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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 84, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/84\hfil Existence of two positive solutions]
{Existence of two positive solutions for a singular Neumann problem}

\author[J.-F. Liao, J. Liu, C.-L. Tang, P. Zhang\hfil EJDE-2014/84\hfilneg]
{Jia-Feng Liao, Jiu Liu, Chun-Lei Tang, Peng Zhang}  % in alphabetical order

\address{Jia-Feng Liao \newline
School  of  Mathematics  and  Statistics, Southwest University,
Chongqing 400715, China. \newline
School of Mathematics and Computational Science,
 Zunyi Normal College, Zunyi 563002, China}
\email{liaojiafeng@163.com}

\address{Jiu Liu \newline
School  of  Mathematics  and  Statistics, Southwest University,
Chongqing 400715, \newline China}
\email{jiuliu2011@163.com}

\address{Chun-Lei Tang (corresponding author) \newline
School  of  Mathematics  and  Statistics, Southwest University,
Chongqing 400715, \newline China}
\email{tangcl@swu.edu.cn,  Tel +86 23 68253135, fax +86 23 68253135}

\address{Peng Zhang \newline
School of Mathematics and Computational Science,
 Zunyi Normal College, \newline Zunyi  563002,  China}
\email{gzzypd@sina.com}

\thanks{Submitted January 23, 2014. Published March 28, 2014.}
\subjclass[2000]{35B09, 35J20, 35J75}
\keywords{Neumann problem; singularity; positive solution;  Nehari method}

\begin{abstract}
 We obtain two positive solutions for Neumann boundary problems with
 singularity and subcritical term, by using the Nehari method.
\end{abstract}

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

\section{Introduction and main result}


 In this article, we consider the  Neumann problem
\begin{equation}\label{e1.1}
\begin{gathered}
-\Delta u+u=\lambda P(x)u^{p}+Q(x)u^{-\gamma}, \quad\text{in }\Omega,\\
u>0,\quad\text{in }\Omega,\\
\frac{\partial u}{\partial \nu}=0, \quad\text{on }\partial \Omega,
\end{gathered}
\end{equation}
where $\Omega\subset R^{N}$ $(N\geq 3)$ is a bounded domain with smooth
boundary $\partial \Omega$ and $\lambda$ is a positive parameter.
The exponent $p$ of the superlinear satisfies $1<p<2^{*}-1$, where
$2^{*}=\frac{2N}{N-2}$ is the critical Sobolev
exponent for the embedding of $H^{1}(\Omega)$ into $L^{q}(\Omega)$
for every $q\in [1,\frac{2N}{N-2}]$. The exponent $\gamma$ of the
singular term satisfies $0<\gamma<1$. The coefficient functions
$P\in L^{r_1}(\Omega),Q\in L^{r_2}(\Omega)$ are
nonzero and nonnegative, where $r_{1}>\frac{2^{*}}{2^{*}-p-1}$ and
$r_{2}>\frac{2^{*}}{2^{*}+\gamma-1}$ are two constants.

A function $u\in H^{1}(\Omega)$ is called a weak solution of problem \eqref{e1.1}
if $u(x)>0$ in $\Omega$ satisfies
\begin{equation}\label{e1.10}
\int_{\Omega}\big((\nabla u,\nabla \phi)+u\phi-\lambda
P(x)u^{p}\phi-Q(x)u^{-\gamma}\phi\big)dx=0,\quad \forall \phi\in
H^{1}(\Omega),
\end{equation}
where $H^{1}(\Omega)$ is a Sobolev space equipped
with the norm $\|u\|=[\int_{\Omega}(|\nabla u|^{2}+u^{2})dx]^{1/2}$.
This is the space we work on in this paper.

The  Dirichlet boundary value problem
\begin{equation}\label{e1.2}
\begin{gathered}
-\Delta u=u^{p}+\lambda u^{-\gamma}, \quad\text{in }\Omega,\\
u>0, \quad\text{in }\Omega,\\
u=0,  \quad\text{on }\partial \Omega,
\end{gathered}
\end{equation}
have been extensively studied in
\cite{MMC,CP,MAP,LM,AA,LZ,NSS,SL,SL08,SW,SWL01,WZZ,HY,Z,ZL,ZY}.
In particular, in \cite{CP}
it has been shown that problem \eqref{e1.2} possesses at least
one solution for $\lambda>0$ small enough, and has no solution when
$\lambda$ is large. This result has been extended in
\cite{MAP,NSS,SL,SL08,SW,SWL01,WZZ,HY,Z,ZL,ZY}.
When the
exponent satisfies $0<p<1$, similar results of \cite{CP}
have been obtained in \cite{LZ,SY,Z,ZL,ZY}.
Especially, Shi and Yao in \cite{SY}  studied the case where the
coefficient of the singular term changes sign.
Using sub-supersolution method, they proved that
problem \eqref{e1.2} has at least one solution for $\lambda$ large
enough and has no solution for $\lambda$ small enough. When the
exponent satisfies $1<p<2^{*}-1$, the multiplicity of positive
solutions has been considered in \cite{SWL01} and \cite{SL08}. They  obtained
two positive solutions for problem \eqref{e1.2} when $\lambda>0$ is small enough
by the Nehari manifold. When the
exponent is the critical exponent, the existence and the multiplicity of
solutions have been studied
in \cite{NSS,SL,SW,WZZ,HY}.


Recently, Chabrowski in \cite{JC}  studied the Neumann
problems with singular superlinear nonlinearities; that is,
\begin{gather*}
-\Delta u=P(x)u^{p}+\lambda Q(x)u^{-\gamma}, \quad\text{in }\Omega,\\
u>0,\quad\text{in }\Omega,\\
\frac{\partial u}{\partial \nu}=0, \quad\text{on }\partial \Omega,
\end{gather*}
where $P\in C(\overline{\Omega})$ changes sign on $\Omega$ and satisfies
$$
\int_{\Omega}P(x)dx<0,
$$
and $Q\in C(\overline{\Omega})$ with $Q>0$.
When $1<p<2^{*}-1$ and $0<\gamma<\min\{p-1,1\}$, he has obtained two positive
solutions for $\lambda>0$ small enough by approximation
and variational methods.

Inspired by \cite{SWL01} and \cite{JC}, we study problem \eqref{e1.1} with
$1<p<2^{*}-1$ and $0<\gamma<1$, and obtain two positive solutions
when $\lambda>0$ is small by the Nehari method. Moreover, we obtain uniform
lower bounds for $\lambda$, namely $T_{p,\gamma}$.

We denote by $|\cdot|_{q}$ the usual $L^{q}$-norm. Let $S$ be the
best Sobolev constant and $T_{p,\gamma}$ be a constant, respectively
\begin{gather}\label{e1.0}
S:=\inf\big\{\frac{\int_{\Omega}(|\nabla
u|^{2}+u^{2})dx}{(\int_{\Omega}|u|^{2^{*}}dx)^\frac{2}{2^{*}}}: u\in
H^{1}(\Omega), u\neq0\big\},
\\
T_{p,\gamma}=\frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}\big)^{\frac{p+\gamma}{1+\gamma}}
\frac{S^{\frac{p+\gamma}{1+\gamma}}}{|P|_{r_1}|Q|_{r_2}^{\frac{p-1}{1+\gamma}}}
|\Omega|^{-\frac{r_{1}r_{2}(p+\gamma)(2^{*}-2)-2^{*}
[r_{1}(p-1)+r_{2}(1-\gamma)]}{2^{*}r_{1}r_{2}(1+\gamma)}}. \nonumber
\end{gather}
For all $u\in H^{1}(\Omega)$, we define
\begin{equation*}
I_{\lambda}(u)
=\frac{1}{2}\int_{\Omega}(|\nabla u|^{2}+|u|^{2})dx
-\frac{\lambda}{p+1}\int_{\Omega}P(x)|u|^{p+1}dx
-\frac{1}{1-\gamma}\int_\Omega Q(x)|u|^{1-\gamma}dx.
\end{equation*}
It is well known that the singular term leads to the functional
$I_{\lambda}\not\in C^1(H^{1}(\Omega), R)$.
However, we may obtain the multiplicity of
solutions for problem \eqref{e1.1} by
investigating suitable minimization problems for the functional
$I_{\lambda}$.
Notice that $u$ is a weak solution of problem \eqref{e1.1}, then $u>0$
in $\Omega$ and satisfies the  equation
\begin{equation*}
\int_{\Omega}(|\nabla u|^{2}+u^{2})dx
-\lambda\int_{\Omega}P(x)u^{p+1}dx-\int_{\Omega}Q(x)u^{1-\gamma}dx=0.
\end{equation*}
So if such a solution exists then it must lie in Nehari manifold $\Lambda$,
which is defined by
\begin{equation*}
\Lambda=\big\{u\in H^{1}(\Omega): \int_{\Omega}(|\nabla u|^{2}+u^{2}
-\lambda P(x)|u|^{p+1}-Q(x)|u|^{1-\gamma})dx=0\big\}.
\end{equation*}
To obtain the multiplicity of positive solutions, we split
 $\Lambda=\Lambda^+\cup\Lambda^0\cup\Lambda^-$ where
\begin{gather*}
\Lambda^+=\big\{u\in \Lambda: (1+\gamma)
\int_{\Omega}(|\nabla u|^{2}+u^{2})dx-\lambda(p+\gamma)
\int_{\Omega}P(x)|u|^{p+1}dx>0\big\},
\\
\Lambda^0=\big\{u\in \Lambda: (1+\gamma)\int_{\Omega}(|\nabla u|^{2}+u^{2})dx
-\lambda(p+\gamma)\int_{\Omega}P(x)|u|^{p+1}dx=0\big\},
\\
\Lambda^-=\big\{u\in \Lambda: (1+\gamma)\int_{\Omega}(|\nabla u|^{2}+u^{2})dx
-\lambda(p+\gamma)\int_{\Omega}P(x)|u|^{p+1}dx<0\big\}.
\end{gather*}
When $\lambda\in(0,T_{p,\gamma})$, we can prove that $\Lambda^{\pm}\neq\emptyset$
and $\Lambda^{0}=\{0\}$. Then we can find two minimizers of
$I_{\lambda}$ on $\Lambda^{+}$ and $\Lambda^{-}$ respectively, which are local
minimizers of $I_{\lambda}$ on $\Lambda$. Finally, we prove that a local
minimizer of $I_{\lambda}$ on $\Lambda$ is indeed a positive solution of
 \eqref{e1.1}.

The main result can be described as follows.

\begin{theorem} \label{thm1.1}
 Suppose $P\in L^{r_1}(\Omega),Q\in L^{r_2}(\Omega)$ are
nonzero and nonnegative, $1<p<2^{*}-1$ and $0<\gamma<1$, then problem \eqref{e1.1}
 has at least two positive solutions for all $\lambda\in(0,T_{p,\gamma})$,
where $r_{1}>\frac{2^{*}}{2^{*}-p-1}$ and $r_{2}>\frac{2^{*}}{2^{*}+\gamma-1}$
are two constants.
\end{theorem}


To the best knowledge, up to now there is no study of the exact estimate of
$\lambda$ such that problem \eqref{e1.1} has at least two positive solutions.
For the case $1<p<2^{*}-1$, Chabrowski
obtained two positive solutions restricting the exponent of singular term with
$0<\gamma<\min\{p-1,1\}$ in \cite{JC}. Moreover, we overcome
the difficulty of the singular term by Nehari manifold, while \cite{JC}
used perturbation method to conquer this difficulty.

This article is organized as follow: in Section 2, we give some
preliminaries which will be used to prove out main result, and the
proof of Theorem \ref{thm1.1} is given in Section 3.

\section{Preliminaries}

In this section, we give some lemmas in preparation for the proof
of our main result.

\begin{lemma} \label{lem2.1}
 Suppose $\lambda\in(0,T_{p,\gamma})$, then $\Lambda^{\pm}\neq\emptyset$
and $\Lambda^{0}=\{0\}$.
Moreover, $\Lambda^{-}$ is closed for all $0<\lambda<T_{p,\gamma}$.
\end{lemma}

\begin{proof}
 According to the assumptions on $P$ and $Q$, there exists $u\in H^{1}(\Omega)$
such that $\int_{\Omega}P(x)|u|^{p+1}dx>0$ and
$\int_{\Omega}Q(x)|u|^{1-\gamma}dx>0$.
Let $\Phi\in C(R^+,R)$ satisfy
$$
\Phi(t)=t^{1-p}\|u\|^2-t^{-\gamma-p}\int_{\Omega}Q(x)|u|^{1-\gamma}dx,
$$
then
$$
\Phi'(t)=(1-p)t^{-p}\|u\|^2+(p+\gamma)t^{-\gamma-p-1}
\int_{\Omega}Q(x)|u|^{1-\gamma}dx.
$$
Let $\Phi'(t)=0$, we can verify
$$
t_{\rm max}=\Big[\frac{(p+\gamma)\int_{\Omega}Q(x)|u|^{1-\gamma}dx}{(p-1)\|u\|^{2}}
\Big]^{1/(1+\gamma)}.
$$
Easy computations show that $\Phi'(t)>0$ for all $0<t<t_{\rm max}$ and
$\Phi'(t)<0$ for all $t>t_{\rm max}$. Thus $\Phi(t)$ attains
its maximum at $t_{\rm max}$, that is,
\begin{equation*}
\Phi(t_{\rm max})=\frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}
\big)^{\frac{p+\gamma}{1+\gamma}}\frac{\|u\|
^{\frac{2(p+\gamma)}{1+\gamma}}}
{\big(\int_{\Omega}Q(x)|u|^{1-\gamma}dx\big)^{\frac{p-1}{1+\gamma}}}.
\end{equation*}
From \eqref{e1.0}, we have
\begin{equation}\label{e2.0}
S|u|^{2}_{2^{*}}< \|u\|^{2},
\end{equation}
and by H\"older's inequality, one has
\begin{gather}\label{e2.1}
\int_{\Omega} P(x)|u|^{p+1}dx\leq|P|_{r_1}|u|_{2^{*}}^{p+1}
|\Omega|^{\frac{r_{1}(2^{*}-p-1)-2^{*}}{r_{1}2^{*}}},
\\
\label{e2.2}
\int_{\Omega} Q(x)|u|^{1-\gamma}dx\leq|Q|_{r_2}|u|_{2^{*}}^{1-\gamma}
|\Omega|^{\frac{r_{2}(2^{*}+\gamma-1)-2^{*}}{r_{2}2^{*}}}.
\end{gather}
Then from \eqref{e2.0}-\eqref{e2.2}, one gets
\begin{equation}\label{e2.3}
\begin{aligned}
&\Phi(t_{\rm max})-\lambda \int_{\Omega}P(x)|u|^{p+1}dx\\
&>\frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}\big)^{\frac{p+\gamma}{1+\gamma}}
\frac{(S|u|^{2}_{2^{*}})^{\frac{p+\gamma}{1+\gamma}}}{(|Q|_{r_2}|u|_{2^{*}}^{1-\gamma}|\Omega|^{\frac{r_{2}(2^{*}+\gamma-1)-2^{*}}{r_{2}2^{*}}})
^{\frac{p-1}{1+\gamma}}}\\
&\quad -\lambda|P|_{r_1}|u|_{2^{*}}^{p+1}|\Omega|
 ^{\frac{r_{1}(2^{*}-p-1)-2^{*}}{r_{1}2^{*}}}\\
&=\Big[\frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}\big)
 ^{\frac{p+\gamma}{1+\gamma}}
\frac{S^{\frac{p+\gamma}{1+\gamma}}}
{\big(|Q|_{r_2}|\Omega|^{\frac{r_{2}(2^{*}+\gamma-1)-2^{*}}{r_{2}2^{*}}}\big)
^{\frac{p-1}{1+\gamma}}}\\
&\quad -\lambda|P|_{r_1}|\Omega|^{\frac{r_{1}(2^{*}-p-1)-2^{*}}{r_{1}2^{*}}}\Big]|u|_{2^{*}}^{p+1}\\
&= |P|_{r_1}|\Omega|^{\frac{r_{1}(2^{*}-p-1)-2^{*}}{r_{1}2^{*}}}
 (T_{p,\gamma}-\lambda)|u|_{2^{*}}^{p+1}
> 0,
\end{aligned}
\end{equation}
for all $\lambda\in(0,T_{p,\gamma})$. Consequently, there exist $t^{+}_{0}$
and $t^{-}_{0}$ satisfying
$0<t^{+}_{0}<t_{\rm max}<t^{-}_{0}$ such that
$$
\Phi(t^{+}_{0})=\lambda \int_{\Omega}P(x)|u|^{p+1}dx=\Phi(t^{-}_{0})
$$
and
$$
\Phi'(t^{+}_{0})<0<\Phi'(t^{-}_{0});
$$
that is, $t^{+}_{0}u\in \Lambda^{+}$ and $t^{-}_{0}u\in \Lambda^{-}$.
Thus $\Lambda^{\pm}$ are non-empty whenever $\lambda\in(0,T_{p,\gamma})$.

Next, we prove that $\Lambda^{0}=\{0\}$ for all $\lambda\in(0,T_{p,\gamma})$.
By contradiction, suppose that there exists $u_{0}\in \Lambda^{0}$ and
$u_0\neq0$. Then it follows that
$$
(1+\gamma)\|u_0\|^{2}-\lambda(p+\gamma)\int_{\Omega}P(x)|u_0|^{p+1}dx=0,
$$
and consequently
\begin{align*}
0&=\|u_0\|^{2}-\lambda\int_{\Omega}P(x)|u_{0}|^{p+1}dx
 -\int_{\Omega}Q(x)|u_{0}|^{1-\gamma}dx\\
&= \frac{p-1}{p+\gamma}\|u_{0}\|^{2}-\int_{\Omega}Q(x)|u_{0}|^{1-\gamma}dx.
\end{align*}
From \eqref{e2.3}, we have
\begin{align*}
0&<\Big[\frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}\big)
 ^{\frac{p+\gamma}{1+\gamma}}
\frac{S^{\frac{p+\gamma}{1+\gamma}}}
 {\big(|Q|_{r_2}|\Omega|^{\frac{r_{2}(2^{*}+\gamma-1)-2^{*}}{r_{2}2^{*}}}\big)
^{\frac{p-1}{1+\gamma}}}\\
&\quad -\lambda|P|_{r_1}|\Omega|^{\frac{r_{1}(2^{*}-p-1)-2^{*}}{r_{1}2^{*}}}\Big]
 |u_{0}|_{2^{*}}^{p+1}\\
&< \frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}\big)^{\frac{p+\gamma}{1+\gamma}}
 \frac{\|u_0\|^{\frac{2(p+\gamma)}{1+\gamma}}}
 {\big(\int_{\Omega}Q(x)|u_0|^{1-\gamma}dx\big)^{\frac{p-1}{1+\gamma}}}
 -\lambda\int_{\Omega}P(x)|u_{0}|^{p+1}dx\\
&= \frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}\big)^{\frac{p+\gamma}{1+\gamma}}
 \frac{\|u_0\|^{\frac{2(p+\gamma)}{1+\gamma}}}
 {\big(\frac{p-1}{p+\gamma}\|u_{0}\|^{2}\big)^{\frac{p-1}{1+\gamma}}}
-\frac{1+\gamma}{p+\gamma}\|u_{0}\|^{2}
=0,
\end{align*}
for all $\lambda\in(0,T_{p,\gamma})$, which is impossible.
Thus $\Lambda^{0}=\{0\}$ for $\lambda\in(0,T_{p,\gamma})$.

Finally, we prove that $\Lambda^{-}$ is closed for all $0<\lambda<T_{p,\gamma}$.
That is, suppose $\{u_{n}\}\subset \Lambda^{-}$ such that
$u_{n}\to  u$ in $H^{1}(\Omega)$ as $n\to \infty$, then $u\in \Lambda^{-}$.
Since $\{u_{n}\}\subset\Lambda^{-}$, from the definition of $\Lambda^{-}$,
 one has
\begin{gather}
\|u_{n}\|^2-\lambda \int_{\Omega}P(x)|u_{n}|^{p+1}dx
-\int_{\Omega}Q(x)|u_{n}|^{1-\gamma}dx=0, \nonumber \\
\label{e2.4}
(1+\gamma)\|u_{n}\|^2-\lambda(p+\gamma)\int_{\Omega}P(x)|u_{n}|^{p+1}dx<0,
\end{gather}
and consequently
\begin{gather*}
\|u\|^2-\lambda \int_{\Omega}P(x)|u|^{p+1}dx
-\int_{\Omega}Q(x)|u|^{1-\gamma}dx=0, \\
(1+\gamma)\|u\|^2-\lambda(p+\gamma)\int_{\Omega}P(x)|u|^{p+1}dx\leq0,
\end{gather*}
thus $u\in\Lambda^{0}\cup\Lambda^{-}$. If $u\in\Lambda^{0}$, combining
$\Lambda^{0}=\{0\}$ it follows that $u=0$.
However, from \eqref{e2.0}, \eqref{e2.1} and \eqref{e2.4}, one gets
\begin{equation}\label{e2.00}
|u_n|_{2^{*}}\geq \Big[\frac{S(1+\gamma)}{\lambda(p+\gamma)|P|_{r_1}}
|\Omega|^{\frac{r_{1}(2^{*}-p-1)-2^{*}}{r_{1}2^{*}}}\Big]^{1/(p-1)},
\quad \forall u_{n}\in\Lambda^{-},
\end{equation}
which contradicts $u=0$. Thus $u\in\Lambda^{-}$ for $\lambda\in(0,T_{p,\gamma})$.
Hence the proof is complete.
\end{proof}

\begin{lemma} \label{lem2.2}
 Given $u\in\Lambda^{-}$ (respectively $\Lambda^{+}$) with $u>0$,
for all $\varphi\in H^{1}(\Omega)$, $\varphi>0$, there exist
$\varepsilon>0$ and a continuous function $t=t(s)>0$, $s\in\mathbb{R}$,
$|s|<\varepsilon$ satisfying
\begin{equation*}
t(0)=1,\quad t(s)(u+s\varphi)\in\Lambda^{-}\; \text{ (respectively $\Lambda^{+}$)},
\quad \forall s\in\mathbb{R},\; |s|<\varepsilon.
\end{equation*}
\end{lemma}

\begin{proof} We define $f: \mathbb{R}\times\mathbb{R}\to  R$ by:
\begin{align*}
f(t,s)&= t^{\gamma+1}\int_{\Omega}\big[|\nabla(u+s\varphi)|^{2}+(u+s\varphi)^{2}\big]
 dx-\lambda t^{p+\gamma}\int_{\Omega}P(x)(u+s\varphi)^{p+1}dx\\
&\quad -\int_{\Omega}Q(x)(u+s\varphi)^{1-\gamma}dx.
\end{align*}
Then
\begin{align*}
f_{t}(t,s)&=(\gamma+1)t^{\gamma}\int_{\Omega}\big[|\nabla(u+s\varphi)|^{2}
 +(u+s\varphi)^{2}\big]dx\\
&\quad-\lambda(p+\gamma)t^{p+\gamma-1}\int_{\Omega}P(x)(u+s\varphi)^{p+1}dx,
\end{align*}
is continuous in $\mathbb{R}\times\mathbb{R}$. Since
$u\in \Lambda^{-}\subset \Lambda$, we have $f(1,0)=0$, and moreover
\begin{equation*}
f_{t}(1,0)=(1+\gamma)\int_{\Omega}(|\nabla u|^{2}+u^{2})dx
-\lambda ({p+\gamma})\int_{\Omega}P(x)u^{p+1}dx<0.
\end{equation*}
Then by applying the implicit function theorem to $f$ at the point
$(1,0)$, we obtain $\overline{\varepsilon}>0$ and a continuous function
$t=t(s)>0$, $s\in\mathbb{R}$,
$|s|<\overline{\varepsilon}$ satisfying that
\begin{equation*}
t(0)=1,\quad t(s)(u+s\varphi)\in\Lambda,\quad \forall s\in\mathbb{R},\;
|s|<\overline{\varepsilon}.
\end{equation*}
Moreover, taking $\varepsilon>0$ possibly smaller
($\varepsilon<\overline{\varepsilon}$), we obtain
\begin{equation*}
t(s)(u+s\varphi)\in\Lambda^{-},\quad \forall s\in\mathbb{R},\; |s|<\varepsilon.
\end{equation*}
The case $u\in\Lambda^{+}$ may be obtained in the same way.
Thus the proof  is complete.
\end{proof}

\section{Proof of main theorem}

 For all $u\in\Lambda$, we have
\begin{align*}
I_{\lambda}(u)
&= \frac{1}{2}\|u\|^{2}-\frac{\lambda}{p+1}\int_{\Omega}P(x)|u|^{p+1}dx
 -\frac{1}{1-\gamma}\int_\Omega Q(x)|u|^{1-\gamma}dx\\
&=\big(\frac{1}{2}-\frac{1}{p+1}\big)\|u\|^{2}-\big(\frac{1}{1-\gamma}
 -\frac{1}{p+1}\big)\int_\Omega Q(x)|u|^{1-\gamma}dx.
\end{align*}
Since $1<p<2^{*}-1$ and $0<\gamma<1$, from \eqref{e2.2} and \eqref{e2.0},
we obtain that $I_{\lambda}$ is coercive and bounded below on $\Lambda$.
According to Lemma \ref{lem2.1} for all $\lambda\in(0,T_{p,\gamma})$
$$
m^{+}=\inf_{u\in\Lambda^{+}}I_{\lambda}(u),\quad
m^{-}=\inf_{u\in\Lambda^{-}}I_{\lambda}(u)
$$
are well defined. Moreover, for all $u\in\Lambda^{+}$, it follows that
$$
(1+\gamma)\|u\|^{2}-\lambda(p+\gamma)\int_{\Omega}P(x)|u|^{p+1}dx>0,
$$
and consequently, since $2<p+1<2^{*}$, $0<\gamma<1$ and $u\not\equiv0$, we have
\begin{align*}
I_{\lambda}(u)
&= \frac{1}{2}\|u\|^{2}-\frac{\lambda}{p+1}\int_{\Omega}P(x)|u|^{p+1}dx
 -\frac{1}{1-\gamma}\int_\Omega Q(x)|u|^{1-\gamma}dx\\
&= \big(\frac{1}{2}-\frac{1}{1-\gamma}\big)\|u\|^{2}
 +\lambda\big(\frac{1}{1-\gamma}-\frac{1}{p+1}\big)\int_{\Omega}P(x)|u|^{p+1}dx\\
&< -\frac{1+\gamma}{2(1-\gamma)}\|u\|^{2}+\frac{1+\gamma}{(1-\gamma)(p+1)}\|u\|^{2}\\
&= -\frac{1+\gamma}{1-\gamma}\big(\frac{1}{2}-\frac{1}{p+1}\big)\|u\|^{2}<0.
\end{align*}
Thus $m^{+}=\inf_{u\in\Lambda^{+}}I_{\lambda}(u)<0$ for all
$\lambda\in(0,T_{p,\gamma})$.


\begin{proof}[Proof of Theorem \ref{thm1.1}]
 Let $\lambda\in(0,T_{p,\gamma})$. The following two steps
complete the proof of Theorem \ref{thm1.1}.

\noindent\textbf{Step 1.}
 We prove that there exists a positive solution of \eqref{e1.1}
in $\Lambda^{+}$. Applying Ekeland's variational principle to the
minimization problem $m^{+}=\inf_{u\in
\Lambda^{+}}I_{\lambda}(u)$, there exists a sequence
$\{u_{n}\}\subset \Lambda^{+}$ with the following properties:
\begin{itemize}
\item[(i)] $I_{\lambda}(u_{n})<m^{+}+\frac{1}{n}$,
\item[(ii)] $I_{\lambda}(u)\geq I_{\lambda}(u_{n})-\frac{1}{n}\|u-u_{n}\|$,
for all $u\in \Lambda^{+}$
\end{itemize}
Since $I_{\lambda}(u)=I_{\lambda}(|u|)$, we can assume from the beginning
that $u_{n}(x)\geq0$ for all $x\in\Omega$. Obviously, $\{u_{n}\}$ is
bounded in $H^{1}(\Omega)$, going if necessary to a subsequence,
still denoted by $\{u_n\}$,  there exists $u_{*}\geq0$ such that
\begin{gather*}
u_n\rightharpoonup u_*, \quad \text{weakly in } H^{1}(\Omega),\\
u_n\to  u_*,\quad \text{strongly in } L^{s}(\Omega),\; 1\leq s<2^*,\\
u_n(x)\to  u_{*}(x),\quad\text{a.e. in }\Omega,
\end{gather*}
as $n\to \infty$. Now we will prove that $u_{*}$ is a positive solution
of problem \eqref{e1.1}.

Firstly, we prove that $u_{*}(x)\not\equiv0$ in $\Omega$.
By Vitali's theorem (see \cite[pp. 133]{RW}), we claim that
\begin{equation}\label{e3.1}
\lim_{n\to \infty}\int_{\Omega}Q(x)|u_n|^{1-\gamma}dx
=\int_{\Omega}Q(x)|u_*|^{1-\gamma}dx.
\end{equation}
Indeed, we only need to prove that $\{\int_{\Omega}Q(x)|u_n|^{1-\gamma}dx,n\in N\}$
is equi-absolutely-continuous. Note that $\{u_{n}\}$ is bounded, by the
Sobolev embedding theorem, so exists a constant $C>0$ such that
$|u_{n}|_{2^{*}}\leq C<\infty$. From \eqref{e2.2}, for every
$\varepsilon>0$, setting
$$
\delta=\Big(\frac{\varepsilon}{|Q|_{r_2}C^{1-\gamma}}\Big)
^{\frac{r_{2}2^{*}}{r_{2}(2^{*}+\gamma-1)-2^{*}}},
$$
 when $E\subset\Omega$ with $mes E<\delta$, we have
\begin{align*}
\int_{E} Q(x)|u_{n}|^{1-\gamma}dx
&\leq |Q|_{r_2}|u|_{2^{*}}^{1-\gamma}\big( \operatorname{meas} E
 \big)^{\frac{r_{2}(2^{*}+\gamma-1)-2^{*}}{r_{2}2^{*}}}\\
&\leq |Q|_{r_2}C^{1-\gamma}\delta^{\frac{r_{2}(2^{*}+\gamma-1)-2^{*}}{r_{2}2^{*}}}
<\varepsilon.
\end{align*}
Thus, our claim is true. Similarly,
\begin{equation}\label{e3.2}
\lim_{n\to \infty}\int_{\Omega}P(x)|u_n|^{p+1}dx=\int_{\Omega}P(x)|u_*|^{p+1}dx.
\end{equation}
By the weakly lower semicontinuity of the norm, combining \eqref{e3.1} and
\eqref{e3.2}, we have
\begin{align*}
I_{\lambda}(u_*)
&= \frac{1}{2}\|u_*\|^{2}-\frac{\lambda}{p+1}\int_{\Omega}P(x)|u_*|^{p+1}dx
-\frac{1}{1-\gamma}\int_\Omega Q(x)|u_{*}|^{1-\gamma}dx\\
&\leq \liminf_{n\to \infty}\Big[\frac{1}{2}\|u_n\|^{2}-\frac{\lambda}{p+1}\int_{\Omega}P(x)|u_n|^{p+1}dx\\
&\quad -\frac{1}{1-\gamma}\int_\Omega Q(x)|u_n|^{1-\gamma}dx\Big]\\
&=\liminf_{n\to \infty}I_{\lambda}(u_n)=m^{+}<0,
\end{align*}
which implies that $u_{*}(x)\not\equiv0$ in $\Omega$.

Secondly, we prove that $u_{*}(x)>0$ a.e. in $\Omega$.
From $u_{n}\in\Lambda^{+}$, we can claim that there exists a constant
$C_1>0$ such that
\begin{equation}\label{e3.3}
(1+\gamma)\|u_n\|^{2}-\lambda(p+\gamma)\int_{\Omega}P(x)|u_n|^{p+1}dx\geq C_1.
\end{equation}
In fact, \eqref{e3.3} is equivalent to
\begin{equation}\label{e3.4}
(1+\gamma)\int_\Omega Q(x)|u_{n}|^{1-\gamma}dx
-\lambda(p-1)\int_{\Omega}P(x)|u_n|^{p+1}dx\geq C_1.
\end{equation}
Since $u_{n}\in\Lambda^{+}$, one has
$$
(1+\gamma)\int_\Omega Q(x)|u_{n}|^{1-\gamma}dx
-\lambda(p-1)\int_{\Omega}P(x)|u_n|^{p+1}dx>0,
$$
and consequently, from \eqref{e3.1} and \eqref{e3.2} it follows that
\begin{align*}
&\lim_{n\to \infty}\Big[(1+\gamma)\int_\Omega Q(x)|u_{n}|^{1-\gamma}dx
 -\lambda(p-1)\int_{\Omega}P(x)|u_n|^{p+1}dx\Big]\\
&=(1+\gamma)\int_\Omega Q(x)|u_{*}|^{1-\gamma}dx-\lambda(p-1)
 \int_{\Omega}P(x)|u_{*}|^{p+1}dx
\geq0.
\end{align*}
Thus we only need to prove that
\begin{equation}\label{e3.00}
(1+\gamma)\int_\Omega Q(x)|u_{*}|^{1-\gamma}dx
 -\lambda(p-1)\int_{\Omega}P(x)|u_{*}|^{p+1}dx>0.
\end{equation}
By contradiction, we assume that
\begin{equation}\label{e3.03}
(1+\gamma)\int_\Omega Q(x)|u_{*}|^{1-\gamma}dx
-\lambda(p-1)\int_{\Omega}P(x)|u_{*}|^{p+1}dx=0.
\end{equation}
Since
\begin{equation}\label{e3.000}
\|u_n\|^{2}-\lambda\int_{\Omega}P(x)|u_n|^{p+1}dx
-\int_\Omega Q(x)|u_{n}|^{1-\gamma}dx=0,
\end{equation}
by the weakly lower semicontinuity of the norm, and combining
\eqref{e3.1}-\eqref{e3.2} and \eqref{e3.03}, we have
\begin{equation}\label{e3.0}
\begin{aligned}
0&\geq \|u_{*}\|^{2}-\lambda\int_{\Omega}P(x)|u_{*}|^{p+1}dx
 -\int_\Omega Q(x)|u_{*}|^{1-\gamma}dx\\
&= \|u_{*}\|^{2}-\frac{p+\gamma}{p-1}\int_\Omega Q(x)|u_{*}|^{1-\gamma}dx\\
&= \|u_{*}\|^{2}-\frac{\lambda(p+\gamma)}{1+\gamma}\int_{\Omega}P(x)|u_{*}|^{p+1}dx,
\end{aligned}
\end{equation}
and consequently, from \eqref{e2.3} one has
\begin{align*}
0&<\Big[\frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}
 \big)^{\frac{p+\gamma}{1+\gamma}}
\frac{S^{\frac{p+\gamma}{1+\gamma}}}{(|Q|_{r_2}|\Omega|
 ^{\frac{r_{2}(2^{*}+\gamma-1)-2^{*}}{r_{2}2^{*}}})
 ^{\frac{p-1}{1+\gamma}}}\\
&\quad -\lambda|P|_{r_1}|\Omega|^{\frac{r_{1}(2^{*}-p-1)-2^{*}}
 {r_{1}2^{*}}}\Big]|u_{*}|_{2^{*}}^{p+1}\\
&< \frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}\big)^{\frac{p+\gamma}{1+\gamma}}
 \frac{\|u_{*}\|^{\frac{2(p+\gamma)}{1+\gamma}}}
 {\big(\int_{\Omega}Q(x)|u_{*}|^{1-\gamma}dx\big)^{\frac{p-1}{1+\gamma}}}
 -\lambda\int_{\Omega}P(x)|u_{*}|^{p+1}dx\\
&= \frac{1+\gamma}{p-1}\big(\frac{p-1}{p+\gamma}\big)^{\frac{p+\gamma}{1+\gamma}}
 \frac{\|u_{*}\|^{\frac{2(p+\gamma)}{1+\gamma}}}
{\big(\frac{p-1}{p+\gamma}\|u_{*}\|^{2}\big)^{\frac{p-1}{1+\gamma}}}
 -\frac{1+\gamma}{p+\gamma}\|u_{*}\|^{2}=0
\end{align*}
for all $\lambda\in(0,T_{p,\gamma})$, which is impossible.
So \eqref{e3.00} is obtained and our claim is true.
Applying  Lemma \ref{lem2.2} with $u=u_{n}$, and
$\varphi\in H^{1}(\Omega),\ \varphi\geq0$, $ t>0$ small enough,
we find a sequence of continuous functions $t_{n}=t_{n}(s)$ such
that $t_{n}(0)=1$ and $t_{n}(s)(u_{n}+s\varphi)\in\Lambda^{+}$.
Noting that $t_{n}(s)(u_{n}+s\varphi)\in\Lambda^{+}$ and
$u_{n}\in\Lambda^{+}$, one has
\begin{align*}
& t_{n}^{2}(s)\|u_{n}+s\varphi\|^{2}
 -\lambda t_{n}^{p+1}(s)\int_{\Omega}P(x)|u_n+s\varphi|^{p+1}dx\\
&-t_{n}^{1-\gamma}(s)\int_{\Omega}Q(x)(u_{n}+s\varphi)^{1-\gamma}dx=0,
\end{align*}
consequently,  from \eqref{e3.000} it follows that
\begin{align*}
0&=[t_{n}^{2}(s)-1]\|u_{n}+s\varphi\|^{2}+(\|u_{n}+s\varphi\|^{2}-\|u_{n}\|^{2})\\
&\quad -\lambda[t_{n}^{p+1}(s)-1]\int_{\Omega}P(x)|u_{n}+s\varphi|^{p+1}dx\\
&\quad -\lambda\int_{\Omega}P(x)(|u_{n}+s\varphi|^{p+1}-|u_{n}|^{p+1})dx\\
&\quad -[t_{n}^{1-\gamma}(s)-1]\int_{\Omega}Q(x)(u_{n}+s\varphi)^{1-\gamma}dx\\
&\quad -\int_{\Omega}Q(x)[(u_{n}+s\varphi)^{1-\gamma}-|u_{n}|^{1-\gamma}]dx\\
&\leq [t_{n}^{2}(s)-1]\|u_{n}+s\varphi\|^{2}+(\|u_{n}+s\varphi\|^{2}-\|u_{n}\|^{2})\\
&\quad -\lambda[t_{n}^{p+1}(s)-1]\int_{\Omega}P(x)|u_{n}+s\varphi|^{p+1}dx\\
&\quad -\lambda\int_{\Omega}P(x)(|u_{n}+s\varphi|^{p+1}-|u_{n}|^{p+1})dx\\
&\quad -[t_{n}^{1-\gamma}(s)-1]\int_{\Omega}Q(x)(u_{n}+s\varphi)^{1-\gamma}dx,
\end{align*}
then dividing by $s>0$, we have
\begin{equation}\label{e3.01}
\begin{aligned}
0&\leq \Big[(t_{n}(s)+1)\|u_{n}+s\varphi\|^{2}
 -\lambda\frac{t_{n}^{p+1}(s)-1}{t_{n}(s)-1}
 \int_{\Omega}P(x)|u_{n}+s\varphi|^{p+1}dx\\
&\quad -\frac{t_{n}^{1-\gamma}(s)-1}{t_{n}(s)-1}
 \int_{\Omega}Q(x)(u_{n}+s\varphi)^{1-\gamma}dx\Big]
 \frac{t_{n}(s)-1}{s}+s\|\varphi\|^{2}\\
&\quad +2\int_{\Omega}((\nabla u_{n},\nabla \varphi)+u_{n}\varphi)dx
-\lambda\int_{\Omega}P(x)\frac{|u_{n}+s\varphi|^{p+1}-|u_{n}|^{p+1}}{s}dx.
\end{aligned}
\end{equation}
Let
\begin{equation}\label{e3.5}
A_{n}(s)=\frac{t_{n}(s)-1}{s},
\end{equation}
\begin{align*}
K_{1,n}(s)&=(t_{n}(s)+1)\|u_{n}+s\varphi\|^{2}
 -\lambda\frac{t_{n}^{p+1}(s)-1}{t_{n}(s)-1}\int_{\Omega}P(x)|u_{n}
 +s\varphi|^{p+1}dx\\
&-\frac{t_{n}^{1-\gamma}(s)-1}{t_{n}(s)-1}
 \int_{\Omega}Q(x)(u_{n}+s\varphi)^{1-\gamma}dx,
\end{align*}
and
\begin{align*}
K_{2,n}(s)&= s\|\varphi\|^{2}+2\int_{\Omega}((\nabla u_{n},\nabla \varphi)
 +u_{n}\varphi)dx\\
&\quad -\lambda\int_{\Omega}P(x)\frac{|u_{n}+s\varphi|^{p+1}-|u_{n}|^{p+1}}{s}dx.
\end{align*}
Then, according to \eqref{e3.000} and \eqref{e3.3} we have
\begin{align*}
\lim_{s\to 0^{+}}K_{1,n}(s)
&=2\|u_{n}\|^{2}-\lambda(p+1)\int_{\Omega}P(x)u_{n}^{p+1}dx
-(1-\gamma)\int_{\Omega}Q(x)u_{n}^{1-\gamma}dx\\
&=(1+\gamma)\|u_{n}\|^{2}-\lambda(p+\gamma)\int_{\Omega}P(x)u_{n}^{p+1}dx\\
&=:  K_{1,n}\geq C_{1}>0,
\end{align*}
and
$$
\lim_{s\to 0^{+}}K_{2,n}(s)=2\int_{\Omega}((\nabla u_{n},\nabla \varphi)
+u_{n}\varphi)dx-\lambda(p+1)\int_{\Omega}P(x)u_{n}^{p}\varphi dx
=: K_{2,n}.
$$
Thus, from \eqref{e3.01} and the continuity of $K_{1,n}(s)$, one obtains
$$
A_{n}(s)\geq\frac{-K_{2,n}(s)}{K_{1,n}(s)},
$$
for $s>0$ small. Since $\{u_{n}\}$ is
bounded in $H^{1}(\Omega)$ there exists a positive constant $C_{2}$ such that
$|K_{2,n}|<C_{2}$ for all $n\in N^{+}$.
Therefore,
\begin{equation}\label{e3.05}
\liminf_{s\to 0^{+}}A_{n}(s)\geq\frac{-K_{2,n}}{K_{1,n}}
\geq\frac{-|K_{2,n}|}{K_{1,n}}\geq-\frac{C_{2}}{C_{1}}
\end{equation}
By the subadditivity of norm we have
$$
\|t_{n}(s)(u_{n}+s\varphi)-u_{n}\|\leq|t_{n}(s)-1|\cdot\|u_{n}\|
+st_{n}(s)\|\varphi\|.
$$
Thus from condition (ii) it follows that
\begin{align*}
&|t_{n}(s)-1|\frac{\|u_{n}\|}{n}+st_{n}(s)\frac{\|\varphi\|}{n}\\
&\geq I_{\lambda}(u_{n})-I_{\lambda}[t_{n}(s)(u_{n}+s\varphi)]\\
&= -\frac{1+\gamma}{2(1-\gamma)}\|u_{n}\|^{2}
 +\lambda\frac{p+\gamma}{(p+1)(1-\gamma)}\int_{\Omega}P(x)u_{n}^{p+1}dx\\
&\quad +\frac{1+\gamma}{2(1-\gamma)}t_{n}^{2}(s)\|u_{n}+s\varphi\|^{2}-\lambda\frac{p
+\gamma}{(p+1)(1-\gamma)}t_{n}^{p+1}(s)\int_{\Omega}P(x)|u_{n}+s\varphi|^{p+1}dx\\
&= \frac{1+\gamma}{2(1-\gamma)}(\|u_{n}+s\varphi\|^{2}-\|u_{n}\|^{2})
 +\frac{1+\gamma}{2(1-\gamma)}[t_{n}(s)-1]\|u_{n}+s\varphi\|^{2}\\
&\quad -\lambda\frac{p+\gamma}{(p+1)(1-\gamma)}t_{n}^{p+1}(s)
 \int_{\Omega}P(x)(|u_{n}+s\varphi|^{p+1}-|u_{n}|^{p+1})dx\\
&\quad -\lambda\frac{p+\gamma}{(p+1)(1-\gamma)}[t_{n}^{p+1}(s)-1]
  \int_{\Omega}P(x)u_{n}^{p+1}dx.
\end{align*}
Then dividing by $s>0$, it follows that
\begin{equation}\label{e3.04}
\begin{aligned}
&\frac{|t_{n}(s)-1|}{s}\frac{\|u_{n}\|}{n}+t_{n}(s)\frac{\|\varphi\|}{n}\\
&\geq \frac{1}{1-\gamma}\Big[\frac{1+\gamma}{2}\|u_{n}+s\varphi\|^{2}\\
&\quad -\lambda\frac{p+\gamma}{p+1}\frac{t_{n}^{p+1}(s)-1}{t_{n}(s)-1}
 \int_{\Omega}P(x)u_{n}^{p+1}dx\Big]\frac{t_{n}(s)-1}{s}\\
&\quad+ \frac{1+\gamma}{2(1-\gamma)}\frac{\|u_{n}+s\varphi\|^{2}-\|u_{n}\|^{2}}{s}\\
&\quad -\lambda\frac{p+\gamma}{(p+1)(1-\gamma)}t_{n}^{p+1}(s)\int_{\Omega}P(x)
 \frac{|u_{n}+s\varphi|^{p+1}-|u_{n}|^{p+1}}{s}dx.
\end{aligned}
\end{equation}
Let
$$
K_{3,n}(s)=\frac{1+\gamma}{2}\|u_{n}+s\varphi\|^{2}
-\lambda\frac{p+\gamma}{p+1}\frac{t_{n}^{p+1}(s)-1}{t_{n}(s)-1}
\int_{\Omega}P(x)u_{n}^{p+1}dx,
$$
and
\begin{align*}
K_{4,n}(s)
&= \frac{1+\gamma}{2(1-\gamma)}\frac{\|u_{n}+s\varphi\|^{2}-\|u_{n}\|^{2}}{s}\\
&\quad -\lambda\frac{p+\gamma}{(p+1)(1-\gamma)}t_{n}^{p+1}(s)
\int_{\Omega}P(x)\frac{|u_{n}+s\varphi|^{p+1}-|u_{n}|^{p+1}}{s}dx.
\end{align*}
Then from \eqref{e3.000} and \eqref{e3.3}, one has
$$
\lim_{s\to 0^{+}}K_{3,n}(s)=(1+\gamma)\|u_{n}\|^{2}
-\lambda(p+\gamma)\int_{\Omega}P(x)u_{n}^{p+1}dx
=K_{1,n}\geq C_{1}>0,$$
and
$$
\lim_{s\to 0^{+}}K_{4,n}(s)=\frac{1+\gamma}{1-\gamma}\int_{\Omega}((\nabla
u_{n},\nabla \varphi)+u_{n}\varphi)dx-\lambda\frac{p+\gamma}{1-\gamma}
\int_{\Omega}P(x)u_{n}^{p}\varphi dx=: K_{4,n}.
$$
From \eqref{e3.04} we have
$$
|A_{n}(s)|\frac{\|u_{n}\|}{n}+t_{n}(s)\frac{\|\varphi\|}{n}
\geq K_{3,n}(s)A_{n}(s)+K_{4,n}(s).
$$
If $A_{n}(s)\geq0$, then
$$
A_{n}(s)\leq\frac{t_{n}(s)\frac{\|\varphi\|}{n}-K_{4,n}(s)}{K_{3,n}(s)
-\frac{\|u_{n}\|}{n}}
\leq\frac{t_{n}(s)\frac{\|\varphi\|}{n}+|K_{4,n}(s)|}{K_{3,n}(s)
-\frac{\|u_{n}\|}{n}}.
$$
If $A_{n}(s)<0$, then
$$
A_{n}(s)\leq\frac{t_{n}(s)\frac{\|\varphi\|}{n}-K_{4,n}(s)}{K_{3,n}(s)
 +\frac{\|u_{n}\|}{n}}
\leq\frac{t_{n}(s)\frac{\|\varphi\|}{n}+|K_{4,n}(s)|}{K_{3,n}(s)
 +\frac{\|u_{n}\|}{n}}.
$$
Hence
$$
A_{n}(s)\leq\frac{t_{n}(s)\frac{\|\varphi\|}{n}+|K_{4,n}(s)|}{K_{3,n}(s)
-\frac{\|u_{n}\|}{n}},
$$
and consequently, for $n$ large enough we have
\begin{equation}\label{e3.06}
\limsup_{s\to 0^{+}}A_{n}(s)\leq\frac{\frac{\|\varphi\|}{n}+|K_{4,n}|}{K_{1,n}
-\frac{\|u_{n}\|}{n}}
\leq2\frac{1+|K_{4,n}|}{K_{1,n}}\leq2\frac{1+C_{3}}{C_{1}},
\end{equation}
where $C_{3}>0$ is a constant such that $|K_{4,n}|<C_3$ by the boundedness
of $\{u_n\}$. Thus, according to \eqref{e3.05} and \eqref{e3.06},
there exists a positive constant $C_4$ such that
\begin{equation}\label{e3.07}
\limsup_{s\to 0^{+}}|A_{n}(s)|\leq C_4
\end{equation}
for $n$ large enough.

By the subadditivity of norm, from (ii), we obtain
\begin{align*}
&\frac{1}{n}[|t_{n}(s)-1|\cdot\|u_{n}\|+st_{n}(s)\|\varphi\|]\\
&\geq \frac{1}{n}\|t_{n}(s)(u_{n}+s\varphi)-u_{n}\|\\
& \geq I_{\lambda}(u_{n})-I_{\lambda}[t_{n}(s)(u_{n}+s\varphi)]\\
&= -\frac{t_{n}^{2}(s)-1}{2}\|u_{n}\|^{2}+\lambda\frac{t_{n}^{p+1}(s)-1}{p+1}
\int_{\Omega}P(x)(u_{n}+s\varphi)^{p+1}dx\\
&\quad +\frac{t_{n}^{1-\gamma}(s)-1}{1-\gamma}\int_{\Omega}Q(x)(u_{n}+s\varphi)^{1-\gamma}dx
+\frac{t_{n}^{2}(s)}{2}\big(\|u_{n}\|^{2}-\|u_{n}+s\varphi\|^{2}\big)\\
&\quad +\frac{\lambda}{p+1}\int_{\Omega}P(x)
 \big[(u_{n}+s\varphi)^{p+1}-u_{n}^{p+1}\big]dx\\
&\quad +\frac{1}{1-\gamma}\int_{\Omega}Q(x)
 \big[(u_{n}+s\varphi)^{1-\gamma}-u_{n}^{1-\gamma}\big]dx,
\end{align*}
and dividing by $s>0$, we have
\begin{equation}\label{e3.9}
\begin{aligned}
&\frac{1}{n}(|A_{n}(s)|\cdot\|u_{n}\|+\|\varphi\|)\\
&\geq -\Big[\frac{t_{n}(s)+1}{2}\|u_{n}\|^{2}
-\lambda\frac{t_{n}^{p+1}(s)-1}{(p+1)(t_{n}(s)-1)}\int_{\Omega}P(x)(u_{n}+s\varphi)^{p+1}dx\\
&\quad -\frac{t_{n}^{1-\gamma}(s)-1}{(1-\gamma)(t_{n}(s)-1)}\int_{\Omega}Q(x)(u_{n}+s\varphi)^{1-\gamma}dx\Big]A_{n}(s)\\
&\quad +\frac{t_{n}^{2}(s)}{2}\frac{\|u_{n}\|^{2}-\|u_{n}+s\varphi\|^{2}}{s}\\
&\quad +\frac{\lambda}{p+1}\int_{\Omega}P(x)\frac{(u_{n}+s\varphi)^{p+1}-u_{n}^{p+1}}{s}dx\\
&\quad +\frac{1}{1-\gamma}\int_{\Omega}Q(x)\frac{(u_{n}+s\varphi)^{1-\gamma}-u_{n}^{1-\gamma}}{s}dx.
\end{aligned}
\end{equation}
Let
\begin{align*}
K_{5,n}(s)&=\frac{t_{n}(s)+1}{2}\|u_{n}\|^{2}
 -\lambda\frac{t_{n}^{p+1}(s)-1}{(p+1)(t_{n}(s)-1)}
\int_{\Omega}P(x)(u_{n}+s\varphi)^{p+1}dx\\
&\quad -\frac{t_{n}^{1-\gamma}(s)-1}{(1-\gamma)(t_{n}(s)-1)}
 \int_{\Omega}Q(x)(u_{n}+s\varphi)^{1-\gamma}dx,
\end{align*}
and
\begin{equation*}
K_{6,n}(s)=\frac{t_{n}^{2}(s)}{2}\frac{\|u_{n}\|^{2}-\|u_{n}+s\varphi\|^{2}}{s}
+\frac{\lambda}{p+1}\int_{\Omega}P(x)\frac{(u_{n}+s\varphi)^{p+1}-u_{n}^{p+1}}{s}dx.
\end{equation*}
Then from \eqref{e3.000}, we have
\begin{equation*}
\lim_{s\to 0^{+}}K_{5,n}(s)=\|u_{n}\|^{2}-\lambda\int_{\Omega}P(x)u_{n}^{p+1}dx
-\int_{\Omega}Q(x)u_{n}^{1-\gamma}dx=0.
\end{equation*}
and
\begin{equation*}
\lim_{s\to 0^{+}}K_{6,n}(s)=-\int_{\Omega}((\nabla u_{n},\nabla \varphi)+u_{n}\varphi)dx+
\lambda\int_{\Omega}P(x)u_{n}^{p}\varphi dx.
\end{equation*}
Thus from \eqref{e3.9} we deduce
\begin{equation}\label{e3.08}
\begin{aligned}
&\frac{1}{1-\gamma}\int_{\Omega}Q(x)\frac{(u_{n}+s\varphi)^{1-\gamma}
 -u_{n}^{1-\gamma}}{s}dx\\
&\leq|K_{5,n}(s)|\cdot|A_{n}(s)|-K_{6,n}(s)+\frac{|A_{n}(s)|\cdot\|u_{n}\|
 +\|\varphi\|}{n}.
\end{aligned}
\end{equation}
Since
$$
Q(x)[(u_{n}+s\varphi)^{1-\gamma}-u_{n}^{1-\gamma}]\geq0,\quad  \forall
x\in\Omega,\; \forall s>0,
$$
then by Fatou's Lemma we have
$$\int_{\Omega}Q(x)u_{n}^{-\gamma}\varphi dx
\leq \liminf_{s\to  0^{+}}\frac{1}{1-\gamma}
\int_{\Omega}Q(x)\frac{(u_{n}+s\varphi)^{1-\gamma}-u_{n}^{1-\gamma}}{s}dx.
$$
Consequently, combining with \eqref{e3.08} and \eqref{e3.07}, it follows that
\begin{align*}
\int_{\Omega}Q(x)u_{n}^{-\gamma}\varphi dx
&\leq\int_{\Omega}((\nabla u_{n},\nabla \varphi)+u_{n}\varphi)dx-
\lambda\int_{\Omega}P(x)u_{n}^{p}\varphi dx\\
&\quad +\frac{C_{4}\|u_{n}\|+\|\varphi\|}{n}
\end{align*}
for $n$ large enough which implies that
$$
\liminf_{n\to \infty}\int_{\Omega}Q(x)u_{n}^{-\gamma}\varphi dx\leq
\int_{\Omega}((\nabla u_{*},\nabla \varphi)+u_{*}\varphi)dx
-\lambda\int_{\Omega}P(x)u_{*}^{p}\varphi dx.
$$
Then applying Fatou's Lemma again, one obtains
$$
\int_{\Omega}Q(x)u_{*}^{-\gamma}\varphi dx\leq \int_{\Omega}((\nabla
u_{*},\nabla \varphi)+u_{*}\varphi)dx-\lambda\int_{\Omega}P(x)u_{*}^{p}\varphi dx;
$$
that is,
\begin{equation}\label{e3.10}
\int_{\Omega}((\nabla u_{*},\nabla \varphi)+u_{*}\varphi -\lambda
P(x)u_{*}^{p}\varphi-Q(x)u_{*}^{-\gamma}\varphi)dx\geq0,
\end{equation}
for all $\varphi\in H^{1}(\Omega)$, $\varphi\geq0$.
This means $u_{*}$ satisfies in the weak sense that
$$
-\Delta u_{*}+u_{*}\geq0, \forall x\in\Omega.
$$
Since $u_{*}\geq0$ and
$u_{*}\not\equiv0$ in $\Omega$, by the strong maximum principle
we have
\begin{equation}\label{e3.11}
u_{*}(x)>0,\quad \text{a.e. } x\in\Omega.
\end{equation}

Thirdly, we prove that $u_{*}\in\Lambda^{+}$.
On one hand, from \eqref{e3.11}, choosing $\varphi=u_{*}$ in \eqref{e3.10}, one has
$$
\|u_{*}\|^{2}\geq\lambda\int_{\Omega}P(x)u_{*}^{p+1}dx
+\int_{\Omega}Q(x)u_{*}^{1-\gamma}dx.
$$
On the other hand, it follows from \eqref{e3.0} that
$$
\|u_{*}\|^{2}\leq\lambda\int_{\Omega}P(x)u_{*}^{p+1}dx
+\int_{\Omega}Q(x)u_{*}^{1-\gamma}dx.
$$
Thus
\begin{equation}\label{e3.12}
\|u_{*}\|^{2}=\lambda\int_{\Omega}P(x)u_{*}^{p+1}dx
+\int_{\Omega}Q(x)u_{*}^{1-\gamma}dx,
\end{equation}
and this implies $u_{*}\in \Lambda$. Moreover from \eqref{e3.000}, one gets
$$
\lim_{n\to \infty}\|u_{n}\|=\lambda\int_{\Omega}P(x)u_{*}^{p+1}dx
+\int_{\Omega}Q(x)u_{*}^{1-\gamma}dx.
$$
Hence according to \eqref{e3.12}, we have $u_n\to  u_*$ in $H^{1}(\Omega)$
as $n\to \infty$.
In particular, combining \eqref{e3.12} with \eqref{e3.00}, we obtain
$$
(1+\gamma)\|u_*\|^{2}-\lambda(p+\gamma)\int_{\Omega}P(x)|u_*|^{p+1}dx>0,
$$
and therefore $u_{*}\in\Lambda^{+}$.

Finally, we prove that $u_*$ is a solution of problem \eqref{e1.1};
 that is, $u_*$ satisfies \eqref{e1.10}.
In fact, we only need prove that \eqref{e3.10} is true for all
$\varphi\in H^{1}(\Omega)$. Our proof is inspired by \cite{SWL01}.
For the convenience of the reader, we sketch the main steps here.
Suppose $\phi\in H^{1}(\Omega)$ and $t>0$. We define
$\Psi\in H^{1}(\Omega)$ by
$$
\Psi\equiv(u_{*}+t\phi)^{+}
$$
where $(u_{*}+t\phi)^{+}=\max \{u_{*}+t\phi,0\}$.
Obviously, $\Psi\geq0$, so we can replace
$\varphi$ with $\Psi$ in \eqref{e3.10}. Combining with \eqref{e3.12} we deduce that
\begin{align*}
0&\leq \int_{\Omega}\big((\nabla
u_{*},\nabla \Psi)+u_{*}\Psi-\lambda P(x)u_{*}^{p}\Psi-Q(x)u_{*}^{-\gamma}\Psi
\big)dx \\
&=\int_{\{x\mid u_{*}+t\phi\geq0\}}\Big[(\nabla u_{*},\nabla (u_{*}+t\phi))
+u_{*}(u_{*}+t\phi)-\lambda P(x)u_{*}^{p}(u_{*}+t\phi)\Big]dx\\
&\quad -\int_{\{x\mid u_{*}+t\phi\geq0\}}Q(x)u_{*}^{-\gamma}(u_{*}+t\phi)dx\\
&=\Big(\|u_{*}\|^{2}-\lambda P(x)u_{*}^{p+1}-\int_\Omega
Q(x)|u_{*}|^{1-\gamma}dx\Big)\\
&\quad + t\int_{\Omega}\big((\nabla
u_{*},\nabla \phi)+u_{*}\phi-\lambda P(x)u_{*}^{p}\phi-Q(x)u_{*}^{-\gamma}\phi\big)dx\\
&\quad -\int_{\{x\mid u_{*}+t\phi<0\}}\big[(\nabla u_{*},\nabla (u_{*}+t\phi))
-\lambda P(x)u_{*}^{p}(u_{*}+t\phi)\big]dx\\
&\quad +\int_{\{x\mid u_{*}+t\phi<0\}}Q(x)u_{*}^{-\gamma}(u_{*}+t\phi)dx\\
&= t\int_{\Omega}\big((\nabla
u_{*},\nabla \phi)+u_{*}\phi-\lambda P(x)u_{*}^{p}\phi-Q(x)u_{*}^{-\gamma}\phi\big)dx\\
&\quad -\int_{\{x\mid u_{*}+t\phi<0\}}\big[(\nabla u_{*},\nabla (u_{*}+t\phi))
-\lambda P(x)u_{*}^{p}(u_{*}+t\phi)\big]dx\\
&\quad +\int_{\{x\mid u_{*}+t\phi<0\}}Q(x)u_{*}^{-\gamma}(u_{*}+t\phi)dx\\
&\leq t\int_{\Omega}\big((\nabla u_{*},\nabla
\phi)+u_{*}\phi-\lambda P(x)u_{*}^{p}\phi-Q(x)u_{*}^{-\gamma}\phi\big)dx\\
&\quad - t\int_{\{x\mid u_{*}+t\phi<0\}}(\nabla u_{*},\nabla \phi)dx.\\
\end{align*}
Since the measure of the domain of integration
$\{x: u_{*}+t\phi<0\}$ tends to zero as $t\to  0^{+}$, it follows
that $\int_{\{x\mid u_{*}+t\phi<0\}}(\nabla u_{*},\nabla \phi)
dx\to 0$ as $t\to  0^{+}$. Dividing by $t$ and letting
$t\to  0^{+}$, we deduce that
$$
\int_{\Omega}\big((\nabla u_{*},\nabla \phi)
+u_{*}\phi-\lambda P(x)u_{*}^{p}\phi-u_{*}^{-\gamma}\phi\big)dx\geq0.
$$
We note that $\phi\in H^{1}(\Omega)$ is arbitrary, which implies that $u_{*}$
is a positive solution of problem \eqref{e1.1}.

\noindent\textbf{Step 2.}
 We prove that there exists a positive solution of problem \eqref{e1.1}
in $\Lambda^{-}$.
Similarly to Step 1, applying Ekeland's variational principle to the
minimization problem $m^{-}=\inf_{u\in
\Lambda^{-}}I_{\lambda}(u)$, there exists a sequence
$\{w_{n}\}\subset
\Lambda^{-}$ with the following properties:
\begin{itemize}
\item[(i)] $I_{\lambda}(w_{n})<m^{-}+\frac{1}{n}$,
\item[(ii)]   $I_{\lambda}(w)\geq I_{\lambda}(w_{n})-\frac{1}{n}\|w-w_{n}\|$,
   for all $w\in \Lambda^{-}$.
\end{itemize}
Since $I_{\lambda}(u)=I_{\lambda}(|u|)$, we may assume
that $w_{n}(x)\geq0$ for all $x\in\Omega$. Obviously, $\{w_{n}\}$ is
bounded in $H^{1}(\Omega)$, going if necessary to a subsequence, still 
denoted by $\{w_n\}$,  there exists $u_{**}\geq0$ such that
\begin{gather*}
w_n\rightharpoonup u_{**},\quad\text{weakly in } H^{1}(\Omega),\\
w_n\to  u_{**},\quad\text{strongly in } L^{s}(\Omega),\; 1\leq s<2^*,\\
w_n(x)\to  u_{**}(x),\quad\text{a. e. in }\Omega,
\end{gather*}
as $n\to \infty$. Now we will prove that $u_{**}$ is a positive solution 
of problem \eqref{e1.1}.

First, we prove that $u_{**}(x)\not\equiv0$ in $\Omega$.
From \eqref{e2.00}, one gets
\begin{equation*}
|w_{n}|_{2^{*}}\geq\Big[\frac{S(1+\gamma)}{\lambda(p+\gamma)|P|_{r_1}}
|\Omega|^{\frac{r_{1}(2^{*}-p-1)-2^{*}}{r_{1}2^{*}}}\Big]^{1/(p-1)},
\end{equation*}
and we obtain $u_{**}\geq0$ and $u_{**}\not\equiv0$ in $\Omega$.

Second, we prove that $u_{**}(x)>0$ a.e. in $\Omega$.
Similarly to the arguments in Step 1, we claim that
\begin{equation}\label{e3.13}
(1+\gamma)\|w_{n}\|^{2}-\lambda(p+\gamma)\int_{\Omega}P(x)|w_{n}|^{p+1}dx
\leq -C_{5}, n=1,2,\cdots,
\end{equation}
where $C_5>0$ is a constant.
Since $w_{n}\in \Lambda$, thus \eqref{e3.13} is  to
\begin{equation}\label{e3.14}
(1+\gamma)\int_\Omega Q(x)|w_{n}|^{1-\gamma}dx-\lambda(p-1)
\int_{\Omega}P(x)|w_n|^{p+1}dx\leq -C_5.
\end{equation}
From $w_{n}\in \Lambda^{-}$, we have
$$
(1+\gamma)\int_\Omega Q(x)|w_{n}|^{1-\gamma}dx
-\lambda(p-1)\int_{\Omega}P(x)|w_n|^{p+1}dx<0,
$$
and combining with \eqref{e3.1} and \eqref{e3.2}, it follows that
\begin{align*}
&\lim_{n\to \infty}\Big[(1+\gamma)\int_\Omega Q(x)|w_{n}|^{1-\gamma}dx
 -\lambda(p-1)\int_{\Omega}P(x)|w_n|^{p+1}dx\Big]\\
&=(1+\gamma)\int_\Omega Q(x)|u_{**}|^{1-\gamma}dx
 -\lambda(p-1)\int_{\Omega}P(x)|u_{**}|^{p+1}dx\leq0.
\end{align*}
Thus we only need prove that
$$
(1+\gamma)\int_\Omega Q(x)|u_{**}|^{1-\gamma}dx
-\lambda(p-1)\int_{\Omega}P(x)|u_{**}|^{p+1}dx<0.
$$
By repeating the proof of \eqref{e3.00} in Step 1.

From Lemma \ref{lem2.2}, choosing $u=w_{n}$, and
$\varphi\in H^{1}(\Omega),\ \varphi\geq0$, $ t>0$ small enough,
we find a sequence of continuous functions $t_{n}=t_{n}(s)$ such
that $t_{n}(0)=1$ and $t_{n}(s)(w_{n}+s\varphi)\in\Lambda^{-}$. 
Similarly to the arguments in Step 1, we also obtain that there exists a constant
$C_{6}>0$, such that
\begin{equation}\label{e3.15}
\limsup_{s\to 0^{+}}|A_{n}(s)|\leq C_{6}
\end{equation}
for $n$ large enough. Here $A_{n}(s)$ is also defined by \eqref{e3.5}.
In the same manner in Step 1, applying (ii) and \eqref{e3.15}, we have
\begin{equation}\label{e3.16}
\int_{\Omega}(\nabla u_{**}\nabla \varphi+u_{**}\varphi
-\lambda P(x)u_{**}^{p}\varphi-Q(x)u_{**}^{-\gamma}\varphi)dx\geq0,
\end{equation}
for all $\varphi\in H^{1}(\Omega),\varphi\geq0$,
which means $u_{**}$ satisfies in the weak sense that 
$$
-\Delta u_{**}+u_{**}\geq0, \quad \forall x\in\Omega.
$$
Since $u_{**}\geq0$ and
$u_{**}\not\equiv0$ in $\Omega$, by the strong maximum principle,
one has
\begin{equation}\label{e3.17}
u_{**}(x)>0,\quad \text{a.e.}  x\in\Omega.
\end{equation}

Finally, according to \eqref{e3.16} and \eqref{e3.17}, we can
repeat the arguments of Step 1, and obtain that $u_{**}\in\Lambda^{-}$ 
is a positive solution of problem \eqref{e1.1}. This complete the proof
of Theorem \ref{thm1.1}.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation of 
China (No. 11071198), the Natural Science Foundation of Education of Guizhou  
Province (No. 2010086), the Science and Technology Foundation of Guizhou 
Province (No. LKZS[2011]2117, No. LKZS[2012]11, No. LKZS[2012]12),
the Fundamental Research Funds for the Central Universities (No. XDJK2014D043).
The authors would like to thank the anonymous referees for their valuable 
suggestions.


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\end{document}