\documentclass[reqno]{amsart}
\usepackage[notref,notcite]{showkeys}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 35, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/35\hfil Existence of solutions]
{Existence of solutions for the p-Laplacian involving a Radon measure}

\author[N. Belhaj Rhouma, W. Sayeb \hfil EJDE-2012/35\hfilneg]
{Nedra Belhaj Rhouma, Wahid Sayeb}  % in alphabetical order

\address{Nedra Belhaj Rhouma \newline
Departement de Math\'ematiques, Facult\'edes sciences de Tunis,
Universit\'e Tunis El Manar, Campus Universitaire 2092, Tunis, Tunisia}
\email{nedra.belhajrhouma@fst.rnu.tn}

\address{Wahid Sayeb \newline 
Departement de Math\'ematiques, Facult\'edes sciences de Tunis,
Universit\'e Tunis El Manar, Campus Universitaire 2092, Tunis, Tunisia}
\email{wahid.sayeb@yahoo.fr}

\thanks{Submitted June 11, 2011. Published February 29, 2012.}
\subjclass[2000]{34B15, 34B18, 35A01, 35A02}
\keywords{Dirichlet problem; $p$-Laplacian; genus function;  eigenfunction;
\hfill\break\indent nonlinear eigenvalue problem; Palais-Smale condition;
 mountain-pass theorem; critical point}

\begin{abstract}
 In this article we study the existence of solutions to eigenvalue problem
 \begin{gather*}
 -\operatorname{div} (|\nabla u|^{p-2}\nabla u)-\lambda |u|^{p-2}u\mu=f \quad
 \text{in }\Omega,\\
 u=0\quad\text{on }\partial\Omega
 \end{gather*}
 where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ and $\mu$ is a
 nonnegative Radon measure.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
%\newtheorem{definition}[theorem]{Definition}

\allowdisplaybreaks

\section{Introduction}

In this article  study the existence of weak solutions of the quasilinear elliptic problem
\begin{equation}  \label{ePlm}
 \begin{gathered}
 -\Delta_{p} u-\lambda|u|^{p-2}u\mu=f(x),\quad\text{in }\Omega,\\
 u=0\quad \text{on }\partial\Omega,
 \end{gathered}
\end{equation}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$,
$N\geq2$, $1<p<N$, and $\mu$ is a nonnegative bounded measure on $\Omega$.

Singular nonlinear problems were studied in
\cite{H-B1,H-B2,cgrad,G-R,K-R,P-R,VRhan,VR,VRW}.
Some recent papers \cite{J-G1,J-G,D-A,H-B,H-B.J-V,S-F.M,E-D}
studied  functional
$$
\frac{1}{p}\int_{\Omega}|\nabla u|^pdx-\frac{\lambda}{p}\int_{\Omega}
\frac{|u|^p}{|x|^p}dx-\int_{\Omega}f(x)u(x)dx,
$$
where $f$ belongs to $L^{p'}(\Omega) $ and $\lambda$ is a real positive number
sufficiently small. This functional is coercive, and one can expect
that there exists a global minimum.
Since the Nemitski operator $u(x)\mapsto \frac{u(x)}{|x|}$ from
$W^{1,p}_0(\Omega)$ in $L^p(\Omega) $ is continuous but not compact, it is not
clear if we can obtain directly the weak lower semicontinuity of the functional
on $W^{1,p}_0(\Omega)$ by using De Giorgi Theorem  \cite{B-D,E-G}, so that it
seems that we cannot apply the direct methods of the calculus of variations.
In \cite{J-G}, using a critical point technique based on the coercivity
and the homogeneity of the functional, it is shown  the existence of a global minimum
(for $\lambda$ belonging to the set in which the functional is coercive) without
using the direct methods of the calculus of variations.
Reference \cite{D-A} treats more general problems with an interesting
nonvariational method, which does not require homogeneity, but only coercivity
of the quadratic form associated to the equation. In any case, both papers
leave open the question of whether the functional is weakly lower semicontinuous.
In \cite{Eg-M}, the author proved that the functionals
$$
\mathcal{H}_{\lambda}(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^pdx
-\frac{\lambda}{p}\int_{\Omega}\frac{|u|^p}{|x|^p}dx
$$
and
$$
\mathcal{S}_{\lambda}(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^pdx
-\frac{\lambda}{p}\Big(\int_{\Omega}|u|^{p^{\star}}dx\Big)^{p/p^*}
$$
are weakly lower semicontinuous in $W^{1,p}_0(\Omega)$,
provided $\lambda$ belongs to the
set of $\mathbb{R}$ in which the functionals are coercive. Note that both functionals
have a nonlinear term which is continuous but not compact on $W^{1,p}_0(\Omega)$.
The author showed the following result.

\begin{theorem} \label{thm1.1}
For all $\lambda\in [0,1/C]$ and all $f\in L^{p'}(\Omega)$, $1<p<N$,
the problem
 \begin{gather*}
 -\Delta_{p} u=\lambda\frac{|u|^{p-2}u}{|x|^p}+f(x),\quad \text{in }\Omega,\\
 u=0\quad \text{on }\partial\Omega,
 \end{gather*}
has a weak solution $u\in W^{1,p}_0(\Omega)$, where
 $C=(\frac{p}{N-p})^p$ is the best constant satisfying
$$
\int_{\Omega}\frac{|u|^p}{|x|^p}dx\leq C\int_{\Omega}|\nabla u|^pdx.
$$
\end{theorem}

For $p=2$, Dupaigne \cite{L-D1} showed that the problem 
 \begin{gather*}
 -\Delta u-\frac{c}{|x|^2}u=f \quad\text{in }\Omega,\\
 u=0\quad\text{on } \partial\Omega,
 \end{gather*}
has a unique solution for all $0<c<(p-2)^2/4$ and $f\in H^{-1}(\Omega)$.
Moreover Peral \cite{I-P} showed that the problem
\begin{gather*}
 -\Delta_{p} u-\frac{\lambda}{|x|^p}|u|^{p-2}u=f\quad\text{in } \Omega,\\
 u=0 \quad\text{on }\partial\Omega,\\
\end{gather*}
has at least one solution in $W^{1,p}_0(\Omega)$ for all $0<\lambda<\frac{1}{C}$
and $f\in W^{-1,p'}(\Omega)$. For the proof of this result, the author used
the convergence Theorem by Boccardo and Murat \cite{B-M}.

\begin{remark} \rm
When $p>2$, the uniqueness is in general not true, see \cite{M-E-R}.
However, the uniqueness in the case $1<p<2$ seems to be an open problem.
\end{remark}

In this paper we assume that the measure $\mu$ is a nonnegative Radon measure
 satisfying the following assumptions.
\begin{itemize}
\item[(H0)]    For each Borel set $A\subset\Omega$, $\mu(A)=0$ implies $|A|=0$,
 where $|\cdot|$ denotes the Lebesgue measure.

\item[(H1)] There exists a constant $C>0$ such that
\[
\int_{\Omega}|u|^pd\mu\leq C\int_{\Omega}|\nabla u|^pdx, \quad
\forall u\in C_0^{\infty}(\Omega).
\]

\item[(H2)]   There exists $(\mu_{n})_{n}\subset \mathcal{M}(\Omega)$ such that
for each integer $n$, the embedding
$ W^{1,p}_0(\Omega,dx)\hookrightarrow L^p(\Omega,\mu_{n})$
is compact, where $\mathcal{M}(\Omega)$ is the set of bounded Radon measures.

\item[(H3)] $\mu_{n}\nearrow\mu$ in $\mathcal{M}(\Omega)$;
i.e., ${\int_{\Omega}\varphi d\mu_{n}\to\int_{\Omega}\varphi d\mu}$,
for all $\varphi\in C_0^{\infty}(\Omega)$.
\end{itemize}

\begin{remark} \rm
When $d\mu(x)=(1/|x|^p)dx$, (H1) is the classical Hardy inequality for $p>1$,
 where the constant $C=(\frac{p}{N-p})^p$ is optimal.
\end{remark}

\begin{remark} \rm
Let ${d\mu(x)=\frac{1}{(\delta(x))^p}dx}$, where $\delta(x)$ is the distance
function to the boundary, the following inequality holds true
(see \cite{S-H.N-T,J-F-P.J-P-G.F-d-T}).
$$
\int_{\Omega}\frac{|u|^p}{(\delta(x))^p}dx
\leq C_{n,p}(\Omega)\int_{\Omega}|\nabla u|^pdx, \quad \forall u\in C_0^{\infty}(\Omega).
$$
Moreover, we will show that ${d\mu(x)=\frac{1}{|x|^p}dx}$ and
 ${d\mu(x)=\frac{1}{(\delta(x))^p}dx}$ are special cases of measures satisfying (H2)
and (H3).
\end{remark}

\begin{theorem} \label{thm0}
The measure ${d\mu(x)=\frac{1}{|x|^p}dx}$ and ${d\mu(x)=\frac{1}{(\delta(x))^p}dx}$
satisfy conditions {\rm (H2)} and {\rm (H3)}.
\end{theorem}

We define the problem
\begin{equation} \label{ePlmn}
\begin{gathered}
   -\operatorname{div} (|\nabla u|^{p-2}\nabla u)-\lambda |u|^{p-2}u\mu_{n}=f
 \quad\text{in }\Omega,\\
   u=0\quad\text{on }\partial\Omega.
   \end{gathered}
\end{equation}

Let $f\in L^{p'}(\Omega)$. We shall say that $u\in W^{1,p}_0(\Omega)$ is a weak
solution of \eqref{ePlmn} (resp. \eqref{ePlm}) if $u$ satisfies
\begin{equation}\label{eq0}
\int_{\Omega}|\nabla u|^{p-2}\nabla u \nabla \varphi dx
-\lambda\int_{\Omega}|u|^{p-2}u \varphi d\mu_{n}=\int_{\Omega}f\varphi dx,
\end{equation}
respectively,
\begin{equation}
\int_{\Omega}|\nabla u|^{p-2}\nabla u \nabla \varphi dx
-\lambda\int_{\Omega}|u|^{p-2}u \varphi d\mu
=\int_{\Omega}f\varphi dx,\;\;\;\forall\varphi \in W^{1,p}_0(\Omega).
\end{equation}

Notice that assumption (H3) ensures that the integral $\int_{\Omega}|u|^{p-2}u\varphi d\mu$
makes sense whenever $u$ and $\varphi$ are in $W^{1,p}_0(\Omega)$.
We prove the following results.
\begin{theorem} \label{thm}\label{thm2}
Let $f\in L^{p'}(\Omega)$, $1<p<N$ and $C$ satisfying
(H1). Then for all $0<\lambda<\frac{1}{C}$, the problem
\eqref{ePlm} has at least a weak solution $u\in
W^{1,p}_0(\Omega)$.
\end{theorem}

\begin{theorem} \label{thm3}
Consider the Dirichlet problem
\begin{equation} \label{ePlam}
\begin{gathered}
-\Delta_{p}u-\lambda|u|^{p-2}u\mu=|u|^{\alpha-2}u \quad \text{in }W^{1,p}_0(\Omega),\\
u=0\quad \text{on }\partial(\Omega).
\end{gathered}
\end{equation}
For every $0<\lambda<1/C$ and $p<\alpha<p^{\star}=Np/(N-p)$,
there exists a nontrivial solution $u\in W^{1,p}_0(\Omega)$.
\end{theorem}

Note that problem \eqref{ePlam} has been studied by Peral \cite{I-P} when
${d\mu(x)=\frac{dx}{|x|^p}}$.\\
Next, we prove an auxiliary result.

\begin{theorem}\label{thm5}
For every $n\in \mathbb{N}$, the problem
\begin{equation} \label{eQlmn}
 \begin{gathered}
 -\Delta_{p} u=\lambda|u|^{p-2}u\mu_{n}\quad \text{in }\Omega,\\
 u=0\quad \text{on }\partial\Omega.
 \end{gathered}
\end{equation}
has a sequence of eigenvalues $(\lambda_k)_{k\in\mathbb{N}}$, such that
${\lim_{k\to\infty}}\lambda_k=+\infty$.
 Moreover, the first eigenvalue $\lambda_1(n)$ is simple, isolated and is defined by
 \begin{equation}\label{eq3}
\lambda_1(n)=\inf\Big\{\|\nabla u\|^p_{L^p}:
u\in W_0^{1,p}(\Omega)\text{ and }\int_{\Omega}|u|^pd\mu_{n}=1\Big\}.
\end{equation}
\end{theorem}

\subsection*{Notation}
for $p>1$, we denote by $p'$ the real number satisfying $\frac{1}{p}+\frac{1}{p'}=1$.
As usual $W^{1,p}(\Omega)$ is the Sobolev space equipped with the norm
$$
\|u\|_{W^{1,p}(\Omega)}=\big(\|u\|_{L^p}^p+\|\nabla u \|_{L^p}^p\big)^{1/p};
$$
$W_0^{1,p}(\Omega)$ is the Sobolev space equipped with the norm
$$
\|u\|=\|u\|_{W_0^{1,p}(\Omega)}=\big(\|\nabla u \|_{L^p}^p\big)^{1/p}.
$$
For a positive Radon measure, we set
$$
L^p(\Omega,\mu)=\{u: u \text{ is measurable and }
\int_{\Omega}|u|^pd\mu<\infty\}.
$$
When $d\mu=dx$, we set $L^p=L^p(\Omega,dx)$.


\begin{proof}[Proof of Theorem \ref{thm0}]
We start by proving that ${d\mu=\frac{1}{|x|^p}dx}$ and
${d\mu=\frac{1}{(\delta(x))^p}dx}$ satisfy conditions (H2) and (H3).
For each $n\in\mathbb{N}^{\star}$, we define $w_{n}(x)=\min(n,|x|^{-p})$ and
$d\mu_{n}(x)=w_{n}(x)dx$.

Case ${d\mu(x)=\frac{1}{|x|^p}dx}$. Since $d\mu_{n}(x)\leq ndx$, (H2) is obvious.
To prove (H3), let $f\in C_0^{\infty}(\Omega)$. Using the fact that $p<N$, we obtain
\[
\int_{\Omega}\frac{f}{|x|^p}dx
\leq \|f\|_{\infty}\int_{\Omega}\frac{1}{|x|^p}dx<+\infty.
\]
On the other hand, since $(fw_{n})_{n}$ converges to $f.\frac{1}{|x|^p}$,
 by Dominated Convergence Theorem we obtain,
$$
\lim_{n\to\infty}\int_{\Omega}f w_{n}dx=\int_{\Omega}\frac{f}{|x|^p}dx.
$$

Case: ${d\mu(x)=\frac{1}{(\delta(x))^p}dx}$.
For  $n\in\mathbb{N}^{\star}$, we define $w_{n}(x)=\min(n,(\delta(x))^{-p})$ and
 $d\mu_{n}(x)=w_{n}(x)dx$.
As in the example from above, (H2) is obvious.
Now using the fact that
$|fw_{n}|\leq\frac{|f|}{(\delta(x))^p}$ and
$$
\int_{\Omega}\frac{f}{(\delta(x))^p}dx\leq C\int_{\Omega}|\nabla f|^pdx,
\quad \forall f\in C_0^{\infty}(\Omega),
$$
we obtain (H3).
\end{proof}

\section{Proof of Theorem \ref{thm5}}

The proof is rather straightforward adaptation of
\cite[Theorem 3.2]{S-H.N-T}
with $d\mu_{n}=v(x)dx$, where the weight $v$ is in $L^{r}$ with
$r=r(p,N)$ satisfying the following conditions
\[
r\begin{cases} >\frac{N}{p} &\text{if }1<p<N,\\
 =1& \text{if } p>N,\\
>p & \text{if }p=N.
 \end{cases}
\]
Let $n\in \mathbb{N}$ fixed and define
$G:W_0^{1,p}(\Omega) \to \mathbb{R}$ and 
$F: W_0^{1,p}(\Omega) \to \mathbb{R}$ by
\[
G(u)= \frac{1}{p}\int_{\Omega}|\nabla u|^p\,dx,\quad
F(u)= \frac{1}{p}\int_{\Omega}|u|^pd\mu_{n}.
\]
 In the sequel we consider the functional
\begin{alignat*}{2}
\phi:& W_0^{1,p}(\Omega)&\to& \mathbb{R}\\
   & u&\mapsto&  (G(u))^2-F(u).
 \end{alignat*}

 \begin{proposition}
The functionals $G$ and $F$ are of class $C^{1}$ on
$W^{1,p}_0(\Omega)$. Moreover
$$
\langle DG(u),v\rangle=\int_{\Omega}|\nabla u|^{p-2} \nabla u \nabla v dx,
$$
and
$$
\langle DF(u),v\rangle = \int_{\Omega}| u|^{p-2} u v d\mu_{n},\quad
\forall v \in W^{1,p}_0(\Omega).
$$
\end{proposition}

\begin{proof}
 We only consider $F$, the proof for $G$ is similar.
 Let $u$ and $\varphi\in W^{1,p}_0(\Omega)$.
 \begin{align*}
 \lim_{t\to0^{+}}\frac{F(u+t\varphi)-F(u)}{t}
 &=\frac{1}{p}\frac{d}{dt}F(u+t\varphi)|_{t=0}\\
 &=\frac{1}{p}\frac{d}{dt}\int_{\Omega}|u+t\varphi|^p|_{t=0}d\mu_{n}\\
 &=\frac{1}{p}\int_{\Omega}\frac{\partial}{\partial t}|u+t\varphi|^p|_{t=0}d\mu_{n}\\
 &=\int_{\Omega}|u|^{p-2}u\varphi d\mu_{n}=\langle DF(u),\varphi\rangle.
 \end{align*}
 The differentiation under the integral is allowed since, if $|t|<1$ then
 \begin{align*}
\|u+t\varphi|^{p-2}(u+t\varphi)\varphi|
&\leq (|u|+|t\|\varphi|)^{p-1}|\varphi|\\
&\leq (|u|+|\varphi|)^{p-1}|\varphi|\in L^{1}(\Omega,\mu_{n}).
\end{align*}
Next, we show that $DF(u)$ is continuous. Indeed, by H\"{o}lder inequality
and using hypotheses (H1)--(H3), we obtain
\begin{align*}
|\langle DF(u),\varphi\rangle|
&=|\int_{\Omega}|u|^{p-2}u\varphi d\mu_{n}|\\
&\leq \int_{\Omega}|u|^{p-1}|\varphi| d\mu_{n}\leq\|u\|^{p-1}_{L^p(\Omega,\mu_{n})}\|\varphi\|_{L^p(\Omega,\mu_{n})}\\
&\leq C \|u\|^{p-1}\|\varphi\|.
\end{align*}
\end{proof}


 \begin{lemma}
 The eigenvalues and eigenfunctions associated to the problem \eqref{eQlmn}
are entirely determined by the nontrivial critical values of $\phi$.
 \end{lemma}

\begin{proof}
 Let $u\not\equiv0$ be a critical point of $\phi$ associated with a critical value $c$,
which means that $\phi(u)=c$ and $\phi'(u)=0$.
Hence
 $$
2G'(u)G(u)=F'(u).
$$
With the condition
$G'(u)\neq0$ we obtain $G(u)=\frac{1}{2\lambda}=\lambda F(u)$ thus
$$
F(u)=\frac{1}{2\lambda^2}
$$
so $c=-G^2(u)$.
Therefore,
\begin{gather*}
\langle G'(u),v\rangle =\frac{1}{2\sqrt{-c}} \langle F'(u),v\rangle
 \quad\text{for all }v\in C_{c}^{\infty}(\Omega),\\
\langle \phi'(u),v\rangle=\frac{1}{2\sqrt{-c}}\langle F'(u),v\rangle \quad
\text{for all }v\in C^{\infty}_{c}(\Omega).
\end{gather*}
Thus, we deduce that $\lambda=1/(2\sqrt{-c})$ is a positive eigenvalue
of \eqref{eQlmn} and $u$ is its associated eigenfunction.
 Conversely, let $(u\not\equiv0,\lambda)$ be a solution of
\eqref{eQlmn}. Then, for every $\beta\in\mathbb{R}^{\star}$, $\beta u$
 is also an eigenfunction associated to $\lambda$.
In particular for 
$\beta=1/(2\lambda G(u))^{1/p}$, the function
 $v=(2\lambda G(u))^{-1/p}u$ is an eigenfunction associated to
$\lambda=1/(2\sqrt{-c})$, which proves that $v$ is a critical point
associated to the critical value $c=-1/(4\lambda^2)$.
\end{proof}

 Next, we recall the Genus function defined as follows
$\gamma: \Sigma \to \mathbb{N}\cup\{\infty\}$,
where $\Sigma=\{A\subset W^{1,p}_0(\Omega):A \text{ is closed },A=-A\}$ by
 $$
\gamma(A)=\min\{i\in\mathbb{N}: \exists\;\varphi\in C(A,\mathbb{R}^{i}\backslash\{0\}),
\;\varphi(x)=-\varphi(-x)\}.
$$
 Let us now consider the sequence
 \begin{equation} \label{q3}
 c_k=\inf_{K\in A_k}\sup_{v\in K}\phi(v),
\end{equation}
 where for $k\geq1$, and
$$
A_k=\{K\subset W^{1,p}_0(\Omega):
 K \text{ is compact symmetric and }\gamma(K)\geq k\}.
$$

 \begin{proposition}
 The values $c_k$ defined by \eqref{q3} are the critical values of $\phi$.
 Moreover $c_k<0$ for $k\geq1$ and $\lim_{k\to\infty}c_k=0$.
 \end{proposition}

\begin{proof}
The proof is based on the fundamental theorem of multiplicity and the
 approximation of Sobolev imbedding by operators of finite rank.
 We first show that for all $k\geq1$, $c_k$ is a critical value of $\phi$ and $c_k<0$.
 Since $\phi$ is even and is $C^{1}$ on $W_0^{1,p}(\Omega)$, then the result
follows from the fundamental theorem of multiplicity if
 $\phi$ satisfies the following conditions:
 \begin{enumerate}
 \item $\phi$ is bounded below.
 \item $\phi$ verify the Palais-Smale condition (P-S).
 \item For all $k\geq1$, there exists a compact symmetric subset $K$ such
that $\gamma(K)=k$ and $\sup_{v\in K}\phi(v)<0$.
 \end{enumerate}
 Let us verify assertion (1). Indeed, condition (H1) implies
that
$$
\phi(u)\geq \frac{1}{p^2}\|u\|^p(\|u\|^p-Cp),\quad
\forall u\in W^{1,p}_0(\Omega),
$$
which proves that $\phi$ is bounded below and $\phi(u)\to+\infty$ as $\|u\|\to+\infty$.

Assertion (2). We show that $\phi$ verify the Palais-Smale condition.
Let $(u_k)_k$ be a sequence in $W_0^{1,p}(\Omega)$ such that $(\phi(u_k))_k$ is
bounded and $(\phi'(u_k))_k\to0$ in $(W_0^{1,p}(\Omega))'$.
Since $\phi$ is coercive then $(u_k)_k$ is bounded in $W_0^{1,p}(\Omega)$.
Thus, there exists a subsequence still denoted by $(u_k)_k$ such that
$(\nabla u_k)_k$ converges to $\nabla u$ weakly in $L^p$, and
$(u_k)_k$ converges to  strongly in $L^p$. By (H3), we obtain that
$(u_k)_k$ converges strongly in $L^p(\Omega,\mu_{n})$.
Suppose that $(\|u_k\|)_k$ converges to some constant $\alpha\geq0$.
We distinguish two cases.

 Case 1:  $\alpha=0$. Since $(u_k)_k\rightharpoonup u$ in $W_0^{1,p}(\Omega)$
and $\|u_k\| \to0$, then $(u_k)_k\to0$ in $W_0^{1,p}(\Omega)$.
Consequently, the condition (P-S) is satisfied.

Case 2: $\alpha>0$. For $k\geq1$ we have
$$
\phi'(u_k)=2G(u_k)G'(u_k)-F'(u_k)
$$
which yields
$$
G'(u_k)=\frac{1}{2G(u_k)}(\phi'(u_k)+F'(u_k));
$$
i.e.,
$$
\frac{p}{2}\frac{(\phi'(u_k)+F'(u_k))}{\|\nabla u_k\|^p_{L^p}}=G'(u_k).
$$
Since $u\mapsto |u|^{p-2}u$ is strongly continuous in
$L^p(\Omega,\mu_{n})$, $\|u_k\|\to\alpha>0$ and $(\phi'(u_k))_k\to0$,
then the expression 
$$
V_k=\frac{p}{2}\frac{(\phi'(u_k)+F'(u_k))}{\|\nabla u_k\|^p_{L^p}}
$$
converges strongly in $(W_0^{1,p}(\Omega))'$. However, $G'$ is continuous,
thus $u_k=(G')^{-1}V_k$ converge strongly in $W_0^{1,p}(\Omega)$, from where
the (P-S) condition holds.

Next, we prove (3). Indeed, by (H0), there exists a family of balls
$(B_i)_{1\leq i\leq k}$ in $\Omega$ such that $B_i\cap B_{j}=\emptyset$ if
$i\neq j$ and $\mu_{n}(\Omega\cap B_i)\neq0$.
We define
\[
v_i=  \begin{cases}
 u_i(x) \big( \int_{B_i}|u_i|^pd\mu_{n}\big) ^{-1/p}&\text{if }x\in B_i,\\
0 &\text{if } x\in\Omega\backslash B_i.
 \end{cases}
\]
 Let $X_k$ denote subspace of $W_0^{1,p}(\Omega)$ spanned by
 $\{v_1, v_{2},\dots ,v_k\}$. Since the $v_i$'s are linearly independent,
we have that $\dim X_k=k$.
For each $v\in X_k$, $v={\sum_{i=1}^{i=k}}\alpha_iv_i$, we obtain
$F(v)={\sum_{i=1}^{i=k}}|\alpha_i|^p$.

Thus $u\mapsto (F(u))^{1/p}$ defines a norm on $X_k$. Then there exists $c>0$
such that
$$
cF(u)\leq G(u)\leq\frac{1}{c}F(u)\quad \forall u\in X_k.
$$
Let $K$ be defined as
$$
K=\{u\in W_0^{1,p}(\Omega)\text{ such that } \frac{c^2}{3}\leq F(u)\leq\frac{c^2}{2}\}.
$$
It is clear that $K_1=K\cap X_k\neq\emptyset$ and
$\sup_{u\in K_1}\phi(u)<-c/12<0$. Since $X_k$ is isomorphic
to $\mathbb{R}^{k}$, one can identify $K_1$ to a crown $K'_1$ of $\mathbb{R}^{k}$
such that $S^{k-1}\subset K'_1\subset\mathbb{R}^{k}\backslash\{0\}$ where
 $S^{k-1}$ is the unit sphere of $\mathbb{R}^{k}$.
Then $\gamma(K_1)=k$ and the result follows.

Finally, we shall prove that $\lim_{k\to+\infty}c_k=0$.
Consider $\{E_i\}$ sequence of linear subspaces in $W_0^{1,p}(\Omega)$, such that
\begin{itemize}
\item $E_i \subset E_{i+1}$,
\item $\overline{\cup E_i} = W_0^{1,p}(\Omega)$,
\item  $\dim (E_i) = i$.
\end{itemize}
   Define
   $$
\widetilde{c_k}=\inf_{K\in A_k}\sup_{v\in K\cap E_{i-1}^c}\phi(v)
$$
where $E_i^c$ is the linear topological complementary of $E_i$.
Obviously $\widetilde{c_k}\leq c_k < 0$. So, it is sufficient to prove that
$$
\lim_{k\to+\infty} \widetilde{c_k} =0.
$$
Assume, by contradiction, that there exists a constant $\alpha < 0$ such that
$\widetilde{c_k} < \alpha < 0$ for all $k\in\mathbb{N}$,
then for each $k\in\mathbb{N}$, there exists $K_k$ such that
$\widetilde{c_k} < \sup_{u\in K_k\cap E_{i-1}^c}\phi(u)<\alpha$
and there exists $u_k \in K_k\cap E^c_{i-1}$
such that $\widetilde{c_k}<\phi(u_k)<\alpha$. In this way,
$\phi$ is bounded, hence for some subsequence still denoted $(u_k)$,
\begin{gather*}
u_k\rightharpoonup u\quad \text{in } W_0^{1,p}(\Omega),\\
u_k \to u\quad \text{in } L^p(\Omega,\mu_{n}).
\end{gather*}
Hence $\phi(u) <\alpha<0$,
which is a contradiction with the fact that $u\equiv 0$ because
$u_k \in E^c_{i-1}$.
\end{proof}

\begin{remark} \rm
It is clear that the sequence $(\lambda_k)_k$ defined by the
formula\\$\lambda_k=\frac{1}{2\sqrt{-c_k}}\to+\infty$ as
$k\to+\infty$.
\end{remark}
\begin{remark} \rm
We consider $\lambda_k={\inf_{K\in\Gamma_k}\sup_{u\in K}}G(u)$, where
 $\Gamma_k$ is define by
$$
\{K\subset W^{1,p}_0(\Omega)\backslash\{0\}:K
\text{ is compact, symmetric }\gamma(K)\geq k, \|u\|_{L^p(\Omega,\mu_{n})}=1\}.
$$
\end{remark}

Particulary
\[
\lambda_1(n)=\inf\big\{ \|\nabla u\|^p_{L^p}:
u\in W_0^{1,p}(\Omega)\text{ and }\|u\|_{L^p(\Omega,\mu_{n})}=1\big\}.
\]
Moreover, using \cite[Theorem 4.11]{L-M.W-P-Z}, we obtain the following result.

\begin{theorem} \label{thm3.1}
If $u\in W^{1,p}_0(\Omega)$ is an eigenfunction of \eqref{eq0}, then $u$
is continuous in $\Omega$.
\end{theorem}

In what follows we will use the so-called Picone's identity proved in \cite{W-A}.
We recall it here for completeness.

\begin{theorem}[Picone's identity]\label{thm4}
Let $u>0$, $v>0$ be two continuous functions in $\Omega$, differentiable a.e..
Denote
\begin{gather*}
L(u,v)=|\nabla u|^p+(p-1)\frac{u^p}{v^p}|\nabla v|^p
 -p\frac{u^{p-1}}{v^{p-1}}|\nabla v|^{p-2}-\nabla u\nabla v,\\
R(u,v)=|\nabla u|^p-|\nabla v|^{p-2}\nabla\Big(\frac{u^p}{v^{p-1}}\Big)\nabla v.
\end{gather*}
Then
\begin{itemize}
\item[(i)]   $L(u,v)=R(u,v)$,
\item[(ii)]  $L(u,v) \geq 0$ a.e.,
\item[(iii)] $L(u,v)=0$ a.e. in $\Omega$ if and only if $u=kv$ for some $k\in\mathbb{R}$.
\end{itemize}
\end{theorem}

We will show that the first eigenvalue $\lambda_1(n)$ of
\eqref{eQlmn} defined by \eqref{eq3} is simple and isolated, and only
 eigenfunctions associated with $\lambda_1(n)$ do not change sign.

\begin{proposition}\label{p:p3}
The first eigenvalue $\lambda_1(n)$ is simple.
\end{proposition}

\begin{proof}
Let $u, v$ be two eigenfunctions associated to $\lambda_1(n)$ and fixed $\epsilon>0$.
 We can assume without restriction that $u$ and $v$ are positive in $\Omega$.
From Picone's identity we have
\begin{align*}
\int_{\Omega}L(u,v+\epsilon)dx
&=\int_{\Omega}R(u,v+\epsilon)dx\\
&=\lambda_1(n)\int_{\Omega}u^pd\mu_{n}
 -\int_{\Omega}|\nabla v|^{p-2}\nabla(\frac{u^p}{(v+\epsilon)^{p-1}})\nabla vdx.
\end{align*}
The functional $u^p/(v+\epsilon)^{p-1}$ belongs to $W^{1,p}_0(\Omega)$
and then it is admissible for the weak formulation of
$-\Delta_{p}u=\lambda_1(n)|u|^{p-2}u\mu_{n}$. It follows that
$$
0\leq \int_{\Omega} L(u,v+\epsilon)dx
=\lambda_1(n)\int_{\Omega}u^p(1-\frac{v^{p-1}}{(v+\epsilon)^{p-1}})d\mu_{n}.
$$
Letting $\epsilon\to0$, we obtain $L(u,v)=0$, a.e. in $\Omega$, and therefore
using (iii), we obtain $u=kv$.
\end{proof}

\begin{proposition}\label{prop1}
Let $u\in W^{1,p}_0(\Omega)$ be a nonnegative weak solution of \eqref{eQlmn},
then either $u\equiv0$ or $u(x)>0$ for all $x\in\Omega$.
\end{proposition}

The proof of the above proposition is a direct consequence of Harnack's inequality,
see \cite{M-S,W-M}.

\begin{theorem} \label{thm3.3}
Let $(u,\lambda)\in W^{1,p}_0(\Omega)\times\mathbb{R}_{+}$ be an eigensolution
of \eqref{ePlm}. Then $u\in L^{\infty}(\Omega,\mu_{n})$.
\end{theorem}

The proof of the above theorem is rather a straightforward adaptation of
\cite[Theorem 4.1]{A-L} with $d\mu_{n}=dx$.

\begin{theorem} \label{thm3.4}
Let $u$ be an eigenfunction of  \eqref{eQlmn} associated to an eigenvalue
$\lambda\neq\lambda_1(n)$ and $1\leq q<p$. We define
$$
I=\min\big\{\int_{\Omega}|u|^{q}d\mu_{n}, u\in L^p(\Omega,\mu_{n}),\,
\int_{\Omega}|u|^pd\mu_{n}=1\big\}.
$$
Then
\begin{equation}
\min(\mu_{n}(\Omega^{-}),\mu_{n}(\Omega^{+}))
\geq((C\lambda)^{-1/p}I)^{\frac{pq}{p-q}},
\end{equation}
where $\Omega^{+}=\{x\in\Omega,\,u(x)>0\}$ and $\Omega^{-}=\{x\in\Omega,\,u(x)<0\}$.
\end{theorem}

\begin{proof}
Let $u$ be an eigenfunction associated to $\lambda$, then
\begin{equation}
\int_{\Omega}|\nabla u|^{p-2}\nabla u \nabla v dx
=\lambda\int_{\Omega}|u|^{p-2}uv d\mu_{n},\quad
\forall v\in W^{1,p}_0(\Omega).
\end{equation}
For $\lambda\neq\lambda_1(n)$, $u$ changes sign i.e., $u^{+}\neq 0$ and $u^{-}\neq0$.
Since $u^{+}\in W^{1,p}_0(\Omega)$ we have
$$
\int_{\Omega}|\nabla u^{+}|^pdx=\lambda\int_{\Omega}|u^{+}|^pd\mu_{n}.$$
For $1\leq q<p$, we have:
\begin{align*}
\int_{\Omega}|u^{+}|^{q}d\mu_{n}
&\leq \Big(\int_{\Omega}|u^{+}|^pd\mu_{n}\Big)^{q/p}(\mu_{n}(\Omega^{+}))^{1-\frac{q}{p}}\\
&\leq C^{q/p}\mu_{n}(\Omega^{+})^{1-\frac{q}{p}}
\Big(\int_{\Omega}|\nabla u^{+}|^pdx\Big)^{q/p}\\
&\leq (\lambda C)^{q/p}\mu_{n}(\Omega^{+})^{1-\frac{q}{p}}
 \Big(\int_{\Omega}| u^{+}|^pd\mu_{n}\Big)^{q/p}
\end{align*}
and
\[
\|u^{+}\|^q_{L^{q}(\Omega,\mu_{n})}
\leq (\lambda C)^{q/p}\mu_{n}(\Omega^{+})^{\frac{p-q}{pp}}
\|u^{+}\|_{L^p(\Omega,\mu_{n})}.
\]
Finally
\begin{equation}\label{eq12}
\mu_{n}(\Omega^{+})\geq ((C\lambda)^{-1/p}I)^{\frac{pq}{p-q}}.
\end{equation}
\end{proof}

Now, we establish the isolation of the first eigenvalue.

\begin{theorem} \label{thm3.5}
Let $\lambda_1(n)$ be the first eigenvalue of the problem \eqref{eQlmn}.
Then $\lambda_1(n)$ is isolated.
\end{theorem}

\begin{proof}
 Our approach is related to the method of \cite{A-N1,A-N}.
Let $\lambda>0$ be an eigenvalue of \eqref{eQlmn} and let $v$ be the corresponding
eigenfunction. By \eqref{eq3}, it follows that $\lambda_1(n)<\lambda$
and so $\lambda_1(n)$ is left-isolated.
To prove that $\lambda_1(n)$ is right-isolated, we argue by contradiction.
We suppose that there exists a sequence of eigenvalues $(\lambda_k)_{k\in\mathbb{N}}$,
such that $\lambda_k\neq \lambda_1(n)$ and $\lambda_k\to\lambda_1(n)$.
Let $(u_k)_{k\in\mathbb{N}}$ be the corresponding sequence of eigenfunctions such that
\begin{equation}\label{eq10}
\int_{\Omega}|\nabla u_k|^pdx=1,\quad \forall\;k\in\mathbb{N}.
\end{equation}
There exists a subsequence, denoted again by $(u_k)_k$ and a function
$u \in W^{1;p}_0(\Omega)$ such that
\begin{gather*}
u_k\rightharpoonup u \quad\text{on } W^{1,p}_0(\Omega)\\
u_k\to u \quad\text{on } L^p(\Omega,\mu_{n}).
\end{gather*}
Our next aim is to show that $u$ is the eigenfunction corresponding to $\lambda_1(n)$.
First, since $-\Delta_{p}$ is a continuous and one-to-one operator from
$W^{1,p}_0(\Omega)$ into $W^{-1,p'}_0(\Omega)$ and so is its inverse operator
$(-\Delta_{p})^{-1}$ defined from $W^{-1,p'}_0(\Omega)$ into $W^{1,p}_0(\Omega)$
(see \cite{I-P}). Thus,
$$
u_k = (-\Delta_{p})^{-1} (\lambda_k|u_k|^{p-2}u\mu_{n}).
$$
By Vitali's Theorem, we have
$$
\lambda_k u_k^{p-2} u_k  \to \lambda |u|^{p-2} u\quad
\text{strongly in } L^{\frac{p}{p-1}}(\Omega,\mu_{n})\hookrightarrow W^{-1,p'}(\Omega).
$$
The continuity property of $(-\Delta_{p})^{-1}$ implies that
$$
u_k \to u\quad \text{strongly in } W^{1,p}_0 (\Omega).
$$
Hence, $u$ is an eigenfunction of \eqref{eQlmn}, corresponding to
$\lambda_1(n)$. Using Vitali's Theorem, again, we have
\begin{gather*}
|\nabla u_k|^{p-2}\nabla u_k\to |\nabla u|^{p-2}\nabla u\quad
\text{strongly in } L^{1}. \\
\int_{\Omega}|u_k|^{p-2}u_kvd\mu_{n}
\leq\Big(\int_{\Omega}|u_k|^pd\mu_{n}\Big)^{\frac{p-1}{p}}.
\Big(\int_{\Omega}|v|^pd\mu_{n}\Big)^{1/p}\leq\|v\|.
\end{gather*}
It should appear a constant $\eta_{\epsilon} > 0$ for every $\epsilon>0$ and
 $\Omega_{\epsilon}\subset\Omega$ such that
\begin{equation}
\mu_{n}(\Omega\backslash\Omega_{\epsilon})\leq\frac{\epsilon}{2}
\quad\text{and}\quad
u(x)\geq 2\eta_{\epsilon}\quad \text{for every } x\in\Omega_{\epsilon}.
\end{equation}
Let us denote
\begin{gather}\label{e:d}
\Omega^{+}_k=\{x\in\Omega,\, u_k(x)>0\},\\
\label{e:f}\Omega^{-}_k=\{x\in\Omega,\,u_k(x)<0\}.
\end{gather}
Moreover, by Egorov's Theorem,
there exists $\Omega'_{\epsilon}\subset\Omega$ such that
$$
\mu_{n}(\Omega\backslash\Omega'_{\epsilon})\leq\frac{\epsilon}{2}
$$
and $u_k$ converges uniformly to $u$. On the other hand, there exists $N_{\epsilon} > 0$
such that for every $k > N_{\epsilon}$, we have
$$
\Omega_{\epsilon}\cap\Omega'_{\epsilon}\subset\Omega^{+}_k
$$
and then
$$
\mu_{n}(\Omega_k^{+})\geq\mu_{n}(\Omega_{\epsilon}\cap\Omega'_{\epsilon})
\geq\mu_{n}(\Omega)-(\mu_{n}(\Omega\backslash\Omega'_{\epsilon})
+\mu_{n}(\Omega\backslash\Omega_{\epsilon}))\geq\mu_{n}(\Omega)-\epsilon.
$$
Hence, it follows that $\mu_{n}(\Omega_k^{+})$ and $\;\mu_{n}(\Omega_k^{-})\geq K$,
 where $K=((C\lambda)^{-1/p}I)^{\frac{pq}{p-q}}$.
If we choose $\epsilon= \frac{K}{2}$, we obtain
$$
\mu_{n}(\Omega)=\mu_{n}(\Omega_k^{-})+\mu_{n}(\Omega^{+}_k)\geq\mu_{n}(\Omega)
-\epsilon+K=\mu_{n}(\Omega)+\epsilon>\mu_{n}(\Omega),
$$
which is a contradiction. Therefore $\lambda_1(n)$ is isolated.
\end{proof}

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

\begin{lemma}
Let $\lambda_1(n)$  be the first eigenvalue associated to \eqref{eQlmn}.
Then, $\lambda_1(n)\geq\frac{1}{C}$ and ${\lim_{n\to\infty}}\lambda_1(n)=\frac{1}{C}$.
\end{lemma}

\begin{proof}
 Notice that
 $$
\lambda_1(n)={\inf_{\Omega}}\frac{\int_{\Omega}|\nabla u|^pdx}{\int_{\Omega}|u|^pd\mu_{n}}
\geq{\inf_{\Omega}}\frac{\int_{\Omega}|\nabla u|^pdx}{\int_{\Omega}|u|^pd\mu}
\geq\frac{1}{C}.
$$
Since  $(\lambda_1(n))_{n}$ is a non increasing sequence, we have to prove that
the limit can not be larger than $\frac{1}{C}$.
Assume by contradiction that ${\lim_{n\to\infty}\lambda_1(n)}=\frac{1}{C}+\delta$,
for some $\delta>0$.
Then we can choose $\phi\in C^{\infty}_0(\Omega)$ such that
 $$
\frac{\int_{\Omega}|\nabla \phi|^pdx}{\int_{\Omega}|\phi|^pd\mu}
<\frac{1}{C}+\frac{\delta}{2}.
$$
Which gives us
$$
\lambda_1(n)\leq\frac{\int_{\Omega}|\nabla \phi|^pdx}{\int_{\Omega}|\phi|^pd\mu_{n}}
\leq\frac{1}{C}+\frac{\delta}{2}
$$
for $n$ large enough.
\end{proof}

In the sequel for $\lambda>0$ let us denote by $\mathfrak{L}_{\lambda}^{\mu_{n}}$
the operator defined on $W^{1,p}_0(\Omega)$ by
$$
\mathfrak{L}_{\lambda}^{\mu_{n}}u=-\Delta_{p}u-\lambda |u|^{p-2}u\mu_{n}.
$$
The first result in this section is an easy consequence of the Hardy's inequality.

\begin{lemma}
 If $0<\lambda<\frac{1}{C}$, then $\mathfrak{L}_{\lambda}^{\mu_{n}}$
is a positive operator.
 \end{lemma}

\begin{proof}
 From assumption (H1) we have
$$
\langle \mathfrak{L}_{\lambda}^{\mu_{n}}u,u\rangle
\geq (1-\lambda C)\|u\|^p\geq0
$$
whenever $0<\lambda<1/C$.
\end{proof}

Next we recall a formula from \cite{J}.

\begin{lemma}\label{lem6}
Let $a, b \in\mathbb{R}^{N}$ and $\langle.,.\rangle$ be the standard scalar
product in $\mathbb{R}^{N}$. Then
\[
 \langle|a|^{p-2}a-|b|^{p-2}b,(a-b)\rangle
\geq \begin{cases}
C_{p}|a-b|^p & \text{if } p\geq2\\
C_{p}\frac{|a-b|^2}{(|a|+|b|)^{2-p}}&\text{if }  1<p<2.
\end{cases}
\]
\end{lemma}

\begin{lemma} 
The operator $\mathfrak{L}_{\lambda}^{\mu_{n}}:W_0^{1,p}(\Omega)\to W^{-1,p'}(\Omega)$ 
is uniformly continuous on bounded sets.
\end{lemma}

\begin{proof}
Assume $p>2$ and consider $K\subset W^{1,p}_0(\Omega)$ be a
bounded set; i.e., there exists $M>0$ such that  
$$
\|u\|\leq M,\quad \forall u\in K.
$$ 
Then, using Lemma \ref{lem6} and H\"older inequality,  for
 $u,v\in K$ and $\phi\in W^{1,p}_0(\Omega)$, we obtain
\begin{align*}
&|\langle\mathfrak{L}_{\lambda}^{\mu_{n}}u-\mathfrak{L}_{\lambda}^{\mu_{n}}v,\phi
 \rangle|\\
&\leq \int_{\Omega}(|\nabla u|^{p-2}+|\nabla v|^{p-2})|\nabla u-\nabla v| |\nabla \phi|dx
+\lambda\int_{\Omega}(|u|^{p-2}+|v|^{p-2})|u-v\|\phi|d\mu_{n}\\
&\leq 2c_{p}M^{p-2}\|\nabla u-\nabla v\|_{L^p}
 +2\lambda c_{p}M^{p-2}\|u-v\|_{L^p(\Omega,\mu_{n})}\\
&\leq 2c_{p}M^{p-2}(\min\{1,C\lambda\})\|u-v\|.
\end{align*}
The same process is applied for $1<p<2$.
\end{proof}

\begin{lemma}\label{lem2} 
The operator  $\mathfrak{L}_{\lambda}^{\mu_{n}}:W_0^{1,p}(\Omega)\to W^{-1,p'}(\Omega)$
is pseudo-monotone.
 \end{lemma}

\begin{proof}
Let $(u_k)_{k\geq1}\subset W^{1,p}_0(\Omega)$ such that 
$u_k\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and
$$
\limsup_{k\to\infty}<\mathfrak{L}_{\lambda}^{\mu_{n}}u_k,u_k-u>\leq0.
$$
We want to prove that 
$$\liminf\langle \mathfrak{L}_{\lambda}^{\mu_{n}}u_k,u_k-v\rangle
\geq\langle \mathfrak{L}_{\lambda}^{\mu_{n}}u,u-v\rangle \quad
\text{for all } v\in W^{1,p}_0(\Omega).
$$
Since $u_k\rightharpoonup u$ in $W^{1,p}_0(\Omega)$, it follows that
\begin{equation}\label{eq13}
\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla(u_k-u)dx\to0,\quad
\text{as }k\to+\infty.
\end{equation}
We estimate
$$
\int_{\Omega}|u_k|^{p-2}u_k(u_k-u)d\mu_{n}
\leq\|u_k\|^{p-1}_{L^p(\Omega,\mu_{n})}\|u_k-u\|_{L^p(\Omega,\mu_{n})}.
$$
Through $u_k\to u$ in $L^p(\Omega,\mu_{n})$  and  $u_k$ is bounded in 
$L^p(\Omega,\mu_{n})$ then
$$
\int_{\Omega}|u_k|^{p-2}u_k(u_k-u)d\mu_{n}\to0,\quad \text{as } k\to+\infty.
$$
So 
$$
\limsup_{k\to\infty} \Big(\int_{\Omega}|\nabla u_k|^{p-2}\nabla u_k\nabla(u_k-u)dx
+(-\lambda\int_{\Omega}|u_k|^{p-2}u_k(u_k-u)d\mu_{n})\Big)\leq0,
$$
which yields
\begin{equation}\label{eq14}
\limsup_{k\to\infty}\int_{\Omega}|\nabla u_k|^{p-2}\nabla u_k\nabla(u_k-u)dx\leq0.
\end{equation}
Combining \eqref{eq13}, \eqref{eq14} and Lemma \ref{lem6} we obtain that
$(u_k)_k\to u$ in $W^{1,p}_0(\Omega)$. The proof is complete.
\end{proof}

\begin{proposition} \label{p:p2}
For every $0<\lambda<1/C$, the operator $\mathfrak{L}_{\lambda}^{\mu_{n}}$ is coercive.
\end{proposition}

\begin{proof} 
Using (H1)--(\H3), we have
\[
\langle \mathfrak{L}_{\lambda}^{\mu_{n}}u,u \rangle
= \int_{\Omega}|\nabla u|^pdx-\lambda\int_{\Omega}|u|^pd\mu_{n}
\geq(1-\lambda C)\int_{\Omega}|\nabla u|^pdx,
\]
which implies that $\mathfrak{L}_{\lambda}^{\mu_{n}}$ is coercive,
 whenever $0<\lambda<\frac{1}{C}$.
\end{proof}

By Proposition \ref{p:p2} and Lemma \ref{lem2} the operator
$\mathfrak{L}_{\lambda}^{\mu_{n}}:W_0^{1,p}(\Omega)\to W^{-1,p'}(\Omega)$ is coercive, 
bounded from below and pseudo-monotone. Hence, by \cite[Theorem 4.11]{L-M.W-P-Z}, 
it is onto. Thus we have the following result.

\begin{theorem} \label{thm6}
For every $f\in L^{p'}$, there exists $u_{n}\in W^{1,p}_0(\Omega)$ which is a 
solution of  \eqref{ePlmn}.
\end{theorem}

\begin{lemma}\label{lem5}
For each $n\in\mathbb{N}$, let $u_{n}$ be a solution of the Dirichlet problem \eqref{ePlmn}.
Then the sequence $(u_{n})_{n}$ is bounded in $W^{1,p}_0(\Omega)$.
\end{lemma}

\begin{proof} 
Since
$$
\int_{\Omega}|\nabla u_{n}|^pdx-\lambda\int_{\Omega}|u_{n}|^pd\mu_{n}
=\int_{\Omega}fu_{n}dx,
$$
and using (H1), we obtain
\begin{gather*}
(1-\lambda C)\|u_{n}\|^p\leq \|f\|_{L^{p'}}\|u_{n}\|_{L^p},\\
\|u_{n}\|\leq \Big(C_1\frac{\|f\|_{L^{p'}}}{(1-\lambda C)}\Big)^{\frac{1}{p-1}},
\end{gather*}
where $C_1$ is the positive constant of the continuous of Sobolev embedding satisfied.
\end{proof}

\begin{lemma}\label{lem4}
Let $(u_{n})_{n}$ be the sequence as defined in Theorem \ref{thm6}. 
Then $(u_{n})_{n}$ converges to a weak solution $u$ of \eqref{ePlm}.
\end{lemma}

\begin{proof}
By Lemma \ref{lem5}, since $u_{n}$ is bounded in $W^{1,p}_0(\Omega)$, we have
\begin{equation}\label{eqx}
\begin{gathered}
u_{n}\rightharpoonup u\quad  \text{in }W^{1,p}_0(\Omega),\\
u_{n}\rightharpoonup u\quad \text{in }L^p(\Omega,\mu),\\
u_{n}\to u\quad \text{in }L^p.
\end{gathered}
\end{equation}
Then
$$
\int_{\Omega}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \varphi dx 
\to |\nabla u|^{p-2}\nabla u\nabla \varphi dx,\quad \text{for all }
\varphi\in W^{1,p}_0(\Omega).
$$
Next, we show that
$$
\int_{\Omega}|u_{n}|^{p-2}u_{n}\varphi d\mu_{n}\to\int_{\Omega}|u|^{p-2}u\varphi d\mu,
\quad\text{for all }\varphi\in C^{\infty}_0(\Omega).
$$
 Indeed, we have
\begin{align*}
&\big|\int_{\Omega}|u_{n}|^{p-2}u_{n}\varphi d\mu_{n}
 -\int_{\Omega}|u|^{p-2}u\varphi d\mu\big|\\
&=\big|\int_{\Omega}|u|^{p-2}u\varphi d(\mu-\mu_{n})
 -\int_{\Omega}(|u_{n}|^{p-2}u_{n}-|u|^{p-2}u)\varphi d\mu_{n}\big|\\
&\leq \|u\|^{p-1}_{L^p(\Omega,\mu-\mu_{n})}\|\varphi\|_{L^p(\Omega,\mu-\mu_{n})}
 +|\int_{\Omega}(|u_{n}|^{p-2}u_{n}-|u|^{p-2}u|)\varphi|d\mu_{n}|.
\end{align*}
So, by (H3) the first integral converges to $0$, as $n \to\infty$; respectively
by the weak convergence in \eqref{eqx}, the second integral converges to $0$,
as $n\to\infty$.
Therefore, $u$ is a solution of our problem in the sense of distributions.
Moreover by density argument and taking into account that $u\in W^{1,p}_0(\Omega)$,
we conclude that $u$ is solution in the sense of $W^{1,p}_0(\Omega)$.
\end{proof}


\section{Proof of Theorem \ref{thm3}}

Some recent papers \cite{G-R,K-R,P-R,R-P,VR,J-S}  considered a class of functionals 
with the minimax method. We will use again the variational approach to study 
the case of unbounded functionals, more precisely the existence of solution via 
the mountain-pass theorem. For instance the following result holds.
For $0<\lambda<\frac{1}{C}$, let
$$
J(u)=\frac{1}{p}\langle \mathfrak{L}_{\lambda}^{\mu_{n}}u,u\rangle
-\frac{1}{\alpha}\|u\|^{\alpha}_{L^{\alpha}},\quad u\in W^{1,p}_0(\Omega).
$$
To obtain a nontrivial critical point of the functional $J$, we apply the 
following version of the mountain-pass theorem from \cite{J-M-B} with the
 usual Palais-Smale compactness condition. So the critical points of the functional 
$J$ are a weak solutions for \eqref{ePlam}. 

\begin{theorem} \label{thmx}
Let $E$ be a real Banach space and $J\in C^{1}(E,\mathbb{R})$ satisfying Palais-Smale 
condition. Suppose that $J(0)=0$ and for some $\sigma,\rho>0$ and $e\in E$,
 with $\|e\|>\rho$, one has $\sigma\leq{\inf_{\|u\|=\rho}}J(u)$ and $J(e)<0$. 
Then $J$ has a critical value $c\geq\sigma$ characterized by 
$$
c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]} J(\gamma(t)),
$$ 
where
$$
\Gamma=\{\gamma\in C([0,1],E):\gamma(0)=0,\,\gamma(1)=e\}.
$$
\end{theorem}
The proof of the above theorem follows from the following lemma.

\begin{lemma}\label{lem1}
The functional $J$ satisfies the Palais-Smale condition.
\end{lemma}

\begin{proof}
Let $(u_k)_k\in W^{1,p}_0(\Omega)$ be a Palais-Smale sequence. Set 
$$
c={\lim_{k\to\infty}} J(u_k),\quad 
J'(u_k)=\epsilon_k
$$ 
such that $(\epsilon_k)_k\to0$. Thus 
$$
|J'(u_k)w|\leq\epsilon_k\|w\|,\text{for all }\; w\in W^{1,p}_0(\Omega).
$$ 
For $k$ large enough, we will have 
\begin{align*}
c+1&\geq J(u_k)-\frac{1}{\alpha}\langle J'(u_k),u_k\rangle
+\frac{1}{\alpha}\langle J'(u_k),u_k\rangle,\\
&\geq (\frac{1}{p}-\frac{1}{\alpha})(1-\lambda C)\|u_k\|^p-\frac{1}{\alpha}\|u_k\|\epsilon_k,\\
&\geq (\frac{1}{p}-\frac{1}{\alpha})(1-\lambda C)\|u_k\|^p-\frac{1}{\alpha}\|u_k\|.
\end{align*}
Hence, the sequence $(u_k)_k$ is bounded in $W^{1,p}_0(\Omega)$. 
By compactness argument we can assume that
\begin{gather*}
u_k\rightharpoonup u\quad \text{in } W^{1,p}_0(\Omega), \\
u_k\to u\quad \text{in } L^{\alpha}(\Omega),\quad \text{for } p<\alpha<p^{\star}.
\end{gather*}
Using (H2), we obtain that $(u_k)_k$ converges to $u$ in $L^p(\Omega,\mu_{n})$.
 It follows that $|u_k|^{p-2}u_k\to|u|^{p-2}u$ in $L^{p'}(\Omega,\mu_{n})$,
 hence in $W^{-1,p'}(\Omega)$. Let us denote by 
$V_k=|u_k|^{p-2}u_k\mu_{n}-|u_k|^{\alpha-2}u_k$
and $V=|u|^{p-2}u\mu_{n}-|u|^{\alpha-2}u$. 
Since $(-\Delta_{p})^{-1}$ is continuous, we conclude that
$$
u_k=(-\Delta_{p})^{-1}(V_k)\;\text{converges to }\;(-\Delta_{p})^{-1}(V)=u.
$$
Therefore, $|u_k|^{p-2}u_k$ converges to $|u|^{p-2}u$ in $W^{1,p}_0(\Omega)$.
\end{proof}

\begin{lemma}\label{lem3}
The functional $J$ satisfies the conditions for the mountain-pass theorem.
\end{lemma}

\begin{proof}
Let $\delta_1=\alpha\delta$.
First, we show that there exist positive constants $\rho$ and $\alpha_1$ such that 
$$
J(u)\geq\alpha,\quad \text{if }\|u\|=\rho,
$$
and there exists $\varphi\in W^{1,p}_0(\Omega)$ such that
$J(t\varphi)\to-\infty$, as $t\to\infty$.
Indeed, for $u\in W^{1,p}_0(\Omega)$, we have
\[
J(u)=\frac{1}{p}\langle \mathfrak{L}_{\lambda}^{\mu_{n}}u,u\rangle
-\frac{1}{\alpha}\|u\|^{\alpha}_{L^{\alpha}}
\geq \frac{1}{p}(1-\lambda C)\|u\|^p-\frac{\delta_1}{\alpha}\|u\|^{\alpha}.
\]
Since $\lambda<1/C$ and $p<\alpha$, we can set
\[
\rho=\Big(\frac{(1-\lambda C)\alpha S^{\alpha/p}}
{|\Omega|^{1-\frac{\alpha}{p^{\star}}}}\big)^{1/\alpha-p)},
\quad
\alpha_1=\Big(\frac{(1-\lambda C)^{\alpha}}{(\delta_1)^p}\Big)
^{1/(\alpha-p)} \big(\frac{1}{p}-\frac{1}{\alpha}\big),
\]
such that $J(u)\geq\alpha_1$ if $ \|u\|=\rho$.

Let us prove the second assertion.
Let $t>0$ large enough, and choose $\varphi\in W^{1,p}_0(\Omega)\backslash\{0\}$ 
satisfying
$$
J(t\varphi)=\frac{1}{p}t^p\langle \mathfrak{L}_{\lambda}^{\mu_{n}}\varphi,\varphi\rangle
-\frac{1}{\alpha}t^{\alpha}\|\varphi\|^{\alpha}_{L^{\alpha}}\to-\infty\quad
\text{as }t\to+\infty.
$$
Thus, we have $J(t\varphi)<0$, for sufficiently large $t$.

So, we can conclude that $J$ has a critical value $c\geq\alpha_1$, which can be 
characterized by
$$
c={\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}} J(\gamma(t)),
$$
where 
$$
\Gamma=\{\gamma\in C([0,1],W^{1,p}_0(\Omega)),\,\gamma(0)=0,\;\gamma(1)=e\}.
$$

Next, we shall prove the positivity of the solution.
Multiply the equation $-\Delta_{p}u-\lambda|u|^{p-2}u\mu_{n}=|u|^{\alpha-2}u$ 
by $u^{-}$ and integrate over $\Omega$, we find
$\|u^{-}\|=0$ and so $u$ is a positive solution of $(P^{\mu_{n}}_{\alpha,\lambda})$
the proof is complete.
\end{proof}

For the proof of Theorem \ref{thm3}, we need the following results.

\begin{lemma}
Let $(u_{n})_{n}$ be a sequence of weak solutions of  \eqref{ePlam} with $\mu_n$ instead of $\mu$.
 Then, $(u_{n})_{n}$ is bounded in $W^{1,p}_0(\Omega)$.
\end{lemma}

\begin{proof}
As $u_{n}$ is a weak solution of \eqref{ePlam} with $\mu_n$ instead of $\mu$, then $u_{n}$ is a critical point
of the functional $J$.
Since $J$ satisfies the Palais-Smale condition, then $(u_{n})_{n}$ 
is bounded in $W^{1,p}_0(\Omega)$.
\end{proof}

\begin{lemma}
Let $(u_{n})_{n}$ be a sequence of weak solutions of the problem 
\eqref{ePlam} with $\mu_n$ instead of $\mu$. Then $(u_{n})_{n}$ converges to a weak solutions $u$ of 
 \eqref{ePlam}.
\end{lemma}

\begin{proof}
By Lemma \ref{lem5}, since $(u_{n})_{n}$ is bounded in $W^{1,p}_0(\Omega)$, it follows that
\begin{equation}\label{eqy}
\begin{gathered}
u_{n}\rightharpoonup u\quad \text{in } W^{1,p}_0(\Omega), \\
u_{n}\rightharpoonup u\quad  \text{in } L^p(\Omega,\mu), \\
u_{n}\to u\quad \text{in } L^{\alpha}\quad p<\alpha<p^{\star}.
\end{gathered}
\end{equation}
Hence
$$
\int_{\Omega}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \varphi dx 
\to |\nabla u|^{p-2}\nabla u\nabla \varphi dx,\quad \text{for all }
\varphi\in W^{1,p}_0(\Omega).
$$
By the compactness of Sobolev embedding, we obtain
\[
\int_{\Omega}|u_{n}|^{\alpha-2}u_{n}\varphi dx
\to\int_{\Omega}|u|^{\alpha-2}u\varphi dx,\quad \text{for all }
\varphi\in W^{1,p}_0(\Omega).
\]
Next, we show that
\[
\int_{\Omega}|u_{n}|^{p-2}u_{n}\varphi d\mu_{n}\to\int_{\Omega}|u|^{p-2}u\varphi d\mu,
\quad \text{for all }
\varphi\in C^{\infty}_0(\Omega).
\]
Indeed, we have
\begin{align*}
&\big|\int_{\Omega}|u_{n}|^{p-2}u_{n}\varphi d\mu_{n}
-\int_{\Omega}|u|^{p-2}u\varphi d\mu\big|\\
&=\big|\int_{\Omega}|u|^{p-2}u\varphi d(\mu-\mu_{n})
-\int_{\Omega}(|u_{n}|^{p-2}u_{n}-|u|^{p-2}u)\varphi d\mu_{n}\big|\\
&\leq \|u\|^{p-1}_{L^p(\Omega,\mu-\mu_{n})}\|\varphi\|_{L^p(\Omega,\mu-\mu_{n})}
+|\int_{\Omega}(|u_{n}|^{p-2}u_{n}-|u|^{p-2}u|)\varphi|d\mu_{n}|.
\end{align*}
So, using (H3) the first integral converges to $0$, as $n \to\infty$ 
respectively by the weak convergence in \eqref{eqy}, the second integral 
converges to $0$, as $n\to\infty$.
Therefore, $u$ is a solution of our problem in the sense of distribution.
Moreover by density argument and taking into account that 
$u\in W^{1,p}_0(\Omega)$,
we conclude that $u$ is solution in the sense of $W^{1,p}_0(\Omega)$.
\end{proof}

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