Abstract
We study the following Choquard type equation in the whole plane
where \(I_2\) is the Newton logarithmic kernel, V is a bounded Schrödinger potential and the nonlinearity f(x, u), whose primitive in u vanishing at zero is F(x, u), exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev–Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to (C).
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1 Introduction
Consider the following class of nonlocal equations
where \(V\ge 0\) is the external Schrödinger potential, F is the primitive function of the nonlinearity f vanishing at zero, the kernel \(I_\alpha \) is defined for \(x\in \mathbb {R}^N\setminus \{0\}\), \(N\ge 2\) by
where \(\Gamma (\cdot )\) denotes Euler’s Gamma function. Notice that passing from \(\alpha <N\) to the limiting case \(\alpha =N\) the kernel is no longer of one sign and does not decay at infinity which sets the problem in a quite different framework. By introducing the function \(\phi :=I_\alpha *F(u)\) one has that (1.1) is equivalent to the following system
which in the case \(\alpha =2\) it turns out to be the so-called Schrödinger–Poisson system which has an Hamiltonian structure and which turns out to be relevant in applications, see [5] and references therein. An extensive literature has been devoted to the higher dimensional case \(N\ge 3\) and we refer to [11, 14, 20, 27, 37] for an up to date, though non exhaustive bibliography. On the contrary, just a few results are available in the planar case. Existence results in the case \(\alpha <N=2\) have been proved in [3] in the case of power-like nonlinearities and in [2] in the case of exponential growth. However, in dimension two the equivalence between (1.1) and (1.2) in the Schrödinger–Poisson case \(\alpha =2\), carries over as long as the logarithmic kernel is taken into account. Existence and qualitative properties of solutions in the case of logarithmic kernel have been obtained in [7, 15] for power-like nonlinearities. On the other hand, the polynomial growth somehow downplays the main feature of dimension two which allows finite energy solutions to have arbitrary polynomial growth up to the exponential.
The main purpose of this paper is to tackle the problem in which one has the logarithmic kernel and the exponential growth, namely the limiting case \(\alpha =N=2\) which is in turn the Schrödinger–Poisson case. As we are going to see, the main difficulty arises in the competing presence of a too loose singular behavior of the logarithmic kernel compared with the exponential growth of the nonlinearity within the convolution, for which the problem demands for a proper function space setting. Here we develop a suitable framework in which we can prove the existence of mountain pass solutions. Let us finally mention that nonlinear terms with exponential growth outside the convolution, which cast the problem in a quite different context, have been recently considered in [1].
We will focus on the following non-autonomous problem
On the Schrödinger potential V we make the following assumptions:
- \((V_1)\):
-
\(V(x)\ge V_0>0\) in \(\mathbb {R}^2\) for some \(V_0>0\);
- \((V_2)\):
-
V(x) is a 1-periodic continuous function.
With a slight abuse of notation, we assume the nonlinearity \(f(x,s)=c(x)f(s)\) where c(x) is a strictly positive, 1-periodic continuous function and f(s) a differentiable function whose primitive vanishing at zero is F(s) and such that:
- \((f_1)\):
-
\(\displaystyle f(s)\ge 0\) for all \(s \ge 0\), \(\displaystyle f(s) \le Cs^{p} e^{4\pi s^2}\) as \(s\rightarrow +\infty \) for some \(p>0\) and \(f(s)\asymp s^{q-1}\) for some \(q > 2\) as \(s\rightarrow 0\);
- \((f_2)\):
-
\(\exists C>\delta >0\) such that \(\delta \le \frac{F(s) f'(s)}{f^2(s)}\le C \quad \forall \, s> 0\);
- \((f_3)\):
-
\(\displaystyle \lim _{s\rightarrow +\infty }\frac{F(s) f'(s)}{f^2(s)}= 1\), or equivalently \(\displaystyle \lim _{s\rightarrow +\infty }\frac{d}{d s}\frac{F(s)}{f(s)}=0\);
- \((f_4)\):
-
\(\displaystyle \lim _{s\rightarrow +\infty } \frac{s^3f(s)F(s)}{e^{8\pi s^2}}\ge \beta > \mathcal {V}\), where \(\mathcal {V}\) will be explicitly given in Sect. 5.
Since we look for positive solutions, we may also assume \(f(s)=0\) for \(s\le 0\).
Let us make a few comments on our assumptions:
-
\((f_1)\) gives the following
$$\begin{aligned} 0\le F(s)\le C\cdot \left\{ \begin{array}{ll} s^{q}, &{} s\le s_0\\ s^{p-1}e^{4\pi s^2}, &{} s>s_0 \end{array} \right. \quad {\hbox { for some }} s_0>1 \end{aligned}$$(1.4)observing that \(\int _{s_0}^s t^pe^{4\pi t^2}dt\le s^{p-1}\int _1^s te^{4\pi t^2}dt\) for any \(s>1\);
-
\((f_2)\) implies f(s) is monotone increasing in s, so that \(F(s)=\int _0^sf(\tau )d\tau \le sf(s)\). Hence, the quantity \(\frac{F(x,s)}{f(x,s)}=\frac{c(x)F(s)}{c(x)f(s)}=\frac{F(s)}{f(s)}\) is well defined and vanishes only at \(s=0\). Furthermore,
$$\begin{aligned} \frac{\partial }{\partial s}\left( \frac{F(x,s)}{f(x,s)}\right) =\frac{d}{d s}\left( \frac{F(s)}{f(s)}\right) =\frac{f^2(s)-F(s)f'(s)}{f^2(s)}\le 1 -\delta \end{aligned}$$(1.5)which implies \(F(x,s)\le (1-\delta ) sf(x,s)\) and this improves the previous Ambrosetti–Rabinowitz condition \(F(x,s)\le sf(x,s)\);
-
\((f_3)\) yields a fine estimate from below on the quotient \(\displaystyle \frac{Ff'}{f^2}\), as \(s\rightarrow +\infty \). Indeed, for any \(\varepsilon >0\) there exists \(s_\varepsilon >0\) such that:
$$\begin{aligned} \frac{Ff'}{f^2}(s)\ge \left\{ \begin{array}{ll} \delta s, &{} s\le s_\varepsilon \\ (1-\varepsilon )s, &{} s>s_\varepsilon \ ; \end{array} \right. \end{aligned}$$(1.6) -
\((f_4)\) is in the spirit of the de Figueiredo–Miyagaki–Ruf condition [16] which in dimension two turns out to be a key ingredient in order to prove compactness. Loosely speaking, it plays the role of the upper bound of the energy in terms of the Sobolev constant \((1/N)S^{N/2}\) in higher dimensions. The role of condition \((f_4)\) will be detailed in Sect. 5;
-
functions \(F_i(s)\) satisfying our set of assumptions are for instance given by:
$$\begin{aligned} \ \ F_1(s)= & {} \left\{ \begin{array}{ll} s^q, &{} s\le s_0\\ e^{4\pi s^2}-1, &{} s>s_0 \end{array}, \ \forall \, q>2; \right. \\ F_2(s)= & {} s^pe^{4\pi s^2},\ \forall \, p> 2;\\ F_3(s)= & {} \left\{ \begin{array}{ll} s^q, &{} s\le s_0\\ cs^{p}e^{4\pi s^2}, &{} s>s_0 \end{array}, \ \forall \, q>2,\, p> 1. \right. \end{aligned}$$
Accordingly to our assumptions on the nonlinearity, we will consider functionals whose asymptotic behavior near the origin is given by \(s^q\), with \(q>2\).
Consider the following weighted Sobolev space \(H_{w}^1(\mathbb {R}^2)\) which is the completion of smooth compactly supported functions with respect to the norm
Theorem 1.1
The weighted Sobolev space \(H_{w}^1(\mathbb {R}^2)\) embeds into the weighted Orlicz space \(L_{\phi }(\mathbb {R}^2, \log (e+|x|)dx)\) where \(\phi \) is the n-function \(\phi (t)=e^{t^2}-1\). More precisely, we have
for any \(u\in H^1_{w}(\mathbb {R}^2)\) and any \(\alpha >0\). Moreover, the following uniform bound holds
For \(q>2\) let us define the weighted Sobolev space \(H^1L^q_{w}(\mathbb {R}^2)\) as the completion of smooth compactly supported functions with respect to the norm
Furthermore, let
and denote with \(H^{1}_V(\mathbb {R}^2)\) the set of all functions with bounded \(\Vert \cdot \Vert _V\) norm. Let also \(w_0(x):=\log (1+|x|)\), and
and consider \(H^1_V L^q_{w_0}(\mathbb {R}^2)\) as the completion of smooth compactly supported functions with respect to the norm \(\Vert \cdot \Vert _{q,V,{w_0}}\), see (4.1) below.
In the more general case \(q>2\), we prove a Trudinger type inequality similar to Theorem 1.1, see Sect. 3. As a consequence, we have our main result
Theorem 1.2
Suppose the nonlinearity f satisfies \((f_1)\)–\((f_4)\) and that the potential V enjoys \((V_1)\)–\((V_2)\). Then, problem (1.3) possesses a nontrivial finite energy solution in the space \(H^1_VL^q_{w_0}(\mathbb {R}^2)\).
1.1 Overview
Equation (1.1) has a long history, heritage of the early studies on Polarons iniziated by Fröhlich [22] and then has been revealed a good model also in completely different contexts such as plasma physics [26] and quantum gravity [30]. We refer the interested reader to the survey [27] and references therein for more on Physical aspects of the problem.
Formally, the energy associated to problem (1.3) is given by
In order to have the energy well defined in presence of a logarithmic kernel, the authors in [15, 34], restrict the space \(H^1\) introducing a further constraint, eventually setting the problem in an intersection space in which the energy turns out to be well defined by the Hardy–Littlewood–Sobolev inequality. Our approach here is different, from one side we look for a proper function space setting in which such a natural constraint turns out to be automatically satisfied and on the other side, we wonder if this can be done by allowing the nonlinearity to exhibit exponential growth which is what we expect in dimension two, since the seminal work of Pohozaev [31] and Trudinger [36]. Indeed, we prove that the Sobolev space \(H^1\) with a logarithmic weight on the \(L^2\) mass term of the norm gives the proper function space setting in which the energy in well defined up to the natural exponential growth in the nonlinearity. Our argument throws light on the fact that, roughly speaking, as concentration phenomena in the Moser functional are controlled by the \(L^2\) norm of the gradient whereas vanishing phenomena are controlled by the \(L^2\) norm, here we prove that a suitable logarithmic weight in the \(L^2\) component of the \(H^1\) norm is enough to obtain a functional inequality which at the end yields a natural function space framework where to set up the problem. We think this result is of independent interest and that could be useful elsewhere. As pointed out also in [1, 15] an extra difficulty is given here by the lack of invariance by translations of the energy which forces to prove a priori bounds of eventually vanishing Palais–Smale sequences. Our method seems to be more natural also in this respect, as starting from any PS sequence we can prove the existence of a weak \(H^1\)-limit with no need to establish a priori bounds.
For convenience of the reader, some preliminary material is recalled in Sect. 2. In Sect. 3 we establish the fundamental embedding inequality which will provide the function space framework of Sect. 4 and that will be used throughout the paper. Section 5 is devoted to show the underlying mountain pass geometry for the energy functional and to prove mountain pass level estimates which in this case is a delicate matter. In Sect. 6 we prove compactness results by carefully analyzing the behavior of PS sequences and finally, we conclude in Sect. 7 the proof of Theorem 1.2.
2 Preliminaries
In this section we recall some well known results which will be used in the sequel.
Let \(H^1_0(\Omega )\) be the classical Sobolev space, completion of smooth compactly supported functions with respect to the Dirichlet norm \(\Vert \nabla \cdot \Vert _2\), when \(\Omega \) is a bounded subset of \(\mathbb {R}^N\), and with respect to the complete Sobolev norm \((\Vert \nabla \cdot \Vert ^2_2+\Vert \cdot \Vert _2^2)^{1/2}\), when the domain is unbounded and in particular for \(\Omega =\mathbb {R}^N\).
If \(N\ge 3\), the classical Sobolev embedding theorem reads as follows
where \(2^*:=\frac{2N}{N-2}\) is the critical Sobolev exponent and the constant S in (2.1) is the best possible [35].
When \(N=2\) is the so-called Sobolev limiting case. One has the embedding \(H^1_0(\Omega )\hookrightarrow L^p(\Omega )\) for all \(1\le p < \infty \) (see also [13] for related best constants estimates), though \(H^1_0(\Omega ) \not \subset L^{\infty }(\Omega )\). The maximal degree of summability for functions in \(H^1_0(\Omega )\) was established independently by Pohožaev [31] and Trudinger [36] (see also [38]) and is of exponential type, in a suitable Orlicž class of functions, namely
Starting from the seminal work of Moser [28] in which a sharp version of (2.2) is established, the Pohozaev–Trudinger embedding has been further developed during the last fifty years, in particular the first extension of (2.2) to unbounded domains appears in [9] for functions with bounded Sobolev’s norm in the following form
Thereafter, several sharp versions have been proved and extensions in many directions for which we refer to [12, 32]. In particular, the borderline case in which \(\alpha =4\pi \) remained uncovered until Ruf in [32] established the following inequality which is sharp in the sense of Moser [28] (so-called Trudinger–Moser type inqualities):
Remark 2.1
Note that in Ruf’s inequality (2.4) the constraint is defined through the complete Sobolev norm \(\Vert \nabla \cdot \Vert _2^2+\Vert \cdot \Vert _2^2\). As one may realize by Cao’s result, a closer inspection of the proof reveals that Ruf’s inequality still holds, at least in the subcritical case \(\alpha <4\pi \), replacing the \(L^2\) norm with any weighted \(L^2\) norm, provided the weight is bounded and also bounded away from the origin.
Remark 2.2
As an application of (2.2), consider the following functional
which is continuous on \(H^1(\mathbb {R}^2)\), a consequence of \((f_1)\), (1.4) and Holder’s inequality. Indeed, note first that for any \(t, s >0\)
so that, if \(u_n\rightarrow u\) in \(H^1(\mathbb {R}^2)\), as \(n\rightarrow \infty \), then
and the same holds for the functional \(u\longmapsto \int _{\mathbb {R}^2}uf(x,u)\, dx\).
The main feature of the equation (1.1) is the nonlocal term defined through a convolution product. This turns out to be well defined in view of the following Hardy–Littlewood–Sobolev inequalities, which we state in \(\mathbb {R}^N\) for any \(N\ge 1\), see [25] and also [6] for the interpolation spaces approach.
Proposition 2.3
(HLS inequality). Let \(s, r>1\) and \(0<\mu <N\) with \(1/s+\mu /N+1/r=2\), \(f\in L^s(\mathbb {R}^N)\) and \(g\in L^r(\mathbb {R}^N)\). There exists a constant \(C(s,N,\mu ,r)\), independent of f, h, such that
Remark 2.4
Note that the Sobolev inequality (2.1) is equivalent, by duality, to a special case of the HLS inequality (2.3) (see [4]). Actually, take \(\mu =N-2\) and \(s=r=2^*\): then (2.3) says that the inclusion \(L^{\frac{2N}{N+2}}(\mathbb {R}^N)\hookrightarrow H^{-1}(\mathbb {R}^N)\) is continuous, and so, by duality, its counterpart, \(H^1(\mathbb {R}^N)\hookrightarrow L^{2^*}(\mathbb {R}^N)\).
By exploiting a limiting procedure as \(\mu \rightarrow 0\), one can prove the so-called logarithmic Hardy–Littlewood–Sobolev inequality, whose main feature is the presence of a sign-changing logarithmic kernel, see [4, 10, 17].
Proposition 2.5
(Logarithmic HLS inequality). Let f, g be two nonnegative functions belonging to \(L\ln L(\mathbb {R}^N)\), such that \(\int f \log (1+|x|)<\infty , \int g \log (1+|x|)<\infty \) and \(\Vert f\Vert _1=\Vert g\Vert _1=1\). There exists a constant \(C_N\), independent of f, g, such that
Remark 2.6
Let us stress the feature of the log kernel, which has variable sign, and it is unbounded both in 0 and at \(+\infty \). This justifies the presence of the additional condition, \(f, g \in L\ln L(\mathbb {R}^N)\) in order to have the inequality (2.5) well defined, and in particular that no cancellation of infinities occurs. However, this does not imply the boundedness of \(\int f\log f, \int g\log g\), but only of the positive parts \(\int f\log _+f\), \(\int g\log _+g\). The further weight conditions \(\int f\log (1+|x|),\int g\log (1+|x|)<\infty \) make both the two sides of inequality (2.5) finite.
3 A log-mass weighted Pohozaev–Trudinger type inequality
This section is devoted to prove a Pohozaev–Trudinger type inequality in the whole plane \(\mathbb {R}^2\), with a logarithmic weight which appears only in the mass component of the energy. Here, the logarithmic weight plays a role only as \(|x|\rightarrow +\infty \), for which we consider as prototype weight \(w=\log (e+|x|)\).
On the other hand, it is well known from [12, 23, 32], how the growth near zero is a key ingredient in proving Pohozaev–Trudinger type inequalities on unbounded domains, since it is strictly related to vanishing phenomena. Here we aim at proving a fundamental inequality which will provide a suitable variational setting for (1.3). Let us point out that the presence of an increasing weight prevents one to use rearrangement arguments.
3.1 Proof of Theorem 1.1
Let us first perform a change of variables which enables one to pass from \(H_{w}^1(\mathbb {R}^2)\) to functions in \(H^1(\mathbb {R}^2)\). Note that the inverse transformation does not turn out to be explicit and this is why we can not expect to prove directly our inequality.
Let us use polar coordinates in \(\mathbb {R}^2\):
We perform the change of variable
which acts only on the radial part of any point in \(\mathbb {R}^2\), equivalently
In order to simplify the notation, set \(r=|x|\) and \(s=|y|\), so that the transformation becomes \(s=T(r)=r\sqrt{\log (e +r)}\). Note that
and thus T is invertible on \(\mathbb {R}^2\), though the inverse map is not explicitly known.
Define
or, equivalently
Then, by a direct calculation, if
we have
so that
Now,
so that
Then,
Noting that
we eventually get
On the other hand,
Since
we conclude that
and, in turn
Finally,
We have then proved that the map
is an invertible, continuous and with continuous inverse map. Then,
by [32] for any \(\alpha >0\). The uniform bound (1.8) follows directly from (3.1), as for any \(u\in H^1_{w}\) and \(\alpha \le 2\pi \) one has
again by [32].
A consequence of this embedding result is the continuity of the weighted Pohozaev–Trudinger functional on \(H^1_{w}(\mathbb {R}^2)\), namely we have
Corollary 3.1
For any \(\alpha >0\) the functional
is continuous.
Remark 3.2
The value \(2\pi \) in (1.8) is not sharp and the problem of establishing a sharp version of (1.8) in the spirit of Moser [28] remains essentially open.
3.2 The case \(q>2\)
Let us now consider the more general case of a growth function \(F(s)\asymp s^q\) as \(s\rightarrow 0\) with \(q>2\). In this case, the natural weighted Sobolev space turns out to be \(H^1L^q_{w}(\mathbb {R}^2)\), defined as the completion of smooth compactly supported functions with respect to the norm
Theorem 3.3
Let F(s) satisfy assumption \((f_1)\). Then, the space \(H^1L^q_{w}(\mathbb {R}^2)\) embeds into the weighted Orlicz space \(L_{F}(\mathbb {R}^2, \log (e+|x|)dx)\). More precisely,
for any \(u\in H^1L^q_{w}(\mathbb {R}^2)\) and any \(\alpha >0\).
Furthermore, for any \(\alpha \le \frac{1}{\sqrt{q}}\) one has
Proof
The result will follow from the following estimate
Indeed, let \(u\in H^1L^q_{w}\) be such that \(\Vert u\Vert _{w}\le 1\) and define
Then
Hence
where we have applied Theorem 1.1 and embeddings for weighted Sobolev spaces. \(\square \)
Clearly, a corollary similar to 3.1 holds also in this case
Corollary 3.4
For any \(\alpha >0\) the functional
is continuous, where F is a function satisfying assumption \((f_1)\).
In analogy to our case, weighted Pohozev–Trudinger inequalities allowing increasing monomial weights have been proved in [19, 29]. Finally, let us mention that related inequalities in Sobolev spaces with respect to log-weighted measures can be found in [8], where the authors consider a logarithmic weight in the full Sobolev norm.
4 The functional framework
The energy functional related to (1.3) is given by
where
and
Thanks to the assumptions on the potential V, \(\Vert u\Vert _V\) is equivalent to the standard Sobolev norm. However, the energy functional \(I_V\) is not well defined on \(H^1(\mathbb {R}^2)\) due to the nonlocal term \(\mathfrak {F}(u)\). In view of the logarithmic HLS inequality (2.5), Stubbe first proposed in the unpublished paper [34] to set the problem in the intersection space adding the integrability condition for which the energy is well defined. Let \(H^1_VL^q_{w_0}(\mathbb {R}^2)\) be the completion of smooth compactly supported functions with respect to the norm
which in the case \(q=2\) is induced by the scalar product
Accordingly to [24, Theorem 1.11], \(H^1_VL^q_{w_0}\) is a Banach space whose dual can be characterized as follows [33, Theorem 14.9]
Next, the Orlicz type embedding established in Sect. 3 will enable us to apply the logarithmic version of the HLS inequality in order to have the energy functional well defined on \(H^1_VL^q_{w_0}\) and sufficiently smooth for variational purposes.
In what follows we will use extensively the following elementary identity
Furthermore, since we suppose that the nonlinearity f(x, s) has the form \(f(x,s)=c(x)f(s)\) where c(x) is a strictly positive, 1-periodic continuous function, from now on we may reduce to the case \(c(x)=1\) (the general case immediately follows).
Let us introduce the following bilinear forms:
Since \(\log (1+t)\le t\) for any \(t>0\), thanks to (2.3) with \(\mu =1, N=2\) we have that \(\mathfrak {B}_2(u,v)\) is well defined on \(L^{4/3}(\mathbb {R}^2)\times L^{4/3}(\mathbb {R}^2)\), and
Since \(\log (1+|x-y|)\le \log (1+|x|+|y|)\le \log (1+|x|)+\log (1+|y|)\) we also have
where for simplicity we wriwe \(\Vert \cdot \Vert _{L^1(w)}\) in place of \(\Vert \cdot \Vert _{L^1(wdx)}\). Evaluating the bilinear forms \(\mathfrak {B}_i(\cdot ,\cdot )\) on the field (F(u), F(u)) we obtain the functionals:
Reasoning as done above for the bilinear forms \(\mathfrak {B}_i\), one has that \(\mathfrak {F}_2(u)\) is well defined on \(H^1_VL^q_{w_0}(\mathbb {R}^2)\) (actually, on the larger Sobolev space \(H^1(\mathbb {R}^2))\) by assumption \((f_1)\) and recalling (2.2)). Whereas for \(\mathfrak {F}_1(u)\), observe that the quantity \(\Vert F(x,u)\Vert _{L^1(w_0dx)}\) is always finite, for any \(u\in H^1_VL^q_{w_0}\), by combining (1.4) with Theorem 1.1 and Theorem 3.3.
Remark 4.1
Note that for instance when \(q=2\), the weight in the \(L^2\) component of the norm is given by \(V(x)+\log (1+|x|)\) for which we have \(C_1[V(x)+\log (1+|x|)]\le \log (e+|x|)\le C_2[V(x)+\log (1+|x|)]\) for some positive constants \(C_i\). Thus the norms involved turn out to be equivalent and this extends to \(q>2\) as the norm \(\Vert \cdot \Vert _V\) is equivalent to the Sobolev norm.
4.1 Regularity of the energy functional
The main goal of this section is to prove regularity of the energy functional \(I_{V}\). We have the following
Lemma 4.2
The functionals \(\mathfrak {F}_1, \mathfrak {F}_2, \mathfrak {F}\) and \(I_V\) are \(\mathcal {C}^1\) on \(H_V^1L^q_{w_0}(\mathbb {R}^2)\).
Proof
Consider \(\mathfrak {F}_1\) and let \(\{u_n\}\) be a sequence in \(H^1_VL^q_{w_0}(\mathbb {R}^2)\) converging to some u. Then
Now the terms \(I_i\) tend to 0, as \(n\rightarrow \infty \) thanks to the continuity of the functional \(\int _{\mathbb {R}^2}F(u)dx\) on \(H^1\) and of the functional \(\int _{\mathbb {R}^2}F(u)\log (1+|x|)dx\) on \(H^1_VL^q_{w_0}\), as a consequence of (1.4) and Corollaries 3.1, 3.4. For instance for \(I_1\) we have
For any \(u\in H^1_VL^q_{w_0}\) the Gâteaux derivative of \(\mathfrak {F}_1\) at \(u\in H^1_VL^q_{w_0}\) is given by
Since
thanks to Theorems 1.1, 3.3, so that \(\mathfrak {F}_1'(u) \in H^{-1}_VL^q_{w_0}\). The fact that \( \mathfrak {F}_1'(u_n) \rightarrow \mathfrak {F}_1'(u)\) in \(H^{-1}_VL^q_{w_0}\) if \(u_n \rightarrow u\) in \(H^{1}_VL^q_{w_0}\) follows by similar arguments, we just sketch the proof:
We conclude by applying Holder’s inequality and Theorems 1.1 and 3.3, together with Corollaries 3.1 and 3.4, which guarantee the integrals involved are continuously bounded, thus
where \(\mathrm{{o}}_n(1)\) tends to 0 together with \(u_n \rightarrow u\) in \(H^1_VL^q_{w_0}\), as \(n\rightarrow \infty \).
Consider \(\mathfrak {F}_2\) and let \(\{u_n\}\) be a sequence in \(H^1_VL^q_{w_0}(\mathbb {R}^2)\) converging to some u and assume \(u\ne 0\). Since \(u_n\rightarrow u\) in \(H^1(\mathbb {R}^2)\), by [18] we have
Hence
The last two integrals tend to 0, thanks to (4.2), the continuity of the functional \(\int F(u)dx\) on \(H^1\) and the HLS inequality. In the case \(u=0\) the proof is similar.
For any \(u\in H^1_VL^q_{w_0}\) the Gâteaux derivative of \(\mathfrak {F}_2\) at \(u\in H^1_VL^q_{w_0}\) is given by
Since
thanks to (2.2) and HLS inequality, one has \(\mathfrak {F}_2'(u) \in H^{-1}_VL^q_{w_0}\). The fact that \( \mathfrak {F}_2'(u_n) \rightarrow \mathfrak {F}_2'(u)\) in \(H^{-1}_VL^q_{w_0}\) if \(u_n \rightarrow u\) in \(H^{1}_VL^q_{w_0}\) is similar to previous cases.
Clearly from \(\mathfrak {F}=\mathfrak {F}_2-\mathfrak {F}_1\) one has \(F\in \mathcal {C}^1\) on \(H^1_VL^q_{w_0}\). \(\square \)
5 The variational framework
As we are going to see, the variational framework for problem (1.3) is non-standard as we will exploit the de Figuereido–Miyagaki–Ruf type condition \((f_4)\) to prove estimates of the mountain pass level for the energy functional \(I_V\). Let us first establish the Mountain Pass geometry in the next
Lemma 5.1
The energy functional \(I_V\) satisfies:
-
(1)
there exist \(\rho , \delta _0>0\) such that \(I_V|_{S_{\rho }}\ge \delta _0>0\) for all
$$\begin{aligned} u \in S_{\rho }=\{u\in H^1_VL^q_{w_0}: \Vert u\Vert _{q, V, w_0}=\rho \}; \end{aligned}$$ -
(2)
there exists \(e\in H^1_VL^q_{w_0}\) with \(\Vert e\Vert _{q, V, w_0}>\rho \) such that \(I_V(e)<0\).
Proof
To shorten the notation, we assume \(f(x,s)=f(s)\). From \(\Vert u\Vert _V^2\le \Vert u\Vert ^2_{q, V, w_0}\), if \(\rho \) is small then the \(H^1\) norm \(\Vert u\Vert _V\) is also small. As a consequence of the logarithmic HLS inequality,
By (2.3) and recalling estimates (1.4), since the Sobolev norm is small enough we have that the \(L^1\) norm of F(u) can be bounded as follows
Moreover, on one side
for some \(r>2\) and \(\alpha >4\pi \). On the other side, since \(q>2\) and \(\Vert F(u)\Vert _1\le C \Vert u\Vert ^2_V\),
provided the Sobolev norm is small and in turn we obtain
and thus
where \(\delta _0\) depends only on \(\rho \). This yields the first claim of the Lemma.
Let now e be a smooth function, compactly supported in a small ball, say \(B_{1/4}\). Then
since \(F(e(x))F(e(y))\ne 0\) only for \(|x|, |y|<1/4\), which implies \(|x-y|<1/2\) and thus \(|x-y|^{-1}>2\). Similarly,
since F has exponential growth, as \(t\rightarrow +\infty \). \(\square \)
By the Ekeland Variational Principle [21], there exists a Palais–Smale sequence (PS in the sequel) \(\{u_n\}\subset H^1_VL^q_{w_0}(\mathbb {R}^2)\) such that
where the Mountain Pass level \(m_{V}\) can be characterized by
where
The next energy level estimate will be crucial in the sequel, in particular in proving compactness, for which 1/2 turns out to be a substitute of the Sobolev level \((1/N)S^{N/2}\) in higher dimensions.
Lemma 5.2
The mountain pass level \(m_{V}\) satisfies
Proof
It is enough to prove that there exists a function \(w\in H^1_VL^q_{w_0}\), with \(\Vert w\Vert _{q,V, w_0}=1\), such that
Let us introduce the following Moser type functions with support in \(B_{\rho }\)
where \(\rho \) will be fixed later on. One has that
The two integral terms in the previous expression can be estimated as follows. On the one hand
as well as
On the other hand, since q may be integer or not, a rough estimate reads as follows
so that eventually
where
Setting \(w_n={\overline{w}}_n/\sqrt{1+\delta _n}\), we get \(\Vert w_n\Vert _{q, V, w_0}\le 1\). We claim that there exists n such that
Let us argue by contradiction and suppose this is not the case, so that for all n let \(t_n>0\) be such that
Then \(t_n\) satisfies
and
Note that in (5.5) we have an inequality instead of the equality since we know \(\Vert w_n\Vert _{q,V, w_0}^2=1\), whereas in the energy functional it appears the equivalent, though smaller norm \(\Vert w_n\Vert _V\). Actually the two norms differ for a quantity which is \(O(1/\log n)\).
From now on let us suppose \( \rho \le 1/2 \). This will simplify a few estimates, since for any \((x,y)\in {\text {supp }}w_n\times {\text {supp }}w_n\) we will have \(|x-y|>1\), and in turn \(\log (1/|x-y|)>0\). Let us now proceed in three steps:
Step 1. The following holds \(\limsup _{n \rightarrow +\infty } t_n^2 \ge 1\).
Let us assume by contradiction that \(\limsup _{n}t^2_n<1\): this implies that, up to a subsequence, there exists a positive constant \(\delta _0\) such that \(t_n^2\le 1-\delta _0\) for n large enough. Since \(\rho \le \frac{1}{2}\), for any \(|x|< \rho \), the set \(\{y: |x-y|>1, |y|<\rho \}\) is empty. Recalling that the functions \(w_n\) are compactly supported in \(B_{\rho }\) we have
and thus a contradiction with (5.6).
Step 2. The following holds \(\liminf _{n \rightarrow +\infty } t_n^2 \le 1\).
Let us suppose by contradiction that \(\liminf _{n \rightarrow +\infty } t_n^2>1\). Hence, up to a subsequence, there exists a constant \(\delta _0>0\) such that
as \(n \rightarrow +\infty \). Let us estimate from below the right hand side of (5.5) (taking into account the possible negative sign of the logarithmic function):
Thanks to \((f_4)\) we have for any \(\varepsilon >0\) (here we choose \(\varepsilon =\beta /2\)),
By the very definition of \(w_n\) and since \(|x-y|< 2\rho /n<1\), we can estimate, for n large enough, \(I_1\) as follows
The last integral can be estimated as follows
As a consequence, we obtain
for any \(n\ge n(\rho , \beta )\). Note that since \(\rho \le 1/2\) we have
Now, combining (5.5), (5.7) and (5.9) yields
which is a contradiction, either if \(t_n\rightarrow +\infty \) or \(t_n\) stays bounded with \(t_n^2\ge 1+\delta _0\). The proof of Step 2 is then completed. Observe that, as a consequence of Step 1 and Step 2
Moreover, as a byproduct of (5.10), we also have for some \(C>0\),
that is
Step 3. We are now in the condition of getting a contradiction and determine the quantity \({\mathcal {V}}\) which appears in condition \((f_4)\). We have proved that \(t_n^2\rightarrow 1\). Moreover, we also know that \(t_n^2\ge 1\) by (5.6), since \(\rho \le 1/2 \). By (5.10), recalling definition (5.2) of \(\delta _n\), we have
where \(V_\rho =\max _{|x|\le \rho } V(x)\). Passing to the limit, we obtain
Now set in assumption \((f_4)\)
a quantity which is actually a minimum, since the right hand function is continuous and unbounded as \(|x|\rightarrow 0\). Finally, fix \(\rho \in (0,1/2]\) such that
to get
which contradicts (5.12). \(\square \)
6 Properties of Palais–Smale sequences
In this section we prove that the weak limit in \(H^1_VL^q_{w_0}(\mathbb {R}^2)\) of the PS sequence for \(I_V\) given by the Ekeland Variational Principle, which we know from Sect. 5 is at the energy level \(m_V<1/2\), is actually a weak nontrivial solution of (1.3). As we are going to see, the presence of the sign changing factor \(\log (|x|)\) makes the estimates rather delicate. We start with the following Lemma in which we prove boundedness of PS sequences at any level \(c<1/2\).
Lemma 6.1
Assume that \((V_1)\)–\((V_2)\) and \((f_1)\)–\((f_4)\) hold. Let \(\{u_n\}\subset H^1_VL^q_{w_0}\) be an arbitrary PS sequence for \(I_V\) at level c, namely
Then, the sequence \(u_n\) is bounded in \(H^1(\mathbb {R}^2)\) as well as
Proof
Let \(\{u_n\}\in H^1_VL^q_{w_0}(\mathbb {R}^2)\) be a PS sequence for \(I_V\), namely as \(n\rightarrow \infty \) is,
and
for all \(v\in H^{-1}_VL^q_{w_0}(\mathbb {R}^2)\), where \(\tau _n\rightarrow 0\) as \(n\rightarrow +\infty \). Since \(H^1(\mathbb {R}^2)\hookrightarrow H^{-1}_VL^q_{w_0}(\mathbb {R}^2)\) and \(H^1_VL^q_{w_0}(\mathbb {R}^2)\hookrightarrow H^1(\mathbb {R}^2)\), we can take \(v=u_n\) in (6.2), to obtain
where we have also used the fact that \(\Vert \cdot \Vert _V\) is an equivalent norm to the standard one in \(H^1(\mathbb {R}^2)\). Another suitable choice for test function is given by
where the quantity \(\delta \) is the one appearing in (1.5), \(F(s)\le (1-\delta )sf(s)\). Indeed, since \(f(s)=0\) if and only if \(s=0\), by \((f_2)\) we have that \(0\le v_n\le C u_n \) (actually, it is uniformly bounded) so that \(v_n\) is well defined and in \(L^q(w_0dx)\). Furthermore,
Since the quantity \((F/f)'\) is bounded by \((f_2)\), see also (1.5), we have
so that \(v_n\in H^1_VL^q_{w_0}(\mathbb {R}^2)\). Taking \(v=v_n=\frac{F(u_n)}{f(u_n)}\) in (6.2) and recalling that \(f(s), F(s)=0\) for any \(s\le 0\) yields
Now recall (6.1), namely
Only two cases may occur as \(n\rightarrow +\infty \) (we are not excluding that both the two cases may appear for different subsequence of \(u_n\)):
-
\(\displaystyle \int _{\mathbb {R}^2}\left[ \log \left( \frac{1}{|x|}\right) *F(x,u_n)\right] F(x,u_n) dx\le 0\). In this case we have
$$\begin{aligned} \Vert u_n\Vert _V^2\le 2c+ \text {o}(1)\le 3c, \quad c\ge 0; \end{aligned}$$ -
\(\displaystyle \int _{\mathbb {R}^2}\left[ \log \left( \frac{1}{|x|}\right) *F(x,u_n)\right] F(x,u_n)> 0\). Combining (6.1) and (6.4) yields
$$\begin{aligned}&\int _{\mathbb {R}^2}\left( |\nabla u_n|^2+Vu^2_n\right) dx-2c+\text {o}(1) -\int _{\{u_n>0\}}|\nabla u_n|^2\left( 1-\frac{F(u_n)f'(u_n)}{f^2(u_n)}\right) dx\\&\quad -(1-\delta )\int _{\{u_n\le 0\}}|\nabla u_n|^2dx-\int _{\mathbb {R}^2}V(x)u_n\frac{F(u_n)}{f(u_n)} dx\le \tau _n \Vert u_n\Vert _V. \end{aligned}$$
Since, as a consequence of \((f_2)\), \(F(x,s)\le (1-\delta )sf(x,s)\) (see also (1.5)) we have, as \(n\rightarrow \infty \)
In conclusion we have proved that \( \Vert u_n\Vert _V\le C\ \). From (6.1) and (6.3), we also have
that is our thesis. \(\square \)
Remark 6.2
Let us note that, as a consequence of the previous Lemma 6.1, given any PS sequence at level c there exists always another positive PS sequence at the same level. Indeed, since \({u_n}\) is bounded in \(H^{1}_V L^q_{w_0}(\mathbb {R}^2)\) as proved above, we can test \({\mathcal {I}}'_V(u_n)\) against \(u_n^{-}=\max {(-u_n,0)}\) obtaining (see (6.2))
Hence, the sequence \(\{u_n^{+}\}\) is positive, and it is still a PS sequence at the same level c (since \(F(s)=0\) for \(s\le 0\)).
By Remark 6.2, from now on we will assume positivity of PS sequences.
Differently from standard contexts in which having proved boundedness of a PS sequence brings the conclusion at hand, here it does not allow to employ standard arguments to prove the weak limit is actually a nontrivial solution to the equation. Indeed, the presence of the exponential nonlinearity together with the sign-changing behavior of the logarithmic kernel, prevents the application of standard estimates. Here comes into play the key estimate for the mountain pass level \(m_V<1/2\) established in Lemma 5.2.
Lemma 6.3
Assume \((V_1)-(V_2)\) and \((f_1)-(f_4)\). Let \(\{u_n\}\subset H^1_VL^q_{w_0}\) be a (positive) PS sequence for \(I_V\) at level \(c<1/2\). Then, for any \(1\le \alpha <\frac{1}{2c}\) the following uniform bound holds
Proof
Without loss of generality, we consider \(f(x,s)=f(s)\). By Lemma 6.1 the sequence \(\{u_n\}\) is bounded in \(H^1(\mathbb {R}^2)\) and we may assume \(u_n\rightharpoonup u\) in \(H^1(\mathbb {R}^2)\), \(u_n\rightarrow u\) in \(L^s_{loc}(\mathbb {R}^2)\) for any \(1\le s< \infty \) and \(u_n\rightarrow u\) a.e. in \(\mathbb {R}^2\), with
As in the proof of Lemma 6.1, we will carefully select a suitable test function \(v_n\). Let us introduce the following auxiliary function
which is well defined and \({\mathcal {C}}^1\) thanks to \((f_2)\). Moreover, by Hölder’s inequality we have
Define
then
as \(u_n\) is bounded in \(H^1\) and applying again \((f_2)\). We aim at proving that
as n is large enough. First, note that as \(n\rightarrow +\infty \), one has
In order to estimate the norm \(\Vert v_n\Vert _V^2\) recall (6.1) and (6.4). From \(\Vert u_n\Vert _V^2\rightarrow A^2\ge \Vert u\Vert _V^2\), we have
and
so that
and in turn
by (6.5), as n is large enough.
Once we have estimated the norm of \(v_n\), let us take advantage of this to improve the exponential integrability of the original sequence \(u_n\). By \((f_3)\), for any \(\epsilon >0\) there exists a constant \(t_\epsilon >0\) such that
Next by \((f_2)\) we also have either \(u_n(x)\le t_\epsilon \) or \(u_n(x)\ge t_\epsilon \) which implies
and thus
Hence (hereafter \(C_\epsilon \) may change from line to line)
where, in the last line, we use the following inequality: for any \(T>0\) and for any \(\epsilon >0\) there exists \(C=C_{T, \epsilon }\) such that \(s^{p-1}\le C_{T, \epsilon } e^{4\pi \epsilon s^2}\) for any \(s\ge T\) (with \(T= t_\epsilon \) already fixed as well as \(C=C_\epsilon \)). Moreover, for any \(\epsilon >0\) there exists \(C_\epsilon \) such that \((t+s)^2\le C_\epsilon t^2+ (1+\epsilon )s^2\) for any \(s,t>0\). Then
As byproduct of (6.7), if \(u_n\ge t_\epsilon \) then \(v_n\ge \delta t_\epsilon \). Combining this with (6.9) and (6.8) we obtain
Let us now fix \(0<\epsilon <1\) and set
With these choices we obtain
By (6.6), \(\Vert v_n\Vert _V^2\le 2c+\mathrm {o}(1)\) as n is large enough, so that
Hence,
since the last inequality is equivalent to
In conclusion we have
by the Ruf inequality (2.4) and Remark 2.1. \(\square \)
Proposition 6.4
Assume that conditions \((V_1)\)–\((V_2)\) and \((f_1)\)–\((f_4)\) are satisfied. Let \(\{u_n\}\subset H^1_VL^q_{w_0}\) be a (positive) PS sequence for \(I_V\) at level \(c<1/2\), weakly converging to u in \(H^1\). If \(u\ne 0\), then \(u\in H^1_VL^q_{w_0}\) and \(u_n\rightharpoonup u\) weakly in \(H^1_VL^q_{w_0}\). Furthermore, as \(n\rightarrow \infty \)
and u is a weak solution to (1.3).
Proof
Fix \(\alpha \in (1, 1/2c)\) so that by (6.1) we have
by Proposition 2.3, since \(\frac{2}{\alpha }+4\frac{\alpha -1}{\alpha }\, \frac{1}{2}=2\). Hence, by Lemma 6.3,
as \(n\rightarrow +\infty \), and thus also \(\int _{\mathbb {R}^2\times \mathbb {R}^2}\log \left( 1+|x-y|\right) u^q_n(x)u^q_n(y)dxdy\) is bounded. Since \(u\ne 0\), by Lemma 2.1 in [15] we have
so that \(\Vert u_n\Vert ^2_{q, V, w_0}\) is bounded. Up to a subsequence we have \(u_n\rightharpoonup u\) in \(H^1_VL^q_{w_0}\). Moreover, recall that as \(n\rightarrow +\infty \),
in particular for any \(\varphi \in {\mathcal {C}}^{\infty }_c(\mathbb {R}^2)\). In order to prove that u is a weak solution of (1.3), let us suppose for the moment the following holds true. Claim:
of which we postpone the proof. Now we apply Lemma 2.1 in [16] to the sequence of functions
restricted to any compact domain \(\Omega \): they are \(L^1\) functions since \(u_n, u\in H^1_VL^q_{w_0}\) and, thanks to the claim, \(u_n(y)g(y, u_n(y))\) is uniformly bounded in \(L^1\). Therefore, from [16] we have
as well as
for any \(\varphi \in {\mathcal {C}}^{\infty }_c(\mathbb {R}^2)\), which is a dense subset of \(H^{-1}_VL^q_{w_0}\). This together with (6.11) implies that u is a weak solution of (1.3).
Proof of the Claim. The key ingredient is the uniform bound provided by Lemma 6.3. In order to simplify the notation, let us set
By (6.3)
where \(A=\lim _{n\rightarrow +\infty }\Vert u_n\Vert _V\ge \Vert u\Vert _V\) so that
for n large enough (note that we are assuming \(u\ne 0\), that is, \(A^2>0\)). Hence,
which implies
and thus
Now, we have
Therefore, for any \(\mu >0\), small, there exists a constant \(\delta _\mu \in (0,1)\) sufficiently small such that
The first integral in the last expression is uniformly bounded, as one can see by Lemma 6.3 and Holder’s inequality, recalling that \(\Vert u_n\Vert _V\) is also uniformly bounded. Concerning the second term, by the HLS inequality if \(2/s +\mu /2=1\), one has
Since
we can choose \(\mu \) small enough to apply again Lemma 6.3 and Holder’s inequality, to obtain that \(\Vert F(u_n)\Vert _s\Vert f(u_n)u_n\Vert _s\) stays bounded. Finally, \(\int w^+_nf(u_n)u_n\) is bounded and the same holds for \(\int |w_n|f(u_n)u_n\), that is our claim. \(\square \)
7 Proof of Theorem 1.2
We are now ready to prove Theorem 1.2. From Lemma 5.1, the functional \(I_V\) satisfies the Mountain Pass geometry. Hence, there exists a (positive) (PS) sequence \(\{u_n\}\subset H^1_VL^q_{w_0}(\mathbb {R}^2)\) at level \({m_V}\) and by Lemma 6.3, \(\{u_n\}\) is bounded in \(H^1\) and it weakly converges to some \(u\in H^1\). We have that either \(\{u_n\}\) is vanishing, that is for any \(r>0\)
or there exist \(r, \delta >0\) and a sequence \(\{y_n\}\subset \mathbb {Z}^2\) such that
If \(\{u_n\}\) is vanishing, by Lions’ concentration-compactness result we have
as \(n\rightarrow \infty \). In this case it is standard to show that
for some values of \(\gamma >1\) and close to 1, thanks to the improved exponential integrability given by Lemma 6.3 and the growth assumption \(F(x,t)<tf(x,t)\). Hence, applying the HLS inequality we deduce, similarly to the conclusion of the proof of Proposition 6.4:
as \(n\rightarrow \infty \). Combining (7.2)–(7.3) with (6.1) and (6.11) yields
so that \(m_v\le 0\), which is not possible. Therefore the vanishing case does not occur.
Now set \(v_n:=u_n(\cdot -y_n)\), then
By the periodicity assumption, \(I_V\) and \(I'_V\) are both invariant by the \(\mathbb {Z}^2\)-action, so that \(\{v_n\}\) is still a PS sequence at level \(m_V\). Then \(v_n\rightharpoonup v\) in \(H^1(\mathbb {R}^2)\) with \(v\ne 0\) by using (7.4), since \(v_n\rightarrow v \) in \(L^2_{loc}(\mathbb {R}^2)\). We conclude by Proposition 6.4 that \(v\in H^1_VL^q_{w_0}\) is a nontrivial critical point of \(I_V\) and \(I_V(v)=m_V\), which completes the proof of Theorem 1.2.
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Cassani, D., Tarsi, C. Schrödinger–Newton equations in dimension two via a Pohozaev–Trudinger log-weighted inequality. Calc. Var. 60, 197 (2021). https://doi.org/10.1007/s00526-021-02071-w
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DOI: https://doi.org/10.1007/s00526-021-02071-w