\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 144, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/144\hfil Multiple solutions] {Multiple solutions for a quasilinear\\ $(p,q)$-elliptic system} \author[S. M. Khalkhali, A. Razani \hfil EJDE-2013/144\hfilneg] {Seyyed Mohsen Khalkhali, Abdolrahman Razani} % in alphabetical order \address{Seyyed Mohsen Khalkhali \newline Department of Mathematics, Science and Research branch, Islamic Azad University, Tehran, Iran} \email{sm.khalkhali@srbiau.ac.ir} \address{Abdolrahman Razani \newline Department of Mathematics, Imam Khomeini International University, Qazvin, Iran} \email{razani@ikiu.ac.ir} \thanks{Submitted May 8, 2013. Published June 25, 2013.} \subjclass[2000]{35J50, 35D30, 35J62, 35J92, 49J35} \keywords{Weak solutions; critical points; Dirichlet system; \hfill\break\indent divergence type operator} \begin{abstract} We prove the existence of three weak solutions of a quasilinear elliptic system involving a general $(p, q)$-elliptic operator in divergence form, with $1 < p \leqslant n$, $1 < q \leqslant n$. Our main tool is an adaptation of a three critical points theorem due to Ricceri. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary $\partial\Omega$ and $1n,q>n$ is ensured for suitable $F$. Some other works \cite{Br,K,DV,NM} studied mainly problems involving $p$-Laplacian type elliptic operators in divergence form and related eigenvalue problems \begin{gather*} -\operatorname{div}(a(x,\nabla u))=\lambda f(x,u)\quad\text{in } \Omega\\ u=0\quad \text{on }\partial\Omega \end{gather*} These operators have $p$-Laplacian operator as a simple case; i.e., if $a(x,s)=|s|^{p-2}s$ then for $p\geqslant 2$ we have $\Delta_pu=\operatorname{div}(a(x,\nabla u))$ and moreover they have other important cases, such as the generalized mean curvature operator $\operatorname{div}\big((1+|\nabla u|^2)^\frac{p-2}{2}\nabla u\big)$ which is generated by $a(x,s)=(1+|s|^2)^\frac{p-2}{2}s$ and is used in studying the geometric properties of manifolds especially minimal surfaces. The existence of multiple solutions for this type of nonlinear differential equations was studied in \cite{DP,K}. Many of these results are based on some three critical points theorems of Ricceri and Bonanno established in \cite{R1,B1}. In \cite{R3}, Ricceri developed one of his results, \cite[Theorem 1]{R1} by means of an abstract result, \cite[Theorem 4]{R2}. In this article, we shall give a variant of Ricceri's three critical points theorem \cite{R3} which it seems its verification for some type of elliptic operators like $\operatorname{div}\big(a(x,\nabla u)\big)$ is easier. As an application, we study the existence of at least three weak solutions for \eqref{p}. Our approach in dealing with \eqref{p} is very close to Ricceri's one in \cite{R3} but employs some calculations of \cite{NM} to adjust it to our problem. \section{Preliminaries} In the sequel, for any $\xi=(\xi_1,\xi_2,\ldots,\xi_n)\in\mathbb{R}^n$ by $|\xi|$ we mean the usual Euclidean norm of $\xi$; that is, $|\xi|=\sqrt{\xi_1^2+\xi_2^2+\cdots+\xi_n^2}$ which is produced by the inner product $\xi\cdot\eta=\sum_{i=1}^n\xi_i\eta_i$ in which $\xi,\eta\in\mathbb{R}^n$. Also for every $1\leqslant p<\infty$ and open $\Omega\subset\mathbb{R}^n$ and measurable $u:\Omega\rightarrow\mathbb{R}$ we define $\|u\|_{L^p(\Omega)}=\Big(\int_{\Omega}|u|^pdx\Big)^{1/p}$ and for $p>1$ we assume the reflexive separable Sobolev space $W_0^{1,p}(\Omega)$ is endowed with the norm $\|u\|_p=\Big(\int_\Omega \vert \nabla u\vert^p dx\Big)^{1/p}$ which is equivalent with its usual norm $\|u\|_{W_0^{1,p}(\Omega)} =\Big(\int_\Omega \vert u\vert^p+\vert \nabla u\vert^p dx\Big)^{1/p}.$ By setting $p_1=p$, $p_2=q$, and inspired by De N\'{a}poli and Mariani \cite{NM} and Deng and Pi \cite{DP}, we assume that the $a_i:\overline{\Omega}\times\mathbb{R}^n\to\mathbb{R}^n$, for $i=1,2$, satisfy the following conditions: \begin{itemize} \item[(H1)] There exists continuous function $A_i:\overline{\Omega}\times\mathbb{R}^n\to\mathbb{R}$ such that $A_i(x,\xi)$ has $a_i(x,\xi)$ as its continuous derivative with respect to $\xi$ at every $(x,\xi)\in\overline{\Omega}\times\mathbb{R}^n$ with the following additional properties: \begin{itemize} \item[(a)] $A_i(x,0)=0,\quad \forall x\in\Omega$. \item[(b)] There exists some constant $C_1>0$ such that $a_i$ satisfies the growth condition $$|a_i(x,\xi)|\leqslant C_1 (1+|\xi|^{p_i-1}),\quad\forall\xi\in\mathbb{R}^n. \label{Hb}$$ \item[(c)] $A_i$ is strictly convex: For every $t\in [0,1]$ $$\label{eq.4} A_i\big(x,(1-t)\xi+t\eta\big)\leqslant(1-t)A_i(x,\xi)+t A_i(x,\eta),\quad\forall x\in\Omega,\ \forall\xi,\eta\in\mathbb{R}^n$$ and this inequality is strict if $t\in(0,1)$. \item[(d)] $A_i$ satisfies the ellipticity condition: There exists a constant $C_2>0$ such that $$\label{He} A_i(x,\xi)\geqslant C_2|\xi|^{p_i},\quad\forall x\in\Omega,\ \forall\xi\in\mathbb{R}^n.$$ \end{itemize} \end{itemize} Assumption (H1) has some consequences that will be helpful in this article. From the strict convexity and differentiability of $A_i(x,\xi)$ with respect to $\xi$, and assumption (H1)(c), we have $A_i(x,\eta)\geqslant A_i(x,\xi)+a_i(x,\xi)(\eta-\xi),$ from which it follows that $$\label{eq.3} \big(a_i(x,\xi)-a_i(x,\eta)\big)\cdot(\xi-\eta)\geqslant 0,$$ for every $x\in\Omega$ and $\xi,\eta\in\mathbb{R}^n$. Also, from \eqref{eq.3} we obtain $$\label{eq.5} a_i(x,\xi+t\eta)\eta\geqslant a_i(x,\xi)\eta$$ for every $t>0$ and $\xi,\,\eta\in\mathbb{R}^n$. We say the mapping $F:X\to X^*$ satisfies the $S_+$ condition, if every sequence $\{x_n\}_{n=1}^\infty$ in $X$ such that $x_n\rightharpoonup x$ and $\limsup_{n\to\infty}\langle F(x_n),x_n-xt\rangle\leqslant 0$ has a convergent subsequence $\{x_{n_k}\}_{k=1}^\infty$ such that $x_{n_k}\to x$. \begin{proposition}\label{prop1} Let $X$ be a reflexive Banach space and $F,J:X\to\mathbb{R}$ two $C^1$ functionals on $X$. If the mapping $F':X\to X^*$ satisfies $S_+$ condition and $J':X\to X^*$ is compact and $F+J:X\to\mathbb{R}$ is coercive then $F+J$ satisfies the Palais-Smale condition. \end{proposition} \begin{proof} If $\{x_n\}_{n=1}^\infty$ is a sequence in $X$ such that $|F(x_n)+J(x_n)|0$ and any $n\in\mathbb{N}$ and $\|F'(x_n)+J'(x_n)\|\to 0$ then coercivity of $F+J$ implies boundedness of $\{x_n\}_{n=1}^\infty$ and since $X$ is reflexive, there exists a subsequence $\{x_{n_k}\}_{k=1}^\infty$ of $\{x_n\}_{n=1}^\infty$ and $x\in X$ such that $x_{n_k}\rightharpoonup x$. Now compactness of $J':X\to X^*$ implies there exists $x^*\in X^*$ such that $J(x_{n_k})\to x^*$ up to a subsequence. Then since $\langle J'(x_{n_k}),x_{n_k}-x\rangle =\langle J'(x_{n_k})-x^*,x_{n_k}-x\rangle+\langle x^*,x_{n_k}-x\rangle$ and $\{x_{n_k}\}_{k=1}^\infty$ is bounded and $x_{n_k}\rightharpoonup x$, we have $\langle J'(x_{n_k}),x_{n_k}-x\rangle\to 0$. Therefore, \begin{align*} &\limsup_{n\to\infty}\langle F'(x_{n_k}),x_{n_k}-x\rangle\\ &\leqslant\limsup_{n\to\infty}\langle F'(x_{n_k})+J'(x_{n_k}),x_{n_k}-x\rangle -\lim_{n\to\infty}\langle J'(x_{n_k}),x_{n_k}-x\rangle\\ &\leqslant\limsup_{n\to\infty}\|F'(x_{n_k})+J'(x_{n_k})\|\,\|x_{n_k}-x\|=0. \end{align*} Hence, by $S_+$ condition of $F'$, for a subsequence of $\{x_{n_k}\}_{k=1}^\infty$ without relabeling $x_{n_k}\to x$. \end{proof} \section{Main results} First we give a theorem that is a variant of \cite[Theorem 1]{R3}. \begin{theorem}\label{thm1} Let $X$ be a separable and reflexive real Banach space; $I\subset\mathbb{R}$ an interval; $\Phi:X\to\mathbb{R}$ a weakly sequentially lower semicontinuous $C^1$ functional, bounded on each bounded subset of $X$ and has unique global minimum at $x_0\in X$ and further the mapping $\Phi':X\to X^*$ satisfies $S_+$ condition and for every bounded $E\subset X$ there exist constants $C>0$ and $\nu>0$ such that for every $x\in E$ $\Phi(x)-\Phi(x_0)\geqslant C\|x-x_0\|^\nu.$ Also suppose $J:X\to\mathbb{R}$ be a $C^1$ functional with compact derivative such that for each $\lambda\in I$, the functional $\Phi-\lambda J$ is coercive and has a strict local not global minimum at $x_0$. Then for each compact interval $[a,b]\subset I$, there exists $r>0$ with the following property: for every $\lambda\in[a,b]$ and every $C^1$ functional $\Psi:X\to\mathbb{R}$ with compact derivative, there exists $\delta>0$ such that, for each $\mu\in[0,\delta]$, the equation $\Phi'(x)=\lambda J'(x)+\mu\Psi'(x)$ has at least three solutions whose norms are less than $r$. \end{theorem} To prove the above theorem, we need the following lemma which is a variant of \cite[Theorem C]{R3}. \begin{lemma}\label{lem5} Let $X$ be a separable and reflexive real Banach space, $\Phi:X\to\mathbb{R}$ a functional that has unique global minimum at $x_0\in X$ and furthermore for every bounded $E\subset X$ there exist constants $C>0$ and $\nu>0$ such that for every $x\in E$ $$\label{eq.1} \Phi(x)-\Phi(x_0)\geqslant C\|x-x_0\|^\nu.$$ Let $J:X\to\mathbb{R}$ be a weakly sequentially lower semicontinuous functional. Assume that $\Phi+J$ has a local strict minimum at $x_0$ in the strong topology of $X$ and $\lim_{\|x\|\to\infty}\big(\Phi(x)+J(x)\big)=\infty.$ Then $x_0$ is a strict local minimum of $\Phi+J$ in the weak topology of $X$. \end{lemma} \begin{proof} The main part of the proof is the same as that of \cite[Theorem C]{R3}. We show $x_0$ must be a strict local minimum in the weak topology of $X$. If not, by assumption there exists $\rho>0$ such that $\Phi(x_0)+J(x_0)<\Phi(x)+J(x)$ for every $x\in X$ satisfying $\|x\|>\rho$. Set $B=\{x\in X\ : \|x\|\leqslant\rho\}.$ Since $X$ is separable and reflexive, the set $B$ is metrizable in its weak topology which we denote its metric by $\sigma$. Since we suppose $x_0$ is not a strict local minimum in weak topology of $X$, there exists a sequence $\{x_n\}$ in $X$ such that for every $n\in\mathbb{N}$, $$\label{eq.7} \sigma(x_0,x_n)<\frac{1}{n},\quad \Phi(x_n)+J(x_n)\leqslant\Phi(x_0)+J(x_0).$$ So, $x_n\in B$ and $x_n\rightharpoonup x_0$. Then weakly sequentially lower semicontinuity of $J$ implies \begin{align*} \liminf_{n\to\infty}\Phi(x_n)+J(x_0) &\leqslant\liminf_{n\to\infty}\Phi(x_n) +\liminf_{n\to\infty}J(x_n)\\ &\leqslant\liminf_{n\to\infty}\big(\Phi(x_n)+J(x_n)\big) \leqslant\Phi(x_0)+J(x_0). \end{align*} and therefore, $\liminf_{n\to\infty}\Phi(x_n)\leqslant\Phi(x_0).$ But $\Phi(x_0)$ is the global minimum of $\Phi(x)$ so, for a suitable convergent subsequence of $\Phi(x_n)$ we have $\lim_{n\to\infty}\Phi(x_n)=\Phi(x_0)$ then by \eqref{eq.1} we have $x_n\to x_0$ which contradicts strict local minimality of $\Phi(x_0)+J(x_0)$ in the strong topology of $X$ by \eqref{eq.7}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] Following the arguments in \cite[Theorem 1]{R3}, since any $C^1$ functional with compact derivative on $X$ is weakly sequentially continuous \cite[Corollary 41.9]{Z2}, and in particular, it is bounded on each bounded subset of $X$, so for any compact $[a,b]\subset I$ and $\sigma>\sup_{\lambda\in[a,b]}\big(\Phi(x_0)-\lambda J(x_0)\big)$, \begin{align*} &\cup_{\lambda\in[a,b]}\{x\in X : \Phi(x)-\lambda J(x)<\sigma\}\\ &\subset \{x\in X : \Phi(x)-a J(x)<\sigma\} \cup\{x\in X : \Phi(x)-b J(x)<\sigma\}. \end{align*} By the coercivity assumption, the set on the right is bounded and there exists $\eta>0$ such that $$\label{eq.8} \cup_{\lambda\in[a,b]}\{x\in X : \Phi(x)-\lambda J(x)<\sigma\}\subset B_{\eta}$$ where $B_{\eta}=\{x\in X :\|x\|<\eta\}$. Now, set $c^*=\sup_{B_{\eta}}\Phi+\max\{|a|,|b|\}\sup_{B_{\eta}}|J|$ and choose $r>\eta$ so that \label{eq.10} \cup_{\lambda\in[a,b]}\{x\in X : \Phi(x)-\lambda J(x)0$is deduced such that for each$\mu\in[0,\gamma]$the functional$\Phi-\lambda J-\mu\tilde{\Psi}$has at least two local minimum in$B_\eta$, say$x_1,x_2$. Now, If $\delta=\min\{\gamma,\frac{1}{\sup_{\mathbb{R}}|g|}\}$ then for every$\mu\in[0,\delta]$the functional$\Phi-\lambda J-\mu\tilde{\Psi}$is coercive by assumption and satisfies Palais-Smale condition, by Proposition \ref{prop1}. Set \begin{gather*} \mathcal{S}=\{u\in C([0,1],X): u(0)=x_1,\,u(1)=x_2\},\\ c_{\lambda,\mu}=\inf_{u\in\mathcal{S}}\sup_{t\in[0,1]} \big(\Phi(u(t))-\lambda J(u(t))-\mu\tilde{\Psi}(u(t))\big) \end{gather*} then by the Mountain Pass Theorem \cite[Theorem 8.2]{AM}), there exists$x_3\in X$distinct from$x_1$and$x_2such that $\Phi'(x_3)-\lambda J'(x_3)-\mu\tilde{\Psi}'(x_3)=0,\quad \Phi(x_3)-\lambda J(x_3)-\mu\tilde{\Psi}(x_3)=c_{\lambda,\mu}.$ Now since \begin{align*} c_{\lambda,\mu} &\leqslant\sup_{t\in[0,1]}\Phi(x_1+t(x_2-x_1))-\lambda J(x_1+t(x_2-x_1)) -\mu\tilde{\Psi}(x_1+t(x_2-x_1))\\ &\leqslant\sup_{B_\eta}\Phi+\max\{|a|,|b|\}\sup_{B_\eta}J +\delta\sup_{\mathbb{R}}|g|\leqslant c^*+1, \end{align*} we have\Phi(x_3)-\lambda J(x_3)< c^*+2$and therefore$x_3\in B_r$by \eqref{eq.10}. Since$\Psi(x)=\tilde{\Psi}(x)$for every$x\in B_r$so$\Psi'(x_i)=\tilde{\Psi}'(x_i)$for$i=1,2,3$. Thus$x_1,x_2,x_3$are three solutions of$\Phi'(x)=\lambda J'(x)+\mu\Psi'(x)$in$B_r$\end{proof} Our main tool in studying \eqref{p} is the following Theorem, which in fact is a restatement of \cite[Theorem 2]{R3}. It adopts it to our situation and its proof is the same as that of \cite[Theorem 2]{R3}, except that we use Theorem \ref{thm1} instead of \cite[Theorem 1]{R3}, and remove the phrase$\hat{x}_\lambda=x_0$. Therefore we omit its proof. \begin{theorem}\label{thm2} Let$X$be a separable and reflexive real Banach space;$I\subset\mathbb{R}$an interval;$\Phi:X\to\mathbb{R}$a weakly sequentially lower semicontinuous$C^1$functional that has unique global minimum at$x_0\in X$and for every bounded$E\subset X$there exist some constants$C>0$and$\nu>0$such that for every$x\in E$$\Phi(x)-\Phi(x_0)\geqslant C\|x-x_0\|^\nu.$ Let$J:X\to\mathbb{R}$be a$C^1$functional with compact derivative. Finally, setting $\alpha=\max\big\{0,\,\limsup_{\|x\|\to\infty}\frac{J(x)}{\Phi(x)}, \,\limsup_{x\to x_0}\frac{J(x)}{\Phi(x)}\big\},\quad \beta=\sup\big\{\frac{J(x)}{\Phi(x)}: x\in\Phi^{-1}(]0,\infty[)\big\},$ assume that$\alpha<\beta$. Then, for each compact interval$[a,b]\subset ]\frac{1}{\beta},\frac{1}{\alpha}[$(with the conventions$\frac{1}{0}=\infty$,$\frac{1}{\infty}=0$) there exists$r>0$with the following property: for every$\lambda\in[a,b]$and every$C^1$functional$\Psi:X\to\mathbb{R}$with compact derivative, there exists$\delta>0$such that, for each$\mu\in[a,b]$, the equation $\Phi'(x)=\lambda J'(x)+\mu\Psi'(x)$ has at least three solutions whose norms are less than$r$. \end{theorem} Hereafter we denote by$X$the product real Banach space$W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)$in which$p,q>1$and equip it with the norm $\|(u,v)\|=\|u\|_p+\|v\|_q=(\int_\Omega \vert \nabla u\vert^p dx)^{1/p} +(\int_\Omega \vert \nabla v\vert^q dx)^\frac{1}{q}.$ At every$(u,v)\in X$, define \begin{gather*} \Phi(u,v)=\int_\Omega A_1(x,\nabla u)\,dx+\int_\Omega A_2(x,\nabla v)\,dx,\quad \Psi(u,v)=\int_\Omega F\big(x,u(x),v(x)\big)\,dx,\\ J(u,v)=\int_\Omega\int_0^{u(x)}g_1(x,s)\,ds\,dx +\int_\Omega \int_0^{v(x)}g_2(x,s)\,ds\,dx \end{gather*} in which$g_1,g_2$satisfy the following inequalities for some constant$C>0$, $$\label{eq.12} |g_1(x,\xi)|\leqslant C(1+|\xi|^{\tau-1}),\quad |g_2(x,\xi)|\leqslant C(1+|\xi|^{\kappa-1}),$$ for a.e.$x\in\Omega$where$1<\tau0, \] then there exists a subsequence of $\{(u_n,v_n)\}$ which we denote it by the same notation $\{(u_n,v_n)\}$ for which $$\label{eq.2} \lim_{n\to\infty}\|a_1(x,\nabla u_n)-a_1(x,\nabla u)\|_{L^{p'}(\Omega)} +\|a_2(x,\nabla v_n)-a_2(x,\nabla v)\|_{L^{q'}(\Omega)}>0.$$ Since $(u_n,v_n)\to (u,v)$ in $X$, we have $u_n\to u$ and $v_n\to v$ in $W_0^{1,p}(\Omega)$ and $W_0^{1,q}(\Omega)$ respectively. So there exist subsequences $\{u_{n_k}\}$ and $\{v_{n_k}\}$ of $\{u_n\}$ and $\{v_n\}$ respectively and some functions $g\in L^p(\Omega)$ and $h\in L^q(\Omega)$ such that $|\nabla u_{n_k}(x)|\leqslant g(x)$ and $\nabla u_{n_k}\to\nabla u\ \text{a.e.}$ and $|\nabla v_{n_k}(x)|\leqslant h(x)$ and $\nabla v_{n_k}\to\nabla v$ a.e. as well. Thus for some constant $C$ and a.e. $x\in\Omega$ we have $|a_1(x,\nabla u_{n_k})-a_1(x,\nabla u)| \leqslant C(2+|\nabla u_{n_k}|^{p-1}+|\nabla u|^{p-1}) \leqslant 2C(1+g^{p-1})$ and by a similar argument $|a_2(x,\nabla v_{n_k})-a_1(x,\nabla v)|\leqslant 2C(1+h^{p-1}).$ Now by the Dominated Convergence Theorem $\lim_{k\to\infty}\|a_1(x,\nabla u_{n_k})-a_1(x,\nabla u)\|_{L^{p'}(\Omega)} +\|a_2(x,\nabla v_{n_k})-a_2(x,\nabla v)\|_{L^{q'}(\Omega)}=0,$ which contradicts \eqref{eq.2}. Therefore $\Phi':X\to X^*$ is continuous and \emph{a priori} $\Phi\in C^1(X;\mathbb{R})$. \end{proof} \begin{lemma}\label{lem2} Let $\Phi:X\to\mathbb{R}$ be defined as previously. Then $\Phi':X\to X^*$ satisfies $S_+$ condition \end{lemma} \begin{proof} If $(u_n,v_n)\rightharpoonup (u,v)$ in $X$ and $$\label{eq.6} \limsup_{n\to\infty}\langle\Phi'(u_n,v_n),(u_n-u,v_n-v) \rangle\leqslant 0$$ then since $u_n\rightharpoonup u$ and $v_n\rightharpoonup v$ in $W_0^{1,p}(\Omega)$ and $W_0^{1,q}(\Omega)$ respectively \begin{align*} &\limsup_{n\to\infty}\langle\Phi'(u_n,v_n),(u_n-u,v_n-v)\rangle\\ &=\limsup_{n\to\infty}(\int_\Omega \big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big)(\nabla u_n-\nabla u)\,dx\\ &\quad + \int_\Omega \big(a_2(x,\nabla v_n)-a_2(x,\nabla v)\big) (\nabla v_n-\nabla v)\,dx) \end{align*} and by \eqref{eq.3} and \eqref{eq.6}, $\lim_{n\to\infty}\langle\Phi'(u_n,v_n),(u_n-u,v_n-v) \rangle=0,$ and obviously \begin{gather} \lim_{n\to\infty}\int_\Omega\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big) (\nabla u_n-\nabla u)\,dx=0,\label{eq.17}\\ \lim_{n\to\infty}\int_\Omega \big(a_2(x,\nabla v_n)-a_2(x,\nabla v)\big) (\nabla v_n-\nabla v)\,dx=0.\label{eq.18} \end{gather} We shall prove $u_n\to u$ as a consequence of \eqref{eq.17}, and in a similar way \eqref{eq.18} implies $v_n\to v$. By imitating the proof of \cite[ Lemma 2.3]{DP}, put $$P_n(x)=\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big)\cdot(\nabla u_n-\nabla u).$$ Then \eqref{eq.3} implies $P_n(x)\geqslant 0$ and because \eqref{eq.17}, there exists a subsequence of $\{u_n\}$ still denoted by $\{u_n\}$ for which $\lim_{n\to\infty}P_n(x)=0$ a.e. in $\Omega$. Let $E=\cap_{n\in\mathbb{N}}\{x\in\Omega : \lim_{n\to\infty}P_n(x)=0,\, |\nabla u_n(x)|<\infty,\, |\nabla u(x)|<\infty\}.$ Then $m(\Omega-E)=0$, $\lim_{n\to\infty}P_n(x)=0$ in $E$. If $x_0\in E$ then by the Mean Value Theorem and inequality \eqref{He}, \begin{align*} &|\nabla u_n(x_0)|^p\\ &\leqslant C_2^{-1}A_1\big(x_0,\nabla u_n(x_0)\big)=C_2^{-1} a_1\big(x_0,t_n\nabla u_n(x_0)\big) \cdot\nabla u_n(x_0)\quad \text{for some }t_n\in(0,1)\\ &\leqslant C_2^{-1}a_1\big(x_0,\nabla u_n(x_0)\big) \cdot\nabla u_n(x_0)\quad \text{by \eqref{eq.5}}\\ &\leqslant C_2^{-1}[P_n(x_0)+a_1(x_0,\nabla u_n(x_0))\nabla u(x_0)+ a_1(x_0,\nabla u(x_0))\cdot(\nabla u_n(x_0)-\nabla u(x_0))]\\ &\leqslant C_2^{-1}[P_n(x_0)+C_1(1+|\nabla u_n(x_0)|^{p-1})| \nabla u(x_0)|+C_1(1+|\nabla u(x_0)|^{p-1})|\nabla u_n(x_0)|\\ &\quad +a_1\big(x_0,\nabla u(x_0)\big)\cdot\nabla u(x_0)]\quad \text{by \eqref{Hb}} \end{align*} which implies $|\nabla u_n(x_0)|\leqslant C$ for some constant $C>0$. Because by our assumption $\lim_{n\to\infty}P_n(x_0)=0$, for any polynomial $q(t)=t^p+kt^{p-1}+mt+c$ with $p>1$, $\lim_{t\to\infty}q(t)=\infty.$ Now, if $\nabla u_n(x_0)\nrightarrow\nabla u(x_0)$, then $\{\nabla u_n(x_0)\}$ has a convergent subsequence which is denoted by the same notation $\{\nabla u_n(x_0)\}$ and converges to a vector $v_0\ne\nabla u(x_0)$. Hence $\lim_{n\to\infty}P_n(x_0)= (a_1(x_0,v_0)-a_1\big(x_0,\nabla u(x_0)\big)) \cdot(v_0-\nabla u(x_0))>0,$ which contradicts the assumption $x_0\in E$. Therefore, $\nabla u_n(x)\rightarrow\nabla u(x)$ for every $x\in E$. As a consequence, $P_n(x)\to 0$ a.e. in $\Omega$ and if $g_n(x)=P_n(x)+\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big) \cdot\nabla u+a_1(x,\nabla u)\cdot(\nabla u_n-\nabla u) +a_1(x,\nabla u)\cdot\nabla u$ then above calculations show that $$\label{eq.19} |\nabla u_n(x)|^p\leqslant C_2^{-1}g_n(x);$$ furthermore, $$\label{eq.20} g_n(x)\to a_1(x,\nabla u)\cdot\nabla u$$ a.e. in $\Omega$. By Lemma \ref{lem1}, the hypothesis $(u_n,v_n)\rightharpoonup (u,v)$ implies \begin{gather*} \lim_{n\to\infty}\int_\Omega\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big) \cdot\nabla u\,dx =\lim_{n\to\infty}\langle\Phi'(u_n,v_n)-\Phi'(u,v),(u,0)\rangle=0, \\ \lim_{n\to\infty}\int_\Omega a_1(x,\nabla u)\cdot(\nabla u_n-\nabla u)\,dx =\lim_{n\to\infty}\langle\Phi'(u,v),(u_n-u,0)\rangle=0. \end{gather*} On the other hand, \eqref{eq.17} gives $\lim_{n\to\infty}\int_\Omega P_n(x)\,dx=0,$ and hence $$\label{eq.21} \lim_{n\to\infty}\int_\Omega g_n(x)=\int_\Omega a_1(x,\nabla u)\cdot\nabla u.$$ By \eqref{eq.19}, we obtain \begin{align*} |\nabla u_n(x)-\nabla u(x)|^p&\leqslant 2^{p-1}(|\nabla u_n(x)|^p +|\nabla u(x)|^p) \leqslant 2^{p-1}(C_2^{-1}g_n(x)+|\nabla u(x)|^p) \end{align*} and since $\nabla u_n(x)\to\nabla u(x)$ a.e. in $\Omega$, so \eqref{eq.20} implies $\lim_{n\to\infty}C_2^{-1}g_n(x)+|\nabla u(x)|^p =C_2^{-1}a_1(x,\nabla u)\cdot\nabla u+|\nabla u(x)|^p,$ a.e. in $\Omega$. By \eqref{eq.21} we find \begin{align*} \lim_{n\to\infty}\int_\Omega C_2^{-1}g_n(x)+|\nabla u(x)|^p\,dx &=\int_\Omega C_2^{-1}a_1(x,\nabla u)\cdot\nabla u+|\nabla u(x)|^p\,dx\\ &\leqslant C_2^{-1}\|a_1(x,\nabla u)\|_{L^{p'}(\Omega)}\|u\|_p+\|u\|_p^p. \end{align*} by the H\"{o}lder inequality in which $p'=\frac{p}{p-1}$. Therefore, the Dominated Convergence Theorem implies $\lim_{n\to\infty}\int_\Omega|\nabla u_n(x)-\nabla u(x)|^p\,dx=0,$ and therefore $u_n\to u$ in $W_0^{1,p}(\Omega)$. Similarly we have $v_n\to v$ in $W_0^{1,q}(\Omega)$ and finally $(u_n,v_n)\to (u,v)$ in $X$. \end{proof} \begin{lemma}\label{lem3} The functional $\Phi:X\to\mathbb{R}$ is weakly sequentially lower semicontinuous and the functional $J:X\to\mathbb{R}$ is $C^1$ with compact derivative and $\Phi-\lambda J$ is weakly sequentially lower semicontinuous and coercive for each $\lambda\in\mathbb{R}$. \end{lemma} \begin{proof} If $(u_n,v_n)\rightharpoonup (u,v)$ in $X$ and $\liminf_{n\to\infty}\Phi(u_n,v_n)<\Phi(u,v)$ then there exists a subsequence of $\{(u_n,v_n)\}$ denote it by $\{(u_{n_k},v_{n_k})\}$ such that $\{\Phi(u_{n_k},v_{n_k})\}$ converges and $\lim_{n\to\infty}\Phi(u_{n_k},v_{n_k})<\Phi(u,v)$. Since $\Phi\in C^1(X;\mathbb{R})$ by Lemma \ref{lem1}, the Mean Value Theorem implies the existence of $t_n\in (0,1)$ for every $n\in\mathbb{N}$ such that $\Phi(u_n,v_n)-\Phi(u,v)=\langle\Phi'\big(u+t_n(u_n-u),v+t_n(v_n-v)\big), (u_n-u,v_n-v)\rangle.$ On the other hand, \eqref{eq.5} implies $$\label{eq.11} \langle\Phi'(u,v),(\xi,\eta)\rangle\leqslant\langle\Phi'(u+t\xi,v+t\eta), (\xi,\eta)\rangle$$ for any $t\geqslant 0$ and $(\xi,\eta)\in X$. Therefore, $\langle\Phi'(u,v),(u_n-u,v_n-v)\rangle \leqslant\langle\Phi'\big(u+t_n(u_n-u),v+t_n(v_n-v)\big),(u_n-u,v_n-v)\rangle$ and as a consequence, \begin{align*} &\limsup_{k\to\infty}\langle\Phi'(u,v),(u_{n_k}-u,v_{n_k}-v)\rangle\\ &\leqslant\lim_{k\to\infty}\langle\Phi'\big(u+t_{n_k}(u_{n_k}-u), v+t_{n_k}(v_{n_k}-v)\big),(u_{n_k}-u,v_{n_k}-v)\rangle<0 \end{align*} which contradicts $(u_n,v_n)\rightharpoonup (u,v)$ since $\Phi'(u,v)\in X^*$ by Lemma \ref{lem1}. Thus $\liminf_{n\to\infty}\Phi(u_n,v_n)\geqslant\Phi(u,v)$ and $\Phi:X\to\mathbb{R}$ is weakly sequentially lower semicontinuous. It can be shown easily that $J$ is a $C^1$ functional \cite[Theorem 2.9]{AP} and $\langle J'(u,v),(\xi,\eta)\rangle=\int_\Omega g_1(x,u)\xi+g_2(x,v)\eta\,dx.$ If $\{(u_n,v_n)\}$ is a bounded sequence in $X$ then it has a weakly convergent subsequence by reflexivity of $X$ which we also denote it by $\{(u_n,v_n)\}$ and assume $(u_n,v_n)\rightharpoonup (u,v)$. Since $10$ $|F_u(x,u,v)|\leqslant C(1+|u|^{p-1}+|v|^{q\frac{p-1}{p}}),\quad |F_v(x,u,v)|\leqslant C(1+|u|^{p\frac{q-1}{q}}+|v|^{q-1})$ for every $x\in\Omega$ and $u,v\in\mathbb{R}$. Then $\Psi\in C^1(X;\mathbb{R})$ and its derivative $\Psi':X\to X^*$ is compact. \end{lemma} \begin{proof} Since $F(x,u,v)$ is $C^1$ with respect to $u,v$, then for every $x\in\Omega$ there exist $\gamma(x),\theta(x)$ in $(0,1)$ such that \begin{align*} |F(x,u,v)-F(x,0,0)|&\leqslant |F(x,u,v)-F(x,u,0)|+|F(x,u,0)-F(x,0,0)|\\ &\leqslant |F_u(x,\gamma(x) u,0)||u|+|F_v(x,u,\theta(x) v)||v|\\ &\leqslant C(1+|u|^{p-1})|u|+C(1+|u|^{p\frac{q-1}{q}}+|v|^{q-1})|v|\\ &\leqslant C(1+|u|^p+|v|^q) \end{align*} hence $\Psi(u,v)\in\mathbb{R}$. Also for every $(u,v),(\xi,\mu)$ in $X$ and $t\in\mathbb{R}-\{0\}$, by the Mean Value Theorem, \begin{align*} & \lim_{t\to 0}\frac{\Psi(u+t\xi,v+t\mu)-\Psi(u,v)}{t}\\ &=\lim_{t\to 0}\frac{1}{t}\int_\Omega F\big(x,u(x)+t\xi(x),v(x) +t\mu(x)\big)-F\big(x,u(x),v(x)\big)\,dx\\ &=\lim_{t\to 0}\Big\{\int_\Omega F_u\big(x,u(x) +t\theta(x)\xi(x),v(x)+t\mu(x)\big)\xi(x)\,dx\\ &\quad+\int_\Omega F_v\Big(x,u(x),v(x) +t\gamma(x)\mu(x)\Big)\mu(x)\,dx\Big\}, \end{align*} in which $0<\theta(x),\gamma(x)<1$ for any $x\in\Omega$. But $F_u$ is continuous and $F_u\big(x,u(x)+t\theta(x)\xi(x),v(x)+t\mu(x)\big)\to F_u\big(x,u(x),v(x)\big) \quad\text{as }t\to 0$ and for $|t|<1$, \begin{align*} &\big|F_u\big(x,u(x)+t\theta(x)\xi(x),v(x)+t\mu(x)\big)\xi(x)\big|\\ &\leqslant C\Big(1+(|u(x)|+|\xi(x)|)^{p-1}+(|v(x)|+|\mu(x)|)^{q\frac{p-1}{p}}\Big)|\xi(x)| \end{align*} therefore, the Dominated Convergence Theorem implies $\lim_{t\to 0}\int_\Omega F_u\big(x,u(x)+t\theta(x)\xi(x),v(x)+t\mu(x)\big) \xi(x)\,dx = \int_\Omega F_u\big(x,u(x),v(x)\big)\xi(x)\,dx$ and similarly $\lim_{t\to 0}\int_\Omega F_v\Big(x,u(x),v(x)+t\gamma(x)\mu(x)\Big)\mu(x)\,dx = \int_\Omega F_v\Big(x,u(x),v(x)\Big)\mu(x)\,dx.$ Therefore, \begin{align*} \langle\Psi'(u,v),(\xi,\mu)\rangle &=\lim_{t\to 0} \frac{\Psi(u+t\xi,v+t\mu)-\Psi(u,v)}{t}\\ &=\int_\Omega F_u(x,u,v)\xi+F_v(x,u,v)\mu\,dx \end{align*} and $\Psi$ is G\^{a}teaux differentiable at any $(u,v)\in X$ and for every $(\xi,\mu)\in X$ $\langle\Psi'(u,v),(\xi,\mu)\rangle =\int_\Omega F_u(x,u,v)\xi+F_v(x,u,v)\mu\,dx.$ The continuity and compactness of $\Psi'$ can be proved like the continuity of $\Phi'$ and the compactness of $J'$ respectively. \end{proof} Now we are ready to prove our next main result which deals with the existence of three weak solutions for \eqref{p}, by introducing some controls on the behaviour of antiderivatives of $g_1$ and $g_2$ at zero. \begin{theorem}\label{thm3} Let $g_1,g_2$ satisfy \eqref{eq.12} and suppose $$\label{eq.9} \max\big\{\limsup_{\xi\to 0}\frac{\sup_{x\in\Omega}G_1(x,\xi)}{|\xi|^p},\, \limsup_{\xi\to 0}\frac{\sup_{x\in\Omega}G_2(x,\xi)}{|\xi|^q}\big\}\leqslant 0,$$ where $G_1(x,\xi)=\int_0^{\hspace*{1pt}\xi} g_1(x,s)\,ds,\quad G_2(x,\xi)=\int_0^{\hspace*{1pt}\xi} g_2(x,s)\,ds$ for any $(x,\xi)\in\Omega\times\mathbb{R}$. Also, suppose the function $F:\overline\Omega\times\mathbb{R}^2\to\mathbb{R}$ satisfies all hypotheses of Lemma \ref{lem4} and in addition $\sup\big\{J(u,v): (u,v)\in X\big\}>0.$ Then, if we set $\gamma=\inf\big\{\frac{\Phi(u,v)}{J(u,v)}: (u,v)\in X,\,J(u,v)>0, \,\Phi(u,v)>0\big\}$ for each compact interval $[a,b]\subset]\gamma,\infty[$ there exists $r>0$ such that for every $\lambda\in[a,b]$, there exists $\delta>0$ such that for every $\mu\in[0,\delta]$, the problem \eqref{p} has at least three weak solutions whose norms in $X$ are less than $r$. \end{theorem} \begin{proof} First note that if $p\leqslant q$ then for every bounded $E\subset X$ there exists some constant $C>0$ such that $\Phi(u,v)-\Phi(0,0)\geqslant C_2\big(\|u\|_p^p+\|v\|_q^q\big) \geqslant C\big(\|u\|_p+\|v\|_q\big)^p=C\|(u,v)\|^p$ for every $(u,v)\in E$, and if $p>q$ then $\Phi(u,v)-\Phi(0,0)\geqslant C\|(u,v)\|^q.$ Furthermore every weak solution of \eqref{p} is a solution of $\Phi'(x)=\lambda J'(x)+\mu \Psi'(x)$. Since $1<\tau0$ and $p0$ is arbitrary $\limsup_{(u,v)\to(0,0)}\frac{J(u,v)}{\Phi(u,v)}=0.$ Hence, by \eqref{eq.14} we have $\alpha=0$ in Theorem \ref{thm2} and since all other hypotheses of Theorem \ref{thm2} for the functionals $\Phi$ and $J$ and the point $x_0=(0,0)\in X$ are established in Lemmas \ref{lem1}, \ref{lem2} and \ref{lem3} and the functional $\Psi$ has needed properties by Lemma \ref{lem4}, therefore the result is proved. \end{proof} \begin{thebibliography}{00} \bibitem{AM} Ambrosetti, A.; Malchiodi, A.; \emph{Nonlinear Analysis and Semilinear Elliptic Problems}, Cambridge: Cambridge University Press, 2007. \bibitem{AP} Ambrosetti, A.; Prodi, G.; \emph{A Primer of Nonlinear Analysis}, Cambridge: Cambridge University Press, 1993. \bibitem{BF} Boccardo, L.; Figueiredo, D.; \emph{Some remarks on a system of quasilinear elliptic equations}, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 309-323. \bibitem{B1} Bonanno, G.; \emph{Some remarks on a three critical points theorem}, Nonlinear Anal. 54 (2003), 651-665. \bibitem{DP} Deng, Y.; Pi, H.; \emph{Multiple solutions for $p$-harmonic type equations}, Nonlinear Anal. 71 (2009), 4952-4959. \bibitem{B2} Bonanno, G.; Heidarkhani, S.; O'Regan D.; \emph{Multiple Solutions for a class of Dirichlet Quasilinear Elliptic Systems driven by a $(P,Q)$-Laplacian operator}, Dynamic Systems and Applications, 20 (2011), 89-100. \bibitem{Bo} Bozhkov, Y.; Mitidieri, E.; \emph{Existence of multiple solutions for quasilinear systems via bering method}, J. Differential Equations, 190 (2003), 239-267. \bibitem{Br} Br\'{e}zis, H.; Oswald, L.; \emph{Remarks on sublinear elliptic equations}, Nonlinear Analysis 10 (1986), 55-64. \bibitem{CC} Chun, Li; Chun-Lei, Tang; \emph{Three solutions for a class of quasilinear elliptic systems involving the (p, q)-Laplacian}, Nonlinear Anal., 69 (2008), 3322-3329. \bibitem{NM} De N\'{a}poli, P.; Mariani, M. C.; \emph{Mountain pass solutions to equations of $p$-Laplacian type}, Nonlinear Anal. 54(2003), 1205-1219. \bibitem{DV} Duc, D. M.; Vu, N. T.; \emph{Nonuniformly elliptic equations of $p$-Laplacian type}, Nonlinear Analysis 61 (2005), 1483-1495. \bibitem{K} Krist\'{a}ly, A; Lisei, H.; Varga, C.; \emph{Multiple solutions for p-Laplacian type equations}, Nonlinear Anal. 68 (2008), 1375-1381. \bibitem{R1} Ricceri, B.; \emph{On three critical points theorem}, Arch. Math. 75 (2000), 220-226. \bibitem{R2} Ricceri, B.; \emph{Sublevel sets and global minima of coercive functionals and local minima of their perturbations}, J. Nonlinear Convex Anal., 5 (2004), 157-168. \bibitem{R3} Ricceri, B.; \emph{A further three critical points theorem}, Nonlinear Anal. 71 (2009), 4151-4157. \bibitem{Z} Zeidler, E.; \emph{Nonlinear functional analysis and its applications}, Vol. II/B. Berlin-Heidelberg-New York 1985. \bibitem{Z2} Zeidler, E.; \emph{Nonlinear functional analysis and its applications}, Vol. III. Berlin-Heidelberg-New York 1985. \end{thebibliography} \end{document}