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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 08, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/08\hfil Nonhomogeneous elliptic equations]
{Nonhomogeneous elliptic equations involving critical Sobolev exponent and weight}

\author[M. Bouchekif, A. Rimouche \hfil EJDE-2016/08\hfilneg]
{Mohammed Bouchekif, Ali Rimouche}

\address{Mohammed Bouchekif \newline
Laboratoire Syst\`emes Dynamiques et Applications,
Facult\'e des Sciences,
Universit\'e de Tlemcen BP 119 Tlemcen 13000, Alg\'erie}
\email{m\_bouchekif@yahoo.fr}

\address{Ali Rimouche (corresponding author)\newline
Laboratoire Syst\`emes Dynamiques et Applications,
Facult\'e des Sciences,
Universit\'e de Tlemcen BP 119 Tlemcen 13000, Alg\'erie}
\email{ali.rimouche@mail.univ-tlemcen.dz}

\thanks{Submitted June 30, 2015. Published January 6, 2016.}
\subjclass[2010]{35J20, 35J25, 35J60}
\keywords{Critical Sobolev exponent; Nehari manifold; variational principle}

\begin{abstract}
 In this article we consider the  problem
 \begin{gather*}
 -\operatorname{div}\big(p(x)\nabla u\big)=|u|^{2^{*}-2}u+\lambda  f\quad
  \text{in }\Omega \\
 u=0 \quad \text{on }\partial\Omega
 \end{gather*}
 where $\Omega$ is a bounded domain in $\mathbb{R}^N$, We study the 
 relationship between the behavior of $p$ near its minima on the 
 existence of solutions.
\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 and statement of main results}

In this article we study the existence of solutions to the  problem
\begin{equation}
\begin{gathered}
-\operatorname{div}\big(p(x)\nabla u\big)=|u|^{2^{*}-2}u+\lambda f\quad
 \text{in }\Omega \\
u=0 \quad  \text{on } \partial\Omega,
\end{gathered} \label{eq:0}
\end{equation}
where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $N\geq3$,
$f$ belongs to $H^{-1}=W^{-1,2}(\Omega)\setminus\{0\}$,
$p\in H^{1}(\Omega)\cap C(\bar{\Omega})$ is a  positive function,
$\lambda$ is a real parameter and $2^{*}=\frac{2N}{N-2}$ is the critical
Sobolev exponent for the embedding of $H_0^{1}(\Omega)$ into
$L^{2^{*}}(\Omega)$.

For a constant function $p$,  problem \eqref{eq:0} has been studied by
many authors, in particular by Tarantello \cite{Tarantello-4}. Using
Ekeland's variational principle and minimax principles, she proved the
existence of at least one solution of \eqref{eq:0} with $\lambda=1$
when $f\in H^{-1}$ and satisfies
\begin{equation*}
\int_{\Omega}{fu\,dx}\leq K_{N}
\Big(\int_{\Omega}{|\nabla u|^2}\Big)^{({N+2})/{4}}
\quad\text{for }\int_{\Omega} |u|^{2^{*}}=1,
\end{equation*}
with
\[
 K_{N}=\frac{4}{N-2}(\frac{N-2}{N+2})^{({N+2})/{4}}.
\]
Moreover when the above inequality is strict, she showed the existence of at
least a second solution. These solutions are nonnegative when $f$ is
nonnegative.

The following problem has been considered by several authors,
\begin{equation}
\begin{gathered}
-\operatorname{div}(p(x)\nabla u)=|u|^{2^{*}-2}u+\lambda
u\quad  \text{in }\Omega \\
u>0 \quad \text{in }\Omega \\
u=0 \quad  \text{on }\partial\Omega\,.
\end{gathered} \label{eq:hadiji}
\end{equation}
We quote in particular the celebrate paper by Brezis and Nirenberg
\cite{Br=0000E9zis-1}, and that of Hadiji and Yazidi \cite{Hadiji-3}.
In \cite{Br=0000E9zis-1}, the authors studied the case when $p$ is
constant.

To our knowledge, the case where $p$ is not constant  has been
considered in \cite{Hadiji-3} and \cite{Rimouche}. The authors in
\cite{Hadiji-3} showed that the existence of solutions depending on a
parameter $\lambda$, $N$, and the behavior of $p$ near its minima.
 More explicitly:
when $p\in H^{1}(\Omega)\cap C(\bar{\Omega})$ satisfies
\begin{equation}
p(x)=p_0+\beta_{k}|x-a|^{k}+|x-a|^{k}\theta(x)\ \text{in }B(a,\tau),
\label{eq:p(x)_1}
\end{equation}
where $k,\beta_{k}, \tau$ are positive constants, and $\theta$
tends to $0$ when $x$ approaches $a$, with
$a\in p^{-1}(\{ p_0\} )\cap \Omega$,
$p_0=\min_{x\in\overline{\Omega}} p(x)$, and $B(a,\tau)$
denotes the ball with center $0$ and radius $\tau$,
when $0<k\leq2$, and  $p$ satisfies the  condition
\begin{equation}
k\beta_{k}\leq\frac{\nabla p(x).(x-a)}{|x-a|^{k}}\quad\mathrm{a.e}
\ x\in\Omega.  \label{eq:ad_cond}
\end{equation}
On the one hand, they obtained the existence of solutions to \eqref{eq:hadiji}
 if one of the following conditions is satisfied:
\begin{itemize}
\item[(i)] $N\geq4$, $k>2$ and $\lambda\in\left]0,\lambda_1(p)\right[$;

\item[(ii)] $N\geq4$, $k=2$ and $\lambda\in\left]\tilde{\gamma}(N),\lambda_1(p)
\right[$;

\item[(iii)] $N=3$, $k\geq2$ and $\lambda\in\left]\gamma(k),\lambda_1(p)\right[$;

\item[(iv)] $N\geq3$, $0<k<2$ and $p$ satisfies \eqref{eq:ad_cond},
$\lambda\in] \lambda^{*},\lambda_1(p)[$;
\end{itemize}
where
\[
\tilde{\gamma}(N)=\frac{(N-2)N(N+2)}{4(N-1)}\beta_{2},
\]
$\gamma(k)$ is a positive constant depending on
$k$, and
$\lambda^{*}\in[\tilde{\beta}_{k}\frac{N^2}{4},\lambda_1(p)[$,
with $\tilde{\beta}_{k}=\beta_{k}\min[(\operatorname{diam}\Omega)^{k-2},1]$.

On the other hand, non-existence results are given in the following cases:
\begin{itemize}
\item[(a)] $N\geq3$, $k>0$ and $\lambda\leq\delta(p)$.

\item[(b)] $N\geq3$, $k>0$ and $\lambda\geq\lambda_1(p)$.
\end{itemize}
We denote by $\lambda_1(p)$ the first eigenvalue of
$(-\operatorname{div} (p\nabla.), H)$ and
\[
\delta(p)=\frac{1}{2}\underset{u\in
H_0^1(\Omega)\setminus\{0\}}{\inf}\frac{\int_{\Omega}{\nabla
p(x)(x-a)|\nabla u|^2dx}}{\int_{\Omega}{|u|^2dx}}.
\]
Then we formulate the question:
What happens in \eqref{eq:0} when $p$ is not necessarily a constant function?
A response to this question is given in Theorem \ref{thm:2} below.

\subsection*{Notation}
 $S$ is the best Sobolev constant for the embedding from
$H_0^1(\Omega)$ to $L^{2^*}(\Omega)$.
$\| \cdot\|$ is the norm of $H_0^1(\Omega)$ induced by the product
$(u,v)=\int_{\Omega}{\nabla u\nabla v\,dx}$.
$\| \cdot\| _{-1}$ and $|\cdot |_{p}=(\int_{\Omega}|.|^{p}dx)^{1/p}$
are the norms in $H^{-1}$ and $L^p(\Omega)$ for $1\leq p<\infty$
respectively.
We denote the space $H_0^1(\Omega)$ by $H$ and the
integral $\int_{\Omega}{u\,dx}$ by $\int{u}$.
$\omega_{N}$ is the area of the sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$.

Let $E=\{ u\in H:\int_{\Omega}\tilde{f}(x)u(x)dx>0\} $ and
\[
 \alpha(p):=\frac{1}{2}\inf_{u\in E} \frac{\int_{\Omega}\hat{p}
(x)|\nabla u(x)|^2dx}{\int_{\Omega}\tilde{f}(x)u(x)dx},
\]
with
\[
\tilde{f}(x):=\nabla f(x).(x-a)+\frac{N+2}{2}f(x),\quad
\hat{ p}(x)=\nabla p(x).(x-a).
\]
Put
\begin{equation} \label{eq:B,A_1}
\begin{gathered}
\Lambda_0:=K_{N}\frac{p_0^{1/2}}{\| f\| _{-1}} (S(p))^{N/4}, \quad
A_l=(N-2)^2\int_{\mathbb{R}^N}\frac{|x|^{l+2}}{(1+|x|^2)^N}, \\
B=\int_{\mathbb{R}^N}\frac{1}{(1+|x|^2)^N}, \quad
D:=w_0(a)\int_{\mathbb{R}^N}(1+|x|^2)^{(N+2)/2},
\end{gathered}
\end{equation}
where $l\geq0$ and
\[
S(p):=\inf_{u\in H\setminus\{0\} }
 \frac{\int_{\Omega} p(x)|\nabla u|^2}{
|u|_{2^{*}}^2}\,.
\]

\begin{definition}\label{def:ground} \rm
We say that $u$ is a ground state solution of \eqref{eq:0}
if $J_{\lambda}(u)=\min\{J_{\lambda}(v) :v\text{ is a solution of 
\eqref{eq:0}}\}$. Here $J_{\lambda}$ is the energy functional associate with
\eqref{eq:0}.
\end{definition}

\begin{remark}\label{rmq:1}\rm
By the Ekeland variational principle \cite{Ekeland} we can prove that for
$\lambda\in(0,\Lambda_0)$ there exists a ground state solution to  \eqref{eq:0}
which will be denoted by $ w_0$. The proof is similar to that in
\cite{Tarantello-4}.
\end{remark}

\begin{remark} \rm
Noting that if $u$ is a solution of the problem \eqref{eq:0}, then $-u$ is
also a solution of the problem \eqref{eq:0} with $-\lambda$ instead of
$\lambda$.  Without loss of generality, we restrict our study to
the case  $\lambda\geq 0$.
\end{remark}

Our main results read as follows.

\begin{theorem}\label{thm:3}
Suppose that $\Omega$ is a star shaped domain with respect to $a$
and $p$ satisfies \eqref{eq:p(x)_1}. Then there is no solution of
problem \eqref{eq:0} in $E$ for all $0\leq\lambda\leq\alpha(p)$.
\end{theorem}

\begin{theorem}\label{thm:2}
Let $p\in H^{1}(\Omega)\cap C(\bar{\Omega})$ such that $p_0>0$ and
$p$ satisfies \eqref{eq:p(x)_1} then, for $0<\lambda<\frac{\Lambda_0}{2}$,
 problem \eqref{eq:0} admits at least two solutions in one of the
following condition:
\begin{itemize}
\item[(i)] $k>\frac{N-2}{2}$,
\item[(ii)] $\beta_{(N-2)/2}>\frac{2D}{A_{(N-2)/2}}(\frac{A_0}{B}
)^{(6-N)/4}$.
\end{itemize}
\end{theorem}

This article is organized as follows:
in the forthcoming section, we give some preliminaries.
Section 3 and 4 present the proofs of our main results.


\section{Preliminaries}

A function $u$ in $H$ is said to be a weak solution of \eqref{eq:0} if $u$
satisfies
\begin{equation*}
\int(p\nabla u\nabla v-|u|^{2^{*}-2}uv-\lambda fv)=0\quad\text{for all }v\in H.
\end{equation*}
It is well known that the nontrivial solutions of \eqref{eq:0} are
equivalent to the non zero critical points of the energy functional
\begin{equation}
J_{\lambda}(u)=\frac{1}{2}\int p|\nabla u|^2-\frac{1
}{2^{*}}\int|u|^{2^{*}}-\lambda\int fu\,.  \label{eq:J_1}
\end{equation}
We know that $J_{\lambda}$ is not bounded from below on $H$, but it is
on a natural manifold called Nehari manifold, which is defined by
\begin{equation*}
\mathcal{N}_{\lambda}=\{ u\in H\setminus\{ 0\}
:\langle J_{\lambda}'(u),u\rangle =0\} .
\end{equation*}
Therefore, for $u\in\mathcal{N}_{\lambda}$, we obtain
\begin{equation}
J_{\lambda}(u)=\frac{1}{N}\int p|\nabla u|^2-\lambda\frac{N+2}{
2N}\int fu,  \label{eq:J_N_f}
\end{equation}
or
\begin{equation}
J_{\lambda}(u)=-\frac{1}{2}\int p|\nabla u|^2+\frac{
N+2}{2N}\int|u|^{2^{*}}.  \label{eq:J_N_u}
\end{equation}

It is known that the constant $S$ is achieved by the family of functions
\begin{equation}
U_{\varepsilon}(x)=\frac{\varepsilon^{(N-2)/2}}{
(\varepsilon^2+|x|^2)^{(N-2)/2}}
\quad\varepsilon>0,\quad x\in\mathbb{R}^N,  \label{eq:u.epsilon_1}
\end{equation}

For $a\in\Omega$, we define
$U_{\varepsilon,a}(x)=U_{\varepsilon}(x-a)$
and $u_{\varepsilon,a}(x)=\xi_{a}(x)U_{\varepsilon,a}(x)$,
where
\begin{equation}
\xi_{a}\in C_0^{\infty}(\Omega)\quad
\text{with $\xi_{a}\geq0$ and $\xi_{a}=1$ in a neighborhood of $a$}.  \label{eq:phi_1}
\end{equation}

We start with the following lemmas given without proofs and based essentially
on \cite{Tarantello-4}.

\begin{lemma} \label{lem:J-born=0000E9e-co=0000E9rcive}
The functional $J_{\lambda}$ is coercive and bounded from below on
 $\mathcal{N}_{\lambda}$.
\end{lemma}
Set
\begin{equation}
\Psi_{\lambda}(u)=\langle
J_{\lambda}'(u),u\rangle .  \label{eq:21_1}
\end{equation}

For $u\in\mathcal{N}_{\lambda}$, we obtain
\begin{eqnarray}
\langle \Psi_{\lambda}'(u),u\rangle & = & \int
p|\nabla u|^2-(2^{*}-1)\int|u|^{2^{*}}  \label{eq:3_1} \\
& = & (2-2^{*})\int p|\nabla
u|^2-\lambda(1-2^{*})\int fu.  \label{eq:4_1}
\end{eqnarray}

So it is natural to split $\mathcal{N}_{\lambda}$ into three subsets
corresponding to local maxima, local minima and points of inflection defined
respectively by
\begin{gather*}
\mathcal{N}_{\lambda}^{+}=\{ u\in\mathcal{N}_{\lambda}:\langle
\Psi_{\lambda}'(u),u\rangle >0\} ,\quad
 \mathcal{N}_{\lambda}^{-}=\{ u\in\mathcal{N}_{\lambda}:\langle
\Psi_{\lambda}'(u),u\rangle <0\}, \\
\mathcal{N}_{\lambda}^{0}=\{ u\in\mathcal{N}_{\lambda}:\langle
\Psi_{\lambda}'(u),u\rangle =0\} .
\end{gather*}

\begin{lemma}\label{lem:min-J-sol}
Suppose that $u_0$ is a local minimizer of
$J_{\lambda}$ on $\mathcal{N}_{\lambda}$. Then if
$u_0\notin\mathcal{N}_{\lambda}^{0}$, we have $J_{\lambda}'(u_0)=0$ in
$H^{-1}$.
\end{lemma}

\begin{lemma}\label{lem:N0-vide}
For each $\lambda\in(0,\Lambda_0)$ we have $\mathcal{N}_{\lambda}^{0}=\emptyset$.
\end{lemma}

By Lemma \ref{lem:N0-vide}, we have $\mathcal{N}_{\lambda}=\mathcal{N}
_{\lambda}^{+}\cup\mathcal{N}_{\lambda}^{-}$ for all $\lambda\in(0,
\Lambda_0)$.
For $u\in H\setminus\{ 0\} $, let
\begin{equation*}
t_m=t_{\rm max}(u):=\Big(\frac{\int p|\nabla u|^2}{(2^{*}-1)
\int|u|^{2^{*}}}\Big)^{(N-2)/4}.
\end{equation*}

\begin{lemma}\label{lem:tu}
Suppose that $\lambda\in(0,\Lambda_0)$ and $u\in
H\setminus\{ 0\} $, then

(i) If $\int fu\leq0$, then there exists an unique $t^{+}=t^{+}(u)>t_m$
such that $t^{+}u\in\mathcal{N}_{\lambda}^{-}$ and
\begin{equation*}
J_{\lambda}(t^{+}u)=\sup_{t\geq t_m} J_{\lambda}(tu).
\end{equation*}

(ii) If $\int fu>0$, then there exist unique $t^{-}=t^{-}(u),\;
t^{+}=t^{+}(u)$ such that $0<t^{-}<t_m<t^{+}$, $t^{-}u\in\mathcal{N}
_{\lambda}^{+}$, $t^{+}u\in\mathcal{N}_{\lambda}^{-}$ and
\begin{equation*}
J_{\lambda}(t^{+}u)=\sup_{t\geq t_m}
J_{\lambda}(tu);\quad J_{\lambda}(t^{-}u)
=\inf_{0\leq t\leq t^{+}} J_{\lambda}(tu).
\end{equation*}
\end{lemma}

Thus we put
\begin{equation*}
c=\inf_{u\in\mathcal{N}_{\lambda}} J_{\lambda}(u), \quad
c^{+}=\inf_{u\in\mathcal{N}_{\lambda}^{+}} J_{\lambda}(u),\quad
c^{-}=\inf_{u\in\mathcal{N}_{\lambda}^{-}} J_{\lambda}(u).
\end{equation*}

\begin{lemma} \label{lem:min-J}

(i) If $\lambda\in(0,\Lambda_0)$, then $c\leq c^{+}<0$.

(ii) If $\lambda\in(0,\frac{\Lambda_0}{2})$, then $c^{-}>0$.
\end{lemma}

\section{Nonexistence result}

\subsection*{Some properties of $\alpha(p)$}

\begin{proposition}
(1) Assume that $p\in C^{1}(\Omega)$ and there exists $b\in\Omega$ such that
$\nabla p(b)(b-a)<0$ and $f\in C^{1}$ in a neighborhood of $b$. Then
$\alpha(p)=-\infty$.

(2) If $p\in C^{1}(\Omega)$ satisfying \eqref{eq:p(x)_1} with $k>2$ and
$\nabla p(x)(x-a)\geq0$ for all $x\in\Omega$ and $f\in C^{1}$ in a
neighborhood of $a$ and $f(a)\neq0$, then $\alpha(p)=0$ for all $N\geq3$.

(3) If $p\in H^{1}(\Omega)\cap C(\bar{\Omega})$ and $\nabla
p(x)(x-a)\geq0$ a.e $x\in\Omega$, then $\alpha(p)\geq0$.
\end{proposition}

\begin{proof}
(1) Set $\varphi\in C_0^{\infty}(\mathbb{R}^N)$ such that
\begin{equation}
0\leq\varphi\leq1,\quad\varphi(x)=
\begin{cases}
1 & \text{if }x\in B(0;r) \\
0 & \text{if }x\notin B(0;2r),
\end{cases}\label{eq:1.5}
\end{equation}
where $0<r<1$.

Set $\varphi_{j}(x)=\operatorname{sgn}[\tilde{f}(x)]\varphi(j(x-b))$
for $j\in\mathbb{N}^{*}$. We have
\[
\alpha(p)  \leq  \frac{1}{2}\frac{\int_{B(b,\frac{2r}{j})}\hat{p}
(x)|\nabla\varphi_{j}(x)|^2}{\int_{B(b,\frac{2r}{j})}\tilde{f}
(x)\varphi_{j}(x)}.
\]
Using the change of variable $y=j(x-b)$ and applying the dominated
convergence theorem, we obtain
\begin{equation*}
\alpha(p)\leq\frac{j^2}{2}\Big[\frac{\hat{p}(b)\int_{B(0,2r)}|\nabla
\varphi(y)|^2}{|\tilde{f}(b)|\int_{B(0,2r)}\varphi(y)}+o(1)
\Big],
\end{equation*}
letting $j\to\infty$, we obtain the desired result.
\smallskip

(2) Since $p\in C^{1}(\Omega)$ in a neighborhood $V$ of $a$, we write
\begin{equation}
p(x)=p_0+\beta_{k}|x-a|^{k}+\theta_1(x),  \label{eq:1.3}
\end{equation}
where $\theta_1\in C^{1}(V)$ such that
\begin{equation}
\underset{x\to a}{\lim}\frac{\theta_1(x)}{|x-a|^{k}}=0.
\label{eq:1.4}
\end{equation}
Thus, we deduce that there exists $0<r<1$, such that
\begin{equation}
\theta_1(x)\leq|x-a|^{k},\quad \text{for all } x\in B(a,2r).
\label{eq:1.6}
\end{equation}
Let $\psi_{j}(x)=\operatorname{sgn}[\tilde{f}(x)]\varphi(j(x-a))$,
$\varphi\in C_0^{\infty}(\mathbb{R}^N)$ defined as in \eqref{eq:1.5}, we
have
\begin{equation*}
0\leq\alpha(p)\leq\frac{1}{2}\frac{\int\nabla
p(x).(x-a)|\nabla\psi_{j}(x)|^2}{\int\tilde{f}(x)\psi_{j}(x)}.
\end{equation*}
Using \eqref{eq:1.3}, we obtain
\begin{equation*}
0\leq\alpha(p)
\leq\frac{k\beta_{k}}{2}\frac{\int_{B(a,\frac{2r}{j}
)}|x-a|^{k}|\nabla\psi_{j}(x)|^2}{\int_{B(a,\frac{2r}{j})}\tilde{f}
(x)\psi_{j}(x)}+\frac{1}{2}\frac{\int_{B(a,\frac{2r}{j}
)}\nabla\theta_1(x).(x-a)|\nabla\psi_{j}(x)|^2}{\int_{B(a,\frac{2r}{j})}
\tilde{f}(x)\psi_{j}(x)}.
\end{equation*}
Using the change of variable $y=j(x-a)$, and integrating by parts the second
term of the right hand side, we obtain
\begin{equation*}
0\leq\alpha(p)
\leq\frac{k\beta_{k}}{2j^{k-2}}\frac{\int_{B(0,2r)}|y|^{k}|
\nabla\varphi(y)|^2}{\int_{B(0,2r)}|\tilde{f}(\frac{y}{j}
+a)|\varphi(y)}+\frac{j}{2}\frac{\int_{B(0,2r)}\theta_1(\frac{y}{j}
+a)div(y|\nabla\varphi(y)|^2)}{\int_{B(0,2r)}|\tilde{f}(
\frac{y}{j}+a)|\varphi(y)}.
\end{equation*}

Using \eqref{eq:1.6} and applying the dominated convergence theorem, we obtain
\begin{align*}
0\leq\alpha(p)
&\leq\frac{k\beta_{k}}{(N+2)j^{k-2}}\frac{
\int_{B(0,2r)}|y|^{k}|\nabla\varphi(y)|^2}{|f(a)|\int_{B(0,2r)}
\varphi(y)}\\
&\quad +\frac{1}{(N+2)j^{k-1}}\frac{\int_{B0,2r)}|y|^{k}div(y|
\nabla\varphi(y)|^2)}{|f(a)|\int_{B(0,2r)}\varphi(y)}+o(1).
\end{align*}
Therefore, for $k>2$ we deduce that $\alpha(p)=0$, which completes the proof.
\end{proof}

\subsection*{Proof of Theorem \ref{thm:3}}

Suppose that $u$ is a solution of \eqref{eq:0}.
We multiply \eqref{eq:0} by $\nabla u(x).(x-a)$ and integrate over $\Omega$,
we obtain
\begin{gather}
\int|u|^{2^{*}-1}\nabla u(x).(x-a)=-\frac{N-2}{2}
\int|u(x)|^{2^{*}},  \label{eq:1.7-1}
\\
\lambda\int f(x)\nabla u(x).(x-a)=-\lambda\int(\nabla
f(x).(x-a)+Nf(x))u(x),    \label{eq:1.8-1} \\
\begin{aligned}
&-\int\operatorname{div}(p(x)\nabla u(x))\nabla u(x).(x-a)\\
&= -\frac{N-2}{2}\int p(x)|\nabla u(x)|^2
 -\frac{1}{2}\int\nabla p(x).(x-a)|\nabla u(x)|^2\\
&\quad -\frac{1}{2}\int_{\partial \Omega}p(x)(x-a).\nu|
\frac{\partial u}{\partial\nu}|^2.
\end{aligned} \label{eq:1.9-1}
\end{gather}
Combining \eqref{eq:1.7-1}, \eqref{eq:1.8-1} and \eqref{eq:1.9-1},
 we obtain
\begin{equation}
\begin{aligned}
&-\frac{N-2}{2}\int p(x)|\nabla u(x)|^2
 -\frac{1}{2}\int\nabla p(x).(x-a)|\nabla u(x)|^2\\
& -\frac{1}{2}\int_{\partial \Omega}p(x)(x-a).\nu|\frac{\partial u}{\partial\nu}|^2\\
&=-\frac{N-2}{2}\int|u(x)|^{2^{*}}-\lambda\int(\nabla
f(x).(x-a)+Nf(x))u(x).
\end{aligned} \label{eq:1.11-1}
\end{equation}
Multiplying \eqref{eq:0} by $\frac{N-2}{2}u$ and
integrating by parts, we obtain
\begin{equation}
\frac{N-2}{2}\int p(x)|\nabla u(x)|^2=\frac{N-2}{2}
\int|u(x)|^{2^{*}}+\lambda\frac{N-2}{2}\int f(x)u(x).
\label{eq:1.10-1}
\end{equation}
From \eqref{eq:1.11-1} and \eqref{eq:1.10-1}, we obtain
\begin{equation*}
-\frac{1}{2}\int\nabla p(x).(x-a)|\nabla u(x)|^2-\frac{1}{2}
\int_{\partial\Omega}p(x)(x-a).\nu|\frac{\partial u}{\partial\nu}
|^2+\lambda\int\tilde{f}(x)u(x)=0.
\end{equation*}
Then
\begin{equation}
\lambda>\frac{1}{2}\frac{\int\nabla p(x).(x-a)|\nabla u(x)|^2}{
\int\tilde{f}(x)u(x)}\geq\alpha(p).  \label{eq:nabla_f}
\end{equation}
Hence the desired result is obtained.

\section{Existence of solutions}

We begin by proving that
\begin{equation}
\inf_{u\in\mathcal{N}_{\lambda}^{-}} J_{\lambda}(u)=c^{-}
<c+\frac{1}{N}(p_0S)^{N/2}.  \label{eq:4_4}
\end{equation}
By some estimates in Brezis and Nirenberg \cite{Brezis}, we have
\begin{equation}
\begin{aligned}
|w_0+Ru_{\varepsilon,a}|_{2^{*}}^{2^{*}}
&=|w_0|_{2^{*}}^{2^{*}}+R^{2^{*}}|u_{\varepsilon,a}|_{2^{*}}^{2^{*}}
+2^{*}R\int|w_0|^{2^{*}-2}w_0u_{ \varepsilon,a}\\
&\quad +2^{*}R^{2^{*}-1}\int u_{\varepsilon,a}^{2^{*}-1}w_0
+o(\varepsilon^{(N-2)/2}),
\end{aligned} \label{eq:estimation1_1}
\end{equation}
Put
\begin{gather}
|\nabla u_{\varepsilon,a}|_{2}^2=A_0+O(\varepsilon^{N-2}),\quad
|u_{\varepsilon,a}|_{2^{*}}^{2^{*}}=B+O(\varepsilon^N),  \label{eq:estimation2_1}
\\
S=S(1)=A_0B^{-2/2^{*}}.  \label{eq:S_1}
\end{gather}

\begin{lemma} \label{lem:p(x)}
Let $p\in H^{1}(\Omega)\cap C(\bar{\Omega})$ satisfying \eqref{eq:p(x)_1}
Then we have estimate
\begin{align*}
&\int p(x)|\nabla u_{\varepsilon,a}(x)|^2\\
&\leq\begin{cases}
p_0A_0+O(\varepsilon^{N-2}) & \text{if }N-2<k, \\
p_0A_0+A_{k}\varepsilon^{k}+o(\varepsilon^{k}) & \text{if } N-2>k, \\
p_0A_0+\frac{(N-2)^2}{2}(\beta_{N-2}+M)\omega_{N}
\varepsilon^{N-2}|\ln\varepsilon|+o(\varepsilon^{N-2}|
\ln\varepsilon|)  & \text{if }N-2=k,
\end{cases}
\end{align*}
where $M$ is a  positive constant.
\end{lemma}

\begin{proof}
by calculations,
\begin{align*}
&\varepsilon^{2-N}\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 \\
& =  \int\frac{
p(x)|\nabla\xi_{a}(x)|^2}{(\varepsilon^2+|x-a
|^2)^{N-2}}+(N-2)^2\int\frac{p(x)|\xi_{a}(x)
|^2|x-a|^2}{(\varepsilon^2+|x-a|^2 )^N} \\
&\quad -  (N-2)\int\frac{p(x)\nabla\xi_{a}^2(x)(x-a
)}{(\varepsilon^2+|x-a|^2)^{N-1}}.
\end{align*}
Suppose that $\xi_{a}\equiv1$ in $B(a,r)$ with $r>0$ small enough.
So, we obtain
\begin{align*}
&\varepsilon^{2-N}\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 \\
& =  \int_{\Omega\setminus
B(a,r)}\frac{p(x)|\nabla\xi_{a}(x)|^2}{
(\varepsilon^2+|x-a|^2)^{N-2}}+(N-2
)^2\int\frac{p(x)|\xi_{a}(x)|^2|x-a|^2}{
(\varepsilon^2+|x-a|^2)^N} \\
&\quad -  2(N-2)\int_{\Omega\setminus
B(a,r)}\frac{p(x)\xi_{a}(x)\nabla\xi_{a}(x)(x-a)}{
(\varepsilon^2+|x-a|^2)^{N-1}}.
\end{align*}
Applying the dominated convergence theorem,
\begin{equation*}
\int p(x)|\nabla
u_{\varepsilon,a}(x)|^2=(N-2)^2\varepsilon^{N-2}\int
\frac{p(x)|\xi_{a}(x)|^2|x-a|^2}{
(\varepsilon^2+|x-a|^2)^N}+O(
\varepsilon^{N-2}).
\end{equation*}
Using  expression \eqref{eq:p(x)_1}, we obtain
\begin{align*}
&\varepsilon^{2-N}(N-2)^{-2}\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 \\
& = \int_{B(a,\tau)}\frac{p_0|x-a|^2+\beta_{k}|x-a
|^{k+2}+\theta(x)|x-a|^{k+2}}{(\varepsilon^2+ |x-a|^2)^N} \\
&\quad +  \int_{\Omega\setminus B(a,\tau)}\frac{p(x)|\xi_{a}(x)|^2|x-a|^2}{
(\varepsilon^2+|x-a|^2)^N}+O(\varepsilon^{N-2}).
\end{align*}
Using again the definition of $\xi_{a}$, and applying the dominated
convergence theorem, we obtain
\begin{align*}
&\varepsilon^{2-N}(N-2)^{-2}\int p(x)|\nabla
u_{\varepsilon,a}(x)|^2 \\
& =  p_0\int_{\mathbb{R}^N}\frac{|x-a|^2}{
(\varepsilon^2+|x-a|^2)^N}+\beta_{k}\int_{B(a,
\tau)}\frac{|x-a|^{k+2}}{(\varepsilon^2+|x-a
|^2)^N} \\
&\quad +  \int_{B(a,\tau)}\frac{\theta(x)|x-a|^{k+2}}{(\varepsilon^2+|x-a|^2
)^N}+O(\varepsilon^{N-2}).
\end{align*}
We distinguish three cases:
\smallskip

\noindent\textbf{Case 1.} If $k<N-2$,
\begin{align*}
&\varepsilon^{2-N}(N-2)^{-2}\int p(x)|\nabla
u_{\varepsilon,a}(x)|^2 \\
& =  p_0\int_{\mathbb{R}^N} \frac{|x-a|^2}{(\varepsilon^2+|x-a|^2 )^N}
 + \int_{B(a,\tau)}\frac{\theta(x)|x-a|^{k+2}}{
(\varepsilon^2+|x-a|^2)^N} \\
&\quad +\Big[\int_{\mathbb{R}
^N}\frac{\beta_{k}|x-a|^{k+2}}{(\varepsilon^2+|x-a |^2)^N}
- \int_{\mathbb{R}
^N\setminus B(a,\tau)}\frac{\beta_{k}|x-a|^{k+2}}{
(\varepsilon^2+|x-a|^2)^N}\Big]
+O(\varepsilon^{N-2})
\end{align*}
Using the change of variable $y=\varepsilon^{-1}(x-a)$ and
applying the dominated
convergence theorem, we obtain
\begin{align*}
&(N-2)^{-2}\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 \\
&=p_0B_0+\varepsilon^{k}\int_{\mathbb{R}^N}
\frac{\beta_{k}|y|^{k+2}}{(1+|y|^2)^N}
+\varepsilon^{k}\int_{\mathbb{R}^N}\frac{\theta(a+\varepsilon
y)|y|^{k+2}}{(1+|y|^2)^N}\chi_{B(0,
\frac{\tau}{\varepsilon})}+o(\varepsilon^{k}).
\end{align*}
Since $\theta(x)$ tends to $0$ when $x$ tends to $a$, this gives us
\begin{equation*}
\int p(x)|\nabla u_{\varepsilon,a}(x)|^2=p_0A_0+\beta_k
A_{k}\varepsilon^{k}+o(\varepsilon^{k}).
\end{equation*}


\noindent\textbf{Case 2.} If $k>N-2$,
\begin{align*}
\int p(x)|\nabla u_{\varepsilon,a}(x)|^2
& =   p_0A_0+
(N-2)^2\varepsilon^{N-2}\Big[\int_{B(a,\tau
)}\frac{(\beta_{k}+\theta(x))|x-a|^{k+2}}{
(\varepsilon^2+|x-a|^2)^N}\\
&\quad - \int_{B(a,\tau)\setminus\Omega}\frac{
(\beta_{k}+\theta(x))|x-a|^{k+2}}{(
\varepsilon^2+|x-a|^2)^N}\Big] \\
&\quad + (N-2)^2\varepsilon^{N-2}\int_{B(a,\tau)}\frac{
\theta(x)|x-a|^{k+2}}{(\varepsilon^2+|x-a|^2
)^N}+O(\varepsilon^{N-2}).
\end{align*}
By the change of variable $y=x-a$, we obtain
\begin{align*}
\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 
& =  p_0A_0+(N-2)^2\varepsilon^{N-2}\int_{B(0,\tau
)}\frac{(\beta_{k}+\theta(a+y))|y|^{k+2}}{
(\varepsilon^2+|y|^2)^N}\\
&\quad +  (N-2)^2\varepsilon^{N-2}\int_{B(a,\tau)}\frac{
\theta(a+y)|y|^{k+2}}{(\varepsilon+|y|^2
)^N}+O(\varepsilon^{N-2}).
\end{align*}
Put $M:=\underset{x\in\bar{\Omega}}{\max}\,\theta(x)$ where
$\theta(x)$ is given by \eqref{eq:p(x)_1}. Then
\begin{align*}
&\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 \\
& =  p_0A_0+\varepsilon^{N-2}(N-2)^2(\beta_{k}+M
)\int_{B(0,\tau)}\frac{|y|^{k+2}}{
(\varepsilon^2+|y|^2)^N}dy+O(
\varepsilon^{N-2}).
\end{align*}
Applying the dominated convergence theorem,
\begin{equation*}
\int p(x)|\nabla u_{\varepsilon,a}(x)|^2=p_0A_0+O(\varepsilon^{N-2}).
\end{equation*}

\noindent\textbf{Case 3.} If $k=N-2$,
following the same previous steps, we obtain
\begin{align*}
&\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 \\
& =  p_0A_0+(N-2)^2\varepsilon^{N-2}\int_{B(a,\tau)}
\frac{\theta(x)|x-a|^{k+2}}{(\varepsilon^2+|x-a |^2)^N}\\
&\quad + (N-2)^2\varepsilon^{N-2}\Big[\int_{B(a,\tau
)}\frac{\beta_{N-2}|x-a|^N}{(\varepsilon^2+ |x-a|^2)^N}
- \int_{B (a,\tau)\setminus\Omega}\frac{\beta_{N-2}|x-a|^N
}{(\varepsilon^2+|x-a|^2)^N}\Big]\\
&\quad +O(\varepsilon^{N-2}).
\end{align*}
Therefore,
\begin{align*}
&\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 \\
& =   p_0A_0+(N-2)^2\varepsilon^{N-2}\int_{B(a,\tau
)}\frac{(\beta_{N-2}+\theta(x))|x-a|^N}{
(\varepsilon^2+|x-a|^2)^N}\\
&\quad + (N-2)^2\varepsilon^{N-2}\int_{B(a,\tau)}\frac{
\theta(x)|x-a|^{k+2}}{(\varepsilon^2+|x-a|^2
)^N}+O(\varepsilon^{N-2}).
\end{align*}
Then
\begin{align*}
&\int p(x)|\nabla u_{\varepsilon,a}(x)|^2 \\
&\leq p_0A_0+(N-2)^2\varepsilon^{N-2}(\beta_{N-2}+M)
\int_{B(a,\tau)}\frac{|x-a|^N}{(
\varepsilon^2+|x-a|^2)^N}+O(\varepsilon^{N-2}).
\end{align*}
On the other hand
\begin{align*}
\varepsilon^{N-2}\int_{B(a,\tau)}\frac{
|x-a|^N}{(\varepsilon^2+|x-a|^2)^N}
& =  \omega_{N}\varepsilon^{N-2}\int_0^{\tau}\frac{r^{2N-1}}{
(\varepsilon^2+r^2)^N}dr+O(\varepsilon^{N-2}) \\
& =  \frac{1}{2N}\omega_{N}\varepsilon^{N-2}\int_0^{\tau}\frac{
((\varepsilon^2+r^2)^N)'}{
(\varepsilon^2+r^2)^N}dr+O(\varepsilon^{N-2}),
\end{align*}
and
\begin{equation}
\varepsilon^{N-2}\int_{B(a,\tau)}\frac{
|x-a|^N}{(\varepsilon^2+|x-a|^2)^N}=
\frac{1}{2}\omega_{N}\varepsilon^{N-2}|\ln\varepsilon|+o(
\varepsilon^{N-2}|\ln\varepsilon|),  \label{eq:9_4}
\end{equation}
Therefore,
\begin{equation*}
\int p(x)|\nabla u_{\varepsilon,a}(x)|^2\leq p_0A_0+
\frac{(N-2)^2}{2}(\beta_{N-2}+M)\omega_{N}
\varepsilon^{N-2}|\ln\varepsilon|+o(\varepsilon^{N-2}|
\ln\varepsilon|).
\end{equation*}
\end{proof}

Knowing that $w_0\neq0$, we set $\Omega'\subset\Omega$ as a set of
positive measure such that $w_0>0$ on $\Omega'$.
Suppose that $a\in\Omega'$ (otherwise replace $w_0$ by $-w_0$
and $f$ by $-f$).

\begin{lemma}\label{lem:ewist_fct_extremal}
For each $R>0$ and $2k>N-2$, there exists
$ \varepsilon_0=\varepsilon_0(R,a)>0$ such that
\begin{equation*}
J_{\lambda}(w_0+Ru_{\varepsilon,a})<c+\frac{1}{N}
(p_0S)^{N/2},\quad\text{for all } 0<\varepsilon<\varepsilon_0.
\end{equation*}
\end{lemma}

\begin{proof}
We have
\begin{equation}
\begin{aligned}
J_{\lambda}(w_0+Ru_{\varepsilon,a})
& =  \frac{1}{2}\int
p|\nabla w_0|^2+R\int p\nabla w_0\nabla u_{\varepsilon,a}+\frac{R^2}{
2}\int p|\nabla u_{\varepsilon,a}|^2  \notag \\
&\quad -\frac{1}{2^{*}}\int\vert
w_0+Ru_{\varepsilon,a}\vert^{2^{*}}-\lambda\int fw_0-\lambda R\int
fu_{\varepsilon,a}.
\end{aligned} \label{eq:5_4}
\end{equation}
Using \eqref{eq:estimation1_1}, \eqref{eq:estimation2_1} and the fact that
$w_0$ satisfies \eqref{eq:0}, we obtain
\begin{align*}
&J_{\lambda}(w_0+Ru_{\varepsilon,a}) \\
& \leq  c+\frac{R^2}{2}
\int p|\nabla u_{\varepsilon,a}|^2-\frac{R^{2^{*}}}{2^{*}}
A-R^{2^{*}-1}\int
u_{\varepsilon,a}^{2^{*}-1}w_0+o\big(\varepsilon^{(N-2)/2}\big).
\end{align*}
Taking $w=0$ the extension of $w_0$ by $0$ outside of $\Omega$,
it follows that
\begin{align*}
\int u_{\varepsilon,a}^{2^{*}-1}w_0
& =  \int_{\mathbb{R}
^N}w(x)\xi_{a}(x)\frac{\varepsilon^{(N+2)/2}}{
(\varepsilon^2+|x-a|^2)^{(N+2)/2}} \\
& =  \varepsilon^{(N-2)/2}\int_{\mathbb{R}^N}w(x)\xi_{a}(x)
\frac{1}{\varepsilon^N}\psi(\frac{x}{\varepsilon})
\end{align*}
where $\psi(x)=(1+|x|^2)^{(N+2)/2}\in L^{1}(\mathbb{R}^N)$.
We deduce that
\begin{equation*}
\int_{\mathbb{R}^N}w(x)\xi_{a}(x)\frac{1}{\varepsilon^N}\psi(\frac{x
}{\varepsilon})\to D\quad\text{as } \varepsilon\to0\,.
\end{equation*}
Then
\begin{equation*}
\int u_{\varepsilon,a}^{2^{*}-1}w_0=\varepsilon^{(N-2)/2}D+o(\varepsilon^{(N-2)/2}).
\end{equation*}
Consequently
\begin{equation}
\begin{aligned}
&J_{\lambda}(w_0+Ru_{\varepsilon,a})\\
&\leq c+\frac{R^2}{2}\int
p|\nabla u_{\varepsilon,a}|^2-\frac{R^{2^{*}}}{2^{*}}B-R^{2^{*}-1}
\varepsilon^{(N-2)/2}D+o\big(\varepsilon^{(N-2)/2}\big). 
\end{aligned} \label{eq:10_4}
\end{equation}
Replacing $\int p|\nabla u_{\varepsilon,a}|^2$ by its value in \eqref{eq:10_4},
we obtain
\begin{align*}
&J_{\lambda}(w_0+Ru_{\varepsilon,a})\\
&\leq\begin{cases}
c+\frac{R^2}{2}p_0A_0-\frac{R^{2^{*}}}{2^{*}}B-
\varepsilon^{(N-2)/2}DR^{2^{*}-1}+o(\varepsilon^{(N-2)/2}) &
\text{if }k>\frac{N-2}{2},
\\[4pt]
c+\frac{R^2}{2}p_0A_0-\frac{R^{2^{*}}}{2^{*}}B+\beta_kA_{k}
\varepsilon^{k}+o(\varepsilon^{k}) & \text{if }k<\frac{N-2}{2},
\\[4pt]
c+\frac{R^2}{2}p_0A_0-\frac{R^{2^{*}}}{2^{*}}B-\varepsilon^{(N-2)/2}
\Big(\frac{R^2}{2}\beta_{(N-2)/2}A_{(N-2)/2}\\
-DR^{2^{*}-1}\Big) +o(\varepsilon^{(N-2)/2})  & \text{if }k=\frac{N-2}{2}.
\end{cases}
\end{align*}
Using  that the function $R\mapsto\Phi(R)=\frac{R^2}{2}B-\frac{
R^{2^{*}}}{2^{*}}A_0$ attains its maximum
$\frac{1}{N}(p_0S)^{N/2}$ at the point
$R_1:=(\frac{A_0}{B})^{(N-2)/4}$,  we obtain
\begin{align*}
& J_{\lambda}(w_0+Ru_{\varepsilon,a})\\
&\leq \begin{cases}
c+\frac{1}{N}(p_0S)^{N/2}-\varepsilon^{(N-2)/2}DR_1^{2^{*}-1}+o
(\varepsilon^{(N-2)/2}) & \text{if }k>\frac{N-2}{2}, \\[4pt]
c+\frac{1}{N}(p_0S)^{N/2}+A_{k}\varepsilon^{k}+o(
\varepsilon^{k}) & \text{if }k<\frac{N-2}{2}, \\[4pt]
c+\frac{1}{N}(p_0S)^{N/2}-\varepsilon^{(N-2)/2}\Big(\frac{
R_1^2}{2}\beta_{(N-2)/2}A_{(N-2)/2}\\
-DR_1^{2^{*}-1}\Big) +o(\varepsilon^{(N-2)/2}) & \text{if }k=\frac{N-2}{2}.
\end{cases}
\end{align*}
So for $\varepsilon_0=\varepsilon_0(R,a)>0$ small enough,
$k> \frac{N-2}{2}$ or $k=\frac{N-2}{2}$ and
\[
\beta_{(N-2)/2}>\frac{2DR_1^{2^*-3}}{B_{(N-2)/2}},
\]
 we conclude that
\begin{equation}
J_{\lambda}(w_0+Ru_{\varepsilon,a})<c+\frac{1}{N}(p_0S)^{N/2},  \label{eq:12_4}
\end{equation}
for all $0<\varepsilon<\varepsilon_0$.
\end{proof}

\begin{proposition}\label{prop:1}
Let $\{ u_{n}\} \subset\mathcal{N}_{\lambda}^{-}$ be a minimizing
sequence such that:
\begin{itemize}
\item[(a)] $J_{\lambda}(u_{n})\to c^{-}$ and
\item[(b)] $\| J_{\lambda}'(u_{n})\|_{-1}\to 0$.
\end{itemize}
Then for all $\lambda\in(0,\Lambda_0/2)$,
 $\{u_{n}\} $ admits a subsequence that converges strongly to a point 
$w_1$ in $H$ such that $w_1\in\mathcal{N}_{\lambda}^{-}$ and 
$J_{\lambda}(w_1)=c^{-}$.
\end{proposition}

\begin{proof}
Let $u\in H$ be such that $\| u\| =1$. Then
\begin{equation*}
t^{+}(u)u\in\mathcal{N}_{\lambda}^{-}\quad\text{and}\quad
J_{\lambda}(t^{+}(u)u)=\max_{t\geq t_m}J_{\lambda}(tu).
\end{equation*}
The uniqueness of $t^{+}(u)$ and its extremal property give that 
$u\mapsto t^{+}(u)$ is a continuous function.
We put 
\begin{gather*}
U_1=\{ u=0\text{ or } u\in H\setminus\{0\}:\| u\| <t^{+}(\frac{u}{\| u\| })\},\\
\quad U_{2}=\{ u\in H\setminus\{0\} :\| u\| >t^{+}(\frac{u}{\|u\| })\} .
\end{gather*}
Then $H\setminus\mathcal{N}_{\lambda}^{-}=U_1\cup U_{2}$ and 
$\mathcal{N}_{\lambda}^{+}\subset U_1$.
In particular $w_0\in U_1$.

As in \cite{Tarantello-4}, there exists $R_0>0$ and $\varepsilon>0$ such
that $w_0+R_0u_{\varepsilon,a}\in U_2$.
We put
\begin{equation*}
\mathcal{F}=\{ h:[0,1]\to H \text{ continuous, $h(0)=w_0$ and }
h(1)=w_0+R_0u_{\varepsilon,a}\} .
\end{equation*}

It is clear that $h:[0,1]\to H$ with 
$h(t)=w_0+tR_0u_{\varepsilon,a}$ belongs to $\mathcal{F}$. Thus by Lemma
\ref{lem:ewist_fct_extremal}, we conclude that
\begin{equation}
c_0=\inf_{h\in\mathcal{F}} \max_{t\in[0,1]}
J_{\lambda}(h(t))<c+\frac{1}{N}(p_0S)^{N/2}.
\label{eq:equation_26}
\end{equation}
Since $h(0)\in U_1$, $h(1)\in U_{2}$ and  $h$ is
continuous,  there exists $t_0\in]0,1[$ such that 
$h(t_0)\in\mathcal{N}_{\lambda}^{-}$
Hence
\begin{equation}
c_0\geq c^{-}=\inf_{u\in\mathcal{N}_{\lambda}^{-}} J_{\lambda}(u).
\label{eq:equation_27}
\end{equation}
Applying again the Ekeland variational principle, we obtain a minimizing
sequence ($u_{n})\subset \mathcal{N}^{-}_{\lambda}$ such that
(a) $J_{\lambda}(u_{n})\to c^{-}$ and 
(b) $\| J_{\lambda}'(u_{n})\|_{-1}\to 0$.
Thus, we obtain a subsequence $(u_{n})$ such that
\begin{equation*}
u_{n}\to w_1\text{ strongly in }H.
\end{equation*}
This implies that $w_1$ is a critical point for $J_{\lambda}$, 
$w_1\in\mathcal{N}^{-}_{\lambda}$ and $J_{\lambda}(w_1)=c^{-}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:2}]
From the facts that $w_0\in\mathcal{N}^{+}_{\lambda}$, 
$w_1\in\mathcal{N}^{-}_{\lambda}$ and 
 $\mathcal{N}^{+}_{\lambda}\cap\mathcal{N}^{-}_{\lambda}=\emptyset$ 
for $\lambda\in(0,\,\frac{\Lambda_0}{2})$, we deduce that
 problem \eqref{eq:0} admits at least two distinct solutions in $H$.
\end{proof}

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\end{document}