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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 54, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2011/54\hfil Nonhomogeneous singular elliptic equation]
{Nonhomogeneous elliptic equations with decaying cylindrical potential
and critical exponent}

\author[M. Bouchekif, M. E. O. El Mokhtar\hfil EJDE-2011/54\hfilneg]
{Mohammed Bouchekif, Mohammed El Mokhtar Ould El Mokhtar}  % in alphabetical order

\address{Mohammed Bouchekif \newline
University of Tlemcen, Departement of Mathematics, BO 119, 13 000
Tlemcen, Algeria}
\email{m\_bouchekif@yahoo.fr}

\address{Mohammed El Mokhtar Ould El Mokhtar \newline
University of Tlemcen, Departement of Mathematics, BO 119, 13 000
Tlemcen, Algeria}
\email{med.mokhtar66@yahoo.fr}

\thanks{Submitted February 23, 2011. Published April 27, 2011.}
\subjclass[2000]{35J20, 35J70}
\keywords{Hardy-Sobolev-Maz'ya inequality; Palais-Smale condition;
\hfill\break\indent Nehari manifold; critical exponent}

\begin{abstract}
 We prove the existence and multiplicity of solutions for a
 nonhomogeneous elliptic equation involving decaying cylindrical
 potential and critical exponent.
\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}

\section{Introduction}

In this article, we consider the problem
\begin{equation} \label{eP}
\begin{gathered}
-\operatorname{div}(|y|^{-2a}\nabla u)-\mu |y|^{-2(a+1)}u
= h|y|^{-2_{\ast }b}|u|^{2_{\ast }-2}u+\lambda g\quad
\text{in }\mathbb{R}^N, \quad  y\neq 0\\
u\in \mathcal{D}_0^{1,2},
\end{gathered}
\end{equation}
where each point in $\mathbb{R}^N$ is written as a pair
$(y,z)\in \mathbb{R}^k \times \mathbb{R}^{N-k}$,  $k$ and $N$
are integers such that $N\geq 3$ and $k$ belongs
to $\{1,\dots ,N\}$;
$-\infty <a<(k-2)/2$;  $a\leq b<a+1$;
$2_{\ast }=2N/(N-2+2(b-a))$;
$-\infty <\mu <\bar{\mu}_{a,k}:=((k-2(a+1))/2)^2 $;
$g\in \mathcal{H}_{\mu}'\cap C(\mathbb{R}^N)$;
$h$ is a bounded positive function on $\mathbb{R}^k $ and
 $\lambda $ is real parameter.
Here $\mathcal{H}_{\mu }'$ is
the dual of $\mathcal{H}_{\mu }$, where $\mathcal{H}_{\mu }$ and
$\mathcal{D}_0^{1,2}$ will be defined later.

Some results are already available for \eqref{eP}
in the case $k=N$; see for example \cite{w1,x1}
 and the references therein.
Wang and Zhou \cite{w1} proved that there exist at least two
solutions for \eqref{eP} with $a=0$,
 $0<\mu \leq \bar{\mu}_{0,N}=((N-2)/2)^2 $ and
$h\equiv 1$, under certain conditions on $g$.
Bouchekif and Matallah \cite{b2} showed the existence of two solutions of
\eqref{eP} under certain conditions  on functions $g$
and $h$, when $0<\mu \leq \bar{\mu}_{0,N}$,
$\lambda \in (0,\Lambda_{\ast })$, $-\infty <a<(N-2)/2$ and
$a\leq b<a+1$, with $\Lambda_{\ast }$ a positive constant.

Concerning existence results in the case $k<N$, we cite
\cite{g1,m1} and the references therein.
Musina \cite{m1} considered \eqref{eP} with
$-a/2$ instead of $a$ and $\lambda=0$,
also \eqref{eP} with  $a=0$, $b=0$, $\lambda=0$,
with $h\equiv 1$ and $a\neq 2-k$.
She established the existence of a ground state solution
when $2<k\leq N$ and $0<\mu <\bar{\mu}_{a,k}=((k-2+a)/2)^2 $
for \eqref{eP} with  $-a/2$ instead of $a$  and  $\lambda=0$.
She also  showed that \eqref{eP} with  $a=0$, $b=0$, $\lambda=0$
does not admit ground state solutions.
Badiale et al \cite{b1} studied \eqref{eP} with  $a=0$, $b=0$, $\lambda=0$
and $h\equiv 1$. They proved the existence of at least a nonzero
nonnegative weak solution $u$, satisfying $u(y,z)=u(|y|,z)$
when $2\leq k<N$ and $\mu <0$.
 Bouchekif and El Mokhtar \cite{b3} proved that
\eqref{eP} with  $a=0$, $b=0$ admits two distinct solutions
when $2<k\leq N$, $b=N-p(N-2)/2$ with
$p\in (2,2^{\ast}]$,
$\mu <\bar{\mu}_{0,k}$, and $\lambda \in (0,\Lambda_{\ast })$
where $\Lambda_{\ast }$ is a positive constant.
Terracini \cite{t2} proved that there are no positive solutions
of \eqref{eP} with  $b=0$, $\lambda=0$ when $a\neq 0$, $h\equiv 1$ and
$\mu <0$.
The regular problem corresponding to $a=b=\mu =0$ and $h\equiv 1$ has
been considered on a regular bounded domain $\Omega $
by Tarantello \cite{t1}.
She proved that for $g$ in $H^{-1}(\Omega )$,
the dual of $H_0^1(\Omega )$, not identically zero and satisfying a
suitable condition, the problem considered admits two distinct solutions.

Before formulating our results, we give some definitions and notation.
We denote by
$\mathcal{D}_0^{1,2}=\mathcal{D}_0^{1,2}(\mathbb{R}^k \backslash \{0\}\times
\mathbb{R}^{N-k})$ and
$\mathcal{H}_{\mu }=\mathcal{H}_{\mu }(\mathbb{R}
^k \backslash \{0\}\times \mathbb{R}^{N-k})$, the closure of
$C_0^{\infty }(\mathbb{R}^k \backslash \{0\}\times \mathbb{R}^{N-k})$
with respect to the norms
\[
\| u\|_{a,0}=\Big(\int_{\mathbb{R}^N}|y|^{-2a}|\nabla u|^2 \,dx\Big)^{1/2}
\]
and
\[
\| u\|_{a,\mu }=\Big(\int_{\mathbb{R}^N}(|y|^{-2a}|\nabla u|^2
-\mu |y|^{-2(a+1)}|u|^2 )\,dx\Big)^{1/2},
\]
respectively, with $\mu <\bar{\mu}_{a,k}=((k-2(a+1))/2)^2 $ for
$k\neq 2(a+1)$.

From the Hardy-Sobolev-Maz'ya inequality, it is easy to see that
the norm $\| u\|_{a,\mu }$ is equivalent to $\| u\|_{a,0}$.

Since our approach is variational, we define the functional
$I_{a,b,\lambda,\mu }$ on $\mathcal{H}_{\mu }$ by
\[
I(u):=I_{a,b,\lambda ,\mu }(u):=(1/2)
\| u\|_{a,\mu }^2 -(1/2_{\ast })\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx-\lambda \int_{\mathbb{R}^N}gu\,dx.
\]
We say that $u\in \mathcal{H}_{\mu }$ is a weak solution of \eqref{eP}
if it satisfies
\begin{align*}
\langle I'(u),v\rangle
&= \int_{ \mathbb{R}^N}(|y|^{-2a}\nabla u\nabla v-\mu |
y|^{-2(a+1)}uv-h|y|^{-2_{\ast
}b}|u|^{2_{\ast }-2}uv-\lambda gv)\,dx \\
&= 0,\quad \text{for }v\in \mathcal{H}_{\mu }.
\end{align*}
Here $\langle \cdot,\cdot\rangle $ denotes the product in the duality
$\mathcal{H}_{\mu }',\mathcal{H}_{\mu }$.

Throughout this work, we consider the following assumptions:
\begin{itemize}
\item[(G)] There exist $\nu_0>0$ and $\delta_0>0$ such that
$g(x)\geq \nu_0$, for all $x$ in $B(0,2\delta_0)$;

\item[(H)] $\lim_{|y|\to 0}h(y)=\lim_{|y|\to \infty }h(y)=h_0>0$,
$h(y)\geq h_0$, $y\in \mathbb{R}^k $.

\end{itemize}
Here, $B(a,r)$ denotes the ball centered at $a$ with radius $r$.\bigskip

Under some  conditions on the coefficients of \eqref{eP},
we split $\mathcal{N}$ in two disjoint subsets
$\mathcal{N}^{+}$ and $\mathcal{N}^{-}$, thus we consider the
minimization problems on $\mathcal{N}^{+}$ and $\mathcal{N}^{-}$.

\begin{remark}\label{rmk1} \rm
Note that all solutions of \eqref{eP} are nontrivial.
\end{remark}

We shall state our main results.

\begin{theorem}\label{thm1}
Assume that $3\leq k\leq N$, $-1<a<(k-2)/2$,
$0\leq\mu <\bar{\mu}_{a,k}$, and {\rm (G)} holds, then there exists
$\Lambda_1>0$ such that the \eqref{eP} has at
least one nontrivial solution on $\mathcal{H}_{\mu }$ for all
$\lambda \in (0,\Lambda_1)$.
\end{theorem}

\begin{theorem}\label{thm2}
In addition to the assumptions of the Theorem \ref{thm1}, if
{\rm (H)} holds, then there exists $\Lambda_{2}>0$ such that
\eqref{eP} has at least two nontrivial solutions
on $\mathcal{H}_{\mu }$ for all $\lambda \in (0,\Lambda_{2})$.
\end{theorem}

This article is organized as follows.
In Section 2, we give some preliminaries. Section 3 and 4 are
devoted to the proofs of Theorems \ref{thm1} and \ref{thm2}.

\section{Preliminaries}

We list here a few integral inequalities. The first one that we need is the
Hardy inequality with cylindrical weights \cite{m1}. It states that
\[
\bar{\mu}_{a,k}\int_{\mathbb{R}^N}|y|^{-2(a+1)}v^2 \,dx
\leq \int_{\mathbb{R}^N}|y|^{-2a}|\nabla v|^2 \,dx,\quad
\text{for all }v\in \mathcal{H}_{\mu },
\]
The starting point for studying \eqref{eP} is the
Hardy-Sobolev-Maz'ya inequality that is particular to the
cylindrical case $k<N$ and that was proved by Maz'ya in \cite{g1}.
It states that there exists positive constant $C_{a,2_{\ast }}$
such that
\[
C_{a,2_{\ast }}\Big(\int_{\mathbb{R}^N}|y|^{-2_{\ast }b}|v|^{2_{\ast
}}\,dx\Big)^{2/2_{\ast }}\leq \int_{\mathbb{R}^N}(|y|^{-2a}|\nabla v|^2
-\mu |y|^{-2(a+1)}v^2 )\,dx,
\]
for any $v\in C_{c}^{\infty }((\mathbb{R}^k \backslash \{0\})
\times \mathbb{R}^{N-k})$.

\begin{proposition}[\cite{g1}]\label{prop0}
The value
 \begin{equation}
S_{\mu ,2_{\ast }}=S_{\mu ,2_{\ast }}(k,2_{\ast })
:=\inf_{v\in \mathcal{H}_{\mu }\backslash \{0\}}\frac{
\int_{\mathbb{R}^N}(|y|^{-2a}|\nabla v|
^2 -\mu |y|^{-2(a+1)}v^2 )\,dx}{(\int_{\mathbb{R}^N}|y|^{-2_{\ast }b}
|v|^{2_{\ast}}\,dx)^{2/2_{\ast }}}  \label{nr3}
\end{equation}
is achieved on $\mathcal{H}_{\mu }$, for $2\leq k<N$ and
$\mu \leq \bar{\mu}_{a,k}$.
\end{proposition}

\begin{definition}\label{def1} \rm
Let $c\in \mathbb{R}$, $E$ be a Banach space and
$I\in C^1(E,\mathbb{R})$.

(i) $(u_{n})_{n}$ is a Palais-Smale sequence at
level $c$ (in short $(PS)_{c}$) in $E$ for $I$ if
$I(u_{n})=c+o_{n}(1)$ and
$I'(u_{n})=o_{n}(1)$,
where $o_{n}(1)\to 0$ as $n\to \infty $.

(ii) We say that $I$ satisfies the $(PS)_{c}$
condition if any $(PS)_{c}$ sequence in $E$ for $I$ has a
convergent subsequence.
\end{definition}

\subsection{Nehari manifold}

It is well known that $I$ is of class $C^1$ in $\mathcal{H}_{\mu }$ and
the solutions of \eqref{eP} are the critical points of $I$
which is not bounded below on $\mathcal{H}_{\mu }$.
Consider the Nehari manifold
\[
\mathcal{N}=\{u\in \mathcal{H}_{\mu }\backslash \{0\}:
\langle I'(u),u\rangle=0\},
\]
Thus, $u\in \mathcal{N}$ if and only if
\begin{equation}
\| u\|_{a,\mu }^2 -\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx-\lambda \int_{\mathbb{R}^N}gu\,dx=0.  \label{e13}
\end{equation}
Note that $\mathcal{N}$ contains every nontrivial solution of \eqref{eP}.
 Moreover, we have the following results.

\begin{lemma} \label{lem4}
The functional $I$ is coercive and bounded from below on $\mathcal{N}$.
\end{lemma}

\begin{proof}
If $u\in \mathcal{N}$, then by (\eqref{e13} and the H\"{o}lder inequality,
we deduce that
\begin{equation} \label{e14}
\begin{aligned}
I(u)
&= ((2_{\ast }-2)/2_{\ast }2)
\| u\|_{a,\mu }^2 -\lambda (1-(1/2_{\ast
}))\int_{\mathbb{R}^N}gu\,dx   \\
&\geq ((2_{\ast }-2)/2_{\ast }2)\|
u\|_{a,\mu }^2 -\lambda (1-(1/2_{\ast })
)\| u\|_{a,\mu }\| g\|_{\mathcal{H
}_{\mu }'}   \\
&\geq -\lambda ^2 C_0,
\end{aligned}
\end{equation}
where
\[
C_0:=C_0(\| g\|_{\mathcal{H}_{\mu }'})
=[(2_{\ast }-1)^2 /2_{\ast }2(2_{\ast }-2)]\| g\|_{\mathcal{H}_{\mu }'}^2 >0.
\]
Thus, $I$ is coercive and bounded from below on $\mathcal{N}$.
\end{proof}

Define
\[
\Psi_{\lambda }(u)=\langle I'(u),u\rangle .
\]
Then, for $u\in \mathcal{N}$,
\begin{equation}
\begin{aligned}  \label{e16}
\langle \Psi_{\lambda }'(u),u\rangle
&= 2\| u\|_{a,\mu }^2 -2_{\ast }\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx-\lambda \int_{\mathbb{R}^N}gu\,dx  \\
&= \| u\|_{a,\mu }^2 -(2_{\ast }-1)\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx   \\
&= \lambda (2_{\ast }-1)\int_{\mathbb{R}^N}gu\,dx-(2_{\ast }-2)
\| u\|_{a,\mu }^2.
\end{aligned}
\end{equation}
Now, we split $\mathcal{N}$ in three parts:
\begin{gather*}
\mathcal{N}^{+}=\{u\in \mathcal{N}:\langle \Psi_{\lambda}'(u),u\rangle >0\},\quad
\mathcal{N}^{0}=\{u\in \mathcal{N}\langle \Psi_{\lambda}'(u),u\rangle =0\}, \\
\mathcal{N}^{-}=\{u\in \mathcal{N}:\langle \Psi_{\lambda }'(u),u\rangle <0\}
\end{gather*}
We have the following results.

\begin{lemma}\label{lem5}
Suppose that there exists a local minimizer $u_0$ for $I$ on
$\mathcal{N}$ and $u_0\notin \mathcal{N}^{0}$. Then,
$I'(u_0)=0$ in $\mathcal{H}_{\mu }'$.
\end{lemma}

\begin{proof}
If $u_0$ is a local minimizer for $I$ on $\mathcal{N}$,
then there exists $\theta \in \mathbb{R}$ such that
\[
\langle I'(u_0),\varphi \rangle
=\theta \langle \Psi_{\lambda }'(u_0),\varphi \rangle
\]
for any $\varphi \in \mathcal{H}_{\mu }$.

If $\theta =0$, then the lemma is proved. If not, taking
$\varphi \equiv u_0$ and using the assumption $u_0\in \mathcal{N}$,
we deduce
\[
0=\langle I'(u_0),u_0\rangle =\theta
\langle \Psi_{\lambda }'(u_0),u_0\rangle .
\]
Thus
\[
\langle \Psi_{\lambda }'(u_0),u_0\rangle =0,
\]
which contradicts that $u_0\notin \mathcal{N}^{0}$.
\end{proof}

Let
\begin{equation}
\Lambda_1:=(2_{\ast }-2)(2_{\ast }-1)^{-(
2_{\ast }-1)/(2_{\ast }-2)}[(h_0)
^{-1}S_{\mu ,2_{\ast }}]^{2_{\ast }/2(2_{\ast }-2)
}\| g\|_{\mathcal{H}_{\mu }'}^{-1}.
\label{e20}
\end{equation}

\begin{lemma}\label{lem6}
We have $\mathcal{N}^{0}=\emptyset $ for all
$\lambda \in (0,\Lambda_1)$.
\end{lemma}

\begin{proof}
Let us reason by contradiction.
Suppose $\mathcal{N}^{0}\neq \emptyset $ for some
$\lambda \in (0,\Lambda_1)$. Then, by \eqref{e16} and for $u\in
\mathcal{N}^{0}$, we have
\begin{equation} \label{e18}
\begin{aligned}
\| u\|_{a,\mu }^2
&= (2_{\ast }-1)\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx   \\
&= \lambda ((2_{\ast }-1)/(2_{\ast }-2))\int_{\mathbb{R}
^N}gu\,dx.
\end{aligned}
\end{equation}
Moreover, by (G), the H\"{o}lder inequality and the Sobolev
embedding theorem, we obtain
\begin{equation}
\Big[\big((h_0)^{-1}S_{\mu ,2_{\ast }}\big)^{2_{\ast
}/2}/(2_{\ast }-1)\Big]^{1/(2_{\ast }-2)}
\leq \| u\|_{a,\mu }
\leq \big[\lambda \big((2_{\ast
}-1)\| g\|_{\mathcal{H}_{\mu }'}/(
2_{\ast }-2)\big)\big].  \label{e19}
\end{equation}
This implies that $\lambda \geq \Lambda_1$, which is a contradiction
to $\lambda \in (0,\Lambda_1)$.
\end{proof}

Thus $\mathcal{N}=\mathcal{N}^{+}\cup \mathcal{N}^{-}$ for
$\lambda \in (0,\Lambda_1)$.
Define
\[
c:=\inf_{u\in \mathcal{N}}I(u),\quad
c^{+}:=\inf_{u\in \mathcal{N}^{+}}I(u), \quad
c^{-}:=\inf_{u\in \mathcal{N}^{-}}I(u).
\]
We need also the following Lemma.

\begin{lemma}\label{lem7}
(i) If $\lambda \in (0,\Lambda_1)$, then
$c\leq c^{+}<0$.

(ii) If $\lambda \in (0,(1/2)\Lambda
_1)$, then $c^{-}>C_1$, where
\begin{align*}
C_1 = C_1(\lambda ,S_{\mu ,2_{\ast }}\| g\|_{ \mathcal{H}_{\mu }'})
&=\big((2_{\ast }-2)/2_{\ast }2\big)(2_{\ast }-1)^{2/(2_{\ast }-2)
}(S_{\mu ,2_{\ast }})^{2_{\ast }/(2_{\ast }-2)}\\
&\quad -\lambda (1-(1/2_{\ast }))(2_{\ast }-1)^{2/(2_{\ast }-2)}
\| g\|_{\mathcal{H}_{\mu }'}.
\end{align*}
\end{lemma}

\begin{proof}
(i) Let $u\in \mathcal{N}^{+}$. By \eqref{e16},
\[
[1/(2_{\ast }-1)]\| u\|_{a,\mu
}^2 >\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx
\]
and so
\begin{align*}
I(u)&= (-1/2)\| u\|_{a,\mu }^2 +(1-(1/2_{\ast }))\int_{
\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|^{2_{\ast }}\,dx \\
&< [(-1/2)+(1-(1/2_{\ast }))
(1/(2_{\ast }-1))]\| u\|_{a,\mu }^2  \\
&= -((2_{\ast }-2)/2_{\ast }2)\|u\|_{a,\mu }^2 ;
\end{align*}
we conclude that $c\leq c^{+}<0$.

(ii) Let $u\in \mathcal{N}^{-}$. By \eqref{e16},
\[
[1/(2_{\ast }-1)]\| u\|_{a,\mu}^2
<\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|^{2_{\ast }}\,dx.
\]
Moreover, by Sobolev embedding theorem, we have
\[
\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx\leq (S_{\mu ,2_{\ast }})^{-2_{\ast
}/2}\| u\|_{a,\mu }^{2_{\ast }}.
\]
This implies
\[
\| u\|_{a,\mu }>[(2_{\ast }-1)]
^{-1/(2_{\ast }-2)}(S_{\mu ,2_{\ast }})^{2_{\ast
}/2(2_{\ast }-2)},\quad \text{for all }u\in \mathcal{N}^{-}.
\]
By \eqref{e14},
\[
I(u)\geq ((2_{\ast }-2)/2_{\ast }2)
\| u\|_{a,\mu }^2 -\lambda (1-(1/2_{\ast
}))\| u\|_{a,\mu }\| g\|_{\mathcal{H}_{\mu }'}.
\]
Thus, for all $\lambda \in (0,(1/2)\Lambda_1)$,
we have $I(u)\geq C_1$.
\end{proof}

For each $u\in \mathcal{H}_{\mu }$, we write
\[
t_{m}:=t_{\rm max}(u)=[\frac{\| u\|
_{a,\mu }}{(2_{\ast }-1)\int_{
\mathbb{R}
^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx}]^{1/(2_{\ast }-2)}>0.
\]

\begin{lemma}\label{lem8}
Let $\lambda \in (0,\Lambda_1)$. For each $u\in
\mathcal{H}_{\mu }$, one has the following:

(i) If  $\int_{\mathbb{R}^N}g(x)u\,dx\leq 0$, then there exists
a unique $t^{-}>t_{m}$ such that $t^{-}u\in \mathcal{N}^{-}$ and
\[
I(t^{-}u)=\sup_{t\geq 0}I(tu).
\]

(ii) If  $\int_{\mathbb{R}^N}g(x)u\,dx>0$, then there exist
unique $t^{+}$ and $t^{-}$ such that $0<$ $t^{+}<t_{m}<t^{-}$,
$t^{+}u\in \mathcal{N}^{+}$, $t^{-}u\in\mathcal{N}^{-}$,
\[
I(t^{+}u)=\inf_{0\leq t\leq t_{m}}I(tu)
\text{ and }I(t^{-}u)=\sup_{t\geq 0}I(tu).
\]
\end{lemma}

The proof of the above lemma follows from a proof in \cite{b5},
with minor modifications.

\section{Proof of Theorem \ref{thm1}}

For the proof we need the following results.

\begin{proposition}[\cite{b5}] \label{prop1}
(i) If $\lambda \in (0,\Lambda_1)$, then
there exists a minimizing sequence $(u_{n})_{n}$ in
$\mathcal{N}$ such that
\begin{equation}
I(u_{n})=c+o_{n}(1),\quad I'(u_{n})=o_{n}(1)\quad
\text{in }\mathcal{H}_{\mu}',  \label{r31}
\end{equation}
where $o_{n}(1)$ tends to $0$ as $n$ tends to $\infty $.

(ii) if $\lambda \in (0,(1/2)\Lambda_1)$, then there exists
a minimizing sequence $(u_{n})_{n}$ in $\mathcal{N}^{-}$ such that
\[
I(u_{n})=c^{-}+o_{n}(1),\quad I'(u_{n})=o_{n}(1)\quad
\text{in }\mathcal{H}_{\mu}'.
\]
\end{proposition}

Now, taking as a starting point the work of Tarantello \cite{t1},
we establish the existence of a local minimum for $I$
on $\mathcal{N}^{+}$.

\begin{proposition}\label{prop2}
If $\lambda \in (0,\Lambda_1)$, then $I$ has a
minimizer $u_1\in \mathcal{N}^{+}$ and it satisfies
\begin{itemize}
\item[(i)] $I(u_1)=c=c^{+}<0$,
\item[(ii)] $u_1$ is a solution of \eqref{eP}.
\end{itemize}
\end{proposition}

\begin{proof}
(i) By Lemma \ref{lem4}, $I$ is coercive and bounded below on
$\mathcal{N}$. We can assume that there exists
$u_1\in \mathcal{H}_{\mu }$ such that
\begin{equation} \label{32}
\begin{gathered}
u_{n} \rightharpoonup  u_1\quad \text{weakly in }\mathcal{H}_{\mu }, \\
u_{n} \rightharpoonup  u_1\quad \text{weakly in }
 L^{2_{\ast }}(\mathbb{R} ^N,|y|^{-2_{\ast }b}),   \\
u_{n} \rightarrow u_1\quad \text{a.e in }\mathbb{R}^N.
\end{gathered}
\end{equation}
Thus, by \eqref{r31} and \eqref{32}, $u_1$
is a weak solution of \eqref{eP} since $c<0$ and
$I(0)=0$. Now, we show that $u_{n}$ converges to $u_1$
strongly in $\mathcal{H}_{\mu }$. Suppose otherwise.
Then $\| u_1\|_{a,\mu }<\liminf_{n\to \infty }\|u_{n}\|_{a,\mu }$
and we obtain
\begin{align*}
c &\leq I(u_1)=((2_{\ast }-2)/2_{\ast
}2)\| u_1\|_{a,\mu }^2 -\lambda (1-(
1/2_{\ast }))\int_{\mathbb{R}^N}gu_1\,dx \\
&< \liminf_{n\to \infty }I(u_{n})=c.
\end{align*}
We have a contradiction. Therefore, $u_{n}$ converges
to $u_1$ strongly in $\mathcal{H}_{\mu }$.
Moreover, we have $u_1\in \mathcal{N}^{+}$. If not,
then by Lemma \ref{lem8}, there are two numbers $t_0^{+}$ and $t_0^{-}$,
uniquely defined so that $t_0^{+}u_1\in \mathcal{N}^{+}$ and
$t_0^{-}u_1\in \mathcal{N}^{-}$. In particular, we have
$t_0^{+}<t_0^{-}=1$. Since
\[
\frac{d}{dt}I(tu_1)\big|_t =t_0^{+}=0, \quad
\frac{d^2 }{dt^2 }I(tu_1)\big|t =t_0^{+} >0,
\]
there exists $t_0^{+}<t^{-}\leq t_0^{-}$ such that
$I(t_0^{+}u_1)<I(t^{-}u_1)$. By Lemma \ref{lem8},
\[
I(t_0^{+}u_1)<I(t^{-}u_1)<I(t_0^{-}u_1)=I(u_1),
\]
which is a contradiction.
\end{proof}

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

In this section, we establish the existence of a second solution
of \eqref{eP}. For this, we require the following Lemmas,
with $C_0$ is given in \eqref{e14}.

\begin{lemma}\label{lem9}
 Assume that {\rm (G)} holds and let
$(u_{n})_{n}\subset \mathcal{H}_{\mu }$ be a $(PS)_{c}$
sequence for $I$ for some $c\in \mathbb{R}$ with
$u_{n}\rightharpoonup u$ in $\mathcal{H}_{\mu }$.
Then, $I'(u)=0$ and
\[
I(u)\geq -C_0\lambda ^2 .
\]
\end{lemma}

\begin{proof}
It is easy to prove that $I'(u)=0$, which implies
that $\langle I'(u),u\rangle =0$, and
\[
\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx=\| u\|_{a,\mu }^2 -\lambda \int_{
\mathbb{R}^N}gu\,dx.
\]
Therefore,
\[
I(u)=((2_{\ast }-2)/2_{\ast }2)
\| u\|_{a,\mu }^2 -\lambda (1-(1/2_{\ast
}))\int_{\mathbb{R}^N}gu\,dx.
\]
Using \eqref{e14}, we obtain
\[
I(u)\geq -C_0\lambda ^2 .
\]
\end{proof}

\begin{lemma}\label{lem10}
 Assume that {\rm (G)} holds and for any $(PS)_{c}$ sequence
with $c$ is a real number such that
$c<c_{\lambda }^{\ast }$.
Then, there exists a subsequence which converges strongly.
Here $c_{\lambda }^{\ast }:=((2_{\ast }-2)/2_{\ast
}2)(h_0)^{-2/(2_{\ast }-2)}(S_{\mu
,2_{\ast }})^{2_{\ast }/(2_{\ast }-2)}-C_0\lambda ^2 $.
\end{lemma}

\begin{proof}
Using standard arguments, we get that $(u_{n})_{n}$ is bounded
in $\mathcal{H}_{\mu }$. Thus, there exist a subsequence of
$(u_{n})_{n}$ which we still denote by $(u_{n})_{n}$ and
$u\in \mathcal{H}_{\mu }$ such that
\begin{gather*}
u_{n} \rightharpoonup u\quad \text{weakly in }\mathcal{H}_{\mu }, \\
u_{n} \rightharpoonup u\quad \text{weakly in }L^{2_{\ast }}(
\mathbb{R}^N,|y|^{-2_{\ast }b}). \\
u_{n} \rightarrow u \quad \text{a.e in }\mathbb{R}^N.
\end{gather*}
Then, $u$ is a weak solution of \eqref{eP}. Let
$v_{n}=u_{n}-u$, then by Br\'{e}zis-Lieb \cite{b4}, we obtain
\begin{equation}
\| v_{n}\|_{a,\mu }^2 =\| u_{n}\|_{a,\mu }^2
-\| u\|_{a,\mu }^2 +o_{n}(1)
\label{46}
\end{equation}
and
\begin{equation}
\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|v_{n}|^{2_{\ast }}\,dx
=\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u_{n}|^{2_{\ast }}\,dx
-\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|u|
^{2_{\ast }}\,dx+o_{n}(1).  \label{e47}
\end{equation}
On the other hand, by using the assumption (H), we obtain
\begin{equation}
\lim_{n\to \infty }\int_{\mathbb{R}^N}h(x)|y|^{-2_{\ast }b}|
v_{n}|^{2_{\ast }}\,dx=h_0\lim_{n\to \infty
}\int_{\mathbb{R}^N}|y|^{-2_{\ast }b}|v_{n}|
^{2_{\ast }}\,dx.  \label{e47'}
\end{equation}
Since $I(u_{n})=c+o_{n}(1)$,
$I'(u_{n})=o_{n}(1)$ and by \eqref{46}, \eqref{e47}, and \eqref{e47'}
 we  deduce that
\begin{equation} \label{48}
\begin{gathered}
(1/2)\| v_{n}\|_{a,\mu }^2 -(1/2_{\ast
})\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}|v_{n}|
^{2_{\ast }}\,dx=c-I(u)+o_{n}(1),\\
\| v_{n}\|_{a,\mu }^2 -\int_{
\mathbb{R}^N}h|y|^{-2_{\ast }b}|v_{n}| ^{2_{\ast }}\,dx=o_{n}(1).
\end{gathered}
\end{equation}
Hence, we may assume that
\begin{equation}
\| v_{n}\|_{a,\mu }^2 \to l,\quad
\int_{\mathbb{R} ^N}h|y|^{-2_{\ast }b}|v_{n}|
^{2_{\ast }}\,dx\to l.  \label{50}
\end{equation}
Sobolev inequality gives
$\| v_{n}\|_{a,\mu }^2 \geq (S_{\mu ,2_{\ast }})\int_{\mathbb{R}
^N}h|y|^{-2_{\ast }b}|v_{n}|^{2_{\ast }}\,dx$.
Combining this inequality with \eqref{50},
we obtain
\[
l\geq S_{\mu ,2_{\ast }}(l^{-1}h_0)^{-2/2_{\ast }}.
\]
Either $l=0$ or $l\geq (h_0)^{-2/(2_{\ast }-2)
}(S_{\mu ,2_{\ast }})^{2_{\ast }/(2_{\ast }-2)}$.
Suppose that
\[
l\geq (h_0)^{-2/(2_{\ast }-2)}(S_{\mu ,2_{\ast }})^{2_{\ast }
/(2_{\ast }-2)}.
\]
Then, from \eqref{48}, \eqref{50} and Lemma
\ref{lem9}, we obtain
\[
c\geq ((2_{\ast }-2)/2_{\ast }2)l+I(u)
\geq c_{\lambda }^{\ast },
\]
which is a contradiction. Therefore, $l=0$ and we conclude
that $u_{n}$ converges to $u$ strongly in $\mathcal{H}_{\mu }$.
\end{proof}

\begin{lemma}\label{lem11}
Assume that {\rm (G)} and {\rm (H)} hold.
Then, there exist $v\in \mathcal{H}_{\mu }$ and
$\Lambda_{\ast }>0$ such that for $\lambda \in (0,\Lambda_{\ast })$,
one has
\[
\sup_{t\geq 0}I(tv)<c_{\lambda }^{\ast }.
\]
In particular,
$c^{-}<c_{\lambda }^{\ast }$ for all $\lambda \in (0,\Lambda _{\ast })$.
\end{lemma}

\begin{proof}
Let $\varphi_{\varepsilon }$ be such that
\[
\varphi_{\varepsilon }(x)
= \begin{cases}
\omega_{\varepsilon }(x)& \text{if }g(x)
\geq 0\text{ for all }x\in \mathbb{R}^N \\
\omega_{\varepsilon }(x-x_0) & \text{if }g(
x_0)>0\text{ for }x_0\in\mathbb{R}^N \\
-\omega_{\varepsilon }(x) & \text{if }g(x)
\leq 0\text{ for all }x\text{ }\in
\mathbb{R}^N
\end{cases}
\]
where $\omega_{\varepsilon }$ satisfies \eqref{nr3}. Then, we
claim that there exists $\varepsilon_0>0$ such that
\begin{equation}
\lambda \int_{\mathbb{R}^N}g(x)\varphi_{\varepsilon }(x)\,dx>0\quad
\text{for any }\varepsilon \in (0,\varepsilon_0).  \label{55}
\end{equation}
In fact, if $g(x)\geq 0$ or $g(x)\leq 0$ for all
$x\in \mathbb{R}^N$, \eqref{55} obviously holds.
If there exists $x_0\in \mathbb{R}^N$ such that $g(x_0)>0$,
then by the continuity of $g(x)$, there exists $\eta >0$
such that $g(x)>0$ for all $x\in B(x_0,\eta )$.
Then by the definition of $\omega_{\varepsilon }(x-x_0)$,
it is easy to see that there exists an $\varepsilon_0$ small
enough such that
\[
\lambda \int_{\mathbb{R}^N}g(x)\omega_{\varepsilon }(x-x_0)\,dx>0
,\quad \text{for any }\varepsilon \in (0,\varepsilon_0).
\]
Now, we consider the  functions
\[
f(t)=I(t\varphi_{\varepsilon }),\quad
\tilde{f}(t)=(t^2 /2)\| \varphi_{\varepsilon }\|_{a,\mu }^2
-(t^{2_{\ast }}/2_{\ast })\int_{\mathbb{R}^N}h|y|^{-2_{\ast }b}
|\varphi_{\varepsilon}|^{2_{\ast }}\,dx.
\]
Then,  for all $\lambda \in (0,\Lambda_1)$,
\[
f(0)=0<c_{\lambda }^{\ast }.
\]
By the continuity of $f$, there exists $t_0>0$ small enough such that
\[
f(t)<c_{\lambda }^{\ast },\quad \text{for all }t\in (0,t_0).
\]
On the other hand,
\[
\max_{t\geq 0}\tilde{f}(t)=((2_{\ast
}-2)/2_{\ast }2)(h_0)^{-2/(2_{\ast
}-2)}(S_{\mu ,2_{\ast }})^{2_{\ast }/(2_{\ast
}-2)}.
\]
Then, we obtain
\[
\sup_{t\geq 0}I(t\varphi_{\varepsilon })<((
2_{\ast }-2)/2_{\ast }2)(h_0)^{-2/(
2_{\ast }-2)}(S_{\mu ,2_{\ast }})^{2_{\ast }/(
2_{\ast }-2)}-\lambda t_0\int_{\mathbb{R}^N}
g\varphi_{\varepsilon }\,dx.
\]
Now, taking $\lambda >0$ such that
\[
-\lambda t_0\int_{\mathbb{R}^N}g\varphi_{\varepsilon }\,dx
<-C_0\lambda ^2 ,
\]
and by \eqref{55}, we obtain
\[
0<\lambda <(t_0/C_0)\Big(\int_{
\mathbb{R}
^N}g\varphi_{\varepsilon }\Big),\quad \text{for }\varepsilon
<<\varepsilon_0.
\]
Set
\[
\Lambda_{\ast }=\min \{\Lambda_1,\text{ }(t_0/C_0)
(\int_{\mathbb{R}^N}g\varphi_{\varepsilon })\}.
\]
We deduce that
\begin{equation}
\sup_{t\geq 0}I(t\varphi_{\varepsilon })<c_{\lambda },\quad
\text{for all }\lambda \in (0,\Lambda_{\ast }).  \label{60}
\end{equation}
Now, we prove  that
\[
c^{-}<c_{\lambda }^{\ast },\quad \text{for all }
\lambda \in (0,\Lambda _{\ast }).
\]
By (G) and the existence of $w_{n}$ satisfying
\eqref{nr3}, we have
\[
\lambda \int_{\mathbb{R}^N}gw_{n}\,dx>0.
\]
Combining this with Lemma \ref{lem8} and from the definition
of $c^{-}$ and \eqref{60}, we obtain that there exists
$t_{n}>0$ such that $t_{n}w_{n}\in \mathcal{N}^{-}$ and for
all $\lambda \in (0,\Lambda_{\ast })$,
\[
c^{-}\leq I(t_{n}w_{n})\leq \sup_{t\geq 0}I(
tw_{n})<c_{\lambda }^{\ast }.
\]
\end{proof}

Now we establish the existence of a local minimum of $I$
on $\mathcal{N}^{-}$.

\begin{proposition}\label{prop3}
There exists $\Lambda_{2}>0$ such that for
$\lambda \in (0,\Lambda_{2})$, the functional $I$ has a
minimizer $u_{2}$ in $\mathcal{N}^{-}$ and satisfies
\begin{itemize}
\item[(i)] $I(u_{2})=c^{-}$,

\item[(ii)] $u_{2}$ is a solution of \eqref{eP} in
$\mathcal{H}_{\mu }$,
\end{itemize}
where $\Lambda_{2}=\min \{(1/2)\Lambda_1,\Lambda_{\ast }\}$
with $\Lambda_1$ defined as in \eqref{e20} and $\Lambda_{\ast }$
defined as in the proof of Lemma \ref{lem11}.
\end{proposition}

\begin{proof}
By Proposition \ref{prop1} (ii), there exists a
$(PS)_{c^{-}}$ sequence for $I$, $(u_{n})_{n}$ in
$\mathcal{N}^{-}$ for all $\lambda \in (0,(1/2)\Lambda_1)$.
 From Lemmas \ref{lem10}, \ref{lem11} and \ref{lem7} (ii),
 for $\lambda \in (0,\Lambda_{\ast })$, $I$
satisfies $(PS)_{c^{-}}$ condition and $c^{-}>0$. Then, we get
that $(u_{n})_{n}$ is bounded in $\mathcal{H}_{\mu }$.
Therefore, there exist a subsequence of $(u_{n})_{n}$ still
denoted by $(u_{n})_{n}$ and $u_{2}\in \mathcal{N}^{-}$ such
that $u_{n}$ converges to $u_{2}$ strongly in $\mathcal{H}_{\mu }$
and $I(u_{2})=c^{-}$ for all $\lambda \in (0,\Lambda_{2})$.
Finally, by using the same arguments as in the proof of the
Proposition \ref{prop2}, for all $\lambda \in (0,\Lambda_1)$,
we have that $u_{2}$ is a solution of \eqref{eP}.
\end{proof}

Now, we complete the proof of Theorem \ref{thm2}.
By Propositions \ref{prop2} and \ref{prop3}, we obtain that \eqref{eP}
has two solutions $u_1$ and $u_{2}$ such that
$u_1\in \mathcal{N}^{+}$ and $u_{2}\in \mathcal{N}^{-}$.
Since $\mathcal{N}^{+}\cap \mathcal{N}^{-}=\emptyset $,
this implies that $u_1$ and $u_{2}$ are distinct.

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\end{document}