\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 $\mathbb{R}^{N}$, $N\geq2$, $1
2$, the uniqueness is in general not true, see \cite{M-E-R}. However, the uniqueness in the case $1
0$ 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 $f\in L^{p'}(\Omega)$, $1
1$, 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,\\
>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 $1\leq q 0\}$ 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 $1\leq q 0$ 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 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 \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 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