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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 35, pp. 1--17.\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-2013/35\hfil Quasilinear elliptic systems]
{Existence of solutions for quasilinear elliptic systems involving
critical exponents and \\ Hardy terms}

\author[D. L\"u \hfil EJDE-2013/35\hfilneg]
{Dengfeng L\"u}  % in alphabetical order

\address{Dengfeng L\"u \newline
School of Mathematics and Statistics,
Hubei Engineering University,
Hubei 432000, China}
\email{dengfeng1214@163.com}

\thanks{Submitted February 15, 2012. Published January 30, 2013.}
\subjclass[2000]{35J92, 35J50, 35B33}
\keywords{Quasilinear elliptic system; variational method;
 critical exponent; \hfill\break\indent Hardy term; multiple solutions}

\begin{abstract}
 Using variational methods, including the Ljusternik-Schnirelmann theory,
 we prove the existence of solutions for quasilinear elliptic systems
 with critical  Sobolev exponents and Hardy terms.
\end{abstract}

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

\section{Introduction and statement of main results}

We consider the critical quasilinear elliptic system
\begin{equation}\label{1.1}
\begin{gathered}
-\Delta_p u-\mu\frac{|u|^{p-2}u}{|x|^p}=\frac{1}{p^{*}}F_{u}(u,v)
     +G_{u}(u,v), \quad x\in\Omega,\\
-\Delta_p v-\mu\frac{|v|^{p-2}v}{|x|^p}=\frac{1}{p^{*}}F_{v}(u,v)
 +G_{v}(u,v), \quad x\in\Omega,\\
u=v=0,\quad x\in\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset {\mathbb{R}}^N$ is a bounded domain with
smooth boundary $\partial\Omega$, $0\in \Omega$,
$\Delta_p u= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$
is the $p$-Laplacian operator,
$N\geq p^{2},2\leq p\leq q<p^{*}$,
$p^{*}=\frac{Np}{N-p}$ denotes the Sobolev critical exponent,
$F,G\in C^{1}(\mathbb{R}^{+}\times \mathbb{R}^{+},\mathbb{R})$
are homogeneous functions of degrees $p^{*}$ and $q$, respectively.
$\mathbb{R}^{+}=[0,+\infty)$,
$(F_{u}(u,v),F_{v}(u,v))=\nabla F,(G_{u}(u,v)$,
$G_{v}(u,v))=\nabla G$, $0\leq\mu<\bar{\mu}$,
$\bar{\mu}=(\frac{N-p}{p})^p$ is the
best constant of the Hardy inequality \cite{4}:
$$
\bar{\mu}\int_{\Omega}\frac{|u|^p}{|x|^p}dx\leq
\int_{\Omega}|\nabla u|^pdx,
$$
for all $u\in W^{1,p}_0(\Omega)$, where $W^{1,p}_0(\Omega)$ is
defined as the completion of $C_0^{\infty}(\Omega)$ with respect
to the norm $\|u\|=(\int_{\Omega}|\nabla u|^pdx)^{1/p}$.
For $\mu\in[0,\bar{\mu})$, it follows from the Hardy inequality that
$$
\|u\|_{\mu}=\Big(\int_{\Omega}|\nabla u|^p-\mu\frac{|u|^p}{|x|^p}dx\Big)^{1/p}
$$
defines a norm in $W^{1,p}_0(\Omega)$ equivalent to its usual norm.
The  best Sobolev constant is  defined as
\begin{equation}
S_{\mu}=\inf_{u\in \mathcal{D}^{1,p}(\mathbb{R}^N)\setminus
\{0\}}\frac{\int_{\mathbb{R}^N}(|\nabla
u|^p-\mu\frac{|u|^p}{|x|^p})dx}
{(\int_{\mathbb{R}^N}|u|^{p^{*}}dx)^{p/p^*}}, \quad
\mu\in[0,\bar{\mu}). \label{1.2}
\end{equation}

In recent years, much attention has been focused on singular
problems involving both the Hardy potential and the Sobolev critical
term. For example, see \cite{7,13,16,18,19,20,23,26} and the
references therein.  In \cite{9}, Ding and Xiao
consider the  $p$-Laplacian system
\begin{equation}\label{1.3}
\begin{gathered}
-\Delta_p u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}+\lambda
|u|^{q-2}u, \quad x\in\Omega,\\
-\Delta_p v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v+\delta
|v|^{q-2}v, \quad x\in\Omega,\\
u=v=0,\quad x\in\partial\Omega,
\end{gathered}
\end{equation}
where $p\leq q<p^{*},\alpha,\beta>1,\alpha+\beta=p^{*}$. Using
standard tools of the variational theory and the
Ljusternik-Schnirelmann category theory, in \cite{9} sufficient conditions on
$\lambda, \delta$ are given for  \eqref{1.3}  to have  at least
$\operatorname{cat}_{\Omega}(\Omega)$ positive solutions.
 This result extended the result of
Alves and Ding in \cite{2} where the  single equation case was studied.
Hsu \cite{17} obtained the existence of two
positive solutions for \eqref{1.3} including a sublinear
perturbation of $1<q<p<N$. Recently, Shen and Zhang extended the
results in \cite{25} to a general class of homogeneous functions and
obtained similar results. For similar problems, we refer the reader to
\cite{3,6,8,10,11,12,14,15,21,22,24} and the references therein.

In this paper, motivated by \cite{2,9,17,25}, we shall extend these
results to the case containing a general class of homogeneous
nonlinearities and Hardy terms. To the best of our knowledge,
problem \eqref{1.1} has not been considered before. Thus, it is
necessary for us to investigate the related singular critical
systems.

The following assumptions are used in this article:
\begin{itemize}
\item[(F0)] $F\in C^{1}({\mathbb{R}}^{+}\times {\mathbb{R}}^{+},\mathbb{R})$
and $F(tu,tv)=t^{p^{*}}F(u,v)(t>0)$
holds for all $(u,v)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$,

\item[(F1)] $F_{u}(0,1)=F_{v}(1,0)=0$,

\item[(F2)] $F_{u}(u,v)\geq 0,F_{v}(u,v)\geq 0$ for all
$u,v\geq 0$,

\item[(F3)] the 1-homogeneous function $(u,v)\mapsto
F(u^{\frac{1}{p^{*}}},v^{\frac{1}{p^{*}}})$ is concave for all
$(u,v)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$.

\item[(G0)] $G$ is $q$-homogeneous for some $p\leq q<p^{*}$,

\item[(G1)] $G_{u}(0,1)=G_{v}(1,0)=0$.

\end{itemize}
To present our results, we define
\begin{gather}\label{1.4}
\lambda=\max\{G(u,v):u,v\geq 0,u^{q}+v^{q}=1\},\\
\label{1.5}
\delta=\min\{G(u,v):u,v\geq 0,u^{q}+v^{q}=1\}.
\end{gather}


If $Y$ is a closed subset of a topological space $X$, we denote, by
$\operatorname{cat}_{X}(Y)$, the Ljusternik-Schnirelmann category of $Y$ in
$X$, namely the least number of closed and contractible sets in $X$
which cover $Y$.
We say that a weak solution $(u,v)\in W^{1,p}_0(\Omega)\times W^{1,p}_0(\Omega)$
of problem \eqref{1.1} is nonnegative if $u, v\geq 0$ in $\Omega$.

The main results of this paper are stated in the following two
theorems whose conclusions are new (to the best of our knowledge).

\begin{theorem} \label{thm1}
Suppose {\rm (F0)--(F3), (G0)--(G1)} are satisfied, and one of the
following two conditions holds:
\begin{itemize}
\item[(I)] $\bar{p}<q <p^{*}$ with
$\bar{p}=\max\Big\{p,\frac{N}{b(\mu)},\frac{p(2N-pb(\mu)-p)}{N-p}\Big\}$,
$0\leq\mu<\bar{\mu}$ and $\lambda,\delta> 0$;

\item[(II)] $q=p$, $0\leq \mu\leq \frac{N^{p-1}(N-p^{2})}{p^p}$ and
$\lambda,\delta\in(0,\frac{1}{p}\Lambda_1)$, where $\Lambda_1$
is the first eigenvalue of $(-\Delta_p,W^{1,p}_0(\Omega))$.
\end{itemize}
Then problem \eqref{1.1} has at least one nonnegative solution.
\end{theorem}

\begin{theorem}\label{thm2}
Suppose {\rm (F0)--(F3),  (G0)--(G1)} are satisfied, and one of the
following two conditions  holds:
\begin{itemize}
\item[(I)]  $\bar{p}<q <p^{*}$ with
$\bar{p}=\max\Big\{p,\frac{N}{b(\mu)},\frac{p(2N-pb(\mu)-p)}{N-p}\Big\}$,
$0\leq\mu<\bar{\mu}$;

\item[(II)] $q=p$, $0\leq \mu\leq \frac{N^{p-1}(N-p^{2})}{p^p}$.
\end{itemize}
Then there exists $\Lambda>0$ such that problem \eqref{1.1} has at least
$\operatorname{cat}_{\Omega}(\Omega)$ distinct nonnegative solutions for
$\lambda,\delta\in (0,\Lambda)$.
\end{theorem}

\begin{remark}\label{bl-L2.4}\rm
Our Theorem \ref{thm1} is a generalization of \cite[Theorem 1.1]{16}
from quasilinear elliptic equations to quasilinear
elliptic systems.
 \end{remark}

 \begin{remark} \rm %\label{bl-L2.4}
Theorem 1 in \cite{9} is the special case of our Theorem \ref{thm2}
corresponding to
$\mu=0,F(u,v)=2|u|^{\alpha}|v|^{\beta},\alpha+\beta=p^{*}$ and
$G(u,v)=\lambda|u|^{q}+\delta|v|^{q}$. In this paper, different from
\cite{25}, we can deal with $F(u,v)$ which possesses both coupled
and uncoupled terms. For example, let
$$
F(u,v)=au^{p^{*}}+\sum_{i=1}^{k}b_iu^{\alpha_i}v^{\beta_i}+cv^{p^{*}},
$$
where $a,b_i,c \geq 0$, $\alpha_i,\beta_i>1$,
$\alpha_i+\beta_i=p^{*}$. $F(u,v)$ obviously satisfies
(F0)--(F3).
\end{remark}

This article is organized as follows. In Section 2, some notation and
the mountain pass levels are established and Theorem \ref{thm1} is
proven. We present some technical lemmas which are crucial in the
proof of Theorem \ref{thm2} in Section 3. Theorem \ref{thm2} is
proven in Section 4.

\section{Preliminaries and proof of Theorem \ref{thm1}}

Throughout this paper, $C,C_i$ will denote various positive
constants whose exact values are not important. And $\to$
(respectively $\rightharpoonup$) denotes strong (respectively weak)
convergence. $O(\varepsilon^{t})$ denotes
$|O(\varepsilon^{t})|/\varepsilon^{t}\leq C,o_{ m}(1)$ denotes
$o_{m}(1)\to 0$ as $m\to\infty$. $L^{s}(\Omega), for
(1\leq s<+\infty)$, denotes Lebesgue spaces, the norm $L^{s}$ is
denoted by $|\cdot|_{s}$ for $1\leq s<+\infty$. Let $B_{r}(x)$
denote a ball centered at $x$ with radius $r$. The dual space of a
Banach space $E$ will be denoted by $E^{-1}$. We define the product
space $E:= W^{1,p}_0(\Omega)\times W^{1,p}_0(\Omega)$ endowed
with the norm $ \|(u,v)\|_{E}=\big(\|u\|_{\mu}^p+
\|v\|_{\mu}^p\big)^{1/p}$.

In view of (F1), (G1), we can extend the function $F(u,v)$
and $G(u,v)$ to the whole $\mathbb{R}^{2}$ by considering
$F(u,v)=F(u^{+},v^{+})$, $G(u,v)=G(u^{+},v^{+})$, where
$u^{+}=\max\{u,0\}$ and $v^{+}=\max\{v,0\}$. It is easy to check
that $F(u,v)$ and $G(u,v)\in C^{1}(\mathbb{R}^{2})$. Therefore, we
always consider $F(u,v)$ and $G(u,v)$ as these extensions.

A pair of functions $(u,v)\in E$ is said to be a weak solution of
problem \eqref{1.1} if
 \begin{align*}
& \int_{\Omega}(|\nabla u|^{p-2}\nabla u\nabla\varphi_1
-\mu\frac{|u|^{p-2}u\varphi_1}{|x|^p}
 + |\nabla v|^{p-2}\nabla
 v\nabla\varphi_2-\mu\frac{|v|^{p-2}v\varphi_2}{|x|^p})dx\\
&-   \frac{1}{p^{*}}
    \int_{\Omega}(F_{u}(u,v)\varphi_1+F_{v}(u,v)\varphi_2)dx
- \int_{\Omega}(G_{u}(u,v)\varphi_1+G_{v}(u,v)\varphi_2)dx =0,
\end{align*}
for all $ (\varphi_1,\varphi_2)\in E$. Using (F0)-(G1)
and well-known arguments, we know that the weak solutions of
\eqref{1.1} are precisely the critical points of the
$C^{1}$-functional $I_{\lambda,\delta}: E \to \mathbb{R}$
given by
\begin{align*}
&I_{\lambda,\delta}(u,v)\\
&=\frac{1}{p}\int_{\Omega}(|\nabla
u|^p-\mu\frac{|u|^p}{|x|^p}+|\nabla
v|^p-\mu\frac{|v|^p}{|x|^p})dx-\frac{1}{p^{*}}
\int_{\Omega}F(u,v)dx-\int_{\Omega}
G_{\lambda,\delta}(u,v)dx.
\end{align*}
We notice that, in the definition of $I_{\lambda,\delta}$, we are
denoting $G_{\lambda,\delta}(u,v):= G(u,v)$ for
 $(u,v)\in \mathbb{R}^{2}$. We shall write $G_{\lambda,\delta}$
instead of $G$ to emphasize that the main theorems depend on the value of the
parameters $\lambda$ and $\delta$ defined in \eqref{1.4} and
\eqref{1.5}, respectively.

The functional $I\in C^{1}(E,{\mathbb{R}})$ is said to satisfy the
$(PS)_{c}$ condition if any sequence $\{z_{m}\}\subset E$ such that
as $m\to\infty$, $I(z_{m})\to c$,
$I'(z_{m})\to 0$ strongly in $E^{-1}$ contains a subsequence
converging in $E$ to a critical point of $I$. In this paper, we will
take $I = I_{\lambda,\delta}$ and $E = W^{1,p}_0(\Omega)\times
W^{1,p}_0(\Omega)$.


 In this section, we will
find the range of $c$ where the $(PS)_{c}$ condition holds for the
functional $I_{\lambda,\delta}$. First, let us define
\begin{equation}\label{2.1}
S_{F}=\inf_{(u,v)\in
E\setminus\{(0,0)\}}\Big\{\frac{\int_{\Omega}|\nabla
u|^p-\mu\frac{|u|^p}{|x|^p}+|\nabla
v|^p-\mu\frac{|v|^p}{|x|^p}dx}{(\int_{\Omega}F(u,v)dx)^{p/p^*}}:
\int_{\Omega}F(u,v)dx>0\Big\}.
\end{equation}


\begin{lemma}\label{lem2.1}
 Suppose {\rm (F0)--(F3), (G0)--(G1)} are satisfied, then the
functional $I_{\lambda,\delta}$ satisfies the
 $(PS)_{c}$ condition for all $c<\frac{1}{N}S_{F}^{N/p}$,
 provided either $p<q<p^{*}$ or $q=p$ and the parameter $\lambda$ defined in
 \eqref{1.4}
 belongs to $(0, \frac{1}{p}\Lambda_1)$, where $\Lambda_1>0$
 denotes the first eigenvalue of $(-\Delta_p , W^{1,p}_0(\Omega))$.
 \end{lemma}

 \begin{proof}
Let $\{(u_{m},v_{m})\}\subset E$ such that
 $I'_{\lambda,\delta}(u_{m},v_{m})\to 0$ and
 $I_{\lambda,\delta}(u_{m},v_{m})\to c<\frac{1}{N}S_{F}^{N/p}$.
Now, we firstly prove that $\{(u_{m},v_{m})\}$ is bounded in $E$. If
$p<q<p^{*}$, it suffices to use the definition of
$I_{\lambda,\delta}$ to obtain $C_1>0$ such that
\begin{align*}
c+C_1\|(u_{m},v_{m})\|_{E}+o_{m}(1)
&\geq I_{\lambda,\delta}(u_{m},v_{m})-\frac{1}{q}
\langle I'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle\\
&=\Big(\frac{1}{p}-\frac{1}{q}\Big)\|(u_{m},v_{m})\|_{E}^p
 +\Big(\frac{1}{q}-\frac{1}{p^{*}}\Big)\int_{\Omega}F(u_{m},v_{m})dx\\
&\geq \frac{q-p}{pq}\|(u_{m},v_{m})\|_{E}^p,
\end{align*}
which implies that  $\{(u_{m},v_{m})\}\subset E$ is bounded. When
$q=p$, in this case, it follows that
$$
\int_{\Omega}G_{\lambda,\delta}(u_{m},v_{m})dx \leq
\lambda\int_{\Omega}(|u_{m}|^p+|v_{m}|^p)dx\leq
\frac{\lambda}{\Lambda_1}\|(u_{m},v_{m})\|_{E}^p,
$$
and  therefore,
\begin{align*}
c+C_1\|(u_{m},v_{m})\|_{E}+o_{m}(1)
&\geq I_{\lambda,\delta}(u_{m},v_{m})-\frac{1}{p^{*}}
\langle I'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle\\
 &=\frac{1}{N}\|(u_{m},v_{m})\|_{E}^p
 - \frac{p}{N}\int_{\Omega}G(u_{m},v_{m})dx\\
&\geq \frac{1}{N}\Big(1-\frac{p\lambda}{\Lambda_1}\Big)
\|(u_{m},v_{m})\|_{E}^p.
\end{align*}
Since $p\lambda<\Lambda_1$, the boundedness of $\{(u_{m},v_{m})\}$
follows as in the first case.

So $\{(u_{m},v_{m})\}$ is bounded in $E$. Going if necessary to a
subsequence, we can assume that
\begin{gather*}
      (u_{m},v_{m})\rightharpoonup(u,v), \quad   \text{in }  E,\\
       (u_{m},v_{m})\to (u,v), \quad  \text{a.e.\ in } \Omega,\\
       (u_{m},v_{m})\to (u,v), \quad  \text{in }
        L^{s}(\Omega)\times L^{s}(\Omega),1\leq s<p^{*},
    \end{gather*}
as $m\to\infty$. Clearly, we have that
\begin{equation}\label{2.2}
 \int_{\Omega}G_{\lambda,\delta}(u_{m},v_{m})dx
=\int_{\Omega}G_{\lambda,\delta}(u,v)dx+o_{m}(1).
\end{equation}
Moreover, a standard argument shows that
$I'_{\lambda,\delta}(u,v)=0$. Thus, we obtain
\begin{equation} \label{2.3}
\begin{aligned}
I_{\lambda,\delta}(u,v)
&=\frac{1}{p}\|(u,v)\|_{E}^p-\frac{1}{p^{*}}
\int_{\Omega}F(u,v)dx-\int_{\Omega} G_{\lambda,\delta}(u,v)dx\\
&=\Big(\frac{1}{p}-\frac{1}{q}\Big)\|(u,v)\|_{E}^p+
\Big(\frac{1}{q}-\frac{1}{p^{*}}\Big)\int_{\Omega}F(u,v)dx
\geq 0.
\end{aligned}
\end{equation}

Let $(\tilde{u}_{m},\tilde{v}_{m})=(u_{m}-u, v_{m}-v)$. Then by the
Brezis-Lieb Lemma \cite{5}, we have
\begin{equation}\label{2.4}
\|(\tilde{u}_{m},\tilde{v}_{m})\|_{E}^p=
\|(u_{m},v_{m})\|_{E}^p-\|(u,v)\|_{E}^p+o_{m}(1).
\end{equation}
By the same method as in \cite[Lemma 8]{11}, we obtain
\begin{equation}\label{2.5}
\int_{\Omega}F(u_{m},v_{m})dx
=\int_{\Omega}F(u,v)dx+\int_{\Omega}F(\tilde{u}_{m},\tilde{v}_{m})dx+o_{m}(1).
\end{equation}
By \eqref{2.2},\eqref{2.3},\eqref{2.4},\eqref{2.5} and the weak
convergence of $(u_{m},v_{m})$, we have
\begin{equation} \label{2.6}
\begin{aligned}
c+o_{m}(1)
&= I_{\lambda,\delta}(u,v)+\frac{1}{p}\|(\tilde{u}_{m},\tilde{v}_{m})\|_{E}^p-
\frac{1}{p^{*}}\int_{\Omega}F(\tilde{u}_{m},\tilde{v}_{m})dx \\
&\geq \frac{1}{p}\|(\tilde{u}_{m},\tilde{v}_{m})\|_{E}^p-
\frac{1}{p^{*}}\int_{\Omega}F(\tilde{u}_{m},\tilde{v}_{m})dx.
\end{aligned}
\end{equation}
Using that $I'_{\lambda,\delta}(u_{m},v_{m})\to 0$ and
\eqref{2.2}, \eqref{2.4}, \eqref{2.5}, we obtain
\begin{align*}
o_{m}(1) &= \langle
I'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle
   \\
&=\|(u_{m},v_{m})\|_{E}^p-\int_{\Omega}F(u_{m},v_{m})dx-q\int_{\Omega}G_{\lambda,\delta}
(u_{m},v_{m})dx
   \\
&= \langle
I'_{\lambda,\delta}(u,v),(u,v)\rangle+\|(\tilde{u}_{m},\tilde{v}_{m})\|_{E}^p-
\int_{\Omega}F(\tilde{u}_{m},\tilde{v}_{m})dx.
\end{align*}
Recalling that $I'_{\lambda,\delta}(u,v)=0$, we can use the above
equality and \eqref{2.6} to obtain
$$
\lim_{m\to\infty}\|(\tilde{u}_{m},\tilde{v}_{m})\|_{E}^p=k=
\lim_{m\to\infty}\int_{\Omega}F(\tilde{u}_{m},\tilde{v}_{m})dx,
\quad c\geq \Big(\frac{1}{p}-\frac{1}{p*}\Big)k=\frac{1}{N}k,
$$
where $k\geq 0$.

In view of the definition of $S_{F}$, we deduce that
$$
\|(\tilde{u}_{m},\tilde{v}_{m})\|_{E}^p\geq
S_{F}\Big(\int_{\Omega}F(\tilde{u}_{m},\tilde{v}_{m})dx\Big)^{p/p^*}.
$$
Taking the limit, we obtain $k\geq S_{F}k^{p/p^*}$. So, if
$k> 0$, we conclude that $k\geq S_{F}^{N/p}$ and therefore
$$
\frac{1}{N}S_{F}^{N/p}\leq \frac{1}{N}k\leq
c<\frac{1}{N}S_{F}^{N/p},
$$
which is a contradiction. Hence $k=0$ and therefore
$(u_{m},v_{m})\to (u,v)$ in $E$.
 \end{proof}

For all $\mu\in[0,\bar{\mu})$, we consider the limiting problem
\begin{equation}\label{2.7}
\begin{gathered}
-\Delta_p U-\mu\frac{U^{p-1}}{|x|^p}=
U^{p^{*}-1}, \quad  \text{in }  \mathbb{R}^N\setminus\{0\},\\
     U>0, \quad \text{in }  \mathbb{R}^N\setminus\{0\},\\
     U\to 0,\quad  \text{as }  |x|\to+\infty.
\end{gathered}
\end{equation}
From \cite{1}, we know that problem \eqref{2.7} has a ground state
$U_{p,\mu}$, which is unique up to scaling. That is, all ground
states must be of the form
\begin{equation}\label{2.8}
V_{p,\mu,\varepsilon}(x)
=\varepsilon^{-\frac{N-p}{p}}U_{p,\mu}\Big(\frac{x}{\varepsilon}\Big)
=\varepsilon^{-\frac{N-p}{p}}U_{p,\mu}\Big(\frac{|x|}{\varepsilon}\Big),
\quad \varepsilon>0,
\end{equation}
that satisfy
\begin{equation}\label{2.9}
\int_{\mathbb{R}^N}(|\nabla
V_{p,\mu,\varepsilon}(x)|^p-\mu\frac{|V_{p,\mu,\varepsilon}(x)|^p}{|x|^p})dx
=\int_{\mathbb{R}^N}|V_{p,\mu,\varepsilon}(x)|^{p^{*}}dx
=S_{\mu}^{N/p},
\end{equation}
where $S_{\mu}$ is the best Sobolev constant given in \eqref{1.2}.

Moreover, the ground state $U_{p,\mu}$ is radially symmetric and
decreasing, and the following asymptotic properties at the origin
and infinity for $U_{p,\mu}(r)$ and $U'_{p,\mu}(r)$ hold:
\begin{gather*}
\lim_{r\to 0^{+}}r^{a(\mu)}U_{p,\mu}(r)=c_1>0, \quad
\lim_{r\to 0^{+}}r^{a(\mu)+1}|U'_{p,\mu}(r)|=c_1a(\mu)\geq0,\\
\lim_{r\to +\infty}r^{b(\mu)}U_{p,\mu}(r)=c_2>0, \quad
\lim_{r\to +\infty}r^{b(\mu)+1}|U'_{p,\mu}(r)|=c_2b(\mu)>0,
\end{gather*}
where $c_1$ and  $c_2$ are positive constants depending only on
$N,p,\mu$, and $a(\mu), b(\mu)$, the zeros of the function
$h(t)=(p-1)t^p-(N-p)t^{p-1}+\mu, \ t\geq 0$, which satisfy $0\leq
a(\mu)<b(\mu)\leq \frac{N-p}{p-1}$.

After a direct calculation, we infer that
$t_{\rm min}=\frac{N-p}{p}$ is the unique minimal point of $h(t)$
and $h(\frac{N-p}{p})=-\bar{\mu}+\mu<0$. Moreover, $h'(t)<0$ for all
$0<t<t_{\rm min}$ and $h'(t)>0$ for all $t>t_{\rm min}$. That
is, $h(t)$ is decreasing on the interval $(0, t_{\rm min})$ and
increasing on the interval $(t_{\rm min},+\infty)$. Thus $0\leq
a(\mu)<\frac{N-p}{p}<b(\mu)$.

In addition, using \cite[Lemma 3]{11} and the homogeneity of $F$,
we obtain $A, B>0$ such that
$$
S_{F}=\frac{\|(AV_{p,\mu,\varepsilon},BV_{p,\mu,\varepsilon}\|_{E}^p}
{(\int_{\mathbb{R}^N}
F(AV_{p,\mu,\varepsilon},BV_{p,\mu,\varepsilon})dx)^{p/p^*}}
=\frac{A^p+B^p}
{(F(A,B))^{p/p^*}}\cdot\frac{S_{\mu}^{N/p}}{|V_{p,\mu,\varepsilon}|_{p^{*}}^p},
$$
from this and \eqref{2.9}, we have
\begin{equation}\label{2.10}
S_{F}=\frac{A^p+B^p} {(F(A,B))^{p/p^*}}S_{\mu}.
\end{equation}
 We define a cut-off function $\phi(x)\in C_0^{\infty}(\mathbb{R}^N)$
such that $\phi(x)=1$ if $|x|\leq R$; $\phi(x)=0$ if $|x|\geq 2R$
and $0\leq\phi(x)\leq 1$, where $B_{2R}(0)\subset \Omega$ and set
$u_{\varepsilon}=\frac{\phi(x)V_{p,\mu,\varepsilon}}{|\phi
V_{p,\mu,\varepsilon}|_{p^{*}}}, $ where $V_{p,\mu,\varepsilon}$ was
defined in \eqref{2.8}. So, $|u_{\varepsilon}|_{p^{*}}=1$. Thus, we
can get the following results from \cite[Lemma 2.2]{26}
(or \cite{16}):
\begin{gather}\label{2.11}
\|u_{\varepsilon}\|_{\mu}^p=S_{\mu}+O(\varepsilon^{pb(\mu)+p-N}),
\\
\int_{\Omega}|u_{\varepsilon}|^{\xi}dx\approx
\begin{cases}
\varepsilon^{(b(\mu)-\frac{N-p}{p})\xi},
&  \text{if } 1\leq\xi<\frac{N}{b(\mu)},\\
 \varepsilon^{N-\frac{N-p}{p}\xi}|\ln\varepsilon|,
&  \text{if } \xi=\frac{N}{b(\mu)},\\
\varepsilon^{N-\frac{N-p}{p}\xi},
&  \text{if }   \frac{N}{b(\mu)}<\xi<p^{*},
\end{cases}
\label{2.12}
 \end{gather}
where $A\approx B$ means $C_1 B\leq A\leq C_2 B$.

As $I_{\lambda,\delta}$ is not bounded below on $E$, we need to
study $I_{\lambda,\delta}$ on the Nehari manifold:
$$
\mathcal{N}_{\lambda,\delta}=\big\{(u,v)\in E\setminus
\{(0,0)\}:\langle I'_{\lambda,\delta}(u,v),(u,v)\rangle=0\big\}.
$$
Note that $\mathcal{N}_{\lambda,\delta}$ contains every nonzero
solution of problem \eqref{1.1}, and we define the minimax
$c_{\lambda,\delta}$ as
$$
c_{\lambda,\delta}=\inf_{(u,v)\in
\mathcal{N}_{\lambda,\delta}}I_{\lambda,\delta}(u,v).
$$
Next, we present some properties of $c_{\lambda,\delta}$ and
$\mathcal{N}_{\lambda,\delta}$. Their proofs can be done as
\cite[Theorem 4.2]{27}. First of all, we note that there exists
$\rho >0$ such that
\begin{equation}\label{2.13}
\|(u,v)\|_{E}\geq \rho>0, \ \forall \ (u,v)\in
\mathcal{N}_{\lambda,\delta}.
\end{equation}
It is standard to check that $I_{\lambda,\delta}$ satisfies the
mountain pass geometry, so we can use the homogeneity of $F$ and $G$
to prove that $c_{\lambda,\delta}$ can be alternatively
characterized by
\begin{equation}\label{2.14}
c_{\lambda,\delta}=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}
I_{\lambda,\delta}(\gamma(t))=\inf_{(u,v)\in E\backslash
\{(0,0)\}}\max_{t\geq 0}I_{\lambda,\delta}(t(u,v))>0,
\end{equation}
where $\Gamma=\{\gamma\in C([0,1],E): \gamma(0)=0,
I_{\lambda,\delta}(\gamma(1))<0\}$. Moreover, for each $(u,v)\in E
\backslash \{(0,0)\}$, there exists a unique $t^{*}> 0$ such that
$t^{*}(u,v) \in \mathcal{N}_{\lambda,\delta}$. The maximum of the
function $t\mapsto I_{\lambda,\delta}(t(u,v))$, for $t\geq 0$, is
achieved at $t=t^{*}$.

\begin{lemma}\label{lem2.2}
 Suppose that
$(F_0)-(F_3)$ and $(G_0)-(G_1)$ hold, $\bar{p}<q <p^{*}$
with
$\bar{p}=\max\Big\{p,\frac{N}{b(\mu)},\frac{p(2N-pb(\mu)-p)}{N-p}\Big\}$,
$0\leq\mu<\bar{\mu}$ and $\lambda, \delta$ defined in \eqref{1.4},
\eqref{1.5} are positive, then
$c_{\lambda,\delta}<\frac{1}{N}S_{F}^{N/p}$. The same result
holds if $q=p,0\leq \mu\leq \frac{N^{p-1}(N-p^{2})}{p^p}$ and
$\lambda,\delta\in(0, \frac{1}{p}\Lambda_1)$.
\end{lemma}

\begin{proof}
We can use the homogeneity of $F$ and $G$ to get, for any $t\geq0$,
$$
I_{\lambda,\delta}(tAu_{\varepsilon},tBu_{\varepsilon})=\frac{t^p}{p}
(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p-\frac{t^{p^{*}}}{p^{*}}F(A,B)-t^{q}
G_{\lambda,\delta}(A,B)|u_{\varepsilon}|_{q}^{q}.
$$
We shall denote the right-hand side of the
above equality by $h(t)$ and consider two distinct cases.

\noindent\textbf{Case 1:} $\bar{p}<q <p^{*}$ with
$\bar{p}=\max\Big\{p,\frac{N}{b(\mu)},\frac{p(2N-pb(\mu)-p)}{N-p}\Big\}$.
From the fact that $\lim_{t\to+\infty}h(t)=-\infty$
and $h(t)>0$ when $t$ is close to $0$, there exists
$t_{\varepsilon}>0$ such that
\begin{equation}\label{2.15}
h(t_{\varepsilon})=\max_{t\geq 0}h(t).
\end{equation}
Let
$$
g(t)=\frac{t^p}{p}
(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p-\frac{t^{p^{*}}}{p^{*}}F(A,B),
\quad t\geq 0,
$$
and notice that the maximum value of $g(t)$ occurs at the point
$$
\tilde{t}_{\varepsilon}
=\Big(\frac{(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p}{F(A,B)}\Big)
^{\frac{1}{p^{*}-p}}.
$$
So, for each $t\geq 0$, $$ g(t)\leq g(\tilde{t}_{\varepsilon})
=\frac{1}{N}\Big(\frac{(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p}{(F(A,B))^{p/p^*}}\Big)
^{N/p},
$$
and therefore
\begin{equation}\label{2.16}
h(t_{\varepsilon})\leq\frac{1}{N}\Big(\frac{(A^p+B^p)
\|u_{\varepsilon}\|_{\mu}^p}{(F(A,B))^{p/p^*}}\Big)
^{N/p}-t_{\varepsilon}^{q}G_{\lambda,\delta}(A,B)|u_{\varepsilon}|_{q}^{q}.
\end{equation}
We claim that, for some $C_2>0$, there holds
$$
t_{\varepsilon}^{q}G_{\lambda,\delta}(A,B)\geq C_2.
$$
Indeed, if this is not the case, we have that $t_{\varepsilon_{m}}
\to 0$ for some sequence $\varepsilon_{m}\to 0^{+}$,
then
$$
0<c_{\lambda,\delta}\leq \sup_{t\geq
0}I_{\lambda,\delta}(tAu_{\varepsilon_{m}},tBu_{\varepsilon_{m}})
=I_{\lambda,\delta}(t_{\varepsilon_{m}}Au_{\varepsilon_{m}},t_{\varepsilon_{m}}Bu_{\varepsilon_{m}})
\to 0,
$$
which is a contradiction. So, the claim holds, and we infer from
\eqref{2.16}, \eqref{2.10}, \eqref{2.11} and \eqref{2.12} that
\begin{equation}\label{2.17}
\begin{aligned}
h(t_{\varepsilon})
&\leq \frac{1}{N}\Big(\frac{A^p+B^p
}{(F(A,B))^{p/p^*}}\Big(S_{\mu}+O(\varepsilon^{pb(\mu)+p-N})\Big)\Big)
^{N/p}-C_2|u_{\varepsilon}|_{q}^{q}
   \\
&\leq\frac{1}{N}S_{F}^{N/p}+O(\varepsilon^{pb(\mu)+p-N})
-C_2|u_{\varepsilon}|_{q}^{q}
   \\
&\leq \frac{1}{N}S_{F}^{N/p}+O(\varepsilon^{pb(\mu)+p-N})-
O(\varepsilon^{N-\frac{N-p}{p}q}).
\end{aligned}
\end{equation}
By $\bar{p}<q<p^{*}$, we obtain $pb(\mu)+p-N>N-\frac{N-p}{p}q$.
Thus, from the above inequality we conclude that, for each
$\varepsilon>0$ small, there holds
$$
c_{\lambda,\delta}\leq \sup_{t\geq
0}I_{\lambda,\delta}(tAu_{\varepsilon},tBu_{\varepsilon})
=h(t_{\varepsilon})<\frac{1}{N}S_{F}^{N/p}.
$$

\noindent\textbf{Case 2:} $q=p$ and $0\leq \mu\leq
\frac{N^{p-1}(N-p^{2})}{p^p}$.
In this case, we have that $h'(t)= 0$ if and only if
$$
(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p-pG_{\lambda,\delta}(A,B)|u_{\varepsilon}|_p^p
=t^{p^{*}-p}F(A,B).
$$
Since we suppose $\lambda<\frac{1}{p}\Lambda_1$, we can use
Poincar\'{e} inequality to obtain
\begin{align*}
pG_{\lambda,\delta}(A,B)|u_{\varepsilon}|_p^p
&\leq p\lambda(A^p+B^p)|u_{\varepsilon}|_p^p
   \\
&<\Lambda_1(A^p+B^p)|u_{\varepsilon}|_p^p
 \\
&\leq(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p.
\end{align*}
Thus, there exists $t_{\varepsilon}>0$ satisfying \eqref{2.15}.

Arguing, as in the first case, we conclude that, from \eqref{2.17},
for $\varepsilon>0$ small, there holds
\begin{align*}
h(t_{\varepsilon}) &\leq
\frac{1}{N}S_{F}^{N/p}+O(\varepsilon^{pb(\mu)+p-N})
-C_2|u_{\varepsilon}|_p^p\\
&=\begin{cases}
       \frac{1}{N}S_{F}^{N/p}+O(\varepsilon^{pb(\mu)+p-N})
       -O(\varepsilon^p|\ln\varepsilon|), & b(\mu)=\frac{N}{p},\\
       \frac{1}{N}S_{F}^{N/p}+O(\varepsilon^{pb(\mu)+p-N})
       -O(\varepsilon^p), & b(\mu)>\frac{N}{p}.
    \end{cases}
\end{align*}
If $b(\mu)=N/p$, then $pb(\mu)+p-N=p$, so
$\varepsilon^{pb(\mu)+p-N}=o(\varepsilon^p|\ln\varepsilon|)$. If
$b(\mu)>N/p$, then $pb(\mu)+p-N>p$, so
$\varepsilon^{pb(\mu)+p-N}=o(\varepsilon^p)$. Choosing
$\varepsilon>0$ small enough, we have
$$
c_{\lambda,\delta}\leq \sup_{t\geq
0}I_{\lambda,\delta}(tAu_{\varepsilon},tBu_{\varepsilon})
=h(t_{\varepsilon})<\frac{1}{N}S_{F}^{N/p}.
$$
On the other hand, it is easy to verify that the function
$$
g(t)=(p-1)t^p-(N-p)t^{p-1}+\mu, \quad t\geq 0
$$ has the only minimal point
$\bar{t}=\frac{N-p}{p}$ and is increasing on the interval
$(\bar{t},+\infty)$. Thus, for $N\geq p^{2}$ we deduce that
$$
\frac{N}{p}\leq b(\mu)\Leftrightarrow g(\frac{N}{p})\leq
g(b(\mu))=0\Leftrightarrow 0\leq \mu\leq
\frac{N^{p-1}(N-p^{2})}{p^p}.
$$
This concludes the proof.
\end{proof}

Using Lemmas \ref{lem2.1} and \ref{lem2.2}, we can prove our first
result.

\begin{proof}[Proof of Theorem \ref{thm1}]
Since $I_{\lambda,\delta}$ satisfies the geometric conditions of the
mountain pass theorem, there exists $\{(u_{m},v_{m})\}\subset E$
such that $ I_{\lambda,\delta}(u_{m},v_{m})\to
c_{\lambda,\delta}$,
and $I'_{\lambda,\delta}(u_{m},v_{m})\to 0$.
It follows from Lemmas \ref{lem2.1} and \ref{lem2.2} that
$\{(u_{m},v_{m})\}$ converges, along a subsequence, to a nonzero
critical point $(u,v) \in E$ of $I_{\lambda,\delta}$. If we then
denote, by $u^{-}= \max\{-u,0\}$ and $v^{-}= \max\{-v,0\}$, the
negative part of $u$ and $v$, respectively, we obtain
\begin{align*}
0&= \langle I'_{\lambda,\delta}(u,v),(u^{-},v^{-})\rangle\\
&=-\|(u^{-},v^{-})\|_{E}^p-
\frac{1}{p^{*}}\int_{\Omega}(F_{u}(u,v)u^{-}+F_{v}(u,v)v^{-})dx\\
&\quad -\int_{\Omega}(G_{u}(u,v)u^{-}+G_{v}(u,v)v^{-})dx\\
&\leq-\|(u^{-},v^{-})\|_{E}^p.
\end{align*}
It thus follows that $(u^{-},v^{-})=(0,0)$. Hence, $u, v\geq 0$ in
$\Omega$. The theorem \ref{thm1} is thus proven.
\end{proof}

We finalize this section with the study of the asymptotic behavior
of the minimax level $c_{\lambda,\delta}$ as both the parameters
$\lambda,\delta$ approach zero.

\begin{lemma}\label{lem2.3}
 $\lim_{\lambda,\delta\to 0^{+}}
c_{\lambda,\delta}=c_{0,0}=\frac{1}{N}S_{F}^{N/p}$.
 \end{lemma}

 \begin{proof}
We first prove the second equality. It follows from
$\lambda=\delta= 0$ that $G_{0,0}\equiv 0$. If
 $A, B, u_{\varepsilon}, g_{\varepsilon}$ and $t_{\varepsilon}$
are the same as those in the
proof of Lemma \ref{lem2.2}, we have that
$(t_{\varepsilon}Au_{\varepsilon},t_{\varepsilon}Bu_{\varepsilon})
\in \mathcal{N}_{0,0}$. Thus
\begin{align*}
c_{0,0}&\leq I_{0,0}
(t_{\varepsilon}Au_{\varepsilon},t_{\varepsilon}Bu_{\varepsilon})\\
&=\frac{1}{N}\Big(\frac{A^p+B^p}{(F(A,B))^{p/p^*}}
\|u_{\varepsilon}\|_{\mu}^p\Big)^{N/p}
\\
&= \frac{1}{N}\Big(\frac{A^p+B^p}{(F(A,B))^{p/p^*}}
\Big(S_{\mu}+O\big(\varepsilon^{pb(\mu)+p-N}\big)\Big)\Big)^{N/p}.
\end{align*}
Taking the limit as $\varepsilon\to 0^{+}$ and using
\eqref{2.10}, we conclude that
$c_{0,0}\leq\frac{1}{N}S_{F}^{N/p}$.

In order to obtain the reverse inequality, we consider
$\{(u_{m},v_{m})\}\subset E$ such that
$I_{0,0}(u_{m},v_{m})\to c_{0,0}$ and
$I'_{0,0}(u_{m},v_{m})\to 0$. It is easy to show that the
sequence $\{(u_{m},v_{m})\}$ is bounded in $E$ and therefore
\[
\langle I'_{0,0}(u_{m},v_{m}),(u_{m},v_{m})\rangle
=\|(u_{m},v_{m})\|_{E}^p-\int_{\Omega}F(u_{m},v_{m})dx= o_{m}(1).
\]
It follows that
$$
\lim_{m\to\infty}\|(u_{m},v_{m})\|_{E}^p=l=
\lim_{m\to\infty}\int_{\Omega}F(u_{m},v_{m})dx.
$$
Taking the limit in the inequality
$S_{F}(\int_{\Omega}F(u_{m},v_{m})dx)^{p/p^*}\leq
\|(u_{m},v_{m})\|_{E}^p$, we conclude, as in the proof of Lemma
\ref{lem2.1}, that $Nc_{0,0}=l\geq S_{F}^{N/p}$. Hence,
\begin{align*}
c_{0,0}=\lim_{m\to\infty}I_{0,0}(u_{m},v_{m})
&=\lim_{m\to\infty} \Big(\frac{1}{p}\|(u_{m},v_{m})\|_{E}^p
-\frac{1}{p^{*}}\int_{\Omega}F(u_{m},v_{m})dx\Big)
\\
&=\frac{1}{N}l\geq \frac{1}{N}S_{F}^{N/p},
\end{align*}
and therefore $c_{0,0}=\frac{1}{N}S_{F}^{N/p}$.

We proceed now to the calculation of
$\lim_{\lambda,\delta\to 0^{+}} c_{\lambda,\delta}$.
Let $\{\lambda_{m}\},\{\delta_{m}\}\subset \mathbb{R}^{+}$ such that
$\lambda_{m},\delta_{m}\to 0^{+}$. Since $\delta_{m}$,
defined in \eqref{1.5}, is positive, we have that
$G_{\lambda_{m},\delta_{m}}(u,v)\geq 0$ whenever $(u,v)$ is
nonnegative. Thus, for this kind of function, we have that
$I_{\lambda_{m},\delta_{m}}(u,v)\leq I_{0,0}(u,v)$. It follows that
\begin{align*}
c_{\lambda_{m},\delta_{m}}
&=\inf_{(u,v)\neq (0,0)}  \max_{t\geq 0}I_{\lambda_{m},\delta_{m}}(t(u,v))
\\
&\leq \inf_{(u,v)\neq (0,0),\, (u,v)\geq 0}
 \max_{t\geq 0}I_{\lambda_{m},\delta_{m}}(t(u,v))
 \\
&\leq \inf_{(u,v)\neq (0,0),\, (u,v)\geq 0}
 \max_{t\geq 0}I_{0,0}(t(u,v))=c_{0,0}.
\end{align*}
In the last equality above, we used the infimum $c_{0,0}$, which can
be attained at a nonnegative solution. The above inequality implies
that
\begin{equation}
\limsup_{m\to\infty}c_{\lambda_{m},\delta_{m}}\leq c_{0,0}.
\label{2.18}
\end{equation}

On the other hand, it follows from Theorem \ref{thm1} that there
exists $\{(u_{m},v_{m})\}\subset E$ such that
$$
I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})=c_{\lambda_{m},\delta_{m}},
\quad I'_{\lambda_{m},\delta_{m}}(u_{m},v_{m})\to 0.
$$
Since $c_{\lambda_{m},\delta_{m}}$ is bounded, the same argument
performed in the proof of Lemma \ref{lem2.1} implies that
$\{(u_{m},v_{m})\}$ is bounded in $E$. Since
$(u_{m},v_{m})\geq 0$,
we obtain $0\leq \int_{\Omega}
G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx\leq
\lambda_{m}\int_{\Omega}(|u_{m}|^{q} + |v_{m}|^{q})dx$, from which
it follows that
\begin{equation}
\lim_{m\to\infty}\int_{\Omega}
G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx=0. \label{2.19}
\end{equation}

Let $t_{m}>0$ be such that
$t_{m}(u_{m},v_{m})\in \mathcal{N}_{0,0}$.
Since $(u_{m},v_{m})\in \mathcal{N}_{\lambda_{m},\delta_{m}}$, we have that
\begin{align*} c_{0,0}
&\leq I_{0,0}(t_{m}(u_{m},v_{m}))\\
&=I_{\lambda_{m},\delta_{m}}(t_{m}(u_{m},v_{m}))+t_{m}^{q}\int_{\Omega}
G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx
 \\
 &\leq I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})+t_{m}^{q}\int_{\Omega}
G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx
\\
 &=c_{\lambda_{m},\delta_{m}}+t_{m}^{q}\int_{\Omega}
G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx.
\end{align*}
If $\{t_{m}\}$ is bounded, we can use the above estimate and
\eqref{2.19} to obtain
$$
c_{0,0}\leq \liminf_{m\to\infty}c_{\lambda_{m},\delta_{m}}.
$$
Using this and \eqref{2.18}, we obtain
$$
c_{0,0}\leq
\liminf_{m\to\infty}c_{\lambda_{m},\delta_{m}}\leq
\limsup_{m\to\infty}c_{\lambda_{m},\delta_{m}}\leq c_{0,0}.
$$
Thus, $c_{0,0}=\lim_{m\to\infty }c_{\lambda_{m},\delta_{m}}$.

It remains to check that $\{t_{m}\}$ is bounded. A straightforward
calculation shows that
\begin{equation}
t_{m}=\Big(\frac{\|(u_{m},v_{m})\|_{E}^p}{\int_{\Omega}F(u_{m},v_{m})dx}
\Big)^{\frac{1}{p^{*}-p}}. \label{2.20}
\end{equation}
Since $(u_{m},v_{m})\in \mathcal{N}_{¦Ë\lambda_{m},\delta_{m}}$, we
obtain
\begin{align*}
&\|(u_{m},v_{m})\|_{E}^p\\
&=\int_{\Omega}F(u_{m},v_{m})dx+q\int_{\Omega}
G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx\leq
S_{F}^{-\frac{p^{*}}{p}}\|(u_{m},v_{m})\|_{E}^{p^{*}}+o_{m}(1).
\end{align*}
Hence $\|(u_{m},v_{m})\|_{E}^p\geq C_3>0$, and therefore from
the above expression, it follows that
$\int_{\Omega}F(u_{m},v_{m})dx\geq C_{4}>0$. Thus, the boundedness
of $\{(u_{m},v_{m})\}$ and \eqref{2.20} imply that $\{t_{m}\}$ is
bounded. This completes the proof.
\end{proof}

\section{Some technical lemmas}

In this section, we will recall and prove some lemmas which are
crucial in the proof of Theorem \ref{thm2}. The first lemma is
standard, and its proof follows adapting arguments found in
\cite{27}.

\begin{lemma}\label{lem3.1}
Let $\{(u_{m},v_{m})\}\subset E$ such that
$\int_{\Omega}F(u_{m},v_{m})dx=1$ and
\[
\lim_{m\to\infty}\|(u_{m},v_{m})\|_{E}^p=S_{F}.
\]
 Then there exist $\{r_{m}\}\subset(0,+\infty)$ and $\{y_{m}\}\subset
\mathbb{R}^N$ such that
\begin{equation}\label{3.1}
\omega_{m}(x)=(\omega^{1}_{m}(x),\omega^{2}_{m}(x))=r_{m}^{\frac{N-p}{p}}
(u_{m}(r_{m}x+y_{m}),v_{m}(r_{m}x+y_{m}))
\end{equation}
contains a convergent subsequence, denoted again by
$\{\omega_{m}\}$, such that $\omega_{m}\to \omega$ in
$\mathcal{D}^{1,p}(\mathbb{R}^N)\times
\mathcal{D}^{1,p}(\mathbb{R}^N)$. Moreover, as
$m\to\infty$, we have  $r_{m}\to0$ and $y_{m}\to y\in
\overline{\Omega}$.
 \end{lemma}

Up to translations, we may assume that $0\in\Omega$. Since $\Omega$
is a smooth bounded domain of $\mathbb{R}^N$, we can choose $r>0$
small enough such that $B_{r}=B_{r}(0)=\{x\in
\mathbb{R}^N:d(x,0)<r\}\subset \Omega$ and the sets
$$
\Omega_{r}^{+}=\{x\in \mathbb{R}^N:\operatorname{dist}(x,\Omega)<r\}, \quad
\Omega_{r}^{-}=\{x\in \mathbb{R}^N:\operatorname{dist}(x,\partial\Omega)>r\}
$$
are homotopically equivalent to $\Omega$. Let
\begin{gather*}
W_{0,rad}^{1,p}(B_{r})=\big\{u\in W_0^{1,p}(B_{r}):u \text{ is radial}\big\},\\
E_{\rm rad}(B_{r})=W_{0,{\rm rad}}^{1,p}(B_{r})\times W_{0,{\rm rad}}^{1,p}(B_{r}).
\end{gather*}
We thus define the functional
\begin{align*}
I_{B_{r}}(u,v)
&=\frac{1}{p}\int_{B_{r}}(|\nabla
u|^p-\mu\frac{|u|^p}{|x|^p}+|\nabla
v|^p-\mu\frac{|v|^p}{|x|^p})dx\\
&\quad -\frac{1}{p^{*}}\int_{B_{r}}
F(u,v)dx-\int_{B_{r}}G_{\lambda,\delta}(u,v)dx
\end{align*}
for $(u,v)\in E_{\rm rad}(B_{r})$, and set
$$
m_{\lambda,\delta}=\inf_{(u,v)\in
\mathcal{N}^{B_{r}}_{\lambda,\delta}}I_{B_{r}}(u,v),
$$
where
$$
\mathcal{N}^{B_{r}}_{\lambda,\delta}:=\{(u,v)\in
E_{\rm rad}(B_{r})\setminus \{(0,0)\}:\langle
I'_{B_{r}}(u,v),(u,v)\rangle=0\}.
$$
Clearly, $m_{\lambda,\delta}$ is
nonincreasing in $\lambda,\delta$. Note that $m_{\lambda,\delta}>0$
for all $\lambda,\delta>0$.

Arguing, as in the proof of Lemma \ref{lem2.3} and Theorem
\ref{thm1}, we obtain the following result.

\begin{lemma}\label{lem3.2}
Suppose {\rm (F0)-(F3), (G0)--(G1)} are satisfied.
Then the infimum $m_{\lambda,\delta}$ is attained
by a positive radial function
$(u_{\lambda,\delta},v_{\lambda,\delta})\in E_{\rm rad}$ whenever
$\bar{p}<q < p^{*}$ with
$\bar{p}=\max\Big\{p,\frac{N}{b(\mu)},\frac{p(2N-pb(\mu)-p)}{N-p}\Big\}$,
$0\leq\mu<\bar{\mu}$ and $\lambda,\delta>0$, or $q=p,0\leq \mu\leq
\frac{N^{p-1}(N-p^{2})}{p^p}$ and $\lambda,\delta \in(0,
\frac{1}{p}\Lambda_{1,rad})$, and where $\Lambda_{1,rad}> 0$ is the
first eigenvalue of the operator $(-\Delta _pu,
W^{1,p}_{0,rad}(B_{r}))$. Moreover,
$$
m_{\lambda,\delta}<\frac{1}{N}S_{F}^{N/p}, \quad
\lim_{\lambda,\delta\to 0^{+}}m_{\lambda,\delta}=\frac{1}{N}S_{F}^{N/p}.
$$
 \end{lemma}

We introduce the barycenter map
$\beta:\mathcal{N}_{\lambda,\delta}\to \mathbb{R}^N$ as
$$
\beta(u,v)=S_{F}^{-N/p}\int_{\Omega}F(u,v)x\,dx.
$$
This map has the following property.

\begin{lemma}\label{lem3.3}
If {\rm (F0)--(F3), (G0)--(G1)}, then there
exists $\lambda^{*}>0$ such that $\beta(u,v)\in \Omega_{ r}^{+}$
whenever $(u,v)\in \mathcal{N}_{\lambda,\delta},\lambda,\delta\in(0,\lambda^{*})$
and $I_{\lambda,\delta}(u,v)\leq m_{\lambda,\delta}$.
\end{lemma}

\begin{proof}
Arguing by contradiction, we suppose that there exist
$\{\lambda_{m}\},\{\delta_{m}\}\subset \mathbb{R}^{+}$ and
$\{(u_{m},v_{m})\}\subset \mathcal{N}_{\lambda_{m},\delta_{m}}$ such
that $\lambda_{m},\delta_{m}\to 0^{+}$ as
$m\to\infty, I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})\leq
m_{\lambda_{m},\delta_{m}}$, but
$\beta(u_{m},v_{m})\not\in\Omega_{r}^{+}$.

From $\{(u_{m},v_{m})\}\subset \mathcal{N}_{\lambda_{m},\delta_{m}}$
and $I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})\leq
m_{\lambda_{m},\delta_{m}}$, it follows that  $\{(u_{m},v_{m})\}$ is
bounded in $E$. Moreover,
\begin{align*}
0&=\langle I'_{\lambda_{m},\delta_{m}}(u_{m},v_{m}),(u_{m},v_{m})\rangle\\
&= \|(u_{m},v_{m})\|_{E}^p-\int_{\Omega}F(u_{m},v_{m})dx
-q\int_{\Omega}G_{\lambda_{m},\delta_{m}} (u_{m},v_{m})dx.
\end{align*}
Since $\lambda_{m}\to 0$, we can use the boundedness of
$\{(u_{m},v_{m})\}$ to get
$$
0\leq \int_{\Omega}G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx\leq
\lambda_{m}\int_{\Omega}(|u_{m}|^{q}+|v_{m}|^{q})dx\to 0,
$$
from which it follows that
$$
\lim_{m\to\infty}\|(u_{m},v_{m})\|_{E}^p
= \lim_{m\to\infty}\int_{\Omega}F(u_{m},v_{m})dx=k\geq 0.
$$
Notice that
\begin{align*}
c_{\lambda_{m},\delta_{m}}
&\leq I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})\\
&= \frac{1}{p}\|(u_{m},v_{m})\|_{E}^p-\frac{1}{p^{*}}
 \int_{\Omega}F(u_{m},v_{m})dx-
\int_{\Omega}G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx
\\
&\leq m_{\lambda_{m},\delta_{m}}.
\end{align*}
Recalling that $c_{\lambda_{m},\delta_{m}}$ and
$m_{\lambda_{m},\delta_{m}}$ both converge to
$\frac{1}{N}S_{F}^{N/p}$, we can use the above expression
and
$\int_{\Omega}G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx\to
0$ again to conclude that $k=S_{F}^{N/p}$. That is,
\begin{equation}
\lim_{m\to\infty}\|(u_{m},v_{m})\|_{E}^p=S_{F}^{N/p}=
\lim_{m\to\infty}\int_{\Omega}F(u_{m},v_{m})dx.\label{3.2}
\end{equation}
Let $t_{m}=(\int_{\Omega}F(u_{m},v_{m})dx)^{-1/p^*}>0$ and
notice that $t_{m}(u_{m},v_{m})$ satisfies the hypotheses of Lemma
\ref{lem3.1}. Using Lemma \ref{lem3.1}, there exist sequences
$\{r_{m}\}\subset (0,+\infty)$ and $\{y_{m}\}\subset \mathbb{R}^N$
satisfying $r_{m}\to 0, y_{m}\to y \in
\overline{\Omega}$. We thus have that $\omega_{m} \to
\omega$ in $\mathcal{D}^{1,p}(\mathbb{R}^N)\times
\mathcal{D}^{1,p}(\mathbb{R}^N)$.

The definition of $\beta(u,v)$, \eqref{3.2}, the strong convergence
of $\{\omega_{m}\}$, and Lebesgue's Theorem provide
\begin{align*}
\beta(u_{m},v_{m})
 &=t_{m}^{-p^{*}}S_{F}^{-N/p}\int_{\Omega}F(t_{m}(u_{m},v_{m}))xdx
\\
 &=(1+o_{m}(1))\int_{\Omega}F(t_{m}u_{m},t_{m}v_{m})xdx
\\
 &=(1+o_{m}(1))\int_{\Omega}F(\omega_{m})(r_{m}x+y_{m})dx
 \\
 &=(1+o_{m}(1))\Big(\int_{\Omega}F(\omega)\bar{y}dx+o_{m}(1)\Big).
\end{align*}
Since $\bar{y}\in \overline{\Omega}$ and
$\int_{\Omega}F(\omega)dx=1$, the above expression implies that
$$
\lim_{m\to\infty}\operatorname{dist} \
(\beta(u_{m},v_{m}),\overline{\Omega})=0.
$$
Such contradicts $\beta(u_{m},v_{m})\not\in \Omega_{r}^{+}$.
 \end{proof}

According to Lemma \ref{lem3.2}, for each $\lambda,\delta>0$ small,
the infimum $m_{\lambda,\delta}$ is attained by a nonnegative radial
function
$\sigma_{\lambda,\delta}=(u_{\lambda,\delta},v_{\lambda,\delta})\in
\mathcal{N}^{B_{r}}_{\lambda,\delta}$. We consider
$$
I_{\lambda,\delta}^{m_{\lambda,\delta}}=\{(u,v)\in E:I(u,v)\leq
m_{\lambda,\delta}\}
$$
and define the function $\gamma:\Omega_{r}^{-}\to
I_{\lambda,\delta}^{m_{\lambda,\delta}}$ by setting, for each $y\in
\Omega_{r}^{-}$,
\begin{equation}
\gamma(y)=
\begin{cases}
       \sigma_{\lambda,\delta}(x-y),  &  \text{if }  x\in B_{r}(y),\\
       0, & \text{otherwise}.
    \end{cases}\label{3.3}
    \end{equation}
A change of variables and straightforward calculations show that the
map $\gamma$ is well defined. Since $\sigma_{\lambda,\delta}$ is
radial, we have that $\int_{B_{r}}
F(u_{\lambda,\delta},v_{\lambda,\delta})xdx=0$. Hence, for each
$y\in \Omega_{r}^{-}$, we obtain
\begin{align*} (\beta\circ\gamma)(y)
 &=S_{F}^{-N/p}\int_{\Omega}
 F(u_{\lambda,\delta}(x-y),v_{\lambda,\delta}(x-y))xdx
\\
 &=S_{F}^{-N/p}\int_{\Omega}
 F(u_{\lambda,\delta}(t),v_{\lambda,\delta}(t))(t+y)dt
\\
 &=S_{F}^{-N/p}\int_{\Omega}
 F(u_{\lambda,\delta}(t),v_{\lambda,\delta}(t))ydt
= y\alpha_{\lambda,\delta},
\end{align*}
where $\alpha_{\lambda,\delta}=S_{F}^{-N/p} \int_{\Omega}
F(u_{\lambda,\delta}(t),v_{\lambda,\delta}(t))dt$.

Along the way of proving Lemma \ref{lem3.3}, we can check easily the
following.

\begin{lemma}\label{lem3.4} If $\lambda,\delta\to 0^{+}$,
then $\alpha_{\lambda,\delta}\to 1$.
 \end{lemma}

 \begin{proof} By Lemma \ref{lem3.2}, we have
\begin{align*}
m_{\lambda,\delta}
&= \frac{1}{p}\int_{B_{r}}\Big(|\nabla
u_{\lambda,\delta}|^p+|\nabla
v_{\lambda,\delta}|^p-\mu\frac{|u_{\lambda,\delta}|^p+|v_{\lambda,\delta}|^p}{|x|^p}\Big)dx
\\
 &\quad -\frac{1}{p^{*}}\int_{B_{r}}F(u_{\lambda,\delta},
v_{\lambda,\delta})dx-
\int_{B_{r}}G_{\lambda,\delta}(u_{\lambda,\delta},
v_{\lambda,\delta})dx
\\
& < \frac{1}{N}S_{F}^{N/p}.
\end{align*}
As before, $\int_{B_{r}}G_{\lambda,\delta}(u_{\lambda,\delta},
v_{\lambda,\delta})dx\to 0$. Thus,
 $I'_{B_{r}}(u_{\lambda,\delta}, v_{\lambda,\delta})=0$,
and the above expression and the same
arguments used in the proof of Lemma \ref{lem3.2} imply that
$$
\int_{\Omega}F(u_{\lambda,\delta}, v_{\lambda,\delta})dx\to
S_{F}^{N/p}.
$$
The above equality and the definition of $\alpha_{\lambda,\delta}$
imply that $\alpha_{\lambda,\delta}\to 1$. The lemma is thus
proven.
\end{proof}

Next we define $H_{\lambda,\delta}:[0,1]\times
(\mathcal{N}_{\lambda,\delta}\cap
I_{\lambda,\delta}^{m_{\lambda,\delta}})\to \mathbb{R}^N$
by
$$
H_{\lambda,\delta}(t,(u,v))=\Big(t+\frac{1-t}{\alpha_{\lambda,\delta}}\Big)
\beta(u,v).
$$

\begin{lemma}\label{lem3.5}
Suppose {\rm (F0)--(F3),  (G0)--(G1)} are satisfied. There then
exists $\lambda^{**}>0$ such that
\begin{equation}
H_{\lambda,\delta}\big([0,1]\times (\mathcal{N}_{\lambda,\delta}\cap
I_{\lambda,\delta}^{m_{\lambda,\delta}})\big)\subset \Omega_{r}^{+}
\label{3.4}
    \end{equation}
for all $\lambda,\delta\in (0,\lambda^{**})$.
 \end{lemma}

 \begin{proof}
Arguing by contradiction, we suppose that there exist sequences
$\{\lambda_{m}\}$, $\{\delta_{m}\}\subset \mathbb{R}^{+}$ and
$t_{m}\in [0,1], (u_{m},v_{m})\in (\mathcal{N}_{\lambda,\delta}\cap
I_{\lambda,\delta}^{m_{\lambda,\delta}})$ such that
$\lambda_{m},\delta_{m}\to 0^{+}$ as $m\to\infty$
and
$H_{\lambda_{m},\delta_{m}}(t_{m},(u_{m},v_{m}))\not\in\Omega_{r}^{+}$
for all $m$, up to a subsequence $t_{m}\to t_0\in[0,1]$.
Moreover, the compactness of $\overline{\Omega}$ and Lemma
\ref{lem3.3} imply that, up to a subsequence,
$\beta(u_{m},v_{m})\to y\in
 \overline{\Omega}$. From Lemma \ref{lem3.4}
$\alpha_{\lambda_{m},\delta_{m}}\to 1$, so we can use the
definition of $H_{\lambda,\delta}$ to conclude that
$H_{\lambda_{m},\delta_{m}}(t_{m},(u_{m},v_{m}))\to y\in
\overline{\Omega}$, which is a contradiction. The lemma is proven.
\end{proof}

\section{Proof of Theorem \ref{thm2}}

We begin with the following lemma.

\begin{lemma}\label{lem4.1}
If $(u,v)$ is a critical point of $I_{\lambda,\delta}$ on
$\mathcal{N}_{\lambda,\delta}$, then it is a critical point of
$I_{\lambda,\delta}$ in $E$.
 \end{lemma}

 \begin{proof}
 The proof is almost the same as \cite[Lemma 3.2]{22} and is thus omitted here.
\end{proof}

\begin{lemma}\label{lem4.2}
Suppose {(F0)--(F3), (G0)--(G1)} are satisfied.
Then any sequence $\{(u_{m},v_{m})\}\subset
\mathcal{N}_{\lambda,\delta}$ such that
$I_{\lambda,\delta}(u_{m},v_{m})\to c<\frac{1}{N}S_{F}^{N/p}$ and
$I'_{\lambda,\delta}(u_{m},v_{m})\to 0$ contains a
convergent subsequence for $\lambda,\delta>0$ if $q>p$ and
$\lambda,\delta\in (0,\lambda^{*})$ if $q=p$ for some small
$\lambda^{*}>0$.
 \end{lemma}

 \begin{proof}
By hypothesis, there exists a sequence $\theta_{m}\in \mathbb{R}$
such that
$\|I'_{\lambda,\delta}(u_{m},v_{m})
-\theta_{m}J'_{\lambda,\delta}(u_{m},v_{m})\|_{E}
\to 0$ as $m\to\infty$, where
\[
J_{\lambda,\delta}(u,v)=\int_{\Omega}(|\nabla
u|^p-\mu\frac{|u|^p}{|x|^p}+|\nabla
v|^p-\mu\frac{|v|^p}{|x|^p})dx-\int_{\Omega}F(u,v)dx
-q\int_{\Omega}G_{\lambda,\delta}(u,v)dx.
\]
Thus,
$$
I'_{\lambda,\delta}(u_{m},v_{m})
=\theta_{m}J'_{\lambda,\delta}(u_{m},v_{m})+o_{m}(1).
$$
Recall that for all $(u_{m},v_{m})\in
\mathcal{N}_{\lambda,\delta}$,
$$
\langle J'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle=
(p-p^{*})\int_{\Omega}F(u_{m},v_{m})dx+
(p-q)\int_{\Omega}G_{\lambda,\delta}(u_{m},v_{m})dx\leq 0.
$$
If $\langle
J'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle\to 0$,
we have
$$
\int_{\Omega}F(u_{m},v_{m})dx\to 0, \quad
\int_{\Omega}G_{\lambda,\delta}(u_{m},v_{m})dx\to 0.
$$
Consequently, $\|(u_{m},v_{m})\|_{E}\to 0$.
On the other hand, if $(u_{m},v_{m})\in \mathcal{N}_{\lambda,\delta}$,
it follows that
$$
1\leq C(\lambda\|(u_{m},v_{m})\|_{E}^{q-p}+\delta\|(u_{m},v_{m})\|_{E}^{q-p}+
\|(u_{m},v_{m})\|_{E}^{p^{*}-p})
$$
for some $C>0$. Hence we arrive at a contradiction if
$\lambda,\delta> 0$ and $q>p$ or $\lambda,\delta\in (0,\lambda^{*})$
for small $\lambda^{*}> 0$ when $q=p$. We may thus assume that
$\langle J'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle\to
\ell< 0$. Since
$\langle I'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle=0$, we
conclude that $\theta_{m}=0$ and, consequently,
$I'_{\lambda,\delta}(u_{m},v_{m})\to 0$. Using this fact, we
have
$$
I'_{\lambda,\delta}(u_{m},v_{m})\to
c<\frac{1}{N}S_{F}^{N/p} \quad \text{and} \quad
I'_{\lambda,\delta}(u_{m},v_{m})\to 0.
$$
By Lemma \ref{lem2.1} the proof is completed.
\end{proof}

Hereafter, we denote the restriction of $I_{\lambda,\delta}$ on
$\mathcal{N}_{\lambda,\delta}$ by
$I_{\mathcal{N}_{\lambda,\delta}}$.

\begin{lemma}\label{lem4.3}
If {\rm (F0)--(F3), (G0)--(G1)}  are satisfied.
Let $\Lambda=\min\{\lambda^{*},\lambda^{**}\}>0$,
$\lambda,\delta\in (0,\Lambda)$. Then
$\operatorname{cat}_{I_{\mathcal{N}_{\lambda,\delta}}^{m_{\lambda,\delta}}}
(I_{\mathcal{N}_{\lambda,\delta}}^{m_{\lambda,\delta}})\geq
\operatorname{cat}_{\Omega}(\Omega)$, where $\lambda^{*},\lambda^{**}$ are
given by Lemma \ref{lem3.3} and \ref{lem3.5}, respectively.
 \end{lemma}

 \begin{proof} Suppose that
 $I_{\mathcal{N}_{\lambda,\delta}}^{m_{\lambda,\delta}}
  =A_1\cup A_2\cup\cdots\cup A_{m}$,
  where $A_j, j=1,2,\cdots,m$, are closed and contractible sets in
$I_{\mathcal{N}_{\lambda,\delta}}^{m_{\lambda,\delta}}$, this means
that there exists
$h_j\in C([0,1]\times A_j,I_{\mathcal{N}_{\lambda,\delta}}
^{m_{\lambda,\delta}})$ such that
$$ h_j(0,z)=z, \quad  h_j(1,z)=\vartheta, \quad \text{for all }
z\in A_j,
$$
where $\vartheta\in A_j$ is fixed. Consider
$B_j=\gamma^{-1}(A_j)$, $1\leq j\leq m$. The sets $B_j$ are
closed and
$$
\Omega_{r}^{-}=B_1\cup B_2\cup\cdots\cup B_{m}.
$$
We define the deformation $g_j:[0,1]\times B_j$ by setting
$$
g_j(t,y)=H_{\lambda,\delta}(t,h_j(t,\gamma(y)))
$$
for $\lambda,\delta\in (0,\Lambda)$. Note that
$$
g_j(0,y)=H_{\lambda,\delta}(0,h_j(0,\gamma(y)))=
\frac{(\beta\circ\gamma)(y)}{\alpha_{\lambda,\delta}}
$$
implies
$$
g_j(0,y)=\frac{\alpha_{\lambda,\delta}y}{\alpha_{\lambda,\delta}}=y,
\quad  \text{for all }  y\in B_j,
$$
and $g_j(1,y)=H_{\lambda,\delta}(1,h_j(1,\gamma(y)))
 =\beta(h_j(1,\gamma(y)))$ implies
 $$
g_j(1,y)=\beta(\vartheta)\in \Omega_{r}^{+}.
 $$
 Thus $B_j$ are contractible in
$\Omega_{r}^{+}$. Hence $\operatorname{cat}_{\Omega}(\Omega)
=\operatorname{cat}_{\Omega_{r}^{+}}(\Omega_{r}^{+})\leq
m$.\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
Using Lemma \ref{lem2.1}, Lemma \ref{lem2.2}, and Lemma \ref{lem3.2}
we know that $c_{\lambda,\delta},
m_{\lambda,\delta}<\frac{1}{N}S_{F}^{N/p}$ for
$\lambda,\delta\in (0,\Lambda)$. Moreover, by Lemma \ref{lem4.2},
$I_{\mathcal{N}_{\lambda,\delta}}$ satisfies the $(PS)_{c}$
condition for all $c<\frac{1}{N}S_{F}^{N/p}$. Therefore, by
Lemma \ref{lem4.3}, a standard deformation argument implies that for
$\lambda,\delta\in (0,\Lambda)$, $I_{\mathcal{N}_{\lambda,\delta}}$
contains at least $\operatorname{cat}_{\Omega}(\Omega)$ critical points of
the restriction of $I_{\lambda,\delta}$ on
$\mathcal{N}_{\lambda,\delta}$. Now, Lemma \ref{lem4.1} implies that
$I_{\lambda,\delta}$ has at least $\operatorname{cat}_{\Omega}(\Omega)$
critical points, and therefore at least
$\operatorname{cat}_{\Omega}(\Omega)$ nontrivial solutions of \eqref{1.1}.
As Theorem \ref{thm1}, the obtained solutions are nonnegative in
$\Omega$. The proof is completed.
\end{proof}


\subsection*{Acknowledgments}
The author is indebted to the anonymous referee for the
valuable comments and suggestions. This work was supported by Natural Science
Foundation of Hubei Engineering University and partially by
grant D20122605 from the Educational Commission of Hubei Province of China.


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\end{document}