\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 84, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/84\hfil $p$-Kirchhoff equation] {Multiple solutions of a $p$-Kirchhoff equation with singular and critical nonlinearities} \author[Q. Li, Z. Yang, Z. Feng \hfil EJDE-2017/84\hfilneg] {Qin Li, Zuodong Yang, Zhaosheng Feng} \address{Qin Li \newline School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui 233030, China} \email{294973245@qq.com} \address{Zuodong Yang (corresponding author) \newline School of Teacher Education, Nanjing Normal University, Nanjing, Jiangsu 210097, China} \email{yangzuodong@njnu.edu.cn} \address{Zhaosheng Feng \newline Department of Mathematics, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA} \email{zhaosheng.feng@utrgv.edu} \thanks{Submitted January 6, 2016. Published March 27, 2017.} \subjclass[2010]{35J60, 35B09, 35J91} \keywords{Weak solution; $p$-Kirchhoff equation; $p$-Laplacian; \hfill\break\indent quasilinear elliptic equation; variational principle; concentration compactness principle} \begin{abstract} In this article, we explore the existence of multiple solutions for a $p$-Kirchhoff equation with the nonlinearity containing both singular and critical terms. By means of the concentration compactness principle and Ekeland's variational principle, we obtain two positive weak solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction}\label{sect1} Consider the $p$-Kirchhoff equation \begin{equation}\label{eq1.1} \begin{gathered} -M(\|u\|^p)\Delta_{p}u = \lambda u^{p^{*}-1}+\rho(x)u^{-\gamma}, \quad x\in \Omega,\\ u=0, \quad x\in \partial\Omega, \end{gathered} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $M(s)=a+bs^m$, $\triangle_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator with $1
0$ is a real parameter. Here, $\gamma\in(0,1)$ is a constant, $\rho(x): \Omega\to \mathbb{R}$ is a given non-negative function in $L^p(\Omega)$, and $p^{*}=Np/(N-p)$ is the critical Sobolev exponent. Problem \eqref{eq1.1} displays some meaningful features. It is nonlocal due to the presence of the Kirchhoff-type coefficient $M$ which makes the equation no longer a pointwise identity. Moreover, it involves singular and critical terms. To the best of our knowledge, not much has been known on the Kirchhoff nonlocal structure with the presence of singular and critical nonlinearities in quasilinear elliptic problems. Recently, considerable attention has been given to the existence of positive solutions by variational methods for the problem \cite{pa4,pa5,pa6,pa7}: \begin{equation}\label{eq1.2} \begin{gathered} -M \Big(\int_{\Omega}|\nabla u|^2dx \Big)\Delta u =f(x,u), \quad x\in \Omega,\\ u=0, \quad x\in \partial\Omega, \end{gathered} \end{equation} and the stationary analogue of the Kirchhoff equation \cite{pa1}: \begin{equation}\label{eq3} u_{tt}-M \Big(\int_{\Omega}|\nabla_{x} u|^2dx \Big)\Delta_{x} u =f(x,t), \end{equation} where $M(s)=a+bs$, $a>0$ and $b>0$. Equation \eqref{eq3} was proposed by Kirchhoff [1] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. Sun et al \cite{pa8, pa3} considered the existence of solutions to a related singular elliptic problem. By using the concentration compactness principle \cite{pa9} and Ekeland's variational principle \cite{pa10}, the existence of two positive weak solutions was presented when the parameter $\lambda$ is small enough. Since problem \eqref{eq1.1} contains a critical term, it becomes difficult for us to apply variational methods directly and does not have the compact embedding $W^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)$. It is also noted that the singular term leads to the non-differentiability of the associated functional $I_{\lambda}$ on $W_0^{1,p}(\Omega)$, so the critical point theory becomes invalid. Based on this fact, in this study we attempt to use the concentration compactness principle, Vitali's theorem as well as Ekeland's variational principle to explore the existence of multiple solutions of \eqref{eq1.1}. Following the traditional notation, we let $X=W_0^{1,p}(\Omega)$ be the standard Sobolev space endowed with the norm $$ \|u\|=\Big(\int_{\Omega}|\nabla u|^pdx \Big)^{1/p} $$ and $\|u\|_{\sigma}$ denotes the norm in $L^{\sigma}(\Omega)$ by $$ \|u\|_{\sigma}=\Big(\int_{\Omega}| u|^{\sigma}dx \Big)^{1/\sigma}. $$ Let $S$ be the best Sobolev constant as \begin{equation}\label{eq1.5} S=\inf \Big\{\frac{\|u\|^p}{\|u\|_{p^{*}}^p},\, u\in X\text{ and } u\neq 0 \Big\}. \end{equation} Then, the infimum is never achieved if $\Omega\neq \mathbb{R}^N$. For $u\in X$, we define $I_{\lambda}$: $X\to \mathbb{R}$ by $$ I_{\lambda}(u)=\frac{1}{p}\widehat{M}(\|u\|^p) -\frac{\lambda}{p^{*}}\int_{\Omega}|u|^{p^{*}}dx -\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|u|^{1-\gamma}dx, $$ where $\widehat{M}(s)=\int_0^{s}M(t)dt=as+\frac{b}{m+1}s^{m+1}$. By analyzing the associated minimization problems for the functional $I_{\lambda}$, one can study weak solutions for \eqref{eq1.1}. Note that if $u$ is a weak solution of \eqref{eq1.1}, then $u$ satisfies $$ M(\|u\|^p)\int_{\Omega}|\nabla u|^pdx-\lambda\int_{\Omega}|u|^{p^{*}}dx -\int_{\Omega}\rho(x)|u|^{1-\gamma}dx=0. $$ So we define a set $$ \Lambda=\Big\{ u\in X|M(\|u\|^p)\int_{\Omega}|\nabla u|^pdx-\lambda\int_{\Omega}|u|^{p^{*}}dx-\int_{\Omega}\rho(x)|u|^{1-\gamma}dx=0 \Big\}. $$ We consider $$ h_{u}(t)=\frac{1}{p}\widehat{M}(t^p\|u\|^p) -\frac{\lambda t^{p^{*}}}{p^{*}}\int_{\Omega}|u|^{p^{*}}dx -\frac{t^{1-\gamma}}{1-\gamma}\int_{\Omega}\rho(x)|u|^{1-\gamma}dx. $$ A straightforward calculation gives $$ h'_{u}(t)=M(t^p\|u\|^p)t^{p-1}||u||^p-\lambda t^{p^{*}-1}\int_{\Omega}|u|^{p^{*}}dx -t^{-\gamma}\int_{\Omega}\rho(x)|u|^{1-\gamma}dx $$ and \begin{align*} h''_{u}(t) &=a(p-1)t^{p-2}\|u\|^p+b[p(m+1)-1]t^{p(m+1)-2}\|u\|^{p(m+1)} \\ &\quad -\lambda (p^{*}-1)t^{p^{*}-2}\int_{\Omega}|u|^{p^{*}}dx+\gamma t^{-\gamma-1}\int_{\Omega}\rho(x)|u|^{1-\gamma}dx. \end{align*} So we have $$ h''_{u}(1)=a(p+\gamma-1)\|u\|^p+b[p(m+1)+\gamma-1]\|u\|^{p(m+1)} -\lambda(p^{*}+\gamma-1)\|u\|_{p^{*}}^{p^{*}}. $$ It is natural to split $\Lambda$ into three parts corresponding to the local minima, the local maxima and the point of inflection. Accordingly, we define \begin{gather*} \Lambda^{+}=\{u\in\Lambda|h''(u)(1)>0\},\\ \Lambda^{0}=\{u\in\Lambda|h''(u)(1)=0\},\\ \Lambda^{-}=\{u\in\Lambda|h''(u)(1)<0\}. \end{gather*} Throughout this paper, we make the following assumptions: \begin{itemize} \item[(A1)] $M(s)=a+bs^m$, where $a$, $b$, $m>0$; \item[(A2)] $1
0$ such that if $\|\rho\|_{p}\leq \Theta$, then $\Lambda^{-}\neq\emptyset$. \end{itemize} We summarize our main results as follows. \begin{theorem} \label{thm1.1} Assume that conditions {\rm (A1)--(A3)} hold. Then there exists $\lambda^{*}>0$ small enough such that for any $\lambda\in(0,\lambda^{*})$, there exist at least two weak positive solutions $u^1$, $u^2\in X$ to problem \eqref{eq1.1}. Moreover, $u^1$ is a local minimizer of $I_{\lambda}$ in $X$ with $I_{\lambda}(u^1)<0$, and $u^2$ is a minimizer of $I_{\lambda}$ on $\Lambda^{-}$ with $I_{\lambda}(u^2)\geq0$. \end{theorem} The remainder of this article is organized as follows. In Section 2, we present some preliminary results, and Section 3 is dedicated to the proof of main results. \section{Preliminaries} \begin{lemma}\label{lmm2.1} The energy functional $I_{\lambda}$ has a local minimum $c$ in $X$ with $c<0$. \end{lemma} \begin{proof} By H\"{o}lder's and Sobolev inequalities, there exist positive constants $C_0$ and $C_1$ such that \begin{equation}\label{eq2.1} \begin{aligned} \int_{\Omega}\rho(x)|u|^{1-\gamma}dx &\leq \|\rho\|_{p}||u||_{p^{*}}^{1-\gamma} |\Omega|^{\frac{(p-1)p^{*}-p(1-\gamma)}{pp^{*}}} \nonumber\\ &\leq C_0\|\rho\|_{p}\|u\|_{p^{*}}^{1-\gamma} \nonumber\\ &\leq C_1\|\rho\|_{p}\|u\|^{1-\gamma}. \end{aligned} \end{equation} From \eqref{eq1.5}, we have \begin{equation}\label{eq2.2} \int_{\Omega}|u|^{p^{*}}dx\leq S^{-\frac{p^{*}}{p}} \Big(\int_{\Omega}|\nabla u|^pdx \Big)^{p^*/p},\quad u\in X. \end{equation} Thus, it gives \begin{align*} I_{\lambda}(u) &\geq \frac{a}{p}\|u\|^p+\frac{b}{p(m+1)}\|u\|^{p(m+1)} -\frac{\lambda}{p^{*}}S^{-\frac{p^{*}}{p}}\|u\|^{p^{*}} -C_2\|\rho\|_{p}\|u\|^{1-\gamma}\\ &\geq \frac{2}{p}\sqrt{\frac{a b}{m+1}}\|u\|^{\frac{p(m+2)}{2}} -\frac{\lambda}{p^{*}}S^{-\frac{p^{*}}{p}}\|u\|^{p^{*}}-C_2\|\rho\|_{p}\|u\|^{1-\gamma}. \end{align*} Since $1-\gamma<\frac{p(m+2)}{2}
0$ such that for any $\lambda\in(0,\lambda_1)$,
there are $R$, $\xi>0$ satisfying $I_{\lambda}(u)\geq\xi$ for all
$u\in X$ with $\|u\|=R$ and $I_{\lambda}(u)$ is bounded from below
on $B_R=\{u\in X| \|u\|\leq R\}$. Then, $c=\inf_{u\in
B_R}I_{\lambda}(u)$ is well-defined for the fixed
$\lambda\in(0,\lambda_1)$. Since $0<1-\gamma<1$, we have
$I_{\lambda}(t\sigma)<0$ for all $\sigma\neq0$ and small $t>0$.
Thus, we arrive at $c=\inf_{u\in B_R}I_{\lambda}(u)<0$.
\end{proof}
\begin{lemma}\label{lmm2.2}
There exists $u^1\in B_R$ satisfying $I_{\lambda}(u^1)=c$.
\end{lemma}
\begin{proof}
From Lemma \ref{lmm2.1}, there exists a minimizing sequence
$\{u_k\}\subset B_R$ such that $I_{\lambda}(u_k)\to
c<0$. Since $I_{\lambda}(u_k)=I_{\lambda}(|u_k|)$, we can
assume $u_k\geq 0$. Due to $||u||\leq R$,
there exists a subsequence (still denoted by $\{u_k\}$)
satisfying
$$
u_k\rightharpoonup u^1 \quad\text{in } X.
$$
From \eqref{eq2.2} we know that $u_k$ is
bounded in $L^{p^{*}}(\Omega)$. Since $X$ is self-reflexive, and
$B_R$ is closed and convex, we see $u^1\in B_R$.
By the concentration-compactness principle \cite{pa9},
there exist non-negative bounded measures $\eta$ and $\mu$
such that
$$
|u_k|^{p^{*}}\rightharpoonup \eta \quad \text{and}\quad
|\nabla u_k|^p\rightharpoonup \mu
$$
weakly in the sense of measures.
Furthermore, there exists a countable index set $J$, a
collection of points $\{x_{j}\}_{j\in J}\subset \overline{\Omega}$
and two numbers $\mu_{j}$, $\eta_{j}>0$ such that
$$
\eta=|u^1|^{p^{*}}+\sum_{j\in J}\eta_{j}\delta_{x_{j}} \quad\text{and}\quad
\mu\geq|\nabla u^1|^p+\sum_{j\in J}\mu_{j}\delta_{x_{j}},
$$
where $\delta_{x_{j}}$ is the Dirac measure concentrated at
$x_{j}$, and $\eta_{j}$ and $\mu_{j}$ satisfy
$$
S\eta_{j}^{p/p^*}\leq\mu_{j}.
$$
Letting $k\to\infty$ leads to
\[
\int_{\Omega}|\nabla
u_k|^pdx\to \int_{\Omega}d\mu\geq\int_{\Omega}|\nabla
u^1|^pdx+\sum_{j\in J}S\eta_{j}^{p/p^*},
\]
and
\begin{equation}\label{eq2.3}
\int_{\Omega}|u_k|^{p^{*}}dx\to
\int_{\Omega}d\eta=\int_{\Omega}|u^1|^{p^{*}}dx+\sum_{j\in
J}\eta_{j}.
\end{equation}
By Vitali's theorem, we find
$$
\lim_{k\to\infty}\int_{\Omega}\rho(x)|u_k|^{1-\gamma}dx
=\int_{\Omega}\rho(x)|u^1|^{1-\gamma}dx.
$$
Then, we deduce that
\begin{align*}
c&= \lim_{k\to\infty} \Big\{\frac{1}{p}\widehat{M}(\|u_k\|^p)
-\frac{\lambda}{p^{*}}\int_{\Omega}|u_k|^{p^{*}}dx
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|u_k|^{1-\gamma}dx \Big\}\\
&= \lim_{k\to\infty} \Big\{\frac{a}{p}\|u_k\|^p+\frac{b}{p(m+1)}
\|u_k\|^{p(m+1)}-\frac{\lambda}{p^{*}}\int_{\Omega}|u_k|^{p^{*}}dx \\
&\quad -\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|u_k|^{1-\gamma}dx \Big\}\\
&\geq \frac{a}{p} \Big(\int_{\Omega}|\nabla u^1|^pdx
+\sum_{j\in J}S\eta_{j}^{p/p^*} \Big)
+\frac{b}{p(m+1)} \Big(\int_{\Omega} |\nabla u^1 |^pdx
+\sum_{j\in J}S\eta_{j}^{p/p^*}\Big)^{m+1}\\
&\quad -\frac{\lambda}{p^{*}} \Big( \int_{\Omega} |u^1 |^{p^{*}}dx
+\sum_{j\in J}\eta_{j} \Big)
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x) |u^1 |^{1-\gamma}dx\\
&\geq \frac{1}{p}\widehat{M} \big( \|u^1 \|^p \big)
-\frac{\lambda}{p^{*}}\int_{\Omega} |u^1 |^{p^{*}}dx
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x) | u^1 |^{1-\gamma}dx\\
&\quad +\frac{a}{p}\sum_{j\in J}S\eta_{j}^{p/p^*}
+\frac{b}{p(m+1)} \Big(\sum_{j\in J}S\eta_{j}^{p/p^*} \Big)^{m+1}
-\frac{\lambda}{p^{*}}\sum_{j\in J}\eta_{j}.
\end{align*}
That is,
$$
c\geq\frac{1}{p}\widehat{M}(\|u^1\|^p)
-\frac{\lambda}{p^{*}}\int_{\Omega}|u^1|^{p^{*}}dx
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|u^1|^{1-\gamma}dx.
$$
From the definition of $c$, it gives
$$
c\leq\frac{1}{p}\widehat{M}(\|u^1\|^p)
-\frac{\lambda}{p^{*}}\int_{\Omega}|u^1|^{p^{*}}dx
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|u^1|^{1-\gamma}dx.
$$
Thus, we have
$$
c=\frac{1}{p}\widehat{M}(\|u^1\|^p)
-\frac{\lambda}{p^{*}}\int_{\Omega}|u^1|^{p^{*}}dx
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|u^1|^{1-\gamma}dx.
$$
Suppose that $J\neq\emptyset$. By way of contradiction, from \eqref{eq2.3} we obtain
\begin{gather*}
\int_{\Omega}d\eta>\int_{\Omega} | u^1 |^{p^{*}}dx, \\
\sum_{j \in J}\eta_{j}=\int_{\Omega}d\eta-\int_{\Omega} | u^1 |^{p^{*}}dx>0.
\end{gather*}
On the other hand, one can find that
\begin{align*}
c&\leq \frac{1}{p}\widehat{M} (\|u^1\|^p )
-\frac{\lambda}{p^{*}}\int_{\Omega}|u^1|^{p^{*}}dx
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|u^1|^{1-\gamma}dx\\
&\leq c-\frac{a}{p}\sum_{j\in J}S\eta_{j}^{p/p^*}
-\frac{b}{p(m+1)} \Big(\sum_{j\in J}S\eta_{j}^{p/p^*} \Big)^{m+1}
+\frac{\lambda}{p^{*}}\sum_{j\in J}\eta_{j}\\
&\leq c-\frac{a}{p}\sum_{j\in J}S\eta_{j}^{p/p^*}
+\frac{\lambda}{p^{*}}\sum_{j\in J}\eta_{j}.
\end{align*}
If for all $j\in J$ and $0<\eta_{j}<1$, we have
$\eta_{j}^{p/p^*}>\eta_{j}$ and
\[
c\leq c-\frac{a}{p}\sum_{j\in J}S\eta_{j}^{p/p^*}
+\frac{\lambda}{p^{*}}\sum_{j\in J}\eta_{j}
\leq c+\Big(\frac{\lambda}{p^{*}}-\frac{a S}{p} \Big)
\sum_{j\in J}\eta_{j}.
\]
This yields a contradiction when $\lambda 1>1-\gamma>0$, we see that
$$
\|u\|\leq \Big\{\frac{C_1(p^{*}+\gamma-1)\|\rho\|_{p}}
{2\sqrt{ab(p^{*}-p)[p^{*}-p(m+1)]}} \Big\}^{\frac{2}{p(m+2)-2(1-\gamma)}}.
$$
This yields a contradiction if we choose
\begin{align*}
\lambda<\lambda_3
&=\frac{2\sqrt{ab(p+\gamma-1)[p(m+1)+\gamma-1]}S^{p^*/p}}{p^{*}+\gamma-1} \\
&\quad\times \Big\{\frac{2\sqrt{ab(p^{*}-p)[p^{*}-p(m+1)]}}{C_1(p^{*}
+\gamma-1)\|\rho\|_{p}} \Big\}^{\frac{2p^{*}-p(m+2)}{p(m+2)-2(1-\gamma)}}.
\end{align*}
Consequently, for all $\lambda\in(0,\lambda_3)$, it holds
$\Lambda^{0}=\{0\}$.
\end{proof}
\begin{lemma}\label{lmm2.5}
$\Lambda^{-}$ is closed in $X$.
\end{lemma}
\begin{proof} Let $\{u_{n}\}\subset \Lambda^{-}$ satisfy
$u_{n}\to u$ in $X$. There
exists a subsequence (still denoted by $\{u_{n}\})$ such that
$u_{n}\to u$ a.e. in $\Omega$, and
$\lim_{n\to\infty}\|u_{n}\|_{p^{*}}=\|u\|_{p^{*}}$. By the
definition of $\Lambda^{-}$, it gives
$$
a(p+\gamma-1)\|u_{n}\|^p+b[p(m+1)+\gamma-1]\|u_{n}\|^{p(m+1)}
-\lambda(p^{*}+\gamma-1)\|u_{n}\|_{p^{*}}^{p^{*}}<0.
$$
So we have
\begin{align*}
&\lim_{n\to\infty} \bigg\{a(p+\gamma-1)\int_{\Omega}|\nabla u_{n}|^pdx+b[p(m+1)+\gamma-1] \left(\int_{\Omega}
|\nabla u_{n}|^pdx\right)^{m+1}\\
&-\lambda(p^{*}+\gamma-1)\int_{\Omega}|
u_{n}|^{p^{*}}dx \bigg\} \leq 0.
\end{align*}
Clearly, we see that
$u\in\Lambda^{0}\cup\Lambda^{-}$. If $\Lambda^{-}$ is not closed,
then $u\in\Lambda^{0}$. By Lemma \ref{lmm2.4}, we obtain $u\equiv 0$.
On the other hand, for any $\{u_{n}\}\subset \Lambda^{-}$ we have
\begin{align*}
&\int_{\Omega}|u_{n}|^{p^{*}}dx \\
&> \frac{a(p+\gamma-1)}{\lambda(p^{*}+\gamma-1)}\int_{\Omega}|\nabla
u_{n}|^pdx+\frac{b[p(m+1)+\gamma-1]}{\lambda(p^{*}+\gamma-1)}
\Big(\int_{\Omega} |\nabla u_{n}|^pdx \Big)^{m+1}\\
&\geq \frac{2\sqrt{a b(p+\gamma-1)[p(m+1)+\gamma-1]}}{\lambda(p^{*}+\gamma-1)}
\Big(\int_{\Omega} |\nabla u_{n}|^pdx \Big)^{\frac{m+2}{2}}\\
&\geq \frac{2\sqrt{a b(p+\gamma-1)[p(m+1)+\gamma-1]}}
{\lambda(p^{*}+\gamma-1)}\Big[S \Big(\int_{\Omega}|u_{n}|^{p^{*}}dx \Big)^{p/p^*}
\Big]^{\frac{m+2}{2}}\\
&= \frac{2\sqrt{a b(p+\gamma-1)[p(m+1)+\gamma-1]}}{\lambda(p^{*}+\gamma-1)}
S^{\frac{m+2}{2}}\Big(\int_{\Omega}|u_{n}|^{p^{*}}dx \Big)^{\frac{p(m+2)}{2p^{*}}}.
\end{align*}
That is,
$$
\Big(\int_{\Omega}|u_{n}|^{p^{*}}dx\Big)^{1/p^*}
>\Big\{\frac{2\sqrt{a b(p+\gamma-1)[p(m+1)+\gamma-1]}}
{\lambda(p^{*}+\gamma-1)}S^{\frac{m+2}{2}} \Big\}^{\frac{2}{2p^{*}-p(m+2)}}.
$$
As $n\to\infty$, one can see that
\begin{equation}\label{eq2.7}
\|u\|_{p^{*}}
\geq \Big\{\frac{2\sqrt{a b(p+\gamma-1)[p(m+1)+\gamma-1]}}{\lambda(p^{*}
+\gamma-1)} \Big\}^{\frac{2}{2p^{*}-p(m+2)}}S^{\frac{m+2}{2p^{*}-p(m+2)}}>0.
\end{equation}
This yields a contradiction to the fact $u=0$. Thus, $u\in\Lambda^{-}$ and
$\Lambda^{-}$ is closed in $X$.
\end{proof}
\begin{lemma}\label{lmm2.6}
There exists $\lambda_{4}>0$ such that for any
$u\in\Lambda^{-}$ and any $\lambda\in(0,\lambda_{4})$,
$I_{\lambda}(u)\geq 0$ holds.
\end{lemma}
\begin{proof}
By contradiction, we suppose that there
exists $\widetilde{u}\in\Lambda^{-}$ satisfying
$I_{\lambda}(\widetilde{u})< 0$. That is,
$$
\frac{1}{p}\widehat{M}(\|\widetilde{u}\|^p)
-\frac{\lambda}{p^{*}}\int_{\Omega}|\widetilde{u}|^{p^{*}}dx
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|\widetilde{u}|^{1-\gamma}dx<0.
$$
Note that
$$
\frac{1}{p}\widehat{M}(\|\widetilde{u}\|^p)
>\frac{1}{p(m+1)}[a\|\widetilde{u}\|^p+b\|\widetilde{u}\|^{p(m+1)}]
=\frac{1}{p(m+1)}M(\|\widetilde{u}\|^p)\|\widetilde{u}\|^p.
$$
So we have
$$
\frac{1}{p(m+1)}M(\|\widetilde{u}\|^p)\|\widetilde{u}\|^p
-\frac{\lambda}{p^{*}}\int_{\Omega}|\widetilde{u}|^{p^{*}}dx
-\frac{1}{1-\gamma}\int_{\Omega}\rho(x)|\widetilde{u}|^{1-\gamma}dx<0
$$
and
$$
\lambda \big[\frac{1}{p(m+1)}-\frac{1}{p^{*}}\big]
\int_{\Omega}|\widetilde{u}|^{p^{*}}dx-
\big[\frac{1}{1-\gamma}-\frac{1}{p(m+1)} \big]
\int_{\Omega}\rho(x)|\widetilde{u}|^{1-\gamma}dx<0.
$$
By \eqref{eq2.1}, we obtain
$$
\lambda \big[\frac{1}{p(m+1)}-\frac{1}{p^{*}}\big]
\int_{\Omega}|\widetilde{u}|^{p^{*}}dx
<\big[\frac{1}{1-\gamma}-\frac{1}{p(m+1)} \big]
C_0\|\rho(x)\|_{p}\|\widetilde{u}\|_{p^{*}}^{1-\gamma}.
$$
This leads to
$$\|\widetilde{u}\|_{p^{*}}^{p^{*}+\gamma-1}<\frac{p^{*}[p(m+1)+\gamma-1]C_0\|\rho\|_{p}}{\lambda[p^{*}-p(m+1)](1-\gamma)}.$$
By choosing
\begin{align*}
\lambda_{4}
&= \Big\{\frac{2\sqrt{a b(p+\gamma-1)[p(m+1)+\gamma-1]}}
{p^{*}+\gamma-1}S^{\frac{m+2}{2}} \Big\}^{\frac{2(p^{*}+\gamma-1)}
{p(m+2)-2(1-\gamma)}}\\
&\quad\times \Big\{\frac{[p^{*}-p(m+1)](1-\gamma)}{p^{*}[p(m+1)+\gamma-1]
C_0\|\rho\|_{p}} \Big\}^{\frac{2p^{*}-p(m+2)}{p(m+2)-2(1-\gamma)}},
\end{align*}
for all $\lambda<\lambda_{4}$ we have
$$
\|u\|_{p^{*}}< \Big\{\frac{2\sqrt{a
b(p+\gamma-1)[p(m+1)+\gamma-1]}}{\lambda(p^{*}+\gamma-1)}
\Big\}^{\frac{2}{2p^{*}-p(m+2)}}S^{\frac{m+2}{2p^{*}-p(m+2)}}.
$$
This yields a contradiction to inequality \eqref{eq2.7}.
Hence, the proof of Lemma \ref{lmm2.6} is complete.
\end{proof}
\begin{lemma}\label{lmm2.7}
If $u\in\Lambda^{-}$, then there exist
an $\epsilon>0$ and a differentiable function $f=f(w)>0$, where $w\in
W_0^{1,p}(\Omega)$ and $\|w\|<\epsilon$ such that
$f(0)=1$ and $f(w)(u+w)\in\Lambda^{-}$ for all $w\in W_0^{1,p}(\Omega)$.
\end{lemma}
\begin{proof}
Define $F$: $\mathbb{R}\times
W_0^{1,p}(\Omega)\to \mathbb{R}$ by
\begin{align*}
F(t,w)
&= at^{p+\gamma-1}\int_{\Omega}|\nabla(u+w)|^pdx+bt^{p(m+1)+
\gamma-1}\left(\int_{\Omega}|\nabla(u+w)|^pdx \right)^{m+1}\\
&\quad -\lambda t^{p^{*}+\gamma-1}\int_{\Omega}|u+w|^{p^{*}}dx
-\int_{\Omega}\rho(x)|u+w|^{1-\gamma}dx.
\end{align*}
Since $u\in\Lambda^{-}\subset\Lambda$, we have $F(1,0)=0$, and
\begin{align*}
F_{t}(1,0)
&= a(p+\gamma-1)\|u\|^p+b[p(m+1)+\gamma-1]\|u\|^{p(m+1)}\\
&\quad -\lambda(p^{*}+\gamma-1)\|u\|_{p^{*}}^{p^{*}}
<0.
\end{align*}
According to the implicit function theorem at the point $(1,0)$,
there exist an $\overline{\epsilon}>0$ and a
continuous function $f=f(w)>0$, where $w\in W_0^{1,p}(\Omega)$ and
$\|w\|<\overline{\epsilon}$, such that
$$
f(0)=1\text{ and $f(w)(u+w)\in\Lambda$ for all }w\in W_0^{1,p}(\Omega).
$$
Clearly, we can take $\epsilon>0$
sufficiently small $(<\overline{\epsilon})$ satisfying
$$
f(w)(u+w)\in\Lambda^{-}\quad \forall w\in W_0^{1,p}(\Omega)
\text{ and } \\|w\|<\epsilon.
$$
\end{proof}
\begin{lemma}\label{lmm2.8}
For all $\lambda>0$, problem \eqref{eq1.1} has a weak solution $u^2$ in $X$.
\end{lemma}
\begin{proof}
From (A3) and Lemma \ref{lmm2.3}, we know
$\Lambda^{-}\neq\emptyset$ and
$c_{-}=\inf_{u\in\Lambda^{-}}I_{\lambda}(u)>-\infty$ is well-defined.
By Ekeland's variational principle, there exists a
minimizing sequence $\{v_k\}\subset\Lambda^{-}$ satisfying
$$
I_{\lambda}(v_k)