\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 111, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/111\hfil Multiplicity of positive solutions]
{Multiplicity of positive solutions for  quasilinear  elliptic
 p-Laplacian systems}

\author[A. Aghajani, J.  Shamshiri \hfil EJDE-2012/111\hfilneg]
{Asadollah Aghajani, Jamileh  Shamshiri}  % in alphabetical order

\address{Asadollah Aghajani \newline
School of Mathematics, Iran University of Science and Technology,
P.O. Box 16846-13114, Narmak, Tehran, Iran}
\email{aghajani@iust.ac.ir}

\address{Jamileh  Shamshiri \newline
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran}
\email{jamileshamshiri@gmail.com}

\thanks{Submitted June 19, 2012. Published July 2, 2012.}
\subjclass[2000]{35B38,  34B15, 35J92}
\keywords{Critical points; nonlinear boundary value problems;
\hfill\break\indent 
quasilinear p-Laplacian problem; fibering map; Nehari manifold}

\begin{abstract}
 We study the existence and multiplicity of solutions to
 the elliptic system
 \begin{gather*}
 -\operatorname{div}(|\nabla u|^{p-2} \nabla u)+m_1(x)|u|^{p-2}u
 =\lambda g(x,u) \quad x\in \Omega,\\
 -\operatorname{div}(|\nabla v|^{p-2} \nabla v)+m_2(x)|v|^{p-2}v=\mu
 h(x,v)  \quad x\in \Omega,\\
|\nabla u|^{p-2}\frac{\partial u}{\partial n}=f_u(x,u,v),\quad
 |\nabla v|^{p-2}\frac{\partial v}{\partial n}=f_v(x,u,v),
 \end{gather*}
 where $\Omega\subset \mathbb{{R}}^N$ is
 a bounded and smooth domain.
 Using  fibering maps and  extracting Palais-Smale sequences in the
 Nehari manifold, we prove the existence of at least two
 distinct nontrivial nonnegative solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article, we study the existence and multiplicity of
positive solutions of the  quasilinear elliptic system
\begin{equation} \label{e1.1}
\begin{gathered}
-\operatorname{div}(|\nabla u|^{p-2} \nabla u)+m_1(x)|u|^{p-2}u
 =\lambda g(x,u)   \quad x\in \Omega,\\
-\operatorname{div}(|\nabla v|^{p-2} \nabla v)+m_2(x)|v|^{p-2}v
 =\mu h(x,v)  \quad x\in \Omega,\\
|\nabla u|^{p-2} \frac{\partial u}{\partial n}=f_u(x,u,v),\quad
|\nabla v|^{p-2}\frac{\partial v}{\partial n}=f_v(x,u,v)
x\in\partial\Omega,
\end{gathered}
 \end{equation}
where $\lambda, \mu>0$, $p>2$, $\Omega\subset\mathbb{{R}}^N$
is a bounded domain in  $\mathbb{{R}}^N$ with the smooth boundary $\partial\Omega$,
$\frac{\partial}{\partial n}$ is the outer normal derivative,
$m_1, m_2 \in C(\bar{\Omega})$ are positive bounded functions
together with  the following  assumptions on the functions $f,g$ and $h$:
\begin{itemize}

\item[(A1)] $\frac{\partial^2}{\partial t^2} f(x,t|u|,t|v|)|_{t=1}\in
C(\partial\Omega \times \mathbb{R}^2)$ and for $u, v\in
L^p(\partial\Omega)$, the integral
$\int_{\partial\Omega}\frac{\partial^2}{\partial t^2}
\big(f(x,t|u|,t|v|)\big)dx$ has the same sign for every $t>0$.

\item[(A2)] There exists $C_1>0$ such that 
$$
f(x,u,v)\leq\frac{1}{r} \frac{\partial}{\partial t}
f(x,tu,tv)|_{t=1}\leq\frac{1}{r(r-1)}\frac{\partial^2}{\partial t^2}
f(x,tu,tv)|_{t=1}\leq C_1(u^r+v^r), 
$$ 
where $p<r<p^*$ ($p^*=\frac{pN}{N-p}$ if $N>2$, $p^*=\infty$ if $N\leq p$) for all
$(x,u,v)\in {\partial\Omega} \times \mathbb{R^+}\times
\mathbb{R^+}$.


\item[(A3)]  $\frac{\partial}{\partial t} f(x,tu,tv)|_{t=0}\geq0$ and
$\lim_{t\to\infty}\frac{\frac{\partial}{\partial t}
f(x,tu,tv)}{t^{p-1}}= \eta(x,u,v)$ uniformly respect to $(x,u,v)$,
where $\eta(x,u,v)\in C(\bar{\Omega}\times\mathbb{R}^2)$ and
$|\eta(x,u,v)|>\theta>0$, a.e.
 for all $(x,u,v)\in {\Omega} \times \mathbb{R^+}\times \mathbb{R^+}$.

\item[(A4)] $g(x,u), h(x,v) \in C^1({\Omega} \times \mathbb{R})$ such that
$g(x,0)\geq0$, $ h(x,v)\geq0$, $g(x,0)\not\equiv0$ and there exist
$C_2>0$, $C_3>0$ such that, $|g(x,u)| \leq C_2(1+u^{p-1})$ and
$|h(x,v)|\leq C_3(1+v^{p-1})$, where $x\in \Omega$, $u,v
\in{\mathbb{R^+}}$ and $p>2$.

\item[(A5)] For $u,v\in W^{1,p}(\Omega)$,
$\int_{\Omega}\frac{\partial}{\partial u} g(x,t|u|)u^2dx$ and
$\int_{\Omega}\frac{\partial}{\partial v} h(x,t|v|)v^2dx$ have the
same sign for every  $t>0$ and there exist $C_4>0$, $C_5>0$ such
that $|g_u(x,u)|\leq C_4u^{p-2}$ and $|h_v(x,v)|\leq C_5v^{p-2}$
for all $(x,u,v)\in \Omega\times\mathbb{R^+}\times\mathbb{R^+}$.

\end{itemize}

\begin{remark} \label{rmk1.1} \rm
Equations involving positively homogeneous functions have been
considered in many papers, such as \cite{3, 10, 17, 18, 27, 28}. It
is clear that, if $f(x,u,v)$ is a positively homogeneous function of
degree $r(r>p>2)$, that is, $f(x,tu,tv)=t^rf(x,u,v)$ $(t>0)$,
then it satisfies  conditions (A1)--(A3). Note that for such an $f$
 we have
\[
f_uu+f_vv=rf(x,u,v)\leq r K_f(|u|^r+|v|^r),
\]
where
\[
K_f=\max\{f(x,u,v):(x,u,v)\in\overline{\Omega}\times\mathbb{R}^2,\;
|u|^r+|v|^r=1\}.
\]
\end{remark}

In recent years, there have been many papers concerned with the
existence and multiplicity of positive solutions for the elliptic
equations (systems) with nonlinear boundary conditions. The results
relating to these problems can be found in \cite{1, 4, 6, 9, 11, 12,
13, 16, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31} and the
references therein.
For instance, Drabek and Schindler \cite{15} showed the
existence of positive, bounded and smooth solutions of the following
$p$-Laplacian equation
 \begin{gather*}
 -\Delta_pu+ b |u|^{p-2}u= f(.,u)  \quad\text{in } \Omega,\\
 \Re u=0 \quad\text{on } \partial\Omega ,
 \end{gather*}
where $\Re u=|\nabla u|^{p-2} \frac{\partial u}{\partial \nu}
+b_{0}|u|^{p-2}u$, $\Omega\subset \mathbb{R}^{N}$ is a bounded
domain and $1<p<N$.

 Brown and Wu \cite{7}  considered the  semilinear
elliptic system
\begin{gather*}
-\Delta u+ u= \frac{\alpha}{\alpha+\beta}f(x)|u|^{\alpha-2}u|v|^\beta
  \quad\text{in }\Omega,\\
-\Delta v+ v= \frac{\beta}{\alpha+\beta}f(x)|u|^{\alpha}|v|^{\beta-2}v
 \quad\text{in }\Omega,\\
 \frac{\partial u}{\partial n}=\lambda g(x)|u|^{q-2}u,\quad
 \frac{\partial v}{\partial n}=\mu h(x)|v|^{q-2}v \quad\text{on }
 \partial\Omega ,
\end{gather*}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$,
 $\alpha, \beta>1$, $2<\alpha+\beta<2^*$ and the
functions $f, g, h$ satisfy the following conditions:
\begin{itemize}
\item $f\in C(\overline{\Omega})$ with $\|f\|_\infty=1$ and
$f^+=max\{f,0\}\not\equiv 0$,

\item $g,\,h\in C(\partial\Omega)$ with
$\|g\|_\infty=\|h\|_\infty=1$,
$g^\pm=\max\{\pm g,0\}\not\equiv 0$ and $h^\pm=\max\{\pm h,0\}\not\equiv 0$.
\end{itemize}
They found that the above problem has at least two nonnegative
solutions if the pair $(\lambda, \mu)$ belongs to a certain subset
of $\mathbb{R}^2$.
Note that the function
$\frac{1}{\alpha+\beta}f(x)|u|^\alpha|v|^\beta$ with the above given
conditions is positively homogeneous of degree $r=\alpha+\beta$ and
clearly satisfies our conditions (A1)--(A3).


Recently, Shen and Zhang \cite{28} considered the
 semilinear p-Laplacian system
\begin{gather*}
-\Delta_p u=\frac{1}{p^*}\frac{\partial F(x,u,v)}{\partial
u}+\lambda |u|^{q-2}u \quad\text{in }\Omega,\\
 -\Delta_p v=\frac{1}{p^*}\frac{\partial F(x,u,v)}{\partial v}+\mu
|v|^{q-2}v \quad\text{in } \Omega, \\
 u>0, \quad v>0 \quad \text{in } \Omega,\\
u=v=0 \quad\text{on }\partial\Omega,
\end{gather*}
where $\Omega$ is bounded domain in $\mathbb{R}^N$ with smooth boundary,
$F\in C^1(\overline{\Omega},\times (\mathbb{R}^+)^2)$ is positively
homogeneous of degree  $p^*$, they proved that this system has at least two positive
solutions when the pair of parameters $(\lambda,\mu)$ belongs to
certain subset of $\mathbb{R}^2$.

In this article, the main difficulty will be  the nonlinearity of
$f(x,u,v)$, $g(x,u)$ and $h(x,v)$ in problem \eqref{e1.1} and the lack of
separability. To overcome this difficultly, we need to restrict the
problem \eqref{e1.1} to assumptions (A1) and (A5). Here we present some
examples for $f(x,u,v)$ satisfying the conditions (A1)--(A3).
\begin{gather*}
f_1(x,u,v) \in C^1({{\partial\Omega}} \times \mathbb{R}^2,\mathbb{R}), \\
f_1(x,tu,tv) = t^r f_1(x,u,v) \quad\text{for } (x,u,v)\in {\partial\Omega}
 \times \mathbb{R^+}\times \mathbb{R^+} \text{ and } t>0,\\
f_2(x,u,v)=a_1(x)(-a_2(x)+\sqrt[q]{(a_2(x)^q+u^{qr}+v^{qr})}, \\
 a_i(x)\in C({\partial\Omega}), \quad a_i(x)\geq0, \quad q>1, \quad q\in\mathbb{N},\\
f_3(x,u,v)=b(x)\frac{u^{q+r}+v^{q+r}}{1+u^q+v^q}, \quad
 b(x)\in C({\partial\Omega}), \quad  b(x)\geq0, \quad  r\geq0.
\end{gather*}

Now we present some examples for $g(x,u)$ and $h(x,v)$ satisfying the conditions
(A4) and (A5):
\[
Q_1(x,z)=\frac{-a_1(x)z^{p+r}}{1+a_2(x)z^{2}}+a_3(x)
\]
with $a_i(x)\in C(\overline{\Omega})$,  $a_i(x)\geq 0$,
$a_3(x)\not\equiv 0$, $\max\{2-p,-1\}\leq r\leq 1$.
\[
Q_2(x,z)=b_1(x)\tan^{-1}(b_2(x)z^{p+k})\ln[1+z^{2k}]+b_3(x)
\]
with $ b_i(x)\in C(\overline{\Omega})$,
$b_i(x)\geq 0$, $b_3(x)\not\equiv 0$, $\frac{p}{2}\leq k\in \mathbb{N}$.
\[
Q_3(x,z)=c_1(x)\sqrt[r]{(1+c_2(x)z^{2k})^{p-1}}
\]
with $c_i(x)\geq 0$, $c_i(x)\in C(\overline{\Omega})$,
 $c_1(x)\not\equiv 0$, $k\in\mathbb{N}$, $0<2k\leq r$.
\[
Q_4(x,z)=\frac{-e_1(x)z^{p-1}}{4+\cot^{-1}(e_2(x)z^k)}+e_3(x)
\]
$e_i(x)\in C(\overline{\Omega})$, $e_i(x)\geq0$,
$e_3(x)\not\equiv 0$, $k\geq0$.


Here our main tool is the Nehari manifold method which is similar
to the fibering method by Drabek and Pohozaev \cite{14}.  The
main idea in our proofs lies in dividing the Nehari manifold
associated with the Euler functional for problem \eqref{e1.1} into two
disjoint parts and then considering the infima of this functional
on each part and by extracting Palais-Smale sequences we show that
there exists at least one solution on each part.

Define the Sobolev space
\begin{equation} \label{e1.2}
W:=W^{1,p}(\Omega)\times W^{1,p}(\Omega),
\end{equation}
endowed with the norm
$$
\|(u,v)\|_{W}=\Big(\int_{\Omega}(|\nabla u|^p+m_1(x)|u|^p)dx
+\int_{\Omega}(|\nabla v|^p+m_2(x)|v|^p)dx\Big)^{1/p},
$$
which is equivalent to the standard norm. We use the standard
$\mathrm{L}^r(\Omega)$ spaces whose norms are denoted by
$\|u\|_{r}$.
Throughout this paper, we denote $S_q$ and $\bar{S}_{q}$
 the best Sobolev and the best Sobolev trace constants for
the embedding
 of $W^{1,p}(\Omega)$ into $\mathrm{L}^{q}(\Omega)$ and $W^{1,p}(\Omega)$ into
 $\mathrm{L}^{q}(\partial\Omega)$, respectively. So we have
\begin{equation}\label{e1.3}
 \frac{(\|(u,v)\|^p_{W})^{q}}{(\int_{\partial\Omega}(|u|^{q}+|v|^{q})dx )^p}
 \geq \frac{1}{2^p\bar{S}^{pq}_q} \quad  \text{and} \quad
\frac{(\|(u,v)\|^p_{W})^{q}}{(\int_{\Omega}(|u|^{q}+|v|^{q})dx )^p}
 \geq \frac{1}{2^p{S}^{pq}_{q}}.
\end{equation}

Before stating our main results, we mention the following remarks.

\begin{remark} \label{rmk1.2} \rm
Notice that using conditions (A4) and (A5), for all
$(x,u,v)\in\Omega \times \mathbb{R}^+\times \mathbb{R}^+$, we have
\begin{itemize}
\item[(A6)] $(r-1)g(x,u)-ug_u(x,u)\leq C_6(1+u^{p-1})$ and
$(r-1)h(x,v)-vh_v(x,v)\leq C_7(1+v^{p-1})$.
\item[(A7)] $G(x,u)-\frac{1}{r} g(x,u)u \leq C_8(1+u^p)$ and
$H(x,v)-\frac{1}{r} h(x,v)v \leq C_9(1+v^p)$, where
\begin{equation} \label{e1.4}
 G(x,u)=\int_{0}^{u}{g(x,s)ds}, \quad
 H(x,v)=\int_{0}^{v}{h(x,s)ds}.
\end{equation}
\end{itemize}
\end{remark}

\begin{remark} \label{rmk1.3} \rm
It should be mentioned that using condition {\rm (A3)} we have
$$
|\frac{\partial}{\partial t}f(x,tw_1,tw_2)|\leq
(1+|\eta(x,w_1,w_2)|)t^{r-1}
$$
for $t$ sufficiently large and
$(x,w_1,w_2)\in \bar{\Omega}\times (\mathbb{R^+})^2$, hence taking
$w_1=\frac{|u|}{|u|+|v|}$, $w_2=\frac{|v|}{|u|+|v|}$ and $t=|u|+|v|$
for $|u|$ and $|v|$ sufficiently large we arrive at
\begin{align*}
\big|f_u(x,|u|,|v|)|u|+f_v(x,|u|,|v|)|v|\big|
&\leq \big(1+|\eta(x,\frac{|u|}{|u|+|v|},\frac{|v|}{|u|+|v|})|\big)(|u|+|v|)^r\\
&\leq A_0(|u|^r+|v|^r),
\end{align*}
where $A_0=2^r \max\{1+|\eta(x,|u|,|v|)|:   |u|+|v|=1\}$ and
$(x,u,v)\in \bar{\Omega}\times \mathbb{R}^2$.
 Furthermore, if we assume that
 $f\in C^2 (\bar{\Omega}\times \mathbb{R^+}^2)$,
 then there exists $A_1>0$  such that
\begin{equation} \label{e1.5}
|\frac{\partial}{\partial t}f(x,tu,tv)|_{t=1}
=|f_u(x,u,v)u+f_v(x,u,v)v|
\leq A_1(1+|u|^r+|v|^r),
\end{equation}
where $ (x,u,v)\in \bar{\Omega}\times(\mathbb{R^+})^2$.
  \end{remark}

The purpose of this paper is to prove the following results.

\begin{theorem} \label{thm1.1}
There exists $K^*\subset (\mathbb{R}^+)^2$ such that for
 each $(\lambda,\mu)\in K^*$  problem \eqref{e1.1} has at least one
positive solution.
\end{theorem}

\begin{theorem} \label{thm1.2}
There exists $K^{**}\subset K^*$ such that for
 each $(\lambda,\mu)\in K^{**}$  problem \eqref{e1.1} has at least two
distinct positive solutions.
\end{theorem}

This paper is organized as follows. In section 2 we point out some
notation and preliminary results and give some properties of Nehari
manifold and fibering maps. In section 3 a fairly complete
description of the Nehari manifold and fibering maps associated with
the problem is given, and finally Theorems \ref{thm1.1} and \ref{thm1.2}
 are proved in Section 4.

\section{Preliminaries and auxiliary results}

First, we define the weak solution of problem \eqref{e1.1} as follows.

\begin{definition} \label{def2.1} \rm
 A pair of functions $(u,v)\in W$ ($W$ is given by \eqref{e1.2})
 is said to be a weak solution of \eqref{e1.1}, whenever
\begin{align*}
&\int_{\Omega}{\big(|\nabla u|^{p-2}\nabla u.\nabla
{\varphi}_1}+m_1(x)|u|^{p-2}u{\varphi}_1\big)dx\\
&- \lambda\int_{\Omega}g(x,u)\varphi_1dx-
\int_{\partial\Omega}{f_u(x,u,v){{\varphi}_1}}dx=0,
\\
&\int_{\Omega}{\big(|\nabla v|^{p-2} \nabla v.\nabla
{\varphi}_2}+m_2(x)|v|^{p-2}v{\varphi}_2\big)dx\\
&-\mu\int_{\Omega}h(x,v)\varphi_2dx-
\int_{\partial\Omega}{f_v(x,u,v){{\varphi}_2}}dx =0,
\end{align*}
for all $(\varphi_1,\varphi_2)\in W$.
 \end{definition}


Associated with problem \eqref{e1.1}, we consider the energy functional
$J_{\lambda,\mu}:W\to\mathbb{R}$
\begin{equation} \label{e2.1}
J_{\lambda,\mu}(u,v)=\frac{1}{p} M(u,v)- F(u,v)-\lambda
\int_{\Omega}G(x,|u|)dx-\mu\int_{\Omega}H(x,|v|)dx,
\end{equation}
where $G(x,u)$ and $H(x,v)$ are introduced in \eqref{e1.4} and
 \begin{equation} \label{e2.2}
\begin{gathered}
 M(u,v)=\int_{\Omega}(|\nabla u|^p+m_1(x)
|u|^p)dx+\int_{\Omega}(|\nabla v|^p+m_2(x) |v|^p)dx, \\
 F(u,v)=\int_{\partial\Omega}f(x,|u|,|v|)dx.
\end{gathered}
\end{equation}

If $J_{\lambda,\mu}$ is bounded from below and $J_{\lambda,\mu}$
has a minimizer on $W$, then this minimizer is a critical point of
$J_{\lambda,\mu}$, so it is a solution of \eqref{e1.1}. Since
$J_{\lambda,\mu}$ is unbounded from below on whole space $W$, it
is useful to consider the functional on the Nehari manifold

\begin{equation}  \label{e2.3}
\mathcal{N}_{\lambda,\mu}(\Omega)=\{(u,v)\in
W\setminus\{(0,0)\} : \langle J_{\lambda,\mu}'(u,v),(u,v)
\rangle=0\},
\end{equation}
where $\langle , \rangle$ denotes the usual duality between $W$ and
$W^{-1}$ ( $W^{-1}$ is the dual space of the Sobolev space $W$). We
recall that any nonzero solution of problem \eqref{e1.1} belongs to
$\mathcal{N}_{\lambda,\mu}(\Omega)$. Moreover, by definition, we have
that $(u,v)\in \mathcal{N}_{\lambda,\mu}(\Omega) $ if and only if
\begin{equation} \label{e2.4}
\begin{split}
&M(u,v)- \int_{\partial\Omega}\big(f_u(x,|u|,|v|)|u|+f_v(x,|u|,|v|)|v|\big)dx\\
&-\lambda\int_{\Omega}g(x,|u|)|u|dx-\mu\int_{\Omega}h(x,|v|)|v|dx=0.
\end{split}
\end{equation}
Furthermore, we have the following result.


\begin{theorem} \label{thm2.1}
$J_{\lambda,\mu}$ is coercive and bounded from below on
$\mathcal{N}_{\lambda,\mu}(\Omega)$ for $\lambda$ and $\mu$
sufficiently small.
\end{theorem}

\begin{proof}
 Let $(u, v)\in \mathcal{N}_{\lambda,\mu}(\Omega)$,
then by (A2), (A7), \eqref{e1.3} and \eqref{e2.1}--\eqref{e2.4}, we obtain
\begin{align*}
 J_{\lambda,\mu}(u, v)
&\geq (\frac{1}{p}-\frac{1}{r})M(u,v)
 -\lambda\int_{\Omega}\Big(G(x,|u|)-\frac{1}{r}g(x,|u|)|u|\Big)dx\\
&\quad -\mu\int_{\Omega}\Big(H(x,|v|)-\frac{1}{r}h(x,|v|)|v|\Big)dx\\
&\geq(\frac{1}{p}-\frac{1}{r})\|(u,v)\|_W^p-\lambda\int_{\Omega}C_8(1+|u|^p)dx
-\mu\int_{\Omega}C_9(1+|v|^p)dx\\
&\geq \frac{r-p}{rp}\|(u,v)\|_W^p-(C_8\lambda+C_9\mu)|\Omega|
 -(C_8\lambda+C_9\mu)2{{S}}_p^p \|(u,v)\|_W^p,
\end{align*}
thus $J_{\lambda,\mu}$ is coercive and bounded from
below on $\mathcal{N}_{\lambda,\mu}(\Omega)$ provided that
$0\leq (C_8\lambda+C_9\mu)2{{S}}_p^p<(r-p)/(rp)$.
\end{proof}

It  can be proved that the points in
$\mathcal{N}_{\lambda,\mu}(\Omega)$ correspond to the stationary
points of the fibering map $\phi_{u,v}(t): [0,\infty)\to
\mathbb{R}$ defined by $\phi_{u,v}( t )= J_{\lambda,\mu}(tu,tv)$,
 which were  introduced by Drabek and Pohozaev in \cite{14} and also discussed
  in Brown and Zhang \cite{8}.
Using \eqref{e2.1} for $(u,v)\in W$, we have
\begin{equation} \label{e2.5}
\begin{gathered}
\begin{aligned}
\phi_{u,v}(t)&=J_{\lambda,\mu}(tu,tv)\\
&=\frac{t^p}{p} M(u,v)-F(tu,tv)-\lambda\int_{\Omega}G(x,t|u|)dx-
\mu\int_{\Omega}H(x,t|v|)dx,
\end{aligned}\\
\begin{aligned}
\phi_{u,v}'(t)
&= t^{p-1}M(u,v)- \int_{\partial\Omega}\nabla
f(x,t|u|,t|v|).(|u|,|v|)dx\\
&\quad -\lambda\int_{\Omega}g(x,t|u|)|u|dx-
\mu\int_{\Omega}h(x,t|v|)|v|dx,
\end{aligned}\\
\begin{aligned}
\phi_{u,v}''(t)&= (p-1)t^{p-2}M(u,v)-
\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx\\
&\quad -\lambda\int_{\Omega}g_u(x,t|u|)u^2dx-
\mu\int_{\Omega}h_v(x,t|v|)v^2dx,
\end{aligned}
\end{gathered}
\end{equation}
where
\begin{equation} \label{e2.6}
\begin{gathered}
\nabla f(x,u,v):=(\frac{\partial f}{\partial u},\frac{\partial
f}{\partial v}), \\
\mathcal{F}(x,tu,tv):=\frac{\partial^2}{\partial t^2}
\big(f(x,t|u|,t|v|)=f_{uu}u^2+f_{vv}v^2+2f_{uv}|uv|.
\end{gathered}
\end{equation}
 Hence it is natural to divide
$\mathcal{N}_{\lambda,\mu}$ into three subsets
$\mathcal{N}_{\lambda,\mu}^{+},\, \mathcal{N}_{\lambda,\mu}^{-}\ and
\ \mathcal{N}_{\lambda,\mu}^{0}$ which correspond to local minima,
local maxima and points of inflection of the fibering maps and so we
define
\begin{equation} \label{e2.7}
\begin{gathered}
\mathcal{N}_{\lambda,\mu}^{+}=\{(u,v)\in
\mathcal{N}_{\lambda,\mu}(\Omega):
\phi_{u,v}''(1)> 0\},\\
\mathcal{N}_{\lambda,\mu}^{-}=\{(u,v)\in
\mathcal{N}_{\lambda,\mu}(\Omega):
\phi_{u,v}''(1)< 0\},\\
\mathcal{N}_{\lambda,\mu}^{0}=\{(u,v)\in
\mathcal{N}_{\lambda,\mu}(\Omega): \phi_{u,v}''(1)=0\}.
\end{gathered}
\end{equation}
The following lemma shows that minimizers for
$J_{\lambda,\mu}(u,v)$ on $N_{\lambda,\mu}(\Omega)$ are usually
critical points for $J_{\lambda,\mu}$, as proved by Brown and
Zhang in \cite{8} or in Aghajani et al. \cite{2}.


\begin{lemma} \label{lem2.1}
 Let $(u_{0},v_0)$ be a local minimizer for $J_{\lambda,\mu}(u,v)$
 on $\mathcal{N}_{\lambda,\mu}(\Omega)$. If
$(u_{0},v_0)$ is not in $\mathcal{N}_{\lambda,\mu}^{0}(\Omega)$, then
$(u_{0},v_0)$ is a critical point of $J_{\lambda,\mu}$.
\end{lemma}

Motivated by the above lemma, we give conditions for
$\mathcal{N}_{\lambda,\mu}^{0}= \emptyset$.

\begin{lemma} \label{lem2.2}
There exists $K_0\subset (\mathbb{R^+})^2$ such that for all
$(\lambda ,\mu)\in K_0$, we have $\mathcal{N}_{\lambda,\mu}^{0}=
\emptyset$.
\end{lemma}

\begin{proof}
 Suppose the contrary, that is there exists 
$(\lambda ,\mu)$ such that $\mathcal{N}_{\lambda,\mu}^{0}\neq \emptyset$.
Then for $(u,v)\in \mathcal{N}_{\lambda,\mu}^{0}$ by \eqref{e2.5}--\eqref{e2.7}
we have
\begin{equation}\label{e2.8}
\begin{split}
\phi'_{u,v}(1)&= M(u,v)-\int_{\partial\Omega}\big(\nabla
f(x,|u|,|v|).(|u|,|v|)\big)dx\\
&\quad -\lambda\int_{\Omega}g(x,|u|)|u|dx
-\mu\int_{\Omega}h(x,|v|)|v|dx=0,
\end{split}
\end{equation}
and by \eqref{e2.7} $\phi''_{u,v}(1)=0$, so
\begin{equation}\label{e2.9}
\begin{split}
&(p-1)M(u,v)-\int_{\partial\Omega}\big(f_{uu}u^2+f_{vv}v^2+2f_{uv}|uv|\big)dx\\
&-\lambda\int_{\Omega}g_u(x,|u|)u^2dx
-\mu\int_{\Omega}h_v(x,|v|)v^2dx=0,
\end{split}
\end{equation}
using (A2) in \eqref{e2.9} we obtain
\begin{equation} \label{e2.10}
\begin{split}
&(p-1)M(u,v)-(r-1)\int_{\partial\Omega}\nabla
f(x,|u|,|v|).(|u|,|v|)dx \\
&-\lambda\int_{\Omega}g_u(x,|u|)u^2dx
-\mu\int_{\Omega}h_v(x,|v|)v^2dx\geq0.
\end{split}
\end{equation}
Using \eqref{e1.3}, \eqref{e2.8}, \eqref{e2.10} and condition (A6) we obtain
\begin{align*}
(r-p)M(u,v)
&\leq \lambda\int_{\Omega}\big((r-1)g(x,|u|)-g_u(x,|u|)|u|\big)|u|dx\\
&\quad + \mu\int_{\Omega}\big((r-1)h(x,|v|)-h_v(x,|v|)|v|\big)|v|dx\\
&\leq 2\lambda C_6 \int_{\Omega}(1+|u|^p)dx+2\mu C_7
\int_{\Omega}(1+|v|^p)dx\\
&\leq (2\lambda C_6+2\mu C_7)|\Omega|+(2\lambda C_6+2\mu
C_7)2{S}_p^p\|(u,v)\|_W^p,
\end{align*}
which concludes
\begin{equation} \label{e2.11}
M(u,v)\leq\Big(\frac{(2\lambda C_6+2\mu
C_7)|\Omega|}{(r-p)-(4\lambda C_6+4\mu C_7) {{S}}_p^p}\Big).
\end{equation}
Moreover,  \eqref{e1.3}, \eqref{e2.2} together (A2) imply
\begin{equation} \label{e2.12}
\begin{split}
&\int_{\partial\Omega}\big(f_{uu}u^2+f_{vv}v^2+2f_{uv}|uv|\big)dx\\
&\leq r(r-1)\int_{\partial\Omega} C_1(|u|^r+|v|^r)dx\leq 2r(r-1)C_1
\bar{S}_r^r\|(u,v)\|_W^r,
\end{split}
\end{equation}
hence using \eqref{e2.12} in \eqref{e2.9} and taking into account (A5) and \eqref{e1.3}
we obtain 
\begin{equation} \label{e2.13}
M(u,v)\leq L\|(u,v)\|_W^r+(\lambda L'+\mu L'') M(u,v),
\end{equation}
where
\begin{equation} \label{e2.14}
L=\frac{2r(r-1)C_1 \bar{S}_r^r}{p-1}, \quad
L'=\frac{C_4{S}^p_p}{p-1}, \quad 
L''=\frac{C_5{S}^p_p}{p-1}.
\end{equation}
From \eqref{e2.13} we obtain
\begin{equation} \label{e2.15}
M(u,v)\geq \Big(\frac{1-\lambda L'-\mu L''}{L}\Big)^\frac{p}{r-p},
\end{equation}
so using \eqref{e2.11} we must have
\begin{equation*}
\Big(\frac{1-\lambda L'-\mu
L''}{L}\Big)^\frac{p}{r-p}\leq\Big(\frac{(2\lambda C_6+2\mu
C_7)|\Omega|}{(r-p)-(4\lambda C_6+4\mu C_7) {{S}}_p^p}\Big),
\end{equation*}
which is a contradiction for $\lambda ,\mu $ sufficiently small.
So there exists $K_0\subset(\mathbb{R}^+)^2$ such that for
$(\lambda ,\mu )\in K_0$,  $\mathcal{N}_{\lambda,\mu}^{0}=
\emptyset$.
\end{proof}

\begin{definition} \label{def2.2} \rm
A sequence $y_n=(u_{n},v_n)\subset W$ is called a Palais-Smale
sequence if $I_{\lambda,\mu}(y_{n})$ is bounded
 and $I'_{\lambda,\mu}(y_{n})\to 0$ as $n\to\infty$. 
If  $I_{\lambda,\mu}(y_{n})\to c$ and
$I'_{\lambda,\mu}(y_{n})\to 0$, then $y_n$ is a
 $(PS)_c$-sequence.
 It is said that the functional $I_{\lambda,\mu}$
 satisfies the Palais-Smale condition (or $(PS)_{c}$-condition), if
each Palais-Smale sequence ($(PS)_c$-sequence) has a convergent
subsequence.
\end{definition}

Now we prove the boundedness of Palais-Smale sequences.

\begin{lemma} \label{lem2.3} 
If $\{(u_n,v_n)\}$ is a $(PS)_{c}$-sequence for $J_{\lambda,\mu}$, then  
$\{(u_n,v_n)\}$ is bounded in $W$ provided that $(\lambda,\mu)\in
K_1=\{(\lambda,\mu): r-p-4r(C_8\lambda+C_9\mu){{S}}_p^p>0\}$.
\end{lemma}

\begin{proof}
 Using \eqref{e1.3}, \eqref{e2.5}, (A2) and (A7) we have
\begin{align*}
&J_{\lambda,\mu} (u_n,v_n) -\frac{1}{r}\langle
J_{\lambda,\mu}'(u_n,v_n),(u_n,v_n)\rangle\\
&\geq\frac{r-p}{rp}M(u_n,v_n)
-\lambda\int_{\Omega}(G(x,|u_n|)-\frac{1}{r}g(x,|u_n|)|u_n|)dx\\
&\quad -\mu\int_{\Omega}(H(x,|v_n|)-\frac{1}{r}h(x,|v_n|)|v_n|)dx\\
&\geq\frac{r-p}{rp}
M(u_n,v_n)-\lambda\int_{\Omega}C_8(1+|u_n|^p)dx
-\mu\int_{\Omega}C_9(1+|v_n|^p)dx \\
&\geq \frac{r-p-4r(C_8\lambda
+C_9\mu){{S}}_p^p}{rp}\|(u_n,v_n)\|_W^p-(C_8\lambda+C_9\mu)|\Omega|,
\end{align*}
so for $(\lambda,\mu)\in K_1$, $\{(u_{n},v_n)\}$
is bounded in $W$.
\end{proof}

\begin{lemma} \label{lem2.4} 
There exists $K_2\subset \mathbb{R}^2$ such that if
 $(\lambda,\mu)\in K_2$ and $(u,v)\in N_{\lambda,\mu}^-$, then
$\int_{\partial\Omega}\mathcal{F}(x,u,v)dx>0$, where
$\mathcal{F}(x,u,v)$ is defined by \eqref{e2.6}.
 \end{lemma}

\begin{proof} 
Suppose otherwise, then
$-\int_{\partial\Omega}\mathcal{F}(x,u,v)dx\geq0$ and from \eqref{e2.5}
and \eqref{e2.7} we obtain
\begin{align*}
\phi''_{u,v}(1)&=(p-1)M(u,v)-\int_{\partial\Omega}\mathcal{F}(x,u,v)dx\\
&\quad -\lambda\int_{\Omega}g_u(x,|u|)u^2dx
-\mu\int_{\Omega}h_v(x,|v|)v^2dx<0,
\end{align*}
so by \eqref{e1.3}, \eqref{e2.2}, \eqref{e2.14} and condition (A5) we have
\begin{align*}
\|(u,v)\|_W^p
&\leq\frac{\lambda}{p-1}\int_{\Omega}g_u(x,|u|)u^2dx
+\frac{\mu}{p-1}\int_{\Omega}h_v(x,|v|)v^2dx \\
&\leq(\lambda L ' +\mu L'')\|(u,v)\|_W^p,
\end{align*}
which is a contradiction for  
$(\lambda,\mu)\in K_2=\{(\lambda,\mu): \ \lambda L' +\mu L"<1\}$.
\end{proof}

\section{Properties of Nehari manifold and fibering maps}

To obtain a better understanding of the behavior of fibering
maps, we will describe the nature of the derivative of the fibering
maps for all possible signs of
$\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx$ (by (A1) and \eqref{e2.6},
$\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx$ has the same sign for
every $t>0$). Define the functions $R(t)$ and $S(t)$ as follows
\begin{gather} \label{e3.1}
R(t):=\frac{1}{p} t^pM(u,v)- F(tu,tv) \quad (t>0),\\
\label{e3.2}
S(t):=\lambda\int_{\Omega}G(x,t|u|)dx+\mu\int_{\Omega}H(x,t|v|)dx \quad
(t>0),
\end{gather}
then from \eqref{e2.5} it follows  that $\phi_{u,v}(t)=R(t)-S(t)$. Moreover,
$\phi'_{u,v}(t)=0$ if and only if $R'(t)=S'(t)$, where
\begin{equation} \label{e3.3}
R'(t)=t^{p-1}M(u,v)-\int_{\partial\Omega}\Big(f_u(x,t|u|,t|v|)|u|
+f_v(x,t|u|,t|v|)|v|\Big)dx,
\end{equation}
and
\begin{equation} \label{e3.4}
S'(t)=\lambda\int_{\Omega}g(x,t|u|)|u|dx+\mu\int_{\Omega}h(x,t|v|)|v|dx.
\end{equation}

In the next result we see that, $\phi_{u,v}$ and $\phi'_{u,v}$ take
on positive values for all nonzero $(u,v)\in
 W$ whenever, $\lambda$ and $\mu$ belong to a certain subset of $\mathbb{R}^2$.

\begin{lemma} \label{lem3.1}
There exists $K_3\subset (\mathbb{R}^+)^2$ such that for all
nonzero $(u,v)\in W$, $\phi_{u,v}(t)$ and $\phi'_{u,v}(t)$ take
on positive values whenever $(\lambda,\mu)\in K_3$.
\end{lemma}

\begin{proof}
 First we show that $\phi_{u,v}(t)$ takes on positive
values, for all possible signs of
$\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx$. If
$\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx\leq0$, then by \eqref{e3.1}
$R''(t)\geq0$ and using \eqref{e3.2}, $R(t)>S(t)$ for t sufficiently large,
so $\phi_{u,v}(t)>0$ for t sufficiently large . Now, suppose there
exists $(u,v)\in W$ such that
$\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx\geq0$. Condition (A2)
together \eqref{e3.1} imply that
\begin{equation*}
R(t)\geq \frac{1}{p}
t^pM(u,v)-C_1t^r\int_{\partial\Omega}(|u|^r+|v|^r)dx.
\end{equation*}
Define
\begin{equation} \label{e3.5}
K(t):=\frac{1}{p} t^pM(u,v)-C_1t^r\int_{\partial\Omega}(|u|^r+|v|^r)dx \quad (t>0),
\end{equation}
we obtain $R(t)\geq K(t)$, and by elementary calculus, we see that
$K(t)$ takes a maximum value at
\begin{equation} \label{e3.6}
t_{\rm max}=\Big(\frac{M(u,v)}{
rC_1\int_{\partial\Omega}(|u|^r+|v|^r)dx)}\Big)^{\frac{1}{r-p}},
\end{equation}
then follows by \eqref{e3.1}, \eqref{e3.6}, \eqref{e1.3} and \eqref{e2.2} that
\begin{equation} \label{e3.7}
\begin{split}
R(t_{\rm max})
&\geq K(t_{\rm max})
=\frac{r-p}{rp}\Big(\frac{(\|(u,v)\|^p_{W})^{r}}{\big(rC_1\int_{\partial\Omega}
(|u|^r+|v|^r)dx\big)^p}\Big)^{\frac{1}{r-p}}\\
&\geq\frac{r-p}{rp}\Big(\frac{1}{(2rC_1)^p{\bar{S}}^{rp}_{r}}\Big)^{\frac{1}{r-p}}=\delta_1,
\end{split}
\end{equation}
where $\delta_1$ is independent of $(u,v)$. Now from \eqref{e3.6},
\eqref{e3.7} and \eqref{e1.3} for $1\leq \alpha<p^*$, we deduce
\begin{equation} \label{e3.8}
\begin{split}
 (t_{\rm max})^{\alpha}\int_{\Omega}(|u|^{\alpha}+|v|^\alpha)dx
&\leq2{{S}}_{\alpha}^{\alpha}\Big(\frac{\|(u,v)\|_{W}^p}{
rC_1\int_{\partial\Omega}
(|u|^r+|v|^r)dx}\Big)^{\frac{\alpha}{r-p}}(\|
(u,v)\|^p_{W})^{\frac{\alpha}{p}}\\
&=2{{S}}_{\alpha}^{\alpha}\Big(\frac{(\|(u,v)\|_{W}^p)^r}{\big(rC_1\int_{\partial\Omega}
(|u|^r+|v|^r)dx\big)^p}\Big)^{\frac{\alpha}{p(r-p)}}\\
&\leq
2{{S}}_{\alpha}^{\alpha}\big(\frac{rp}{r-p}\big)^{\frac{\alpha}{p}}(R(t_{\rm max})\big)^{\frac{\alpha}{p}}
=c_1(R(t_{\rm max})\big)^{\frac{\alpha}{p}}.
\end{split}
\end{equation}
Combining (A4), (A7), \eqref{e1.3} and \eqref{e3.8} imply that
\begin{equation} \label{e3.9}
\begin{split}
S(t_{\rm max})
&=\lambda\int_{\Omega}G(x,t_{\rm max}|u|)dx+\mu\int_{\Omega}H(x,t_{\rm max}|v|)dx\\
&\leq \frac{\lambda}{r}\int_{\Omega}rC_8(1+|t_{\rm max}u|^p)
 +C_2(|t_{\rm max}u|+|t_{\rm max}u|^p)dx\\
&\quad +\frac{\mu}{r}\int_{\Omega}rC_9(1+|t_{\rm max}v|^p)+C_3(|t_{\rm max}v|+|t_{\rm max}v|^p)dx\\
&\leq \lambda b_0\int_{\Omega}(1+|t_{\rm max}u|^p)dx+\mu b_1\int_{\Omega}(1+|t_{\rm max}v|^p)dx\\
&\leq \lambda B_0(1+ R(t_{\rm max}))+ \mu B_1(1+R(t_{\rm max})),
\end{split}
\end{equation}
where $B_0$ and $B_1$ are independent of $(u,v)$. Using \eqref{e3.9}
together with \eqref{e3.7} and \eqref{e2.5}, we obtain
\begin{equation} \label{e3.10}
\begin{split}
\phi_{u,v}(t_{\rm max})&= R(t_{\rm max})-S(t_{\rm max})\\
&\geq R(t_{\rm max})\Big(1-(\lambda B_0+\mu B_1)(R(t_{\rm max})^{-1}+1)\Big)\\
&\geq\delta_1\big (1-(\lambda B_0+\mu B_1)(\delta_1^{-1}+1)\big).
\end{split}
\end{equation}
So we conclude that if $2(\lambda B_0+\mu B_1)(1+\delta_1)<\delta_1$, 
then $\phi_{u,v}(t_{\rm max})>0$ for all
nonzero $(u,v)\in W$.


 Now we prove that
 $\phi'_{u,v}(t)$ takes on positive values.
If $\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx\leq0$, then using
\eqref{e3.1}, \eqref{e3.2} $\phi'_{u,v}(t)\geq0$ for t sufficiently large.
Suppose that, there exists $(u,v)\in W$ such that
$\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx\geq0$.
To verify that $\phi'_{u,v}(t)>0$, it is sufficient to show
that $tR'(t)>tS'(t)$.
Using (A2) and \eqref{e3.3} we have
 \begin{equation*}
tR'(t)\geq t^pM(u,v)-rC_1t^r\int_{\partial\Omega}(|u|^r+|v|^r)dx.
\end{equation*}
In view of \eqref{e3.5}, we write
\begin{equation} \label{e3.11}
\bar{K}(t):= t^pM(u,v)-rC_1t^r\int_{\partial\Omega}(|u|^r+|v|^r)dx \
\ (t>0),
\end{equation}
so $tR'(t)>\bar{K}(t)$ and by elementary calculus we can show that
$\bar{K}(t)$ achieves its maximum at
\begin{equation} \label{e3.12}
 \tau_{\rm max}
=\Big(\frac{pM(u,v)}{r^2C_1\int_{\partial\Omega}(|u|^r+|v|^r)dx}\Big)^{\frac{1}{r-p}}.
\end{equation}
Using  \eqref{e1.3}, \eqref{e3.3}, \eqref{e3.11} and \eqref{e3.12}, we arrive at
\begin{equation} \label{e3.13}
\begin{split}
 \tau_{\rm max}R'(\tau_{\rm max})
&=(\frac{p}{r^2C_1})^\frac{p}{r-p}(\frac{r-p}{r})\Big(\frac{(\|
(u,v)\|^p)^{r}}{(\int_{\partial\Omega}(|u|^r+|v|^r)dx)
^p}\Big) ^{\frac{1}{r-p}}\\
&\geq(\frac{p}{r^2C_1})^\frac{p}{r-p}(\frac{r-p}{r})
\Big(\frac{1}{2^p{\bar{S}}_r^{rp}}\Big)^{\frac{1}{r-p}}
=\delta_2>0,
 \end{split}
\end{equation}
where ${\delta}_2$ is independent of $(u,v)$. Using \eqref{e1.3}, \eqref{e3.12}
and \eqref{e3.13}, and by some calculations very Similar to \eqref{e3.8},
 we obtain
\begin{equation} \label{e3.14}
(\tau_{\rm max})^{\beta}\int_{\Omega}(|u|^{\beta}+|v|^\beta)dx \leq
c_2(\tau_{\rm max}R'(\tau_{\rm max})\big)^{\frac{\beta}{p}},
\end{equation}
for $1\leq \beta<2^*$. Then using \eqref{e1.3}, \eqref{e3.4}, \eqref{e3.14} and
condition (A4) we find
\begin{align*}
\tau_{\rm max}S'(\tau_{\rm max})
&=\lambda\tau_{\rm max}\int_{\Omega}g(x,\tau_{\rm max}|u|)|u|dx+
\mu\tau_{\rm max}\int_{\Omega}h(x,\tau_{\rm max}|v|)|v|dx\\
&\leq \lambda\int_{\Omega}C_1(|t_{\rm max}u|+|t_{\rm max}u|^p)dx+
\mu\int_{\Omega}C_3(|t_{\rm max}v|+|t_{\rm max}v|^p)dx\\
&\leq (\lambda e_0+\mu e_1)\Big((t_{\rm max}R'(t_{\rm max}))^\frac{1}{p}
+t_{\rm max}R'(t_{\rm max})\Big),
\end{align*}
where $e_0$ and $e_1$ are independent of $(u,v)$, so from the above
inequality and \eqref{e3.13}, we obtain
\begin{align*}
&\tau_{\rm max} \phi'_{u,v}(\tau_{\rm max})= \tau_{\rm max}R'(\tau_{\rm max})-\tau_{\rm max}S'(\tau_{\rm max})\\
&\geq \tau_{\rm max}R'(\tau_{\rm max})\Big(1- (\lambda e_0+\mu
e_1)\big((\tau_{\rm max}R_\lambda'(\tau_{\rm max}))^\frac{1-p}{p}
+1\big)\Big)\\
 &\geq \delta_2\Big(1- (\lambda e_0+\mu e_1)\big(\delta_2^\frac{1-p}{p}
+1\big)\Big),
\end{align*}
 Clearly for all nonzero $(u,v)\in W$, $\tau_{\rm max}
\phi'_{u,v}(\tau_{\rm max})>0$ provided that 
$2(\lambda e_0+\mu e_1)\big(\delta_2^\frac{1}{p}
+\delta_2\big)<\delta_2$.

Using the above inequality and \eqref{e3.10}, we obtain that if
$(\lambda,\mu)\in K_3$, where
\begin{equation} \label{e3.15}
  K_3=\{(\lambda,\mu): 2(\lambda B_0+\mu B_1)(1+\delta_1)<\delta_1
 \text{ and }  2(\lambda e_0+\mu e_1)\big(\delta_2^\frac{1}{p} 
+\delta_2\big)<\delta_2\}, 
\end{equation}
then $\phi_{u,v}(t)$ and $\phi'_{u,v}(t)$ take on positive values
for all nonzero $(u,v)\in W$ and this completes the proof.
\end{proof}

\begin{corollary} \label{coro3.1}
If $(\lambda,\mu)\in K_2\cap K_3$, then there exists $\varepsilon>0$
such that $J_{\lambda,\mu}(u,v)>\epsilon$ for all $(u,v)\in
\mathcal{N}_{\lambda,\mu}^{-}$.
\end{corollary}

\begin{proof}
If $(u,v)\in N_{\lambda,\mu}^-$, then by lemma \ref{lem2.4},
$\int_{\partial\Omega}\mathcal{F}(x,u,v)dx>0$. Also due to (A1) and
(A5), $\phi_{u,v}$ has a positive global maximum at $t=1$ and so by
\eqref{e2.5}, \eqref{e3.10} and \eqref{e3.15}
$$
J_{\lambda,\mu}(u,v)=\phi_{u,v}(1)\geq \phi_{u,v}(t_{\rm max})
\geq\delta_1\big (1-(\lambda B_0+\mu B_1)(\delta_1^{-1}+1)\big)\geq
\delta_1/2= \varepsilon>0. 
$$
\end{proof}


From (A1) and \eqref{e2.6}, $\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx$
has the same sign for every $t>0$, so we have the following
corollary.

\begin{corollary} \label{coro3.2} 
for $(u,v)\in W\setminus \{(0,0)\}$ we have
\begin{itemize}
\item[(i)]
 If $\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx\leq0$, then
there exists $t_1$ such that $(t_1u,t_1v)\in
N_{\lambda,\mu}^{+}$ and
 $\phi_{u,v}(t_1)<0$.
\item[(ii)]
  If $\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx\geq0$ and $(\lambda,\mu)\in K_3$, then there exist
$0<t_1< t_2$ such that $(t_1u,t_1v)\in N_{\lambda,\mu}^{+}$,
$(t_2u,t_2v)\in N_{\lambda,\mu}^{-}$ and $\phi_{u,v}(t_1)<0$.
\end{itemize}
\end{corollary}
%
%
%
\begin{proof} (i) From \eqref{e2.5}, (A3), (A4) and the assumptions we obtain
$\phi'_{u,v}(0)<0$ and
$\lim_{t\to\infty}\phi'_{u,v}(t)=+\infty$, so by the
intermediate value theorem, there exists  $t_1>0$  such that
 $\phi'_{u,v}(t_1)=0$. Now using (A1) and (A5), for $0<t<t_1$,
 $\phi'_{u,v}(t)<0$  and for
 $t>t_1$, $\phi'_{u,v}(t)>0$, therefore $(t_1u,t_1v)\in N_{\lambda,\mu}^{+}$ and
 $\phi_{u,v}(t_1)<\phi_{u,v}(0)=0$.

 (ii) Using \eqref{e2.5}, (A3), (A5) and the assumption that 
$\int_{\partial\Omega}\mathcal{F}(x,tu,tv)dx\geq0$
we obtain $\lim_{t\to\infty}\phi'_{u,v}(t)=- \infty$,
$\phi'_{u,v}(0)<0$
 and by Lemma \ref{lem3.1} we have  $\phi'_{u,v}(\tau)>0$ for suitable $\tau>0$, so using
again the intermediate value theorem concludes that there exist
$t_1$ and $t_2$ such that $0<t_1<\tau<t_2$, and  $
\phi'_{u,v}(t_1)=\phi'_{u,v}(t_2)=0$. Also using the same
argument as in the proof of (i) and using (A1) and (A5) we have
$(t_1u,t_1v)\in N_{\lambda,\mu}^{+}$, $(t_2u,t_2v)\in
N_{\lambda,\mu}^{-}$ and $\phi_{u,v}(t_1)<\phi_{u,v}(0)=0$.
\end{proof}

\section{Proof of Theorems \ref{thm1.1} and \ref{thm1.2}}

To prove these to theorems, we need to show the
existence of local minimum for $J_{\lambda,\mu}$ on
$N_{\lambda,\mu}^+$ and $N_{\lambda,\mu}^-$. To do this, we need the
Remark \ref{rmk4.1}, below.
Here for simplicity, for a functional $\psi$ defined on a normed
space $E$, and $w\in E$ by $\psi'(w)$ and $\psi''(w)$, we mean
$\frac{\partial}{\partial t} \psi(wt)| _{t=1}$, and 
$\frac{\partial^2}{\partial t^2} \psi(wt)| _{t=1}$, respectively.

\begin{remark} \label{rmk4.1} \rm
From Remark \ref{rmk1.3}, \eqref{e2.6} and (A2) we obtain that 
$$
|\nabla f(x,|u|,|v|).(|u|,|v|)|\leq A_1(1+|u|^r+|v|^r)
$$ 
and   $ |\mathcal{F}(x,u,v)| \leq A_2(1+|u|^r+|v|^r)$,
 also from (A4) and (A5) we obtain
 \begin{gather*}
|g(x,|u|)| \leq C_2(1+|u|^{p-1}), \quad  |g_u(x,|u|)| \leq C_4(1+|u|^{p-1}),\\
|h(x,|v|)| \leq C_3(1+|v|^{p-1}), \quad  |h_v(x,|v|)| \leq C_5(1+|v|^{p-1}),
\end{gather*}
for $r>p\geq2$. Hence from the compactness of the
 embeddings
$W^{1,p}\hookrightarrow L^{\alpha}(\Omega)$ and
$W^{1,p}\hookrightarrow L^{\alpha}(\partial\Omega)$ for
$1\leq \alpha<p^*$ (the Rellich-Kondrachov Theorem \cite{5}) and
the fact that the $g(x,u)$, $h(x,u)$ are continuous and
$f(x,u,v)\in C^2(\partial{\Omega}\times\mathbb{R}^2)$, we conclude
that the functionals
$I_1(u,v)=\int_{\partial\Omega}{f(x,|u|,|v|)dx}$,
 $I_2(u)=\int_{\Omega}{G(x,|u|)dx}$ and
$I_3(A5)=\int_{\Omega}H(x,|v|)dx$ are weakly continuous, i.e. if
$(u_{n},v_n)\rightharpoonup (u,v)$, then $I_1(u_{n},v_n)\to
I_1(u,v)$, $I_2(u_{n})\to I_2(u)$ and $I_3(v_n)\to
I_3(A5)$. Moreover the operators
$I'_1(u,v)=\int_{\partial\Omega}{\nabla f(x,|u|,|v|).|(|u|,|v|)dx}$,
$I'_2(u)=\int_{\Omega}{g(x,|u|)|u|dx}$,
$I'_3(A5)=\int_{\Omega}{h(x,|v|)|v|dx}$,
$I''_1(u,v)=\int_{\partial\Omega}\mathcal{F}(x,u,v)dx$,
$I'_2(u)=\int_{\Omega}{g_u(x,|u|)u^2dx}$ and
$I'_3(A5)=\int_{\Omega}{h_v(x,|v|)v^2dx}$ are weak to strong
continuous, i.e. if $(u_n,v_n)\rightharpoonup(u,v)$ then
$I'_1(u_n,v_n)\to I'_1(u,v)$, $I''_1(u_n,v_n)\to
I''_1(u,v)$, $I'_2(u_n)\to I'_2(u)$, $I''_2(u_n)\to
I''_2(u)$, $I'_3(v_n)\to I'_3(A5)$ and $I''_3(v_n)\to
I''_3(A5)$.
\end{remark}

Now, we establish the existence of local minimum for
$J_{\lambda,\mu}$ on $N_{\lambda,\mu}^+$ and $N_{\lambda,\mu}^-$.
For simplicity let $K^*=K_0\cap K_1\cap K_3$ and $K^{**}=K_0\cap
K_1\cap K_2\cap K_3$, where $K_i$'s $ (i=0,1,2,3)$  are given in the
previous section.

\begin{lemma} \label{lem4.1}
\begin{itemize}
\item[(i)] For $(\lambda,\mu)\in K^*$, there exists
a minimizer of $J_{\lambda,\mu}$ on
$\mathcal{N}_{\lambda,\mu}^{+}(\Omega)$.
\item[(ii)] For $(\lambda,\mu)\in K^{**}$, there exists a minimizer of
$J_{\lambda,\mu}$ on $\mathcal{N}_{\lambda,\mu}^{-}(\Omega)$.
\end{itemize}
\end{lemma}

\begin{proof} (i)  As in Theorem \ref{thm2.1},  $J_{\lambda,\mu}$ is bounded
from below on $\mathcal{N}_{\lambda,\mu}(\Omega)$ and so on
$\mathcal{N}_{\lambda,\mu}^{+}(\Omega)$. Let $\{(u_{n},v_n)\}$ be a
minimizing sequence for $J_{\lambda,\mu}$ on
$\mathcal{N}_{\lambda,\mu}^{+}(\Omega)$; i.e.,
\begin{equation*}
\lim_{n \to \infty}J_{\lambda,\mu}(u_{n},v_n)= \inf_{(u,v)\in
\mathcal{N}_{\lambda,\mu}^+}J_{\lambda,\mu}(u,v).
\end{equation*}
By Ekeland's variational principle \cite{16} we may assume that
\[
\langle J'_{\lambda,\mu} (u_{n},v_n),(u_n,v_n)\rangle \to 0,
\]
combining the compact embedding Theorem \cite{5} and Lemma \ref{lem2.3},
 we obtain that there exists a subsequence $\{(u_n,v_n)\}$ and
 $(u_1,v_1)$ in $W$ such that
\begin{equation} \label{e4.1}
\begin{gathered}
 u_n \rightharpoonup u_1 \quad   \text{weakly in }   W^{1,p}(\Omega),\\
 v_n \rightharpoonup v_1  \quad   \text{weakly in }   W^{1,p}(\Omega),\\
 u_n \to u_1   \quad  \text{strongly in } L^m(\Omega),\; 1\leq m<p^*,\\
 v_n \to v_1  \quad  \text{strongly in } L^m(\partial\Omega),\; 1\leq m<p^*,\\
\end{gathered}
 \end{equation}
and $(u_n(x),v_n(x))\to (u_1(x),v_1(x))$ almost everywhere.


By Corollary \ref{coro3.2} for $(u_1,v_1)\in W\setminus\{(0,0)\}$, there
exists $t_1$ such that
 $(t_1u_1,t_1v_1)\in N_{\lambda,\mu}^{+}$ and so 
$\phi'_{u_1,v_1}(t_1)=0$. Now we
show that $(u_{n},v_n)\to (u_1,v_1)$ in $W$. Suppose this
is false, then
\begin{equation} \label{e4.2}
M(u_1,v_1)< \liminf_{n\to \infty}  M(u_{n},v_n),
\end{equation}
so from \eqref{e2.5}, \eqref{e4.1}, \eqref{e4.2} and Remark \ref{rmk4.1},
$\phi'_{u_{n},v_n}(t_1)> \phi'_{u_1,v_1}(t_1)=0$ for $n$
sufficiently large. Since $\{(u_{n},v_n)\}\subseteq
N^{+}_{\lambda,\mu}(\Omega) $, by considering the possible
fibering maps it is easy to see that, $\phi'_{u_{n},v_n}(t)<0$ for
$0<t<1$ and $\phi'_{u_{n},v_n}(1)=0$ for all $n$. Hence we must
have $t_1>1$, but $(t_1u_1,t_1 v_1)\in N^{+}_{\lambda,\mu}$
and so
\begin{align*}
&J_{\lambda,\mu}(t_1u_1,t_1 v_1)
=\phi_{u_1,v_1}(t_1)<\phi_{u_1,v_1}(1)\\
&<\lim_{n \to \infty}\phi_{u_{n},v_n}(1)
= \lim_{n \to \infty}J_{\lambda,\mu}(u_{n},v_n)
= \inf_{(u,v)\in \mathcal{N}_{\lambda,\mu}^{+}} J_{\lambda,\mu}(u,v),
\end{align*}
which is a contradiction. Therefore, $(u_{n}v_n)\to (u_1,v_1)$
in $W$ and this concludes that
\[
J_{\lambda,\mu}(u_1,v_1)= \lim_{n \to \infty}J_{\lambda,\mu}(u_{n},v_n)=
\inf_{(u,v)\in \mathcal{N}_{\lambda,\mu}^{+}}
J_{\lambda,\mu}(u,v).
\]
Thus $(u_1,v_1)$ is a minimizer for $J_{\lambda,\mu}$ on
$\mathcal{N}_{\lambda,\mu}^{+}(\Omega)$. 

(ii) By Corollary \ref{coro3.1} we have
$J_{\lambda,\mu}(u,v)\geq\varepsilon>0$ for all 
$(u,v)\in \mathcal{N}_{\lambda,\mu}^{-}$, so
\[
\inf_{(u,v)\in\mathcal{N}_{\lambda,\mu}^{-}}J_{\lambda,\mu}(u,v)>0,
\]
hence, there exists a minimizing sequence
 $\{(u_{n},v_n)\}\subseteq \mathcal{N}_{\lambda,\mu}^{-}(\Omega)$ such that
\begin{equation} \label{e4.3}
 \lim_{n \to \infty}J_{\lambda,\mu}(u_{n},v_n)= \inf_{(u,v)\in
\mathcal{N}_{\lambda,\mu}^{-}} J_{\lambda,\mu}(u,v)>0.
\end{equation}
Similar to the argument in the proof of (i) we find that
$\{(u_{n},v_n)\}$ is bounded in $W$ and also the results obtained in
\eqref{e4.1} are satisfied for $\{(u_{n},v_n)\}$ and $\{(u_2,v_2)\}$.

Since $(u_{n},v_n)\in \mathcal{N}_{\lambda,\mu}^{-}(\Omega)$, so by
\eqref{e2.7}, $\phi''_{u_n, v_n}(1)<0$, letting  $n\to\infty$, by
\eqref{e2.5}, Remark \ref{rmk4.1} and the above argument we see that
\begin{equation} \label{e4.4}
\begin{split}
\phi''_{u_2,v_2}(1)
&=M(u_2,v_2)-\int_{\partial\Omega}\mathcal{F}(x,u_2,v_2)dx
 -\lambda\int_{\Omega}g_{u}(x,|u_2|)u_2^2dx\\
&\quad -\mu\int _{\Omega}h_{v}(x,|v_2|)v_2^2dx\leq0.
\end{split}
\end{equation}

On the other hand for $(u_{n},v_n)\in
\mathcal{N}_{\lambda,\mu}^{-}$, by Lemma \ref{lem2.4}, 
$\int_{\partial\Omega}\mathcal{F}(x,u_{n},v_n)dx>0$. Letting
$n\to\infty$, we see that
$\int_{\partial\Omega}\mathcal{F}(x,u_2,v_2)dx\geq0$. 

We claim
that $\int_{\partial\Omega}\mathcal{F}(x,u_2,v_2)dx\neq0$. If
$\int_{\partial\Omega}\mathcal{F}(x,u_2,v_2)dx=0$, then by (A4),
\eqref{e1.3}, \eqref{e2.14} and \eqref{e4.4} we have
\begin{equation*}
M(u_2,v_2)\leq\lambda\int
_{\partial\Omega}g_{u}(x,|u_2|)u_2^2dx+\mu\int
_{\partial\Omega}h_{v}(x,|v_2|)v_2^2dx\leq(\lambda L+\mu
L')M(u_2,v_2),
\end{equation*}
which is a contradiction for $(\lambda,\mu) \in K_2$. So
$\int_{\partial\Omega}\mathcal{F}(x,u_2,v_2)dx>0$ and by 
Corollary \ref{coro3.2} (ii) there exists $t_2>0$ such that 
$ (t_2u_2,t_2v_2)\in \mathcal{N}_{\lambda,\mu}^{-}(\Omega)$. 
We claim that $(u_{n},v_n)\to (u_2,v_2)$ in $W$. Suppose that this is
false, so we have
\begin{equation} \label{e4.5}
M(u_2,v_2)< \liminf_{n\to \infty }M(u_{n},v_n).
 \end{equation}

 However, $(u_{n},v_n)\in \mathcal{N}_{\lambda,\mu}^{-}$ and so
$J_{\lambda,\mu}(u_{n},v_n)\geq J_{\lambda,\mu}(tu_{n},tv_n)$ for
all $t\geq0$. Therefore, considering \eqref{e2.5}, \eqref{e4.3}--\eqref{e4.5} and
Remark \eqref{e4.1}, we can write
\begin{align*}
&J_{\lambda,\mu}(t_2u_2,t_2v_2)\\
&=\frac{t_2^p}{p} M(u_2,v_2)- F(t_2u_2,t_2v_2)-\lambda
\int_{\Omega}H(x,t_2|u_2|)dx-\mu\int_{\Omega}G(x,t_2|v_2|)dx\\
 & <\lim_{n \to \infty}\Big(\frac{t_2^p}{p}
M(u_n,v_n)-F(t_2u_n,t_2v_n)-\lambda
\int_{\Omega}H(x,t_2|u_n|)dx\\ 
&\quad -\mu\int_{\Omega}G(x,t_2|v_n|)dx\Big)\\
& = \lim_{n \to \infty}J
_{\lambda,\mu}(t_2u_{n},t_2v_n)\leq \lim_{n \to \infty}
J_{\lambda,\mu}(u_{n},v_n)= \inf_{(u,v)\in
\mathcal{N}_{\lambda,\mu}^{-}} J_{\lambda,\mu}(u,v),
\end{align*}
which is a contradiction. So, $(u_{n},v_n)\to (u_2,v_2)$ in
$W$ and the proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 By Lemma \ref{lem4.1} (i) there exists
$(u_1,v_1)\in N_{\lambda,\mu}^{+}(\Omega)$ such that
$J_{\lambda,\mu}(u_1,v_1)=\inf_{(u,v) \in N_{\lambda,\mu}^{+}}
J_{\lambda,\mu}(u,v)$ and by Lemmas \ref{lem2.1} and \ref{lem2.2}, $(u_1,v_1)$ is a
critical point of $J_{\lambda,\mu}$ on $W$ and hence is a weak
solution of problem \eqref{e1.1}. On the other hand
$J_{\lambda,\mu}(u,v)=J_{\lambda,\mu}(|u|,|v|)$, so we may assume
that $(u_1,v_1)$ is a positive solution and the proof is
complete.
\end{proof} 

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 By Lemma \ref{lem4.1} there exist
 $(u_1,v_1)\in N_{\lambda,\mu}^{+}(\Omega)$ and 
$(u_2,v_2)\in N_{\lambda,\mu}^{-}(\Omega)$ such that
$$
J_{\lambda,\mu}(u_1,v_1)=\inf_{(u,v) \in N_{\lambda,\mu}^{+}}
J_{\lambda,\mu}(u,v),\quad
J_{\lambda,\mu}(u_2,v_2)=\inf_{(u,v) \in N_{\lambda,\mu}^{-}}J_{\lambda,\mu}(u,v).
$$
 By Lemmas \ref{lem2.1} and \ref{lem2.2}, $(u_1,v_1)$ and $(u_2,v_2)$ are critical points of
$J_{\lambda,\mu}$ on $W$ and hence are weak solutions of problem
\eqref{e1.1}. Similar to the proof of Theorem \ref{thm1.1}, we may assume that
$(u_1,v_1)$ and $(u_2,v_2)$ are positive solutions. Also since
$N_{\lambda,\mu}^{+}\cap N_{\lambda,\mu}^{-}={\emptyset}$, this
implies that $(u_1,v_1)$ and $(u_2,v_2)$ are distinct and the
proof is complete.
\end{proof}

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\end{document}