\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 $1
n,q>n$ 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
\begin{equation}
|a_i(x,\xi)|\leqslant C_1 (1+|\xi|^{p_i-1}),\quad\forall\xi\in\mathbb{R}^n.
\label{Hb}
\end{equation}
\item[(c)] $A_i$ is strictly convex: For every $t\in [0,1]$
\begin{equation}\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
\end{equation}
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
\begin{equation}\label{He}
A_i(x,\xi)\geqslant C_2|\xi|^{p_i},\quad\forall x\in\Omega,\ \forall\xi\in\mathbb{R}^n.
\end{equation}
\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
\begin{equation}\label{eq.3}
\big(a_i(x,\xi)-a_i(x,\eta)\big)\cdot(\xi-\eta)\geqslant 0,
\end{equation}
for every $x\in\Omega$ and $\xi,\eta\in\mathbb{R}^n$.
Also, from \eqref{eq.3} we obtain
\begin{equation}\label{eq.5}
a_i(x,\xi+t\eta)\eta\geqslant a_i(x,\xi)\eta
\end{equation}
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,
\]
then there exists a subsequence of $\{(u_n,v_n)\}$ which we denote it
by the same notation $\{(u_n,v_n)\}$ for which
\begin{equation}\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.
\end{equation}
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
\begin{equation}\label{eq.6}
\limsup_{n\to\infty}\langle\Phi'(u_n,v_n),(u_n-u,v_n-v)
\rangle\leqslant 0
\end{equation}
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
\begin{equation}\label{eq.19}
|\nabla u_n(x)|^p\leqslant C_2^{-1}g_n(x);
\end{equation}
furthermore,
\begin{equation}\label{eq.20}
g_n(x)\to a_1(x,\nabla u)\cdot\nabla u
\end{equation}
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
\begin{equation}\label{eq.21}
\lim_{n\to\infty}\int_\Omega g_n(x)=\int_\Omega a_1(x,\nabla u)\cdot\nabla u.
\end{equation}
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
\begin{equation}\label{eq.11}
\langle\Phi'(u,v),(\xi,\eta)\rangle\leqslant\langle\Phi'(u+t\xi,v+t\eta),
(\xi,\eta)\rangle
\end{equation}
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 $1 0$
\[
|F_u(x,u,v)|\leqslant C(1+|u|^{p-1}+|v|^{q\frac{p-1}{p}}),\quad
|F_v(x,u,v)|\leqslant C(1+|u|^{p\frac{q-1}{q}}+|v|^{q-1})
\]
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
\begin{equation}\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,
\end{equation}
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 $1<\tau 0$ and $p