\documentstyle{amsart} \begin{document} {\noindent\small {\em Electronic Journal of Differential Equations}, Vol.\ {\bf 1999}(1999), No.~14, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)} \thanks{\copyright 1999 Southwest Texas State University and University of North Texas.} \vspace{1.5cm} \title[\hfilneg EJDE--1999/14\hfil Quasilinear diagonal elliptic systems] {Existence of multiple solutions for quasilinear diagonal elliptic systems} \author[Marco Squassina\hfil EJDE--1999/14\hfilneg] {Marco Squassina} \address{ Marco Squassina \hfill\break Dipartimento di Matematica, Milan University, Via Saldini 50, 20133 Milano, Italy } \email{squassin@@ares.mat.unimi.it } \date{} \thanks{Submitted January 4, 1999. Published May 10, 1999.} \subjclass{35D05, 35J20, 35J60} \keywords{Quasilinear elliptic differential systems, \hfill\break\indent Nonsmooth critical point theory.} \begin{abstract} Nonsmooth-critical-point theory is applied in proving multiplicity results for the quasilinear symmetric elliptic system $$-\sum_{i,j=1}^{n}D_j(a^{k}_{ij}(x,u)D_iu_k)+ {\frac 12}\sum_{i,j=1}^{n}\sum_{h=1}^N D_{s_k}a^{h}_{ij}(x,u)D_iu_hD_ju_h=g_k(x,u)\,,$$ for $k=1,..,N$. \end{abstract} \maketitle \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} % Many papers have been published on the study of multiplicity of solutions for quasilinear elliptic equations via nonsmooth-critical-point theory; see e.g. \cite{ab,ag,arioli,c,cvar,cd,cod,pell,s}. However, for the vectorial case only a few multiplicity results have been proven: \cite[Theorem 3.2]{s} and recently \cite[Theorem 3.2]{arioli}, where systems with multiple identity coefficients are treated. In this paper, we consider the following diagonal quasilinear elliptic system, in an open bounded set $\Omega\subset{\mathbb R}^n$ with $n\ge 3$, $$\label{system} \begin{gathered} -\sum_{i,j=1}^{n}D_j(a^{k}_{ij}(x,u)D_iu_k)+ {\frac 12}\sum_{i,j=1}^{n}\sum_{h=1}^N D_{s_k}a^{h}_{ij}(x,u)D_iu_hD_ju_h= \\ = D_{s_k}G(x,u) \quad \text{in } \Omega\,, \end{gathered}$$ for $k=1,..,N$, where $u:\Omega\to{\mathbb R}^N$ and $u=0$ on $\partial\Omega.$ To prove the existence of weak solutions, we look for critical points of the functional $f:H^{1}_{0}(\Omega,{\mathbb R}^{N})\to{\mathbb R}$, $$f(u)=\frac{1}{2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N} a^{h}_{ij}(x,u)D_iu_hD_ju_h\,dx- \int_{\Omega}G(x,u)\,dx\,. \label{Jlambda}$$ This functional is not locally Lipschitz if the coefficients $a^{h}_{ij}$ depend on $u$; however, as pointed out in \cite{ab,c}, it is possible to evaluate $f'$, \begin{eqnarray*} f'(u)(v)& = &\int_\Omega\sum_{i,j=1}^{n} \sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\ && + \frac{1}{2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a^{h}_{ij}(x,u)\cdot vD_iu_hD_ju_h\,dx- \int_\Omega D_sG(x,u)\cdot v\,dx \end{eqnarray*} for all $v\in H^{1}_{0}(\Omega,{\mathbb R}^{N}) \cap L^\infty(\Omega ,{\mathbb R}^N)$. We shall apply the nonsmooth-critical-point theory developed in \cite{cdm,dm,ioffe,ka}. For notation and related results, the reader is referred to \cite{cd}. To prove our main result and to provide some regularity of solutions, we consider the following assumptions. \smallskip \noindent{\bf (A1)} The matrix $\left(a^{h}_{ij}(\cdot,s)\right)$ is measurable in $x$ for every $s\in{\mathbb R}^N$, and of class $C^1$ in $s$ for a.e. $x\in\Omega$ with $$a^{h}_{ij}=a^{h}_{ji}\,.$$ Furthermore, we assume that there exist $\nu >0$ and $C>0$ such that for a.e. $x\in\Omega$, all $s\in{\mathbb R}^N$ and $\xi\in {\mathbb R}^{nN}$$$\label{elliptic} \sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,s){\xi }^{h}_i{\xi }^{h}_j \ge \nu |{\xi }|^2, \enskip\left|a^{h}_{ij}(x,s)\right|\leq C, \enskip\left|D_sa^{h}_{ij}(x,s)\right|\leq C$$ and$$\sum_{i,j=1}^{n}\sum_{h=1}^{N}s\cdot D_sa^{h}_{ij}(x,s) {\xi }^{h}_i{\xi }^{h}_j\ge 0. \label{semipositivity}$$ \smallskip \noindent{\bf (A2)} There exists a bounded Lipschitz function $\psi:{\mathbb R}\to{\mathbb R}$, such that for a.e. $x\in \Omega$, for all $\xi\in {\mathbb R}^{nN}$, $\sigma\in\{-1,1\}^N$ and $r,s\in{\mathbb R}^{N}$$$\label{key} \sum_{i,j=1}^{n}\sum_{h=1}^{N} \left(\frac{1}{2}D_{s}a^{h}_{ij}(x,s) \cdot\exp_\sigma(r,s)+ a_{ij}^{h}(x,s)D_{s_h}(\exp_\sigma(r,s))_h\right)\xi_i^h\xi_j^h\leq 0,$$ where $\left(\exp_{\sigma}(r,s)\right)_i:= \sigma_i\exp[\sigma_i(\psi(r_i)-\psi(s_i))]$ for each $i=1,..,N$. \medskip \medskip \noindent{\bf (G1)} The function $G(x,s)$ is measurable in $x$ for all $s\in{\mathbb R}^N$ and of class $C^1$ in $s$ for a.e. $x\in\Omega$, with $G(x,0)=0$. Moreover for a.e. $x\in\Omega$ we will denote with $g(x,\cdot)$ the gradient of $G$ with respect to $s$. \smallskip \medskip \noindent{\bf (G2)} For every $\varepsilon>0$ there exists $a_{\varepsilon}\in L^{2n/(n+2)}(\Omega)$ such that $$\label{subcritic} |g(x,s)|\leq a_{\varepsilon}(x) + \varepsilon |s|^{(n+2)/(n-2)}$$ for a.e. $x\in\Omega$ and all $s\in{\mathbb R}^N$ and that there exist $q>2$, $R>0$ such that for all $s\in{\mathbb R}^N$ and for a.e. $x\in \Omega$ $$|s|\geq R\Longrightarrow 0< q G(x,s)\le s\cdot g(x,s). \label{g}$$ \noindent{\bf (AG)} There exists $\gamma \in(0,q-2)$ such that for all $\xi\in{\mathbb R}^{nN}$, $s\in{\mathbb R}^N$ and a.e. in $\Omega$ $$\sum_{i,j=1}^{n}\sum_{h=1}^{N}s\cdot D_sa^{h}_{ij}(x,s)\xi^{h}_{i}\xi^{h}_{j}\le\gamma \sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,s)\xi^{h}_{i}\xi^{h}_{j}\,. \label{AIJ}$$ \medskip Under these assumptions we will prove the following. \begin{theorem} \label{main} Assume that for a.e. $x\in\Omega$ and for each $s\in{\mathbb R}^N$ $$a_{ij}^{h}(x,-s)=a_{ij}^{h}(x,s),\quad g(x,-s)=-g(x,s)\,.$$ Then there exists a sequence $(u^{m})\subseteq H^1_0(\Omega,{\mathbb R}^N)$ of weak solutions to {\rm (\ref{system})} such that $$\lim_{m}f(u^{m})=+\infty\,.$$ \end{theorem} The above result is well known for the semilinear scalar problem $$\begin{gathered} -\sum_{i,j=1}^{n}D_j(a_{ij}(x)D_iu)= g(x,u) \quad \text{in \Omega} \\ u=0 \quad \text{on \partial\Omega}\,. \end{gathered}$$ A. Ambrosetti and P. H. Rabinowitz in \cite{ar,ra} studied this problem using techniques of classical critical point theory. The quasilinear scalar problem $$\begin{gathered} -\sum_{i,j=1}^{n}D_j(a_{ij}(x,u)D_iu)+ {\frac 12}\sum_{i,j=1}^{n}D_sa_{ij}(x,u)D_iuD_ju= g(x,u) \quad \text{in }\Omega \\ u=0 \quad \text{on } \partial\Omega \,, \end{gathered}$$ was studied in \cite{c,cvar,cd} and in \cite{pell} in a more general setting. In this case the functional $$f(u)={1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}a_{ij}(x,u) D_iuD_ju\,dx - \int_{\Omega}G(x,u)\,dx$$ is continuous under appropriate conditions, but it is not locally Lipschitz. Consequently, techniques of nonsmooth-critical-point theory have to be applied. In the vectorial case, to my knowledge, problem (\ref{system}) has only been considered in \cite[Theorem 3.2]{s} and recently in \cite[Theorem 3.2]{arioli} for coefficients of the type $a^{hk}_{ij}(x,s)=\delta^{hk}\alpha_{ij}(x,s)$. In \cite{arioli} a new technical condition is introduced to be compared with our (\ref{key}). They assume that there exist $K>0$ and an increasing bounded Lipschitz function $\psi:[0,+\infty[\to[0,+\infty[$, with $\psi(0)=0$, $\psi'$ non-increasing, $\psi(t)\to K$ as $t\to+\infty$ and such that for all $\xi\in {\mathbb R}^{n}$, for a.e. $x\in \Omega$ and for all $r,s\in{\mathbb R}^{N}$$$\label{Arioli} \sum_{i,j=1}^{n}\sum_{k=1}^{N} \left|D_{s_k}a_{ij}(x,s)\xi_i\xi_j\right|\leq 2e^{-4K}\psi'(|s|)\sum_{i,j=1}^{n}a_{ij}(x,s)\xi_i\xi_j\,.$$ The proof itself of \cite[Lemma 6.1]{arioli} shows that condition (\ref{Arioli}) implies our assumption \textbf{(A2)}. On the other hand, if $N\geq 2$, the two conditions look quite similar. However, condition \textbf{(A2)} seems to be preferable, because when $N=1$ it reduces to the inequality $$\left|\sum_{i,j=1}^{n}D_sa_{ij}(x,s)\xi_i\xi_j\right|\leq 2\psi'(s)\sum_{i,j=1}^{n}a_{ij}(x,s)\xi_i\xi_j,$$ which is not so restrictive in view of (\ref{elliptic}), while (\ref{Arioli}) is in this case much stronger. \section{Boundedness of concrete Palais-Smale sequences} % \begin{definition} \label{defncpsc} Let $c\in{\mathbb R}$. A sequence $(u^m)\subseteq H^1_0(\Omega;{\mathbb R}^N)$ is said to be {\em a concrete Palais-Smale sequence at level $c$} ({\em $(CPS)_c-$sequence}, in short) for $f$, if $f(u^m)\to c$, $$\sum_{i,j=1}^{n}\sum_{h=1}^ND_{s_k}a^{h}_{ij}(x,u^m)D_iu^m_hD_ju^m_h\in H^{-1}(\Omega;{\mathbb R}^N)$$ eventually as $m\to\infty$, and $$-\sum_{i,j=1}^{n}D_j(a^{k}_{ij}(x,u^m)D_iu^m_k)+ {\frac 12}\sum_{i,j=1}^{n}\sum_{h=1}^N D_{s_k}a^{h}_{ij}(x,u^m)D_iu^m_hD_ju^m_h-g_k(x,u^m)$$ converges to zero strongly in $H^{-1}(\Omega;{\mathbb R}^N)$. We say that $f$ satisfies {\em the concrete Palais-Smale condition at level $c$} ($(CPS)_c$ in short), if every $(CPS)_c-$sequence for $f$ admits a strongly convergent subsequence in $H^1_0(\Omega;{\mathbb R}^N)$. \end{definition} Next we state and prove a vectorial version of Brezis-Browder's Theorem \cite{bb}. \begin{lemma} \label{bbvect} Let $T\in L_{\rm loc}^1(\Omega,{\mathbb R}^N)\cap H^{-1}(\Omega,{\mathbb R}^N)$, $v\in H^1_0(\Omega,{\mathbb R}^N)$ and $\eta\in L^1(\Omega)$ with $T\cdot v\geq\eta$. Then $T\cdot v\in L^1(\Omega)$ and $$\langle T,v\rangle=\int_{\Omega}T\cdot v\,dx$$ \end{lemma} \begin{pf} Let $(v_h)\subseteq C^{\infty}_c(\Omega,{\mathbb R}^N)$ with $v_h\to v$. Define $\Theta_h(v)\in H^1_0\cap L^{\infty}$ with compact support in $\Omega$ by setting $$\Theta_h(v)=\min\{|v|,|v_h|\}\frac{v}{\sqrt{|v|^2+\frac{1}{h}}}.$$ Since $$\min\{|v|,|v_h|\}\frac{T\cdot v}{\sqrt{|v|^2+\frac{1}{h}}}\geq -\eta^-\in L^1(\Omega),$$ and $$\left\langle T,\min\{|v|,|v_h|\}\frac{v}{\sqrt{|v|^2+\frac{1}{h}}}\right\rangle =\int_\Omega\min\{|v|,|v_h|\}\frac{T\cdot v}{\sqrt{|v|^2+\frac{1}{h}}}\,dx,$$ a variant of Fatou's Lemma implies $\int_\Omega T\cdot v\,dx\leq \left\langle T,v\right\rangle$, so that $T\cdot v\in L^1(\Omega)$. Finally, since $$\left|\min\{|v|,|v_h|\}\frac{T\cdot v}{\sqrt{|v|^2+\frac{1}{h}}}\right|\leq |T\cdot v|,$$ Lebesgue's Theorem yields $$\left\langle T,v\right\rangle=\int_\Omega T\cdot v\,dx,$$ and the proof is complete. \end{pf} The first step for the $(CPS)_c$ to hold is the boundedness of $(CPS)_c$ sequences. \begin{lemma} \label{bound} Assume {\rm\bf{(A1)}}, {\rm\bf{(G1)}}, {\rm\bf{(G2)}} and {\rm\bf{(AG)}}. Then for all $c\in{\mathbb R}$ each $(CPS)_{c}$ sequence of $f$ is bounded in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$. \end{lemma} \begin{pf} Let $a_{0}\in L^{1}(\Omega)$ be such that for a.e. $x\in \Omega$ and all $s\in{\mathbb R}^N$ $$qG(x,s)\leq s\cdot g(x,s)+a_{0}(x).$$ Now let $(u^{m})$ be a $(CPS)_{c}$ sequence for $f$ and let $w^m\to 0$ in $H^{-1}(\Omega,{\mathbb R}^N)$ such that for all $v\in C^{\infty}_{c}(\Omega,{\mathbb R}^{N})$, \begin{eqnarray*} \langle w^{m},v\rangle &=&\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N} a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}D_{j}v_{h}\,dx+ \\ & &+ {1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}^{h}(x,u^{m}) \cdot vD_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx- \int_{\Omega}g(x,u^{m})\cdot v\,. \end{eqnarray*} Taking into account the previous Lemma, for every $m\in{\mathbb N}$ we obtain \begin{eqnarray*} \lefteqn{ -\| w^{m}\|_{H^{-1}(\Omega,{\mathbb R}^{N})}\| u^{m}\|_{H^{1}_{0} (\Omega,{\mathbb R}^{N})}\leq } \\ & \leq & \int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m}) D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx+ \\ && + {1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}^{h} (x,u^{m})\cdot u^m D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx - \int_{\Omega}g(x,u^{m})\cdot u^{m}\,dx\leq \\ &\leq& \int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx+ \\ & &+ {1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}^{h}(x,u^{m}) \cdot u^m D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx+ \\ & &- q\int_{\Omega}G(x,u^{m})\,d x+\int_{\Omega}a_{0}\,dx\,. \end{eqnarray*} Taking into account the expression of $f$ and assumption {\rm\bf{(AG)}}, we have that for each $m\in{\mathbb N}$, \begin{eqnarray*} \lefteqn{ -\| w^{m}\|_{H^{-1}(\Omega,{\mathbb R}^{N})}\| u^{m}\|_{H^{1}_{0} (\Omega,{\mathbb R}^{N})}\leq }\\ & \leq & -\left({q\over 2}-1\right)\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m} D_{j}u_{h}^{m}\,dx+ \\ & &+ {1\over 2}\int_{\Omega}\sum_{i,j=1}^{n} \sum_{h=1}^{N}D_{s}a_{ij}^{h}(x,u^{m})\cdot u^m D_{i}u_h^{m}D_{j}u_h^{m}\,d x+qf(u^{m})+\int_{\Omega}a_{0}\,dx\leq \\\ & \leq & -\left({q\over 2}-1-{{\gamma}\over 2}\right) \int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx+ \\ & &+ qf(u^{m})+\int_{\Omega}a_{0}\,dx\,. \end{eqnarray*} Because of {\rm\bf{(A1)}}, for each $m\in{\mathbb N}$, \begin{eqnarray*} \nu(q-2-\gamma)\| Du^{m}\|^2_2 & \leq & (q-2-\gamma)\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx \\ & \leq & 2\| w^{m}\|_{H^{-1}(\Omega,{\mathbb R}^{N})}\| u^{m}\|_{H^{1}_{0}(\Omega,{\mathbb R}^{N})}+ 2qf(u^{m})+2\int_{\Omega}a_{0}\,dx\,. \end{eqnarray*} Since $(w^{m})$ converges to $0$ in $H^{-1}(\Omega,{\mathbb R}^{N})$, we conclude that $(u^{m})$ is a bounded sequence in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$. \end{pf} \begin{lemma} \label{gcc} If condition {\rm(\ref{subcritic})} holds, then the map \begin{eqnarray*} H^{1}_{0}(\Omega,{\mathbb R}^{N}) & \longrightarrow & L^{2n/(n+2)}(\Omega,{\mathbb R}^{N}) \\ u & \longmapsto & g(x,u) \end{eqnarray*} is completely continuous. \end{lemma} \begin{pf} This is a direct consequence of \cite[Theorem 2.2.7]{cd}. \end{pf} \section{Compactness of concrete Palais-Smale sequences} % The next result is crucial for the $(CPS)_c$ condition to hold for our elliptic system. \begin{lemma} \label{comp} Assume {\rm\bf (A1)} and {\rm\bf(A2)}, let $(u^m)$ be a bounded sequence in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$, and set \begin{eqnarray*} \langle w^m,v\rangle &=& \int_{\Omega}\sum_{i,j=1}^{n} \sum_{h=1}^{N}a^{h}_{ij}(x,u^m)D_iu_{h}^mD_jv_{h}\,dx+ \\ && +\frac 12\int_{\Omega}\sum_{i,j=1}^{n} \sum_{h=1}^{N}D_{s}a^{h}_{ij}(x,u^m)\cdot vD_iu_h^mD_ju_h^m\,dx \end{eqnarray*} for all $v\in C^{\infty}_c(\Omega,{\mathbb R}^N)$. If $(w^m)$ is strongly convergent to some $w$ in $H^{-1}(\Omega,{\mathbb R}^{N})$, then $(u^m)$ admits a strongly convergent subsequence in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$. \end{lemma} \begin{pf} Since $(u^m)$ is bounded, we have $u^m\rightharpoonup u$ for some $u$ up to a subsequence. Each component $u_k^m$ satisfies (2.5) in \cite{bm}, so we may suppose that $D_iu_k^m\rightarrow D_iu_k$ a.e. in $\Omega$ for all $k=1,\dots,N$ (see also \cite{dmm}). We first prove that \begin{multline} \int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_ju_h\,dx+ \\ +{\frac 12}\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a^{h}_{ij}(x,u) \cdot uD_iu_hD_ju_h\,dx=\langle w,u\rangle. \label{tech} \end{multline} Let $\psi$ be as in assumption {\rm\bf{(A2)}} and consider the following test functions \begin{equation*} v^m=\varphi (\sigma _1\exp [\sigma _1(\psi (u_1)-\psi (u_1^m))],\ldots ,\sigma_N\exp [\sigma_N(\psi (u_N)-\psi (u_N^m))]), \end{equation*} where $\varphi\in C^\infty_c(\Omega)$, $\varphi \ge 0$ and $\sigma_l=\pm 1$ for all $l$. Therefore, since we have $$D_jv_k^m=\left(\sigma_kD_j\varphi+ (\psi'(u_k)D_ju_k-\psi'(u_k^m)D_ju_k^m)\varphi\right) \exp[\sigma_k(\psi(u_k)-\psi(u_k^m))],$$ we deduce that for all $m\in{\mathbb N}$, \begin{multline*} \int_\Omega\sum_{i,j=1}^{n} \sum_{h=1}^{N}a^{h}_{ij}(x,u^m)D_iu_h^m(\sigma_hD_j\varphi +\psi '(u_h)D_ju_h\varphi) \exp [\sigma_{h}(\psi (u_h)-\psi (u_h^m))]\,dx+ \\ + \int_\Omega\sum_{i,j=1}^{n}\sum_{h,l=1}^{N} \frac{\sigma_l}{2}D_{s_l}a^{h}_{ij}(x,u^m)\exp [\sigma_l(\psi (u_l)-\psi(u_l^m))]D_iu_h^mD_ju_h^m\varphi\,dx+ \\ - \int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u^m) D_iu_h^mD_ju_h^m\psi'(u_h^m)\exp[\sigma_h(\psi(u_h)-\psi (u_h^m))] \varphi\,dx=\\ =\langle w^m,v^m\rangle\,. \end{multline*} Let us study the behavior of each term of the previous equality as $m\to\infty$. First of all, if $v=(\sigma_1\varphi,\dots ,\sigma_N\varphi)$, we have that $v^m\rightharpoonup v$ implies$$\lim_m\langle w^m,v^m\rangle=\langle w,v\rangle. \label{passlim}$$ Since $u^m\rightharpoonup u$, by Lebesgue's Theorem we obtain \begin{eqnarray} \lefteqn{ \lim_m\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u^m) D_iu_h^m(D_j(\sigma_h\varphi)+ } \nonumber\\ && \hspace{15mm}+\varphi\psi'(u_h) D_ju_h)\exp[\sigma_{h}(\psi (u_h)-\psi (u_h^m))]\,dx= \label{doppia}\\ & = & \int_\Omega\sum_{i,j=1}^{n} \sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_h(D_jv_h+\varphi\psi '(u_h)D_ju_h)\,dx\,.\nonumber \end{eqnarray} Finally, note that by assumption {\rm\bf{(A2)}} we have \begin{eqnarray*} \sum_{i,j=1}^{n}\sum_{h=1}^{N}\Big(\sum_{l=1}^{N} \frac{\sigma_l}{2}D_{s_l}a^{h}_{ij}(x,u^m)\exp [\sigma_l(\psi (u_l) -\psi(u_l^m))]+ && \\ -a^{h}_{ij}(x,u^m)\psi'(u_h^m)\exp[\sigma_h(\psi(u_h)-\psi (u_h^m))]\Big) D_iu_h^mD_ju_h^m&\leq& 0\,. \end{eqnarray*} Hence, we can apply Fatou's Lemma to obtain \begin{eqnarray*} \lefteqn{ \limsup_m\Big\{\frac{1}{2}\int_\Omega \sum_{i,j=1}^{n}\sum_{h,l=1}^{N}D_{s_l}a^{h}_{ij}(x,u^m)\exp [\sigma_l(\psi (u_l)-\psi(u_l^m))] D_iu_h^mD_ju_h^m(\sigma_l\varphi)\,dx+ }\\ \lefteqn{ -\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u^m)D_iu_h^mD_ju_h^m \psi '(u_h^m)\exp[\sigma_h(\psi (u_h)-\psi (u_h^m))]\varphi\,dx\Big\}\leq }\\ &\leq& \frac{1}{2}\int_\Omega\sum_{i,j=1}^{n}\sum_{h,l=1}^{N}D_{s_l}a^{h}_{ij} (x,u)D_iu_hD_ju_h(\sigma_l\varphi)\,dx+ \\ &&-\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_ju_h\psi' (u_h)\varphi\,dx \,, \hspace{48mm} \end{eqnarray*} which, together with (\ref{passlim}) and (\ref{doppia}), yields \begin{multline*} \int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\ +\frac{1}{2}\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N} D_{s}a^{h}_{ij}(x,u)\cdot vD_iu_hD_ju_h\,dx\ge \langle w,v\rangle \end{multline*} for all test functions $v=(\sigma_1\varphi ,\dots ,\sigma_N\varphi )$ with $\varphi\in C_c^\infty(\Omega,{\mathbb R}^N)$, $\varphi \ge 0$. Since we may exchange $v$ with $-v$ we get \begin{multline*} \int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\ +{\frac 12}\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N} D_{s}a^{h}_{ij}(x,u)\cdot vD_iu_hD_ju_h\,dx=\langle w,v\rangle \end{multline*} for all test functions $v=(\sigma _1\varphi ,\dots ,\sigma_N\varphi )$, and since every function $v\in C_c^\infty(\Omega,{\mathbb R}^N)$ can be written as a linear combination of such functions, taking into account Lemma \ref{bbvect}, we infer (\ref{tech}). Now, let us prove that$$\limsup_m\int_\Omega \sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u^m)D_iu_h^mD_ju_h^m\,dx\leq \int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_ju_h\,dx. \label{limsup}$$ Because of~(\ref{semipositivity}), Fatou's Lemma implies that \begin{eqnarray*} \lefteqn{ \int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}u\cdot D_sa^{h}_{ij}(x,u)D_iu_hD_ju_h\,dx\leq }\\ &\leq& \liminf_m \int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}u^m\cdot D_sa^{h}_{ij}(x,u^m)D_iu^m_hD_ju^m_h\,dx\,. \end{eqnarray*} Combining this fact with~(\ref{tech}), we deduce that \begin{eqnarray*} \lefteqn{ \limsup_m \int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N} a^{h}_{ij}(x,u^m)D_iu^m_hD_ju^m_h\,dx= }\\ &=& \limsup_m\Big[-{1\over 2}\int_{\Omega}\sum_{i,j=1}^n \sum_{h=1}^{N}u^m\cdot D_sa^{h}_{ij}(x,u^m) D_iu^m_hD_ju^m_h\,dx+\langle w^m,u^m \rangle \Big]\leq \\ &\leq& -{1\over 2} \int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}u\cdot D_sa^{h}_{ij} (x,u)D_iu_hD_ju_h\,dx+\langle w,u\rangle= \\ &=&\int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}a^{h}_{ij}(x,u) D_iu_hD_ju_h\,dx\,, \end{eqnarray*} so that (\ref{limsup}) is proved. Finally, by (\ref{elliptic}) we have \begin{eqnarray*} \lefteqn{ \nu \| Du^m-Du\|_2 ^2\leq }\\ &\le& \int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N} a^{h}_{ij}(x,u^m)\left(D_iu_h^mD_ju_h^m-2D_iu_h^mD_ju_h+D_iu_hD_ju_h\right)\,dx. \end{eqnarray*} Hence, by (\ref{limsup}) we obtain\begin{equation*} \limsup_m\Vert Du^m-Du\Vert_2\le 0 \end{equation*} which proves that $u^m\to u$ in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$. \end{pf} We now come to one of the main tools of this paper, the $(CPS)_c$ condition for system (\ref{system}). \begin{theorem} \label{exp} Assume {\rm\bf{(A1)}}, {\rm\bf{(A2)}}, {\rm\bf{(G1)}}, {\rm\bf{(G2)}}, {\rm\bf{(AG)}}. Then $f$ satisfies $(CPS)_c$ condition for each $c\in{\mathbb R}$. \end{theorem} \begin{pf} Let $(u^m)$ be a $(CPS)_c$ sequence for $f$. Since $(u^m)$ is bounded in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$, from Lemma \ref{gcc} we deduce that, up to a subsequence, $(g(x,u^m))$ is strongly convergent in $H^{-1}(\Omega,{\mathbb R}^{N})$. Applying Lemma \ref{comp}, we conclude the present proof. \end{pf} \section{Existence of multiple solutions for elliptic systems} % We now prove the main result, which is an extension of theorems of \cite{c,cd} and a generalization of \cite[Theorem 3.2]{arioli} to systems in diagonal form. \vskip5pt \noindent{\it Proof of Theorem} \ref{main}. We want to apply \cite[Theorem 2.1.6]{cd}. First of all, because of Theorem \ref{exp}, $f$ satisfies $(CPS)_{c}$ for all $c\in{\mathbb R}$. Whence, $(c)$ of \cite[Theorem 2.1.6]{cd} is satisfied. Moreover we have \begin{eqnarray*} {\nu\over 2}\int_{\Omega}\vert Du\vert^{2}\,dx-\int_{\Omega}G(x,u)\,dx &\leq& f(u)\leq \\ &\leq& {1\over 2}nNC\int_{\Omega}\vert Du\vert^{2}\,dx- \int_{\Omega}G(x,u)\,dx. \end{eqnarray*} We want to prove that assumptions (a) and (b) of \cite[Theorem 2.1.6]{cd} are also satisfied. Let us observe that, instead of (b) of \cite[Theorem 2.1.6]{cd}, it is enough to find a sequence $\left(W_n\right)$ of finite dimensional subspaces with $\dim(W_n)\to+\infty$ satisfying the inequality of $(b)$ (see also \cite[Theorem 1.2]{marzo}). Let $W$ be a finite dimensional subspace of $H^{1}_{0}(\Omega;{\mathbb R}^{N})\cap L^\infty(\Omega,{\mathbb R}^N)$. From (\ref{g}) we deduce that for all $s\in{\mathbb R}^N$ with $|s|\geq R$ $$G(x,s)\geq {{G\left(x,R{s\over{\vert s\vert}}\right)} \over R^{q}}\vert s\vert^{q}\geq b_0(x)|s|^q,$$ where $$b_0(x)=R^{-q}\inf\{G(x,s): |s|=R\}>0$$ a.e. $x\in\Omega$. Therefore there exists $a_0\in L^1(\Omega)$ such that$$\label{a0} G(x,s)\geq b_0(x)|s|^q-a_0(x)$$ a.e. $x\in\Omega$ and for all $s\in{\mathbb R}^N$. Since $b_0\in L^1(\Omega),$ we may define a norm $\|\cdot\|_G$ on $W$ by $$\|u\|_G=\left(\int_{\Omega}b_0|u|^q\,dx\right)^{1/ q}.$$ Since $W$ is finite dimensional and $q>2$, from (\ref{a0}) it follows $$\lim_{\|u\|_G\to+\infty\atop u\in W}f(u)=-\infty$$ and condition (b) of \cite[Theorem 2.1.6]{cd} is clearly fulfilled too for a sufficiently large $R>0$. Let now $(\lambda_{h},u_{h})$ be the sequence of eigenvalues and eigenvectors for the problem \begin{gather*} \Delta u=-\lambda u \quad \text{in }\Omega \\ u=0 \quad \text{on } \partial\Omega \,. \end{gather*} Let us prove that there exist $h_0,\alpha>0$ such that $$\forall u\in V^{+}: \| Du\|_{2}=1 \Longrightarrow f(u)\geq\alpha,$$ where $V^{+}=\overline{{\rm span}}\left\{u_{h}\in H^1_0(\Omega,{\mathbb R}^N): h\geq h_{0}\right\}$. In fact, given $u\in V^{+}$ and $\varepsilon>0$, we find $$a_{\varepsilon}^{(1)}\in C^{\infty}_{c}(\Omega),\enskip a_{\varepsilon}^{(2)} \in L^{{2n}\over{n+2}}(\Omega),$$ such that $\| a_{\varepsilon}^{(2)}\|_{{2n}\over{n+2}}\leq\varepsilon$ and $$|g(x,s)|\leq a_{\varepsilon}^{(1)}(x)+a_{\varepsilon}^{(2)}(x) +\varepsilon |s|^{n+2\over n-2}.$$ If $u\in V^{+}$, it follows that\begin{eqnarray*} f(u) & \geq & {\nu\over 2}\| Du\|_2^2-\int_{\Omega}G(x,u)\,dx\\ & \geq & {\nu\over 2}\| Du\|_{2}^{2}-\int_{\Omega} \left(\left(a_{\varepsilon}^{(1)}+a_{\varepsilon}^{(2)}\right) \vert u\vert+{n-2\over 2n}\varepsilon\vert u\vert^{{2n}\over{n-2}}\right)\,dx \\ & \geq & {\nu\over 2}\| Du\|_{2}^{2}-\|a_{\varepsilon}^{(1)}\|_2\|u\|_2 -c_1\|a_{\varepsilon}^{(2)}\|_{2n\over n+2}\|Du\|_2 -\varepsilon c_2 \|Du\|_2^{2n\over n-2} \\ & \geq & {\nu\over 2}\| Du\|_{2}^{2}-\|a_{\varepsilon}^{(1)}\|_2\|u\|_2 -c_1\varepsilon\|Du\|_2 -\varepsilon c_2\|Du\|_2^{2n\over n-2}. \end{eqnarray*} Then if $h_0$ is sufficiently large, from the fact that $(\lambda_h)$ diverges, for all $u\in V^{+}$, $\| Du\|_{2}=1$ implies $$\|a_{\varepsilon}^{(1)}\|_2\|u\|_2\leq {\nu\over 6}\,.$$ Hence, for $\varepsilon>0$ small enough, $\| Du\|_{2}=1$ implies that $f(u)\geq {\nu/ 6}$. \medskip Finally, set $V^{-}=\overline{{\rm span}}\left\{u_{h}\in H^{1}_{0}(\Omega,{\mathbb R}^N): h0$ and $q<\frac{n+2}{n-2}$ such that for all $s\in{\mathbb R}^N$ and a.e. in $\Omega$ \begin{eqnarray} \label{subcr} |g(x,s)|\leq c\left(1+|s|^{q}\right). \end{eqnarray} Then it follows that for every $M>0$, there exists $C(M)>0$ such that for a.e. $x\in\Omega$, for all $\xi\in{\mathbb R}^{nN}$ and $s\in{\mathbb R}^N$ with $|s|\leq M$ $$\label{subcrit} \vert b(x,s,\xi)\vert\leq c(M)\left(1+|\xi|^2\right)\,.$$ A nontrivial regularity theory for quasilinear systems (see, \cite[Chapter VI]{giaquinta}) yields the following : \begin{theorem} \label{general} For every weak solution $u\in H^{1}(\Omega,{\mathbb R}^N)\cap L^{\infty}(\Omega,{\mathbb R}^N)$ of the system {\rm(\ref{system})} there exist an open subset $\Omega_{0}\subseteq\Omega$ and $s>0$ such that \begin{gather*} \forall p\in(n,+\infty): u\in C^{0,1-\frac{n}{p}}(\Omega_{0}; {\mathbb R}^{N}), \\ {\cal H}^{n-s}(\Omega\backslash\Omega_{0})=0\,. \end{gather*} \end{theorem} \begin{pf} For the proof, see \cite[Chapter VI]{giaquinta}. \end{pf} We now consider the particular case when $a_{ij}^{hk}(x,s)=\alpha_{ij}(x,s)\delta^{hk},$ and provide an almost everywhere regularity result. \begin{lemma} \label{partic} Assume condition {\rm (\ref{subcrit})}. Then the weak solutions $u\in H^{1}_0(\Omega,{\mathbb R}^N)$ of the system \begin{multline} \int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\ +{\frac 12}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}(x,u) \cdot vD_iu_hD_ju_h\,dx= \int_{\Omega}g(x,u)\cdot v\,dx \end{multline} for all $v\in C^\infty_c(\Omega,{\mathbb R}^N)$, belong to $L^{\infty}(\Omega,{\mathbb R}^N)$. \end{lemma} \begin{pf} By \cite[Lemma 3.3]{s}, for each $(CPS)_c$ sequence $(u^m)$ there exist $u\in H^{1}_0\cap L^{\infty}$ and a subsequence $(u^{m_k})$ with $u^{m_k}\rightharpoonup u$. Then, given a weak solution $u$, consider the sequence $(u^m)$ such that each element is equal to $u$ and the assertion follows. \end{pf} We can finally state a partial regularity result for our system. \begin{theorem} Assume condition {\rm (\ref{subcrit})} and let $u\in H^{1}_0(\Omega,{\mathbb R}^N)$ be a weak solution of the system \begin{multline} \int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\ +{\frac 12}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}(x,u) \cdot vD_iu_hD_ju_h\,dx= \int_{\Omega}g(x,u)\cdot v\,dx \end{multline} for all $v\in C^\infty_c(\Omega,{\mathbb R}^N)$. 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