\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Superquadratic Hamiltonian systems \hfil EJDE--2003/15} {EJDE--2003/15\hfil Jo\~{a}o Marcos do \'{O} \& Pedro Ubilla \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2003}(2003), No. 15, pp. 1--14. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A multiplicity result for a class of superquadratic Hamiltonian systems % \thanks{ {\em Mathematics Subject Classifications:} 35J50, 35J60, 35J65, 35J55. \hfil\break\indent {\em Key words:} Elliptic systems, minimax techniques, Mountain Pass Theorem, \hfil\break\indent Ekeland's variational principle, multiplicity of solutions. \hfil\break\indent \copyright 2003 Southwest Texas State University. \hfil\break\indent Submitted May 15, 2002. Published February 14, 2003.} } \date{} % \author{Jo\~{a}o Marcos do \'{O} \& Pedro Ubilla} \maketitle \begin{abstract} We establish the existence of two nontrivial solutions to semilinear elliptic systems with superquadratic and subcritical growth rates. For a small positive parameter $\lambda$, we consider the system \begin{gather*} -\Delta v = \lambda f(u) \quad \mbox{in } \Omega , \\ -\Delta u = g(v) \quad \mbox{in } \Omega , \\ u = v=0 \quad \mbox{on } \partial \Omega , \end{gather*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ with $N\geq 1$. One solution is obtained applying Ambrosetti and Rabinowitz's classical Mountain Pass Theorem, and the other solution by a local minimization. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[section]{Proposition} \numberwithin{equation}{section} \section{Introduction} The object of this paper is to establish the existence of two nontrivial solutions to a class of elliptic problems using a variational approach. In particular, we consider the problem for Hamiltonian systems $$\begin{gathered} -\Delta v = \lambda f(u) \quad \mbox{in } \Omega , \\ -\Delta u = g(v) \quad \mbox{in } \Omega , \\ u = v=0 \quad \mbox{on } \partial \Omega , \end{gathered} \label{HS}$$ where $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$ with $N\geq 1$, $\lambda$ is a small positive parameter, and $f,\,g:\Omega \to \mathbb{R}$ are given continuous functions satisfying the following hypotheses: \begin{enumerate} \item[(H1)] There exists a positive constant $C$ such that $| f(t)| \leq C(1+| t| ^{r}) , \quad \mbox{for all }t\in \mathbb{R} .$ \item[(H2)] $g$ is odd function which is increasing, $g(0)=0$, and $\lim_{t\to +\infty }\frac{g(t)}{t^{s}}=1 ,$ where $r\geq 0$, $s>0$, and $$\frac{1}{r+1}+\frac{1}{s+1}>\frac{N-2}{N} \,, \label{dmais}$$ when $N\geq 3$. However, for $N=1,2$ there is no restriction. \item[(H3)] There are positive constants $\mu$ and $R$, with $(\mu -1)s>1$, such that for all $| u| \geq R$, $0<\mu F(u)\leq uf(u)\,.$ \end{enumerate} The inequality (\ref{dmais}) expresses the subcritical character of system (\ref{HS}). Its superquadratic behavior is given by the Assumption (H3). In recent years, various results on the existence of solutions for superlinear elliptic systems have been obtained. Among others, de Figueiredo and Felmer \cite{deFigueiredo-Felmer}, and Hulshof and van der Vorst \cite{Hulshof-Vorst} study these problems by means of a variational approach that considers a Lagrangian formulation with strongly indefinite quadratic part and uses the generalized mountain pass theorem in its infinite dimensional setting due to Benci--Rabinowitz \cite{Benci-Rabinowitz}. In system (\ref{HS}), we isolate $v$ in the second equation obtaining the fourth order quasilinear scalar problem $$\label{SP} \begin{gathered} \Delta (g^{-1}(\Delta u) = \lambda f(u) \quad \mbox{in } \Omega ,\\ u = 0\quad \mbox{on } \partial \Omega , \\ \Delta u = 0\quad \mbox{on } \partial \Omega \end{gathered}$$ whose two solutions may be obtained using variational techniques. Therefore, the solutions of system (\ref{HS}) may be obtained by means the critical points of the functional (\ref{F}). One critical point is established using Ambrosetti and Rabinowitz's classical Mountain Pass Theorem (see for example \cite{Ambrosetti-Rabinowitz}). To obtain the other critical point, we combine Ekeland's variational principle with local minimization. It is well known that the solutions of equation (\ref{SP}) are the critical points of the associated functional $$I_{\lambda }(u)=\frac{1}{p}\int_{\Omega }A(| \Delta u| ^p)\,dx-\lambda \int_{\Omega }F(u)\,dx \label{F}$$ defined on the reflexive Banach space $E=W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega )$ endowed with the norm $\| u\| =\| \Delta u\| _{L^p}$ and with Fr\'{e}chet derivative given by $$\langle I_{\lambda }'(u),\phi \rangle =\int_{\Omega }a(| \Delta u| ^p)| \Delta u| ^{p-2}\Delta u\Delta \phi \,dx-\lambda \int_{\Omega }f(u)\phi\,dx , \quad \mbox{for all } \phi \in E , \label{D}$$ where $A(t)=\int_0^{t}a(s)ds \quad \mbox{and} \quad F(t)=\int_0^{t}f(s)ds ,$ and where the function $a$ is defined by $$a(| t| ^p)| t| ^{p-2}t=g^{-1}(t) , \label{a}$$ with $p=(s+1)/s$. By hypotheses (H1) and (H2), and using standard arguments, we see that the expressions (\ref{F}) and (\ref{D}) are well defined, as well as that the functional $I_{\lambda }$ is of class $C^{1}$. (See for example \cite{deFigueiredo} and \cite{Rabinowitz}.) Note that the subcritical condition for equation (\ref{SP}) given by $r2 , \\ \frac{1}{r+1}+\frac{1}{s+1} \leq \frac{N-2}{N}+\frac{1}{(r+1)(s+1)} \end{gather*} and that$ g$is of class$ C^{2} $with$g^{\prime \prime }(t)=O(| t| ^{s-2}) $at infinity. \end{enumerate} To state our main result, which will be proved in Section 3, we assume the almost homogeneity condition at zero: \begin{enumerate} \item[(H5)] There are positive constants$ r_0$and$ s_0$such that$ r_0s_0<1$, $\lim_{t\to 0}\frac{f(t\sigma )}{f(t)}=\sigma ^{r_0}\quad \mbox{\rm and }\quad \lim_{t\to 0}\frac{g(t\sigma )}{g(t)}=\sigma ^{s_0} .$ \end{enumerate} \begin{theorem} \label{Principal} Assume hypotheses (H1), (H2), (H3), and (H5). Then there exists a positive constant$ \lambda ^{\ast } $such that for any$ 0<\lambda <\lambda ^{\ast }$, there exist at least two nontrivial critical points$ u_{\lambda ,1}$,$ u_{\lambda ,2}\in E $of the functional$ I_{\lambda } $such that$ \| u_{\lambda ,1}\| _{E}\to +\infty $and$\| u_{\lambda ,2}\| _{E}\to 0 $as$ \lambda \to 0 $. Moreover, if we assume that condition (H4) holds, then$ (u_{\lambda ,1},g^{-1}(\Delta u_{\lambda ,1})) $and$ (u_{\lambda ,2},g^{-1}(\Delta u_{\lambda ,2})) $are strong solutions of system (\ref{HS}). \end{theorem} Observe that the fact that system (\ref{HS}) is superquadratic does not imply that its two scalar equations are superquadratic. We have an analogous remark for the subcritical behavior of system (\ref{HS}). We should mention that the idea of isolating one variable of system (\ref{HS}) to obtain a scalar equation is similar in spirit to the one proposed in \cite {Clement-Felmer-Mitidieri} for the study of the existence of both positive periodic solutions and homoclinic solutions of a class of Hamiltonian systems. However, the functional analytic framework of \cite {Clement-Felmer-Mitidieri} is different from ours because, for instance, the nonlinearity$ g(v)$is a power. For work on superlinear elliptic systems using a priori estimates and degree theory arguments, see for example \cite{Clement-deFigueiredo-Mitidieri,Souto}) and the references therein. Costa and Magalh\~{a}es \cite{Costa-Magalhaes} discuss other results for systems with Hamiltonian form. This paper is organized as follows. In Section 2, we give an abstract framework in which we establish an abstract critical point theorem used in Section 3 to prove the main result. \section{The Abstract Framework} This section establishes an abstract critical point theorem used in Section 3 to prove our main result. We first recall standard definitions and notation. Let$ X$be a reflexive Banach space endowed with a norm$\| \cdot \| $. We let$ \langle \cdot , \cdot \rangle $denote the duality pairing between$ X$and its dual$ X^{\ast } $. We denote the weak convergence in$ X $by \lq \lq$ \rightharpoonup $\rq \rq , and denote the strong convergence by \lq \lq$ \to $\rq \rq . We say a mapping$ T:X\to X^{\ast } ${\it satisfies the condition$ (S_{+}) $} if for every sequence$ (u_{n})\subset X $, with$ u_{n}\rightharpoonup u $in$X $, and$\; \limsup\nolimits_{n\to +\infty }\langle T(u_{n}),u_{n}-u\rangle \leq 0$, \linebreak then$ u_{n}\to u $in$X$. Let$ I\in C^{1}(X,\mathbb{R})$. We will say$ I${\it satisfies the Palais--Smale condition}, \linebreak denoted$(PS)$condition, if every Palais--Smale sequence of$ I$(or in other words a sequence$ (u_{n})\subset X $such that$ (I(u_{n})) $is bounded and$ I'(u_{n})\to 0 $in the dual space$X^{\ast } $) is relatively compact. \begin{lemma} \label{Isabela} Let$\Phi ,\Psi :X\to \mathbb{R}$be$C^{1}$functionals satisfying for all$ u\in X$, $$\begin{gathered} \mu \Phi (u)-\langle \Phi '(u),u\rangle \geq M\|u\| ^p-N, \\ \mu \Psi (u)-\langle \Psi '(u),u\rangle \leq Q \end{gathered} \label{PS01}$$ with$ \mu >p>1 $, and where$ M,N $and$ Q$are positive constants. Then every Palais--Smale sequence of the functional$ I_{\lambda }(u)=\Phi (u)-\lambda \Psi (u) $is bounded. \end{lemma} \noindent {\bf Proof.}\quad Let$(u_{n})\subset X $be a Palais--Smale sequence, that is, $$\Phi (u_{n})-\lambda \Psi (u_{n})\to c \label{PS1}$$ and $$| \langle \Phi '(u_{n}),v\rangle -\lambda \langle \Psi '(u_{n}),v\rangle | \leq \epsilon _{n}\| v\| \label{PS2}$$ where$ \epsilon _{n}\to 0 $as$ n\to +\infty $. Multiplying ( \ref{PS1}) by$ \mu $, subtracting (\ref{PS2}), with$ v=u_{n}, from the expression obtained, using (\ref{PS01}) we conclude that \begin{align*} 1+\mu c+\epsilon _{n}\| u_{n}\| & \geq \mu \Phi (u_{n})-\Phi '(u_{n})u_{n}+\lambda (\Psi '(u_{n})u_{n}-\mu \Psi (u_{n}))\\ & \geq M\| u_{n}\| ^p-N-\lambda Q . \end{align*} Consequently,(u_{n})$is bounded in$X$, since$p>1$. \hfill$\square$\begin{lemma} \label{PS} Let$ \Phi ,\Psi :X\to \mathbb{R}$be functionals which satisfy the hypotheses of Lemma \ref{Isabela} such that$ \Phi ' $belongs to the class$ (S)_{+} $, and such that for every sequence$ (u_{n}) $in$ X $, with$ u_{n}\rightharpoonup u $, we have$\lim_{n\to \infty }\langle \Psi '(u_{n}),u_{n}-u\rangle =0$. Then the functional$I:X\to \mathbb{R}$given by$ I(u)=\Phi (u)-\lambda \Psi (u) $satisfies the Palais-Smale condition. \end{lemma} \noindent {\bf Proof.}\quad Let$ (u_{n})\subset X $be a Palais-Smale sequence. According to Lemma \ref{Isabela}, we have$ (u_{n}) $is bounded in$ X$. Hence we may take a subsequence, which is denoted by the same index, such that$ u_{n}\rightharpoonup u $, for some$ u$in$X$. Since$ I'(u_{n})\to 0 $in$ X^{\ast }$, we have $| \langle \Phi '(u_{n}),u_{n}-u\rangle -\lambda \langle \Psi '(u_{n}),u_{n}-u\rangle | \leq \epsilon _{n}\| u_{n}-u\|$ where$ \epsilon _{n}\to 0 $as$ n\to \infty $. Therefore,$ \lim_{n\to \infty }\langle \Phi '(u_{n}),u_{n}-u\rangle =0$, because$ \lim_{n\to \infty }\langle \Psi '(u_{n}),u_{n}-u\rangle =0$. Now since$ \Phi ' $belongs to the class$ (S)_{+}$, we conclude$ u_{n}\to u $in$X$. \hfill$\square$\smallskip The main result of this section, which will be proved in Subsection 2.1, is the following. \begin{theorem} \label{MAIN} Let$ \Phi ,\Psi :X\to \mathbb{R}$be functionals satisfying the hypotheses of Lemma \ref{PS}. Further, suppose that the following three conditions are satisfied: \begin{enumerate} \item[(I1)]$C_{1}\| u\| ^p\leq \Phi (u)\leq C_{2}\| u\| ^p+C_{3}$, for all$ u\in X$. \item[(I2)]$C_{4}\| u\| ^{\mu}-C_{5}\leq \Psi (u)\leq C_{6}\| u\| ^{r}+C_{7}$, for all$ u\in X$. \item[(I3)] There is$ v\in X-\{0\} $such that $\lim_{t\to 0}\frac{\Psi (tv)}{\Phi (tv)}=+\infty ,$ where$ r>\mu >p>1 $and$ C_{1},\ldots , C_{7}$are positive constants. Then there exists$ \lambda ^{\ast }>0 $such that for all$ \lambda \in (0,\lambda ^{\ast })$, there exist two nontrivial critical points$ \{u_{\lambda },v_{\lambda}\}$of the functional$ I_{\lambda }(u)=\Phi (u)-\lambda \Psi (u) $such that$ \| u_{\lambda }\| \to +\infty$and$\| v_{\lambda }\| \to 0 $as$\lambda \to 0$. \end{enumerate} \end{theorem} Nest, we present three lemmas which lead to the proof of Theorem \ref{MAIN}. \begin{lemma}\label{Fla} Let$ \Phi ,\Psi :X\to \mathbb{R}$be functionals satisfying for all$u\in X$, $$\begin{gathered} \Phi (u) \geq C_{1}\| u\| ^p, \\ \Psi (u) \leq C_{6}\| u\| ^{r} + C_{7} \end{gathered} \label{Gloria}$$ where$ r>p>1 $and$ C_{1}, C_{6}$and$C_{7}$are positive constants. Then there exist positive constants$ \alpha _{\lambda }$,$ \rho _{\lambda }$such that $\lim_{\lambda \to 0^{+}}\rho _{\lambda }=+\infty$ {\rm and} $I_{\lambda }(u)=\Phi (u)-\lambda \Psi (u)>\alpha _{\lambda } , \quad\mbox{if} \quad \| u\| =\rho _{\lambda } .$ \end{lemma} \noindent {\bf Proof.}\quad By assumptions (\ref{Gloria}) we have $$I_{\lambda }(u)\geq C_{1} \| u\| ^p-\lambda C_{6} \| u\| ^{r}-\lambda C_{7} , \quad \mbox{for all } u\in X . \label{Marina}$$ Choosing$ u\in X $such that \| u\| =\lambda ^{-s},\quad \mbox{with } 0p>1$ and $C_{2}$, $C_{3}$, $C_{4}$, and $C_{5}$ are positive constants. Then $I_{\lambda }(tu)=\Phi (tu)-\lambda \Psi (tu)\to -\infty$ as $t\to +\infty$, for all $u\in X-\{0\}$. \end{lemma} \noindent {\bf Proof.} It follows easily from (\ref{Amalia}) that for all $t>0$, $I_{\lambda }(tu)\leq C_{2}t^p\| u\| ^p+ C_{3}-C_{4}t^{\mu }\lambda \| u\| ^{\mu }+ C_{5} , \quad \mbox{for all } u\in X .$ Since $\mu>p$, the result follows. \hfill$\square$ \begin{lemma} \label{Mini} Let $\Phi ,\Psi :X\to \mathbb{R}$ be $C^{1}$ functionals which satisfy the $(PS)$ condition and the assumptions (\ref{Gloria}). Furthermore, suppose there exits $v\in X-\{0\}$ such that $$\lim_{t\to 0}\frac{\Psi (tv)}{\Phi (tv)}=+\infty . \label{M1}$$ Then there exits $\tilde{\lambda}>0$ such that for all $\lambda \in (0, \tilde{\lambda})$, the functional $I_{\lambda}(u)=\Phi (u)-\lambda \Psi (u)$ has a nontrivial critical point $v_{\lambda }$ such that $\| v_{\lambda }\|\to 0$ as $\lambda \to 0$, provided that $I_{\lambda }(0)=0$. \end{lemma} \noindent {\bf Proof.}\quad Let $\alpha \in \mathbb{R}$ be such that $\alpha p<1$, and let $u\in X$ such that $\| u\| =\lambda^{\alpha }$. According to inequality (\ref{Marina}), there exists $\tilde{\lambda} >0$ such that for all $\lambda \in (0,\tilde{\lambda})$, $I_{\lambda }(u)\geq C_{1}\lambda ^{\alpha p}-C_{6}\lambda ^{1+\alpha r}-\lambda C_{7}\geq 0 .$ It follows from the first inequality of (\ref{Gloria}) and the assumption (\ref{M1}) that there is $\delta >0$ such that $\Psi (tv)>0$, for all $| t| \leq \delta$. Therefore, given $\lambda \in (0,\tilde{\lambda})$, there exists $t_{\lambda }\in (-\delta ,\delta )$ such that $I_{\lambda }(t_{\lambda }v) = \Phi (t_{\lambda }v)-\lambda \Psi (t_{\lambda }v) = \Psi (t_{\lambda }v)\Big( \frac{\Phi (t_{\lambda }v)}{\Psi (t_{\lambda }v)}-\lambda \Big) <0 .$ Hence the infimum of the functional $I_{\lambda }$ in $B_{X}[0,\lambda ^{\alpha }]$ is negative, where $B_{X}[0,R]$ denotes the closed ball with radius $R$ centered at origin of $X$. Applying Ekeland's variational principle, we obtain a sequence $(u_{n})\subset B_{X}[0,\lambda ^{\alpha }]$ such that $I_{\lambda }(u_{n})\to \inf_{B_{X}[0,\lambda ^{\alpha }]}I_{\lambda }$ and $I_{\lambda }'(u_{n})\to 0$. Since $I_{\lambda }$ satisfies the $(PS)$ condition, and since $I_{\lambda }(0)=0$, we obtain a nontrivial minimizer $u_{\lambda }$, which completes the proof of the Lemma. \hfill$\square$ \paragraph{Proof of Theorem \ref{MAIN}} By Lemma \ref{PS}, the functional $I_{\lambda }$ satisfies the $(PS)$ condition. According to Lemmas \ref{Fla} and \ref{Rico}, we may apply the Mountain Pass Theorem. It follows that there exists $\widehat{\lambda } >0$ such that for all $\lambda \in (0,\widehat{\lambda })$, the functional $I_{\lambda }$ has a critical point $u_{\lambda }$ such that $I_{\lambda }(u_{\lambda })>\alpha _{\lambda }>0 \;$ and $\; \| u_{\lambda }\| \geq \rho _{\lambda }=\lambda ^{-s}\to +\infty$ as $\lambda \to 0$. Finally, since the functional $I_{\lambda }$ satisfies the $(PS)$ condition, by Lemma \ref{Mini}, we can take a suitable small $\lambda ^{\ast }$ such that, for all $\lambda \in (0,\lambda ^{\ast })$, the functional $I_{\lambda }$ has another critical point $v_{\lambda }$ such that $I_{\lambda }(v_{\lambda })<0$ and $\| v_{\lambda }\| \leq \lambda ^{\alpha }\to 0$ as $\lambda \to 0$. The proof of Theorem \ref{MAIN} is now complete. \subsection*{On the $(S_{+})$ condition} The next two lemmas are crucial results in our minimax argument. \begin{lemma} \label{Madalena} Let $L:D(L)\to (L^p(\Omega ))^{k}$ be a continuous, injective linear operator defined on the Banach space $D(L)$ endowed with the norm $\| u\| ^p=\int | Lu| ^p ,$ where $|\cdot |$ denotes a norm of $\mathbb{R}^{k}$. Then the derivative of the functional $J:D(L)\to \mathbb{R}$ defined by $J(u)=\frac{1}{p}\int_{\Omega }| Lu| ^p\,dx$ belongs to the class $(S_{+})$. \end{lemma} \noindent {\bf Proof.}\quad Let $(u_{n})\subset D(L)$ be such that $u_{n}\rightharpoonup u \; \mbox{ in } \; D(L) \quad \mbox{ and } \quad \lim_{n\to \infty }\sup \langle J'(u_{n}),u_{n}-u\rangle \leq 0 .$ Note that since $u_{n}\rightharpoonup u$ in $D(L)$, and since $L$ is continuous, we have $$\mathfrak{J}_{n} = \frac{1}{p}\int_{\Omega }| Lu| ^{p-2}Lu(Lu_{n}-Lu)\,dx\to 0 . \label{SM1}$$ Now according to the H\"{o}lder inequality and using the elementary inequality $| x-y| ^p \leq c(p)\{(| x| ^{p-2}x-| y| ^{p-2}y)(x-y)\}^{s/2}\{| x| ^p+| y| ^p\}^{1-s/2},$ where $s=p$ if $p\in (1,2)$, $s=2$ if $p\geq 2$, and $c(p)$ is a positive constant depending only on $p$, we see that \begin{aligned} \langle J'(u_{n}),u_{n}-u\rangle & = \int_{\Omega }|Lu_{n}| ^{p-2}Lu_{n}(Lu_{n}-Lu)\,dx \\ & = \int_{\Omega }[| Lu_{n}| ^{p-2}Lu_{n}-| Lu| ^{p-2}Lu]L(u_{n}-u)\,dx+\mathfrak{J}_{n} \\ & \geq C\left\{ \int_{\Omega }| Lu_{n}| ^p+| Lu| ^p\right\} ^{s/2-1}\int_{\Omega }| L(u_{n}-u)| ^p+\mathfrak{J}_{n} . \end{aligned} \label{SM2} Finally, we combine (\ref{SM1}) and (\ref{SM2}) with the fact that $\lim_{n\to \infty }\sup \langle J'(u_{n}),u_{n}-u\rangle \leq 0$ to obtain $\lim_{n\to \infty }\int_{\Omega }| L(u_{n}-u)| ^p\,dx=0 ,$ which completes the proof of the lemma. \hfill$\square$ \smallskip Next let $a\in C(\mathbb{R}^{+},\mathbb{R})$ and define $A(s)=\int_0^{s}a(t)dt$ such that the following two conditions are satisfied \begin{itemize} \item[(A1)] The function $h(t) = A(| t| ^p)$ is strictly convex. \item[(A2)] There are positive constants $c_0, c_{1}, c_{2}$, and $c_{4}$ such that $c_0t-c_{1}\leq A(t)\leq c_{2}t-c_{3},\quad \mbox{for all }t>0.$ \end{itemize} Consider the functional $\Phi :D(L)\to \mathbb{R}$ given by $\Phi (u)=\frac{1}{p}\int_{\Omega }A(| Lu| ^p)\,dx .$ It is well known that $\Phi$ is well defined and that it is a $C^{1}$ functional \cite{Rabinowitz}, with Fr\'{e}chet derivative given by $\langle \Phi '(u),v\rangle =\int_{\Omega }a(| Lu| ^p) | Lu| ^{p-2}LuLv\,dx .$ The following is an important convergence criterion. \begin{lemma} \label{Nina} The derivative of the functional $\Phi$ belongs to the class $(S_{+})$. \end{lemma} \noindent {\bf Proof.}\quad Let $(u_{n})\subset D(L)$ be such that $u_{n}\rightharpoonup u\mbox{ in }D(L)\quad \mbox{and \quad} \lim_{n\to \infty }\sup \langle \Phi ^{\prime }(u_{n}),u_{n}-u\rangle \leq 0 .$ We will show that $$\lim_{n\to \infty }\sup \int_{\Omega }| Lu_{n}| ^{p-2}Lu_{n}\cdot(Lu_{n}-Lu)\,dx=0 , \label{mflores}$$ and the proof will then follow from Lemma \ref{Madalena}. First note that \begin{align*} &\langle \Phi '(u_{n}),u_{n}-u\rangle \\ & = \int_{\Omega }[a(| Lu_{n}| ^p)| Lu_{n}| ^{p-2}Lu_{n}-a(| Lu| ^p)| Lu| ^{p-2}Lu]\cdot (Lu_{n}-Lu)\,dx + \mathfrak{K}_{n} , \end{align*} where $$\mathfrak{K}_{n}\;=\; \int_{\Omega }a(| Lu| ^p)| Lu| ^{p-2}Lu]\cdot (Lu_{n}-Lu)\,dx .$$ Since $u_{n}\rightharpoonup u$ in $D(L)$, and since $a$ and $L$ are continuous functions, we have $\mathfrak{K}_{n}\to 0$. Also, since the function $h(t)=A(| t| ^p)$ is strictly convex, and since $\lim_{n\to \infty }\sup \langle \Phi '(u_{n}),u_{n}-u\rangle \leq 0$, it follows that $Lu_{n}(x)\to Lu(x)$ almost everywhere in $\Omega$. Now observe that if $x\in \Omega$ is such that $Lu_{n}(x)(Lu_{n}(x)-Lu(x))\leq 0 ,$ then $Lu_{n}(x)Lu_{n}(x)\leq Lu_{n}(x)Lu(x)\leq | Lu_{n}(x)| | Lu(x)| ,$ which implies $| Lu_{n}(x)| \leq | Lu(x)| .$ %\newpage On the other hand, if $x\in \Omega$ is such that $Lu_{n}(x)(Lu_{n}(x)-Lu(x))\geq 0 ,$ then by (A2), there exist positive constants $C$ and $M$ such that $a(| Lu_{n}| ^p)| Lu_{n}| ^{p-2}Lu_{n}(Lu_{n}-Lu)\geq C| Lu_{n}| ^{p-2}Lu_{n}(Lu_{n}-Lu) ,$ when $| Lu_{n}(x)| >M$. Set $\eta _{n}(x)=| Lu_{n}(x)| ^{p-2}Lu_{n}(x)\cdot(Lu_{n}(x)-Lu(x))$ and consider the sets \begin{gather*} {\cal A}_{n} = \left\{ x\in \Omega :| Lu_{n}(x)|\leq M\right\} ,\\ {\cal B}_{n} = \left\{ x\in \Omega :| Lu_{n}(x)| > M\right\} , \\ {\cal C}_{n} = \left\{ x\in \Omega :\eta _{n}(x)\geq 0\right\} , \\ {\cal D}_{n} = \left\{ x\in \Omega :\eta _{n}(x)<0\right\} . \end{gather*} Then $\int_{\Omega }\eta _{n}(x)\,dx=\int_{\Omega }\eta _{n}\chi _{{\cal A}_{n}}\chi _{{\cal C}_{n}}\,dx+\int_{\Omega }\eta _{n}\chi _{{\cal B}_{n}}\chi _{{\cal C} _{n}}\,dx+\int_{\Omega }\eta _{n}\chi _{{\cal D}_{n}}\,dx ,$ where $\chi _{{\cal U}}$ denotes the characteristic function of the a set ${\cal U}\subset \mathbb{R}^{N}$. By the Lebesgue dominated convergence theorem we have $$\lim_{n\to \infty }\int_{\Omega }\eta _{n}\chi _{{\cal A}_{n}}\chi _{ {\cal C}_{n}}\,dx=\int_{\Omega }\eta _{n}\chi _{{\cal D}_{n}}\,dx=0 . \label{JO1}$$ On the other hand, \begin{align*} &\int_{\Omega }\eta _{n}\chi _{{\cal B}_{n}}\chi _{{\cal C}_{n}}\,dx\\ & \leq C\int_{\Omega }a(| Lu_{n}| ^p)\eta _{n}\chi_{{\cal B}_{n}} \chi _{ {\cal C}_{n}} \,dx \\ & = C\int_{\Omega }a(| Lu_{n}| ^p)\eta _{n}\chi _{{\cal B}_{n}}(1-\chi _{{\cal D}_{n}}) \,dx \\ & = C\int_{\Omega }a(| Lu_{n}| ^p)\eta _{n}\chi _{{\cal B} _{n}} \,dx-C\int_{\Omega }a(| Lu_{n}| ^p)\eta _{n}\chi _{{\cal B}_{n}}\chi _{{\cal D}_{n}} \,dx \\ & = C\int_{\Omega }a(| Lu_{n}| ^p)\eta _{n}(1-\chi _{{\cal A} _{n}}) \,dx-C\int_{\Omega }a(| Lu_{n}| ^p)\eta _{n}\chi _{{\cal B} _{n}}\chi _{{\cal D}_{n}} \,dx . \end{align*} These estimates together with the Lebesgue dominated convergence theorem and the fact that $\lim_{n\to \infty }\sup \langle \Phi '(u_{n}),u_{n}-u\rangle \leq 0$ imply $$\lim_{n\to \infty }\sup \int_{\Omega }\eta _{n}\chi _{{\cal B} _{n}}\chi _{{\cal C}_{n}} \,dx=0 . \label{JO2}$$ The equality (\ref{mflores}) now follows from (\ref{JO1}) and (\ref{JO2}). The Lemma results from (\ref{mflores}) and the fact that $u_{n}\to u$ in $D(L)$. \hfill$\square$ \section{Proof of Theorem \ref{Principal}} \subsection*{Existence of critical point for functional $I_{\protect\lambda }$ in (\ref{F})} This part of the proof is an application of Theorem \ref{MAIN}. Consider the functionals $\Phi (u)=\frac{1}{p}\int_{\Omega }A(| \Delta u| ^p)\,dx\quad \mbox{and} \quad \Psi (u)=\int_{\Omega }F(u)\,dx$ defined on the Banach space $E=W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega )$. Next we check that the conditions of Theorem \ref{MAIN} are satisfied. By our assumption (H2) and (\ref{a}), it is standard to prove that assumptions (A1) and (A2) of Lemma \ref{Nina} hold with $c_{1}=0$ and, of course, the condition (I1) of Theorem \ref{MAIN} holds with $L=\Delta$. Thus by Lemma \ref{Nina}, the derivative of the functional $\Phi$ belongs to the class $(S_{+})$. Furthermore, from assumptions (H1), (H2), and (H3) it is easy to see that conditions (\ref{PS01}) and (I2) hold, and using the Sobolev imbedding theorem we have $\lim_{n\to \infty }\langle \Psi ^{\prime }(u_{n}),u_{n}-u\rangle =\lim \int_{\Omega }f(x,u_{n})(u_{n}-u)=0 ,$ for every sequence $(u_{n})$ in $E$ such that $u_{n}\rightharpoonup u$. Then, using assumption (H4), we show that condition (I3) holds, i.e., there exists $v\in E$ such that $$\lim_{t\to 0}\frac{\Psi (tv)}{\Phi (tv)}=+\infty . \label{PU01}$$ First note that by assumption (H5), $$\lim_{t\to 0}\frac{F(t)}{A(t^p)}=+\infty . \label{PU1}$$ Let $v\in C_0^{\infty }(\Omega ,[0,1])\backslash \{0\}$ such that $0\leq \Delta v\leq 1$. We have $$\frac{\Psi (tv)}{\Phi (tv)}=\frac{p\int_{\Omega }\frac{F(tv)}{F(t)}\,dx}{ \int_{\Omega }\frac{A(t^p| \Delta v| ^p)}{A(t^p)}\,dx}\frac{F(t)}{ A(t^p)} \cdot \label{PU2}$$ Also, from (H5), by the Lebesgue dominated convergence theorem we get $\lim_{t\to 0^{+}}\int_{\Omega }\frac{F(tv)}{F(t)}\,dx=\int_{\Omega }v^{r_0+1}\,dx$ and $\lim_{t\to 0^{+}}\int \frac{A(t^p| \Delta v| ^p)}{A(t^p)}% \,dx=\int_{\Omega }| \Delta v| ^p\,dx.$ Therefore, passing to the limit in (\ref{PU2}) and using (\ref{PU1}) we obtain (\ref{PU01}). \subsection*{Regularity and the existence of solutions for system (\ref{HS})} Here we prove that critical points of the functional $I_{\lambda }$ are indeed strong solutions of problem (\ref{SP}). \begin{proposition} \label{Fati} Let $u\in E=W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega )$ be a critical point of the functional $I_{\lambda }$. Then $u\in W^{4,l}(\Omega )$ for some $l>1$. \end{proposition} As a consequence of Proposition \ref{Fati} we see that $u\in W^{4-(1/l),l}(\partial \Omega )$. Hence, integration by parts shows that $u$ satisfies the second boundary condition of problem (\ref{SP}). Therefore the pair $(u,g^{-1}(\Delta u))$ is a strong solution of system (\ref{HS}). \paragraph{Proof of Proposition \ref{Fati}.} Assume that $N>2p$ (the other case is easier). Using the continuous imbedding $W^{2,p}(\Omega )\hookrightarrow L^{q}(\Omega )$ for $q=Np/(N-2p)$ and assumption (H1) we see that $f(u)\in L^{q/r}(\Omega )$. Thus, from standard regularity argument we get that $v=g^{-1}(\Delta u)$ belongs to $W^{2,q/r}(\Omega )$ which is continuously imbedded in the Sobolev space $W^{1,r_{1}}(\Omega )$ with $r_{1}=Nq/(Nr-q)$ (see \cite{Gilbarg-Trudinger}). To obtain $u\in W^{4,l}(\Omega )$ for some $l > 1$, it is enough to prove that $g(v)\in W^{2,l}(\Omega )$, that is, $g'(v)\partial v/\partial x_i\in W^{1,l}(\Omega )$. We separate the proof into two cases. \smallskip \noindent {\bf Case 1. }\quad Assume that $$\begin{gathered} s\leq 2, \\ \frac{3N-2}{2N}\frac{1}{r+1}+\frac{1}{s+1}\geq \frac{N-2}{N} +\frac{1}{(r+1)(s+1)} \end{gathered} \label{Susu}$$ and that $g$ is a differentiable function such that its derivative $g'$ is a Lipschitz continuous function. In this case, it is well known that $g'(v)\in W^{1,r_{1}}(\Omega )$. This fact together with (\ref{Susu}) implies that $g'(v)\partial v/\partial x_i\in W^{1,l}(\Omega )$ for some $l>1$. Hence, Proposition \ref{Fati} is proved in case 1. \smallskip \noindent {\bf Case 2. }\quad Assume that $$\begin{gathered} s>2, \\ \frac{1}{r+1}+\frac{1}{s+1}\leq \frac{N-2}{N}+\frac{1}{(r+1)(s+1)} \end{gathered} \label{Fafa}$$ and that $g$ is of class $C^{2}$ with $g''(t)=O(| t| ^{s-2})$ at infinity. In this case, we use the following result concerning superposition mapping on Sobolev space, due to Marcus and Mizel \cite{Marcus-Mizel}. Let $\mathfrak{M}(\Omega )$ denote the space of real measurable functions in $% \Omega$. Given a Borel function $h:\mathbb{R}\to \mathbb{R}$ we define the superposition mapping $T_{h}:\mathfrak{M}(\Omega )\to \mathfrak{M}% (\Omega )$ by $T_{h}u\doteq h\circ u$. \begin{proposition} Assume that $\eta ,\xi$ are two numbers such that $1<\eta \leq \xi 2$. This condition is equivalent to $N\geq \xi =\frac{Nq}{Nr-q},$ that is, $\frac{N-2}{N}+\frac{1}{(r+1)(s+1)}\geq \frac{1}{r+1}+\frac{1}{s+1}.$ This completes the proof of Proposition \ref{Fati}.\hfill$\square$ \paragraph{Acknowledgments.} Part of this work was done while Jo\~{a}o Marcos do \'{O} was visiting the Universidad de Santiago de Chile. He would like to express his gratitude for the hospitality received there. He also was partially supported by CNPq, PRONEX-MCT/Brazil and Millennium Institute for the Global Advancement of Brazilian Mathematics - IM-AGIMB. Pedro Ubilla was supported by CAPES/Brazil, DICYT--USACH Grant, and FONDECYT Grants 1950605 and 1990183. \begin{thebibliography}{00} \frenchspacing \bibitem{Ambrosetti-Rabinowitz} A. Ambrosetti and P. H. Rabinowitz, {\it % Dual variational methods in critical point theory and applications}, J. Funct. Anal. \textbf{14} (1973), 349-381. \bibitem{Benci-Rabinowitz} V. Benci and P.H. Rabinowitz, {\it Critical Point Theorems for Indefinite Functionals}, Invent. Math. \textbf{52} (1979), 241-273. \bibitem{Clement-deFigueiredo-Mitidieri} Ph. Clement, D. G. deFigueiredo and E. Mitidieri, {\it A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities,} Pitman \ Res. Notes in Math. (1996) 73-91. \bibitem{Clement-Felmer-Mitidieri} Ph. Cl\'{e}ment, P. L. Felmer and E. 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