Uniform $L^{\infty }$-Estimates for Quasilinear Elliptic Systems

Abstract

The aim of this work is to provide uniform $L^{\infty }$-estimates for the solutions of a quite general class of (p, q)-quasilinear elliptic systems depending on two parameters $\alpha$ and $\delta$.

1 Introduction

Let us consider the following autonomous quasilinear system

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} -\text {div} \left( (\alpha +|\nabla u|^2)^{\frac{p-2}{2}}\nabla u\right) = H_s(\delta , u,v) &{} \hbox {in} \ \Omega \\ -\text {div} \left( (\alpha +|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v\right) = H_t(\delta ,u,v) &{} \hbox {in} \ \Omega \\ u=v=0 &{} \hbox {on} \ \partial \Omega \end{array} \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega$ is a smooth bounded domain of $\mathbb {R}^N$, $N\ge 3$, $p,q\in [2,N)$, $\alpha \ge 0$ and $H:I\times \mathbb {R}^2 \rightarrow \mathbb {R}$ is a function, where $I\subset \mathbb {R}$ is an interval and $H(\delta ,\cdot ,\cdot ) \in C^1(\mathbb {R}^2, \mathbb {R})$ for any $\delta \in I$.

Moreover, we assume that

$(*)$:

there are $p'\in (p,p^*)$, $q'\in (q,q^*)$ and $C_0>0$ such that

$$\begin{aligned}{} & {} H_s(\delta ,s,t)\le C_0 \left( |s|^{p'-1} +|t|^{q'\frac{p'-1}{p'}} +1 \right) \\{} & {} H_t(\delta ,s,t)\le C_0 \left( |s|^{p'\frac{q'-1}{q'}}+ |t|^{q'-1} +1 \right) \end{aligned}$$

for any $(\delta ,s,t)\in I\times \mathbb {R}^2$.

Let X be the product space $W_0^{1,p}(\Omega )\times W_0^{1,q}(\Omega )$ endowed with the norm

$$\begin{aligned} \Vert z\Vert = \Vert u\Vert _{1,p} + \Vert v\Vert _{1,q} \end{aligned}$$

where $z=(u,v)\in X$. In what follows we shall denote respectively by $\Vert \cdot \Vert _s$ and $\Vert \cdot \Vert _{1,s}$ the usual norms in $L^s(\Omega )$ and $W^{1,s}_0(\Omega )$.

Weak solutions of problem (1.1) correspond to critical points of the Euler functional $I_{\alpha , \delta }:X\rightarrow \mathbb {R}$ defined as

$$\begin{aligned} I_{\alpha , \delta }(z) =I_{\alpha , \delta }(u,v)&= \frac{1}{p} \int \limits _{\Omega }\left( \alpha + |\nabla u(x)|^2\right) ^{\frac{p}{2}} \ \text {d}x + \frac{1}{q}\int \limits _{\Omega }\left( \alpha + |\nabla v(x)|^2\right) ^{\frac{q}{2}} \ \text {d}x \\&\quad - \int \limits _{\Omega }H(\delta ,u(x),v(x)) \ \text {d}x \qquad \hbox {for any } z=(u,v) \in X. \end{aligned}$$

By $(*)$, the functional $I_{\alpha , \delta }$ is $C^1$ on X and, for any $z_0=(u_0,v_0)$ and $z=(u, v)$ in X, it results

$$\begin{aligned} \langle I'_{\alpha , \delta }(z_0), z \rangle&= \displaystyle \int \limits _{\Omega }({\alpha }+ |\nabla u_0|^2)^{\frac{p-2}{2}}\nabla u_0\nabla u + \int \limits _{\Omega }({\alpha }+ |\nabla v_0|^2)^{\frac{q-2}{2}}\nabla v_0 \nabla v \\&\quad -\displaystyle \int \limits _{\Omega }H_s(\delta ,u_0,v_0) u + H_t(\delta ,u_0,v_0) v. \end{aligned}$$

Systems involving this kind of quasilinear operators model some phenomena in non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology; see [7, 9, 11, 12]. Existence, nonexistence and regularity results for such quasilinear elliptic systems are obtained by various authors, see for instance [1, 3, 6, 8, 14].

More recently we proved that any weak solution of the following system, not depending on $\delta$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} -\text {div} \left( (\alpha +|\nabla u|^2)^{\frac{p-2}{2}}\nabla u\right) = H_s( u,v) &{} \hbox {in} \ \Omega \\ -\text {div} \left( (\alpha +|\nabla v|^{2})^{\frac{q-2}{2}}\nabla v\right) = H_t(u,v) &{} \hbox {in} \ \Omega \\ u=v=0 &{} \hbox {on} \ \partial \Omega \end{array} \end{array}\right. } \end{aligned}$$

is in $\left( L^\infty (\Omega )\right) ^2$ (see [4, Theorem 1.1]).

In this work we want to extend the previous result to the class of systems (1.1) depending also on $\delta$. Moreover here we show carefully that, for any arbitrary $z_0 \in X$ and $r>0$, the $\left( L^\infty (\Omega )\right) ^2$-norm of the weak solutions to (1.1) belonging to $B_r(z_0)$ depends just on r and $z_0$, but is independent on $\alpha$ and $\delta$.

The main result of this work is the following:

Theorem 1.1

If (u, v) is a solution of (1.1) and $(*)$ holds, then $(u,v)\in \left( L^\infty (\Omega )\right) ^2$.

Moreover, for any fixed $(u_0,v_0)\in X$, $r>0,\, \alpha \ge 0$ and $\delta \in I$, denoting by

$$\begin{aligned} D_{r,\alpha , \delta }(u_0,v_0)=\{ (u,v)\in X\,\ \Vert (u,v)-(u_0,v_0)\Vert \le r, \ I'_{\alpha , \delta }(u,v)=0 \}, \end{aligned}$$

there exists $C>0$, depending on r and $(u_0,v_0)$ but independent of $\alpha$ and $\delta$, such that

$$\begin{aligned} \Vert u\Vert _\infty ,\, \Vert v\Vert _\infty \, \le C \qquad \forall \, (u,v)\in D_{r,\alpha , \delta }(u_0,v_0). \end{aligned}$$

This uniform $L^\infty$-estimate will be used in the forthcoming paper [2] in which we derive some crucial existence results about system (1.1), studying the interaction of the spectrum of the quasilinear operators with the nonlinearity H which grows (p, q)-linearly at infinity, in continuity with the Amann–Zehnder type results obtained in [5] for a class of quasilinear elliptic equations.

2 Proof of Theorem 1.1

We first introduce the following result.

Lemma 1.1

Let $s\in (1,N)$ and denote by $s^{*}$ the conjugate Sobolev exponent of s, namely $s^*=sN/(N-s)$. If $r,\varepsilon >0$, $u_0\in W^{1,s}_0(\Omega )$ and $s'\in [1,s^*)$, there is $\sigma >0$ such that

$$\begin{aligned} \int \limits _{\{|u(x)|\,\ge \sigma \}} \hspace{-1mm} |u(x)|^{s'}\, \text {d}x<\varepsilon \end{aligned}$$

for any $u\in B_r(u_0)=\{u\in W^{1,s}_0(\Omega )\,\ \Vert u-u_0\Vert _{1,s}\le r \}$.

Proof

By contradiction, assume that there are $r,\varepsilon >0$, $u_0\in W^{1,s}_0(\Omega )$, $s'<s^{*}$, $h_n\ge n$ and $u_n \in B_r(u_0)$ such that

$$\begin{aligned} \int \limits _{\{|u_n(x)|\,\ge h_n\}} |u_n(x)|^{s'}\, \text {d}x\ge \varepsilon \end{aligned}$$

(2.1)

for any $n\in \mathbb {N}$.

Up to subsequences, $u_n$ strongly converges to some ${\bar{u}}$ in $L^{s'}(\Omega )$.

Moreover, denoting by $E_n=\{x\in \Omega \,\ |u_n(x)|\,\ge h_n\}$, we claim that

$$\begin{aligned} |E_n|\rightarrow 0. \end{aligned}$$

(2.2)

Otherwise, if not, we should have, up to subsequences, $|E_n|\ge \alpha >0$ for any n, hence

$$\begin{aligned} \int \limits _{\Omega }|u_n(x)|^{s'}\, \text {d}x\ge \int \limits _{E_n}|u_n(x)|^{s'}\, \text {d}x\ge \alpha \, n^{s'}\quad \Rightarrow \quad \int \limits _{\Omega }|u_n(x)|^{s'}\, \text {d}x\rightarrow \infty \end{aligned}$$

while $\int _{\Omega }|u_n(x)|^{s'}\, \text {d}x\rightarrow \int _{\Omega }|{\bar{u}}(x)|^{s'}\, \text {d}x$. This proves (2.2), hence the Vitali convergence theorem gives that

$$\begin{aligned} \int \limits _{E_n}|u_n(x)|^{s'}\, \text {d}x \rightarrow 0 \end{aligned}$$

which contradicts (2.1). $\square$

Now, inspired by [4] and [10], we prove the main result.

Proof of Theorem 1.1

For every $\gamma ,\, t,\, k>1$ we define

$$\begin{aligned} h_{k,\gamma }(s)&= {\left\{ \begin{array}{ll} s|s|^{\gamma -1} &{} |s|\le k, \\ \gamma k^{\gamma -1} s + \text {sign}(s)(1-\gamma ) k^{\gamma } &{} |s|>k, \end{array}\right. } \\ \Phi _{k,t,\gamma }(s)&=\int _{0}^{s}\left| h'_{k,\gamma }(r)\right| ^{\frac{t}{\gamma }}\text {d}r. \end{aligned}$$

Observe that $h_{k,\gamma }$ and $\Phi _{k,t,\gamma }$ are $C^{1}$-functions with bounded derivative, depending on $\gamma , t$ and k. Thus if $(u,v)\in X=W^{1,p}_{0}(\Omega )\times W^{1,q}_{0}(\Omega )$, then $\Phi _{k,t,\gamma }(u)\in W_{0}^{1,p}(\Omega )$ and $\Phi _{k,t,\gamma }(v)\in W_{0}^{1,q}(\Omega )$. Moreover, for every $t\ge \gamma$, there exists a positive constant C, depending on $\gamma$ and t but independent of k, such that

$$\begin{aligned}{} & {} |s|^{\frac{t}{\gamma }-1}|\Phi _{k,t,\gamma }(s)|\le C |h_{k,\gamma }(s)|^{\frac{t}{\gamma }} \end{aligned}$$

(2.3)

$$\begin{aligned}{} & {} |\Phi _{k,t,\gamma }(s)|\le C |h_{k,\gamma }(s)|^{\frac{1}{\gamma }(1+t\frac{\gamma -1}{\gamma })} \end{aligned}$$

(2.4)

and

$$\begin{aligned} \left| h_{k,\gamma }\left( |s|^{\frac{q'}{p'}}\right) \right| ^{p'} \le C\left| h_{k^{\frac{p'}{q'}},\gamma }\left( s\right) \right| ^{q'}. \end{aligned}$$

(2.5)

Let us fix $r>0,\, \alpha \ge 0$, $\delta \in I$ and consider an arbitrary ${\bar{z}}=({\bar{u}}, {\bar{v}}) \in D_{r,\alpha , \delta }(u_0,v_0)$.

In particular,

$$\begin{aligned} \langle I'_{\alpha , \delta }({\bar{z}}),\left( \Phi _{k,\gamma p,\gamma }({\bar{u}}),0\right) \rangle =0 \end{aligned}$$

for any $k,\gamma >1$.

So, as $W^{1,p}_{0}(\Omega )\hookrightarrow L^{p'}(\Omega )$, there is $c>0$ such that

$$\begin{aligned}&\left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}} \le c \displaystyle \int \limits _{\Omega }|\nabla h_{k,\gamma }({\bar{u}})|^{p} = c \displaystyle \int \limits _{\Omega }|\nabla \bar{u}|^{p}|h'_{k,\gamma }({\bar{u}})|^{p} \\&\quad \le c \displaystyle \int \limits _{\Omega }(\alpha + |\nabla {\bar{u}}|^2)^{\frac{p-2}{2}} |\nabla \bar{u}|^{2}|h'_{k,\gamma }({\bar{u}})|^{p} = c \displaystyle \int \limits _{\Omega }(\alpha + |\nabla {\bar{u}}|^2)^{\frac{p-2}{2}} \nabla {\bar{u}} \cdot \nabla \Phi _{k,\gamma p,\gamma }({\bar{u}}) \\&\quad = c \displaystyle \int \limits _{\Omega }H_s(\delta ,{\bar{u}},{\bar{v}}) \Phi _{k,\gamma p,\gamma }({\bar{u}}). \end{aligned}$$

By $(*)$, we get

$$\begin{aligned}{} & {} \left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}} \hspace{1cm} \nonumber \\{} & {} \quad \le c\,C_0\left( \displaystyle \int \limits _{\Omega }(|{\bar{u}}|^{p'-1}+1)|\Phi _{k,\gamma p,\gamma }({\bar{u}})| + \displaystyle \int \limits _{\Omega }|{\bar{v}}|^{q'\frac{p'-1}{p'}}|\Phi _{k,\gamma p,\gamma }({\bar{u}})| \right) . \end{aligned}$$

(2.6)

For any $\sigma >1$ and w in $W^{1,p}_{0}(\Omega )$ or w in $W^{1,q}_{0}(\Omega )$, we denote by

$$\begin{aligned} \Omega _{\sigma ,w}=\{x\in \Omega \, \ |w(x)| > \sigma \}. \end{aligned}$$

Therefore, using (2.3), (2.4) and redefining from now on, when necessary, the positive constant C, depending on $\gamma$ but independent of k and $\sigma$, we have

$$\begin{aligned}&\displaystyle \int \limits _{\Omega }(|{\bar{u}}|^{p'-1}+1)|\Phi _{k,\gamma p,\gamma }({\bar{u}})|\\&\quad \le (\sigma ^{p'-1}+1)\displaystyle \int \limits _{\Omega }|\Phi _{k,\gamma p,\gamma }({\bar{u}})| +\int \limits _{\Omega _{\sigma ,{\bar{u}}}} |{\bar{u}}|^{p'-p} |{\bar{u}}|^{p-1} |\Phi _{k,\gamma p,\gamma }({\bar{u}})|\\&\quad \le 2\sigma ^{p'-1}\displaystyle \int \limits _{\Omega }|\Phi _{k,\gamma p,\gamma }({\bar{u}})| +C\int \limits _{\Omega _{\sigma ,{\bar{u}}}} |{\bar{u}}|^{p'-p} |h_{k,\gamma }({\bar{u}})|^p\\&\quad \le C\sigma ^{p'-1}\displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{\frac{p\gamma +1 -p}{\gamma }} +C\int \limits _{\Omega _{\sigma ,{\bar{u}}}} |{\bar{u}}|^{p'-p} |h_{k,\gamma }({\bar{u}})|^p. \end{aligned}$$

Using Hölder inequality we deduce

$$\begin{aligned} \displaystyle \int \limits _{\Omega }(|{\bar{u}}|^{p'-1}+1)|\Phi _{k,\gamma p,\gamma }({\bar{u}})|{} & {} \le C \sigma ^{p'-1}\left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}\frac{\gamma p+1-p}{\gamma p}} \nonumber \\{} & {} \quad +\, C\Vert {\bar{u}} \Vert ^{p'-p}_{L^{p'}(\Omega _{\sigma ,{\bar{u}}})} \left( \displaystyle \int \limits _{\Omega }\left| h_{k,\gamma }({\bar{u}})\right| ^{p'}\right) ^{\frac{p}{p'}}. \end{aligned}$$

(2.7)

We deal with the second integral in (2.6) and similarly, using (2.3), (2.4), (2.5) and the fact that $\Phi _{k,\gamma p,\gamma }$ is non decreasing, we obtain

$$\begin{aligned}&\displaystyle \int \limits _{\Omega }|{\bar{v}}|^{q'\frac{p'-1}{p'}}|\Phi _{k,\gamma p,\gamma }({\bar{u}})|\\&\quad \le \sigma ^{q'\frac{p'-1}{p'}}\displaystyle \int \limits _{\Omega }|\Phi _{k,\gamma p,\gamma }({\bar{u}})| +\int \limits _{\Omega _{\sigma ,{\bar{v}}}\cap \{|{\bar{v}}|^\frac{q'}{p'}\le |{\bar{u}}|\}} |{\bar{v}}|^{\frac{q'}{p'}(p'-p)} |{\bar{u}}|^{p-1} |\Phi _{k,\gamma p,\gamma }({\bar{u}})| \\&\qquad +\int \limits _{\Omega _{\sigma ,{\bar{v}}}\cap \{|{\bar{v}}|^\frac{q'}{p'}\ge |{\bar{u}}|\}} |{\bar{v}}|^{\frac{q'}{p'}(p'-p)} \bigl (|{\bar{v}}|^{\frac{q'}{p'}}\bigr )^{p-1} |\Phi _{k,\gamma p,\gamma }(|{\bar{v}}|^{\frac{q'}{p'}})| \\&\quad \le C\sigma ^{q'\frac{p'-1}{p'}}\displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{\frac{p\gamma +1 -p}{\gamma }} +C \int \limits _{\Omega _{\sigma ,{\bar{v}}}\cap \{|{\bar{v}}|^\frac{q'}{p'}\le |{\bar{u}}|\}} |{\bar{v}}|^{\frac{q'}{p'}(p'-p)}|h_{k,\gamma }({\bar{u}})|^p \\&\qquad +C \int \limits _{\Omega _{\sigma ,{\bar{v}}}\cap \{|{\bar{v}}|^\frac{q'}{p'}\ge |{\bar{u}}|\}} |{\bar{v}}|^{\frac{q'}{p'}(p'-p)} |h_{k,\gamma }\bigl (|{\bar{v}}|^{\frac{q'}{p'}}\bigr )|^p \\&\quad \le C\sigma ^{q'\frac{p'-1}{p'}} \left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}\frac{\gamma p+1-p}{\gamma p}}\\&\qquad +C \Vert {\bar{v}} \Vert ^{\frac{q'}{p'}(p'-p)}_{L^{q'}(\Omega _{\sigma ,{\bar{v}}})} \left( \left( \displaystyle \int \limits _{\Omega }\left| h_{k,\gamma }({\bar{u}})\right| ^{p'}\right) ^{\frac{p}{p'}} +\left( \displaystyle \int \limits _{\Omega }|h_{k^{\frac{p'}{q'}},\gamma }\left( {\bar{v}}\right) |^{q'}\right) ^{\frac{p}{p'}} \right) . \end{aligned}$$

Combining with (2.6) and (2.7), we get

$$\begin{aligned}&\left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}} \\&\quad \le C \sigma ^{p'-1}\left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}\frac{\gamma p+1-p}{\gamma p}} + C\Vert {\bar{u}} \Vert ^{p'-p}_{L^{p'}(\Omega _{\sigma ,{\bar{u}}})} \left( \displaystyle \int \limits _{\Omega }\left| h_{k,\gamma }({\bar{u}})\right| ^{p'}\right) ^{\frac{p}{p'}} \\&\qquad + C\sigma ^{\frac{q'}{p'}(p'-1)} \left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}\frac{\gamma p+1-p}{\gamma p}}\\&\qquad +C \Vert {\bar{v}} \Vert ^{\frac{q'}{p'}(p'-p)}_{L^{q'}(\Omega _{\sigma ,{\bar{v}}})} \left( \left( \displaystyle \int \limits _{\Omega }\left| h_{k,\gamma }({\bar{u}})\right| ^{p'}\right) ^{\frac{p}{p'}} +\left( \displaystyle \int \limits _{\Omega }|h_{k^{\frac{p'}{q'}},\gamma }\left( {\bar{v}}\right) |^{q'}\right) ^{\frac{p}{p'}} \right) . \end{aligned}$$

Through Lemma 2.1, there is $\sigma _1>1$ such that, for any $\sigma \ge \sigma _1$ and for any $k,\gamma >1$:

$$\begin{aligned} \frac{1}{2}\left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}}\le & {} C \Bigl (\sigma ^{p'-1}+\sigma ^{\frac{q'}{p'}(p'-1)} \Bigr ) \left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}\frac{\gamma p+1-p}{\gamma p}}\nonumber \\{} & {} \quad + C\, \Vert {\bar{v}} \Vert ^{\frac{q'}{p'}(p'-p)}_{L^{q'}(\Omega _{\sigma ,{\bar{v}}})} \left( \displaystyle \int \limits _{\Omega }|h_{k^{\frac{p'}{q'}},\gamma }\left( {\bar{v}}\right) |^{q'}\right) ^{\frac{p}{p'}}. \end{aligned}$$

(2.8)

If $\eta \in (0,1)$, using Young inequality we obtain that

$$\begin{aligned} ax^{\eta }\le \frac{x}{4} + (4a)^{1/(1-\eta )} \qquad \forall \, a,x \ge 0. \end{aligned}$$

In particular, as $\frac{\gamma p+1-p}{\gamma p}<1$,

$$\begin{aligned}&C \Bigl (\sigma ^{p'-1}+\sigma ^{\frac{q'}{p'}(p'-1)}\Bigr ) \left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}\frac{\gamma p+1-p}{\gamma p}} \hspace{-4mm}\\&\quad \le \ \frac{1}{4} \left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}}+C\, \Bigl (\sigma ^{p'-1}+\sigma ^{\frac{q'}{p'}(p'-1)}\Bigr )^{\frac{\gamma p}{p-1}} \end{aligned}$$

so that (2.8) becomes

$$\begin{aligned}&\frac{1}{4}\left( \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\right) ^{\frac{p}{p'}}\\&\quad \le C\Bigl (\sigma ^{p'-1}+\sigma ^{\frac{q'}{p'}(p'-1)}\Bigr )^{\frac{\gamma p}{p-1}} + C\, \Vert {\bar{v}} \Vert ^{\frac{q'}{p'}(p'-p)}_{L^{q'}(\Omega _{\sigma ,{\bar{v}}})} \left( \displaystyle \int \limits _{\Omega }|h_{k^{\frac{p'}{q'}},\gamma }\left( {\bar{v}}\right) |^{q'}\right) ^{\frac{p}{p'}}. \end{aligned}$$

Thus there are $C>0$ and $\sigma _1>1$ such that

$$\begin{aligned}{} & {} \ \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'}\nonumber \\{} & {} \quad \le C\Bigl (\sigma ^{p'-1}+\sigma ^{\frac{q'}{p'}(p'-1)}\Bigr )^{\frac{\gamma p'}{p-1}} + C\, \Vert {\bar{v}} \Vert ^{\frac{q'}{p}(p'-p)}_{L^{q'}(\Omega _{\sigma ,{\bar{v}}})} \displaystyle \int \limits _{\Omega }|h_{k^{\frac{p'}{q'}},\gamma }\left( {\bar{v}}\right) |^{q'} \end{aligned}$$

(2.9)

for any $({\bar{u}}, {\bar{v}}) \in D_{r,\alpha , \delta }(u_0,v_0)$, any $k,\gamma >1$ and any $\sigma \ge \sigma _1$.

Reasoning in a similar way and exploiting that $\langle I'_{\alpha , \delta }({\bar{z}}),\left( 0,\Phi _{k,\gamma p,\gamma }({\bar{v}})\right) \rangle =0$, we find $C>0$ and $\sigma _2\ge \sigma _1$ such that

$$\begin{aligned}{} & {} \displaystyle \int \limits _{\Omega }|h_{{\tilde{k}},\gamma }({\bar{v}})|^{q'}\nonumber \\{} & {} \quad \le C \Bigl (\sigma ^{q'-1}+\sigma ^{\frac{p'}{q'}(q'-1)} \Bigr )^{\frac{\gamma q'}{q-1}} + C\, \Vert {\bar{u}} \Vert ^{\frac{p'}{q}(q'-q)}_{L^{p'}(\Omega _{\sigma ,{\bar{u}}})} \displaystyle \int \limits _{\Omega }|h_{{\tilde{k}}^{\frac{q'}{p'}},\gamma }\left( {\bar{u}}\right) |^{p'} \end{aligned}$$

(2.10)

for any $({\bar{u}}, {\bar{v}}) \in D_{r,\alpha , \delta }(u_0,v_0)$, any ${\tilde{k}},\gamma >1$ and any $\sigma \ge \sigma _2$.

Setting $\tilde{k}=k^{\frac{p'}{q'}}$ in (2.10) and substituting in (2.9) we obtain

$$\begin{aligned}&\displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'} \le C \Bigl (\sigma ^{p'-1}+\sigma ^{\frac{q'}{p'}(p'-1)} \Bigr )^{\frac{\gamma p'}{p-1}}\\&\quad + C\, \Vert {\bar{v}} \Vert ^{\frac{q'}{p}(p'-p)}_{L^{q'}(\Omega _{\sigma ,{\bar{v}}})} \left( \Bigl (\sigma ^{q'-1}+\sigma ^{\frac{p'}{q'}(q'-1)} \Bigr )^{\frac{\gamma q'}{q-1}} + \, \Vert {\bar{u}} \Vert ^{\frac{p'}{q}(q'-q)}_{L^{p'}(\Omega _{\sigma ,{\bar{u}}})} \displaystyle \int \limits _{\Omega }|h_{k,\gamma }\left( {\bar{u}}\right) |^{p'} \right) . \end{aligned}$$

Using again Lemma 2.1 and choosing a suitable $\sigma \ge \sigma _2$, we find $C>0$ such that, for any $k,\gamma >1$

$$\begin{aligned} \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{u}})|^{p'} \le C \end{aligned}$$

where C depends on r and $\gamma$ but is independent of k.

Analogously, we can prove that there is $C>0$, independent of k, such that

$$\begin{aligned} \displaystyle \int \limits _{\Omega }|h_{k,\gamma }({\bar{v}})|^{p'} \le C \end{aligned}$$

for any $k,\gamma >1$.

Thus we can use Fatou Lemma and, passing to the limit for $k\rightarrow +\infty$, we get

$$\begin{aligned} \displaystyle \int \limits _{\Omega }|{\bar{u}}|^{\gamma p'}, \displaystyle \int \limits _{\Omega }|{\bar{v}}|^{\gamma q'} \le C \qquad \qquad \forall ({\bar{u}}, {\bar{v}}) \in D_{r,\alpha , \delta }(u_0,v_0) \end{aligned}$$

(2.11)

where C depends on r and $\gamma$.

Since $\gamma >1$ is an arbitrary number, we have that ${\bar{u}}, {\bar{v}} \in L^{t}(\Omega )$ for any $t>1$.

In particular, by $(*)$, we derive that there is $m>\max \{N/p,N/q\}$ such that $H_s(\delta ,{\bar{u}},{\bar{v}}), \, H_t(\delta ,{\bar{u}},{\bar{v}})\in L^m(\Omega )$, for any $({\bar{u}}, {\bar{v}}) \in D_{r,\alpha , \delta }(u_0,v_0)$, and

$$\begin{aligned} \Vert H_s(\delta ,{\bar{u}},{\bar{v}})\Vert _m, \ \Vert H_t(\delta ,{\bar{u}},{\bar{v}})\Vert _m\le C_1 \end{aligned}$$

(2.12)

where the constant $C_1>0$ depends on r but is independent of $\alpha$ and $\delta$.

We want to prove that for any $({\bar{u}}, {\bar{v}}) \in D_{r,\alpha , \delta }(u_0,v_0)$, ${\bar{u}}$ and ${\bar{v}}$ are in $L^\infty (\Omega )$ and

$$\begin{aligned} \Vert {\bar{u}}\Vert _{\infty }, \ \Vert {\bar{v}}\Vert _{\infty }\le C_2 \end{aligned}$$

where the constant $C_2>0$ still depends on r but is independent of $\alpha$ and $\delta$.

Denoting by ${p^*}'=\frac{p^*}{p^*-1}$, from $m>N/p$ we see that

$$\begin{aligned} m>{p^*}' \ \hbox { and } \quad \frac{p^*}{p-1}\left( \frac{1}{{p^*}'}-\frac{1}{m}\right) >1. \end{aligned}$$

Once fixed $({\bar{u}}, {\bar{v}}) \in D_{r,\alpha , \delta }(u_0,v_0)$, for any $k\in \mathbb {N}$, let us denote by $A_k= \left\{ x\in \Omega \,\ |{\bar{u}} (x)|\ge k\right\}$ and by

$$\begin{aligned} G_k(r)&= {\left\{ \begin{array}{ll} 0 &{} \hbox {if }|r|\le k, \\ r-k &{} \hbox {if } r\ge k, \\ r+k &{} \hbox {if } r\le -k. \end{array}\right. } \end{aligned}$$

As $G_k({\bar{u}}) \in W^{1,p}_{0}(\Omega )$ and $\langle I'_{\alpha , \delta }({\bar{u}}, {\bar{v}}),\left( G_k({\bar{u}}),0\right) \rangle =0$, denoting by ${\bar{f}}=H_s(\delta ,{\bar{u}},{\bar{v}})$ and redefining, when necessary, a positive constant C independent on k, we have

$$\begin{aligned}&\left( \displaystyle \int \limits _{\Omega }|G_k({\bar{u}})|^{p^*}\right) ^{p/p^*} \le C\displaystyle \int \limits _{\Omega }|\nabla G_k({\bar{u}})|^p = C\displaystyle \int \limits _{\Omega }|\nabla {\bar{u}}|^p G_k'({\bar{u}})\\&\quad \le C\displaystyle \int \limits _{\Omega }(\alpha +|\nabla {\bar{u}}|^2)^{\frac{p-2}{2}} \bigl ( \nabla {\bar{u}} \cdot \nabla G_k({\bar{u}})\bigr )\\&\quad =C\displaystyle \int \limits _{\Omega }{\bar{f}} G_k({\bar{u}}) =C\int _{A_k}{\bar{f}} G_k({\bar{u}}) \le C\left( \int _{A_k} |{\bar{f}}|^{{p^*}'}\right) ^{1/{p^*}'} \left( \displaystyle \int \limits _{\Omega }|G_k({\bar{u}})|^{p^*}\right) ^{1/p^*} \end{aligned}$$

hence

$$\begin{aligned}{} & {} \left( \displaystyle \int \limits _{\Omega }|G_k({\bar{u}})|^{p^*}\right) ^{(p-1)/p^*}\nonumber \\{} & {} \quad \le C\left( \int _{A_k} |{\bar{f}}|^{{p^*}'}\right) ^{1/{p^*}'} \hspace{-5mm}\le C\left( \int _{A_k} |{\bar{f}}|^m\right) ^{1/m}\hspace{-2mm}|A_k|^{\frac{1}{{p^*}'}-\frac{1}{m}}. \end{aligned}$$

(2.13)

Moreover, for any $h>k$

$$\begin{aligned}&\left( \displaystyle \int \limits _{\Omega }|G_k({\bar{u}})|^{p^*}\right) ^{(p-1)/p^*} \ge \left( \int _{A_h}|G_k({\bar{u}})|^{p^*}\right) ^{(p-1)/p^*} \\&\quad \ge \left( \int _{A_h}(h-k)^{p^*}\right) ^{(p-1)/p^*} =(h-k)^{p-1} |A_h|^{\frac{p-1}{p^*}}, \end{aligned}$$

so that, combining with (2.12) and (2.13),

$$\begin{aligned} |A_h|\le \frac{C}{(h-k)^{p^*}}\ |A_k|^{\frac{p^*}{p-1} (\frac{1}{{p^*}'}-\frac{1}{m})} \end{aligned}$$

where $\frac{p^*}{p-1} (\frac{1}{{p^*}'}-\frac{1}{m})>1$.

Thereby, applying Lemma 4.1 in [13], there is $C_2$, depending just on r, such that

$$\begin{aligned} |A_h|=0 \qquad \forall h\ge C_2 \end{aligned}$$

which means that ${\bar{u}}\in L^\infty (\Omega )$ and $\Vert {\bar{u}}\Vert _\infty \le C_2$. As $m>N/q$, reasoning in the same way, we find that also $\bar{v}\in L^\infty (\Omega )$ and $\Vert \bar{v}\Vert \le C_2$, choosing a suitable $C_2>0$. $\square$