to 1. With this remark, the proof of (6.8) is complete.