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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 32, 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 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/32\hfil Elliptic equations with a concave term]
{Singular elliptic equations involving a concave term and
critical Caffarelli-Kohn-Nirenberg exponent with sign-changing
weight functions}

\author[M. Bouchekif, A. Matallah\hfil EJDE-2010/32\hfilneg]
{Mohammed Bouchekif, Atika Matallah}  % in alphabetical order

\address{Mohammed Bouchekif \newline
Universit\'{e} Aboubekr Belkaid,
 D\'{e}partement de Math\'{e}matiques, 
 BP 119 (13000) Tlemcen - Alg\'{e}rie}
\email{m\_bouchekif@yahoo.fr}

\address{Atika Matallah \newline
Universit\'{e} Aboubekr Belkaid,
 D\'{e}partement de Math\'{e}matiques, 
 BP 119 (13000) Tlemcen - Alg\'{e}rie.}
\email{atika\_matallah@yahoo.fr}

\thanks{Submitted September 28, 2009. Published March 3, 2010.}
\subjclass[2000]{35A15, 35B25, 35B33, 35J60}
\keywords{Variational methods;  critical
Caffarelli-Kohn-Nirenberg exponent; \hfill\break\indent
concave term; singular and sign-changing
weights; Palais-Smale condition}

\begin{abstract}
 In this article we establish the existence of at least two distinct
 solutions to singular elliptic equations involving a concave term
 and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing
 weight functions.
\end{abstract}

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

\section{Introduction}

 This article shows the existence of at least two
solutions to the  problem
\begin{equation} \label{p-lambda-mu}
\begin{gathered}
-\mathop{\rm div}\big(\frac{\nabla u}{|x|^{2a}}\big) -\mu
\frac{u}{|x|^{2(a+1)}}=\lambda h(x) \frac{| u|
^{q-2}u}{|x|^{c}}+k(x)\frac{| u|^{2_{\ast }-2}u}{|x| ^{2_{\ast
}b}} \quad \text{in }\Omega \backslash\{0\}\\
u=0 \quad \text{on}\ \partial \ \Omega
\end{gathered}
\end{equation}
where $\Omega \subset\mathbb{R}^{N}$ is an open bounded domain,
$N\geq 3$, $0\in \Omega $, $a<( N-2) /2$,
$a\leq b<a+1$, $1<q<2$, $c\leq q(a+1)+N(1-q/2)$,
$2_{\ast}:=2N/(N-2+2(b-a))$ is the critical Caffarelli-Kohn-Nirenberg
exponent, $\mu <\bar{\mu }_{a}:=(N-2(a+1))^{2}/4$,
$\lambda$ is a positive parameter and $h$, $k$ are continuous functions
which change sign in $\bar{\Omega}$.

It is clear that degeneracy and singularity occur in problem
\eqref{p-lambda-mu}. In these situations, the
classical methods fail to be applied directly so that the
existence results may become a delicate matter that is closely
related to some phenomena due to the degenerate (or singular)
character of the differential equation.  The starting point
of the variational approach to these problems is the following
Caffarelli-Kohn-Nirenberg inequality in \cite{CK}: there is a
positive constant $C_{a,b}$ such that
\begin{equation}
\Big(\int_{\mathbb{R}^{N}}|x|^{-2_{\ast }b}|u|^{2_{\ast }}dx\Big)
^{1/2_{\ast }}\leq
C_{a,b}\Big(\int_{\mathbb{R}^{N}}|x|^{-2a}
|\nabla u|^{2}dx\Big)^{1/2}\quad \forall u\in
C_{0}^{\infty }( \Omega ),  \label{ee}
\end{equation}
where $-\infty <a<(N-2)/2$, $a\leq b<a+1$,
$2_{\ast}=2N/(N-2+2(b-a))$.
For sharp constants and extremal
functions, see [7,9]. In \eqref{ee}, as $b=a+1$, then
$2_{\ast}=2$ and we have the following weighted Hardy inequality
\cite{CC}:
\begin{equation} \label{e1.2}
\int_{\mathbb{R}^{N}}|x|^{-2(a+1)}u^{2}dx\leq \frac{1}{ \bar{\mu
}_{a}}\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla u|^{2}dx \quad
\text{for all }u\in C_{0}^{\infty }(\Omega ).
\end{equation}
We introduce a weighted Sobolev space $D_{a}^{1,2}(\Omega )$ which
is the completion of the space $C_{0}^{\infty }(\Omega )$  with
respect to the norm
\[
\| u\|_{0,a}=\Big(\int_{\Omega }| x| ^{-2a}|\nabla u|^{2}dx\Big) ^{1/2}.
\]
Define $H_{\mu }$\ as the completion of the space
$C_{0}^{\infty}( \Omega )$ with respect to the  norm
\[
\| u\|_{\mu ,a}:=\Big(\int_{\Omega }\big( | x| ^{-2a}|\nabla u|^{2}-\mu
|x| ^{-2(a+1)}u^{2}\big)dx\Big)^{1/2}\quad \text{for } -\infty <\mu
<\bar{\mu }_{a}.
\]
By weighted Hardy inequality $\| \cdot\|_{\mu ,a}$ is equivalent to
$\| \cdot\|_{0,a}$; i.e.,
\[
\Big(1- \frac{1}{\bar{\mu }_{a}}\max (\mu ,0)\Big) ^{1/2}
\|u\|_{0,a}\leq \| u\|_{\mu ,a}
\leq \Big(1-\frac{1}{\bar{mu}_{a}}
\min (\mu ,0)\Big) ^{1/2}\| u\|_{0,a},
\]
for all $u\in H_{\mu }$.  From the boundedness of $\Omega $ and
the standard approximation
arguments,
it is easy to see that \eqref{ee} hold for any $u\in H_{\mu }$
in the sense
\begin{equation} \label{e1.3}
\Big(\int_{\Omega }|x|^{-c}|u|^{p}dx\Big)^{1/p}\leq C(\int_{\Omega
}|x|^{-2a}|\nabla u|^{2}dx) ^{1/2},
\end{equation}
for $1\leq p\leq 2N/(N-2)$, $c\leq p(a+1)+N( 1-p/2)$, and in
\cite{X.} if $p<2N/(N-2)$ the embedding $H_{\mu }\hookrightarrow
L_{p}(\Omega ,|x|^{-c})$ is compact, where
$L_{p}(\Omega,|x|^{-c})$ is the weighted $L_{p}$ space with norm
\[
|u|_{p,c}=\Big(\int_{\Omega }|x|^{-c}| u|^{p}dx\Big)^{1/p}.
\]

We start by giving a brief historic point of view.
It is known that the number of nontrivial solutions of problem
\eqref{p-lambda-mu} is affected by the concave and
convex terms. This study has been the focus of a great deal of
research in recent years.

The case $h\equiv 1$ and $k\equiv 1$ has been studied extensively
by many authors, we refer the reader to \cite{AB}, \cite{BA},
\cite{C}, \cite{Lin} and the references therein. In \cite{AB}
Ambrosetti et al. studied the problem \eqref{p-lambda-mu}
for $\mu=0$, $a=b=c=0,2_{\ast }=2^{\ast }=2N/(N-2)$ replaced by $p$, where
$1<p\leq 2_{\ast }$. They establish the existence of $\Lambda
_{0}>0$ such that $(\mathcal{P}_{\lambda ,0})$ for $\lambda $
fixed in $(0,\Lambda _{0})$ has at least two positive solutions by
using sub-super method and the Mountain Pass Theorem, problem \eqref{p-lambda-mu} for $\lambda =\Lambda _{0}$ has also
a positive solution  and no positive solution for
$\lambda>\Lambda _{0}$. When $\mu >0$, $a=b=c=0$, Chen \cite{C}
studied the
asymptotic behavior of solutions to problem \eqref{p-lambda-mu}
by using the Moser's iteration. By applying the Ekeland
Variational Principle he obtained a first positive solution, and
by the Mountain Pass Theorem he proved the existence of a second
positive solution.  Recently, Bouchekif and Matallah
\cite{BA} extended the results of \cite{C} to problem
$(\mathcal{P}_{\lambda ,\mu })\ $with $a=c=0$, $0\leq b<1$, they
established the existence of two positive solutions under some
sufficient conditions for $\lambda $ and $\mu $. Lin \cite{Lin}
considered a more general problem \eqref{p-lambda-mu}
with $0\leq a<(N-2)/2$, $a\leq b<a+1$, $c=0,1<q<2$ and
$\mu>0$.

For the case $h\not\equiv 1$ or $k\not\equiv 1$, we refer the
reader to \cite{BA2,HH,T,W} and the
references therein. Tarantello \cite{T} studied the problem
\eqref{p-lambda-mu} for $\mu=0$, $a=b=c=0$, $q=\lambda =1$,
$k\equiv 1$ and $h$ not necessarily equals to $1$, satisfying some
conditions.
Recently, problem \eqref{p-lambda-mu} in $\Omega =
\mathbb{R}^{N}$ with $q=1$ has considered in \cite{BA2}.

Wu \cite{W} showed the existence of multiple positive solutions
for problem \eqref{p-lambda-mu} with $a=b=c=0$,
$1<q<2$, $k\equiv 1,h$ is a continuous function which changes sign in
$\bar{\Omega }$. In \cite{HH}, Hsu and Lin established the
existence of multiple nontrivial solutions to problem
\eqref{p-lambda-mu} with $a=b=c=0$, $1<q<2$, $h$ and
$k$ are smooth functions which change sign in $\bar{\Omega}$.

The operator
$L_{\mu ,a}u:=-\mathop{\rm div}(|x|^{-2a}\nabla u)-\mu |x|^{-2(
a+1)}u$ has been the subject of many papers, we quote, among
others \cite{FG} for $a=0$ and $\mu < \bar{\mu}_{0}$, and \cite{E}
or \cite{X} for general case i.e $-\infty <a<( N-2)/2$ and
$\mu<\bar{\mu}_{a}$.

Xuan et al. \cite{X} proved that under the  conditions
\begin{gather*}
N\geq 3,\quad a<(N-2)/2,\quad 0<\sqrt{\bar{\mu }_{a}}-
\sqrt{\bar{\mu }_{a}-\mu }+a<(N-2)/2,\\
a\leq b<a+1, \quad \mu <\bar{\mu }_{a}-b^{2},
\end{gather*}
for $\varepsilon >0$, the function
\begin{equation} \label{e1.4}
u_{\varepsilon }(x)=C_{0}\varepsilon ^{\frac{2}{2_{\ast }-2}
}\Big(\varepsilon ^{\frac{2\sqrt{\bar{\mu }_{a}-\mu }}{\sqrt{
\bar{\mu }_{a}-\mu }-b}}|x|^{\frac{2_{\ast }-2}{2}
(\sqrt{\bar{\mu }_{a}}-\sqrt{\bar{\mu }_{a}-\mu
})}+|x|^{\frac{2_{\ast }-2}{2}(\sqrt{\bar{\mu }
_{a}}+\sqrt{\bar{\mu }_{a}-\mu })}\Big)^{-\frac{2}{2_{\ast }-2 }}
\end{equation}
with a suitable positive constant $C_{0}$, is a weak solution
of
\[
-\mathop{\rm div}\big(|x|^{-2a}\nabla u\big)
-\mu | x| ^{-2(a+1)}u=|x| ^{-2_{\ast}b}|u|^{2_{\ast }-2}u\quad
\text{in }\mathbb{R}^{N}\backslash \{0\}.
\]
Furthermore,
\begin{equation}
\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla u_{\varepsilon }|^{2}dx
-\mu \int_{\mathbb{R}^{N}}|x|^{-2(a+1)}u_{\varepsilon }^{2}dx
=\int_{\mathbb{R}^{N}}|x|^{-2_{\ast }b}|u_{\varepsilon }|^{2_{\ast }}dx
=A_{a,b,\mu},  \label{0}
\end{equation}
where $A_{a,b,\mu }$ is the best constant,
\begin{equation} \label{e1.6}
A_{a,b,\mu }=\inf_{u\in H_{\mu }\backslash \{0\}}
E_{a,b,\mu }(u)=E_{a,b,\mu }(u_{\varepsilon }),
\end{equation}
with
\[
E_{a,b,\mu }(u):=\frac{\int_{\mathbb{R}^{N}}
|x|^{-2a}|\nabla u|^{2}dx-\mu \int_{\mathbb{R}^{N}}
|x|^{-2(a+1)}u^{2}dx}{( \int_{\mathbb{R}^{N}}
|x|^{-2_{\ast }b}|u|^{2_{\ast }}dx)^{2/2_{\ast }}}.
\]

Also in \cite{K} and \cite{Lin}, they proved that for
$0\leq a<(N-2)/2$, $a\leq b<a+1$, $0\leq \mu <\bar{\mu }_{a}$, the
function defined for $\varepsilon >0$ as
\begin{equation}
v_{\varepsilon }(x)=(2.2_{\ast }\varepsilon ^{2}( \bar{\mu
}_{a}-\mu ))^{\frac{1}{2_{\ast }-2}}\Big(\varepsilon ^{2}| x|^{\frac{(
2_{\ast }-2)(\sqrt{\bar{\mu }_{a}}-\sqrt{\bar{\mu
}_{a}-\mu })}{2} }+| x|^{\frac{2_{\ast }-2}{2}(\sqrt{\bar{\mu
} _{a}}+\sqrt{\bar{\mu }_{a}-\mu })}\Big)^{-\frac{2}{2_{\ast }-2
}}
\end{equation}
is a weak solution of
\[
-\mathop{\rm div}(|x|^{-2a}\nabla u)-\mu | x| ^{-2(a+1)}u
=|x| ^{-2_{\ast}b}|u|^{2_{\ast }-2}u\quad \text{in  }
\mathbb{R}^{N}\backslash \{0\},
\]
and satisfies
\begin{equation}
\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla v_{\varepsilon }|^{2}dx
-\mu \int_{\mathbb{R}^{N}}|x|^{-2(a+1)}v_{\varepsilon }^{2}dx
=\int_{\mathbb{R}^{N}}|x|^{-2_{\ast }b}|v_{\varepsilon }|^{2_{\ast }}dx
=B_{a,b,\mu},
\end{equation}
where $B_{a,b,\mu }$ is the best constant,
\begin{equation}
B_{a,b,\mu }:=\inf_{u\in H_{\mu }\backslash \{0\}}
E_{a,b,\mu }(u)=E_{a,b,\mu }(v_{\varepsilon }).
\end{equation}

A natural question that arises in concert applications is to see what
happens if these elliptic (degenerate or non-degenerate) problems are
affected by a certain singular perturbations. In our work we prove the
existence of at least two distinct nonnegative critical points of energy
functional associated to problem \eqref{p-lambda-mu} by
splitting the Nehari manifold (see for example Tarantello \cite{T}
or Brown and Zhang \cite{BZ}).

In this work we consider the following assumptions:
\begin{itemize}
\item[(H)] $h$ is a continuous function defined in
$\bar{\Omega}$ and there exist $h_{0}$  and $\rho_{0}$
positive such that
$h(x)\geq h_{0}$ for all $x\in B(0,2\rho _{0})$,
where $B(a,r)$ is a ball centered at $a$  with radius $r$;

\item[(K)] $k$ is a continuous function defined in
$\bar{\Omega}$ and satisfies
$k(0)=\max_{x\in \bar{\Omega} }k(x)>0$,
$k(x)=k(0)+o (x^{\beta })$ for $x\in B(0,2\rho _{0})$  with
$\beta>2_{\ast }\sqrt{\bar{\mu }_{a}-\mu }$;

\end{itemize}
and one of the following two assumptions
\begin{itemize}
\item[(A1)] $N >2(|b|+1)$  and
\[
(a,\mu )\in ]-1,0[ \times ] 0, \bar{\mu }_{a}-b^{2}[ \cup
[ 0,\tfrac{N-2}{2}[ \times] a(a-N+2),\bar{\mu }_{a}-b^{2}[ ,
\]
\item[(A2)] $N \geq 3$, $(a,\mu )\in [0,\frac{N-2}{2}[ \times
[ 0,\bar{\mu }_{a}[$.

\end{itemize}
Following the method introduced in \cite{T,HH}, we obtain the
following existence result.

\begin{theorem} \label{thm1}
Suppose that $a<(N-2)/2$, $a\leq b<a+1$, $1<q<2$, $c\leq
q(a+1)+N(1-q/2)$, {\rm (H), (K)} hold and {\rm (A1)} or
{\rm (A2)} are satisfy. Then there exists $\Lambda ^{\ast }>0$ such
that for $\lambda \in ( 0,\Lambda ^{\ast })$
problem \eqref{p-lambda-mu} has at least two  nonnegative solutions in
$H_{\mu }$.
\end{theorem}

This paper is organized as follows.
In section 2 we give some preliminaries.
Section 3 is devoted to the proof of Theorem \ref{thm1}.

\section{Preliminary results}

We start by giving the following definitions.

Let $E$ be a Banach space and a functional $I\in
\mathcal{C}^{1}(E,\mathbb{R) }$. We say that $(u_{n})$ is a Palais
Smale sequence at level $l$  ($(PS)_l$ in short)
 if $I(u_{n})\to l$ \ and $I'(u_{n})\to 0$ in $E'$
(dual of $E$) as $n\to \infty $. We say also that $I$
satisfies the Palais Smale condition at level $l$ if any
$(PS)_l$ sequence has a subsequence converging strongly in $E$.

Define
\begin{equation}
w_{\varepsilon }:= \begin{cases}
u_{\varepsilon } &\text{if }(a,\mu )\in ] -1,0[ \times
] 0,\bar{\mu }_{a}-b^{2}[ \cup [0,\tfrac{N-2}{2} [ \times ] a(a-N+2) ,
\bar{\mu}_{a}-b^{2}[, \\
v_{\epsilon } &\text{if }(a,\mu )\in [ 0,\tfrac{N-2}{2}
[ \times [ 0,\bar{\mu }_{a}[ ,
\end{cases}
  \label{e2.1} %u
\end{equation}
and
\begin{equation}
\begin{aligned}
&S_{a,b,\mu }:=\\
&\begin{cases}
A_{a,b,\mu }&\text{if }(a,\mu )\in ] -1,0[ \times
] 0,\bar{\mu }_{a}-b^{2}[ \cup [0,\tfrac{N-2}{2}[ \times ] a(a-N+2) ,
\bar{\mu}_{a}-b^{2}[, \\
B_{a,b,\mu }&\text{if }(a,\mu )\in [ 0,\tfrac{N-2}{2}[
\times [ 0,\bar{\mu }_{a}[ ,.
\end{cases}  \label{s}
\end{aligned}
\end{equation}

Since our approach is variational, we define the functional
$I_{\lambda ,\mu }$ as
\[
I_{\lambda ,\mu }(u)=\frac{1}{2}\| u\|_{\mu ,a}^{2}-\frac{\lambda
}{q}\int_{\Omega }h(x)| x| ^{-c}|u|^{q}dx-\frac{1}{2_{\ast }}
\int_{\Omega }k(x)|x| ^{-2_{\ast }b}|u|^{2_{\ast }}dx,
\]
 for $u\in H_{\mu }$.
By \eqref{ee} and \eqref{e1.3} we can guarantee that $I_{\lambda ,\mu }$
is well defined in $H_{\mu }$ and
$I_{\lambda ,\mu }\in C^{1}(H_{\mu }, \mathbb{R})$.

$u\in H_{\mu }$ is said to be a weak solution of  \eqref{p-lambda-mu}
if it satisfies
\[
\int_{\Omega }(|x|^{-2a}\nabla u\nabla v-\mu | x|
^{-2(a+1)}uv-\lambda h( x) |x|^{-c}|u|^{q-2}uv-k(x) |x| ^{-2_{\ast
}b}|u|^{2_{\ast }-2}uv)dx=0
\]
for all $v\in H_{\mu }$. By the standard elliptic regularity
argument, we have that $u\in C^{2}(\Omega \backslash \{ 0\} )$.

In many problems as \eqref{p-lambda-mu}, $I_{\lambda ,\mu }$ is
not bounded below on $H_{\mu }$ but is bounded below on an appropriate
subset of $H_{\mu }$ and a minimizer in this set (if it exists) may give
rise to solutions of the corresponding differential equation.

A good candidate for an appropriate subset of $H_{\mu }$ is the so called
Nehari manifold
\[
\mathcal{N}_{\lambda }=\{u\in H_{\mu }\backslash \{0\},
\langle I_{\lambda ,\mu }'( u),u\rangle =0\}.
\]

It is useful to understand $\mathcal{N}_{\lambda }$  in terms of the
stationary points of mappings of the form
\[
\Psi _{u}(t)=I_{\lambda ,\mu }(tu),\text{ \ }t>0,
\]
and so
\[
\Psi _{u}'(t)=\langle I_{\lambda ,\mu }'(tu),u\rangle
=\frac{1}{t}\langle I_{\lambda ,\mu }'(tu),tu\rangle .
\]
An immediate consequence is the following proposition.

\begin{proposition} \label{prop1}
Let $u\in H_{\mu }\backslash \{0\}$ and $t>0$. Then $tu\in
\mathcal{N}_{\lambda }$ if and only if $\Psi _{u}'(t)=0$.
\end{proposition}

Let $u$ be a local minimizer of $I_{\lambda ,\mu }$, then
$\Psi _{u}$ has a
local minimum at $t=1$. So it is natural to split $\mathcal{N}_{\lambda }$
into three subsets $\mathcal{N}_{\lambda }^{+}$, $\mathcal{N}_{\lambda }^{-}$
and $\mathcal{N}_{\lambda }^{0}$ corresponding respectively to local
minimums, local maximums and points of inflexion.

We define
\begin{align*}
\mathcal{N}_{\lambda }^{+}
&=\big\{u\in \mathcal{N}_{\lambda }:
(2-q)\| u\|_{\mu ,a}^{2}-(2_{\ast }-q)\int_{\Omega }k(x)\frac{|u|
^{2_{\ast }}}{|x|^{2_{\ast }b}}dx>0\big\}\\
&=\{u\in \mathcal{N}_{\lambda }:(2-2_{\ast
})\| u\|_{\mu ,a}^{2}+( 2_{\ast }-q)\lambda \int_{\Omega
}h(x)\frac{|u|^{q}}{ |x| ^{c}}dx>0\}.
\end{align*}
Note that $\mathcal{N}_{\lambda }^{-}$ and
$\mathcal{N}_{\lambda }^{0}$ similarly by
replacing $>$ by $<$ and $=$ respectively.
\begin{equation}
c_{\lambda }:=\inf_{u\in \mathcal{N}_{\lambda }}I_{\lambda ,\mu }(
u);\text{ }c_{\lambda }^{+}:=\inf_{u\in \mathcal{N}_{\lambda
}^{+}}I_{\lambda ,\mu }(u);\quad
c_{\lambda }^{-}:=\inf_{u\in
\mathcal{N}_{\lambda }^{-}}I_{\lambda ,\mu }(u).
\end{equation}
The following lemma shows that minimizers on $\mathcal{N}_{\lambda }$ are
critical points for $I_{\lambda ,\mu }$.

\begin{lemma} \label{lem1}
Assume that $u$ is a local minimizer for $I_{\lambda ,\mu }$ on
$\mathcal{N} _{\lambda }$ and that $u\notin \mathcal{N}_{\lambda
}^{0}$. Then $I_{\lambda ,\mu }'(u)=0$.
\end{lemma}

The proof of the above lemma is essentially the same as that of
\cite[Theorem 2.3]{BZ}.


\begin{lemma} \label{lem2}
Let
\[
\Lambda _1:=\Big(\frac{2-q}{2_{\ast }-q}\Big)^{\frac{2-q}{2_{\ast }-q}
}\Big(\frac{2_{\ast }-2}{(2_{\ast }-q)C_1}\Big)| h^{+}| _{\infty
}^{-1}|k^{+}|_{\infty }(S_{a,b,\mu })^{\frac{N( 2-q)}{4(a+1-b)}},
\]
where $\eta ^{+}(x)=\max (\eta (x),0)$, and
$|\eta ^{+}|_{\infty}=\sup_{x\in \Omega }ess|\eta ^{+}(x)|$. Then
$\mathcal{N}_{\lambda }^{0}=\emptyset$ for all
$\lambda \in( 0,\Lambda _1)$.
\end{lemma}

\begin{proof}
Suppose that $\mathcal{N}_{\lambda }^{0}\neq \emptyset $.
Then for $u\in \mathcal{N}_{\lambda }^{0}$, we have
\begin{gather*}
\| u\|_{\mu ,a}^{2}=\frac{2_{\ast }-q}{2-q}\int_{\Omega
}k(x)\frac{|u|^{2_{\ast }}}{|x| ^{2_{\ast }b}}dx, \\
\| u\|_{\mu ,a}^{2}=\lambda \frac{2_{\ast }-q}{2_{\ast }-2}
\int_{\Omega }h(x)\frac{|u|^{q}}{ |x| ^{c}}dx.
\end{gather*}
Moreover by (H), (K), Caffarelli-Kohn-Nirenberg and H\"{o}lder
inequalities, we obtain
\begin{gather*}
\| u\|_{\mu ,a}^{2}\geq \Big(\frac{2-q}{( 2_{\ast }-2)|k^{+}|_{\infty
}}( S_{a,b,\mu }) ^{2_{\ast }/2}\Big)^{2/(2_{\ast }-2)}, \\
\| u\|_{\mu ,a}^{2}\leq \Big(\lambda \frac{2_{\ast }-q}{ 2_{\ast
}-2}(S_{a,b,\mu })^{-q/2}C_1| h^{+}| _{\infty }\Big)^{2/(2-q)}.
\end{gather*}
Thus
$\lambda \geq \Lambda _1$.
 From this, we can conclude that $\mathcal{N}_{\lambda
}^{0}=\emptyset $ if $\lambda \in (0,\Lambda _1)$.
\end{proof}

Thus we conclude that $\mathcal{N}_{\lambda }=\mathcal{N}_{\lambda
}^{+}\cup \mathcal{N}_{\lambda }^{-}$ for all $\lambda \in
(0,\Lambda _1)$.

\begin{lemma} \label{lem3}
Let $c_{\lambda }^{+}$, $c_{\lambda }^{-}$ defined in \eqref{e2.1}.
Then there exists $\delta _{0}>0$ such that
\[
c_{\lambda }^{+}<0\; \forall \lambda \in (0,\Lambda
_1)\quad \text{and}\quad
c_{\lambda }^{-}>\delta _{0}\;\forall
\lambda \in (0,\frac{q}{2}\Lambda _1).
\]
\end{lemma}

\begin{proof}
Let $u\in \mathcal{N}_{\lambda }^{+}$. Then
\[
\int_{\Omega }k(x)\frac{|u|^{2_{\ast }}}{ | x| ^{2_{\ast
}b}}dx<\frac{2-q}{2_{\ast }-q}\| u\|_{\mu ,a}^{2},
\]
which implies
\begin{align*}
c_{\lambda }^{+} &\leq I_{\lambda ,\mu }(u)\\
&= \big(\frac{1}{2}-\frac{1}{q}\big)\| u\|_{\mu
,a}^{2}+\big(\frac{1}{q}-\frac{1}{2_{\ast }}\big)\int_{\Omega }k(x)
\frac{|u|^{2_{\ast }}}{|
x|^{2_{\ast }b}}dx \\
&< -\frac{(2-q)(2_{\ast }-2)}{2.2_{\ast }q}
\| u\|_{\mu ,a}^{2}
< 0.
\end{align*}
Let $u\in \mathcal{N}_{\lambda }^{-}$. Then
\[
\frac{2-q}{2_{\ast }-q}\| u\|_{\mu ,a}^{2}<\int_{\Omega
}k(x)\frac{|u|^{2_{\ast }}}{|x| ^{2_{\ast }b}}dx.
\]
Moreover by (H), (K) and Caffarelli-Kohn-Nirenberg inequality,
we have
\[
\int_{\Omega }k(x)\frac{|u|^{2_{\ast }}}{ | x| ^{2_{\ast
}b}}dx\leq ( S_{a,b,\mu }) ^{-2_{\ast }/2}\| u\|_{\mu ,a}^{2_{\ast
}}| k^{+}|_{\infty }.
\]
This implies
\[
\| u\|_{\mu ,a}>\big(\frac{2-q}{(2_{\ast }-2) |k^{+}|_{\infty
}}\big)^{1/(2_{\ast }-2) }(S_{a,b,\mu })^{2_{\ast }/(2(2_{\ast }-2))}.
\]
On the other hand,
\[
I_{\lambda ,\mu }(u)\geq \| u\|_{\mu
,a}^{q}\Big(\big(\frac{1}{2}-\frac{1}{2_{\ast }}\big)\| u\|_{\mu
,a}^{2-q}-\lambda \frac{2_{\ast }-q}{2_{\ast }q}(S_{a,b,\mu
})^{-q/2}C_1|h^{+}|_{\infty }\Big)
\]
Thus, if $\lambda \in (0,\frac{q}{2}\Lambda _1)$ we get
$I_{\lambda ,\mu }(u)\geq \delta _{0}$,
where
\begin{align*}
\delta _{0}
&:= \Big(\tfrac{2-q}{(2_{\ast }-2)| k^{+}| _{\infty
}}\Big)^{\frac{q}{2_{\ast }-2}}( S_{a,b,\mu })^{\frac{2_{\ast
}q}{2(2_{\ast }-2) }}
\Big(\big(\frac{1}{2}-\frac{1}{2_{\ast }}\big)(S_{a,b,\mu
})^{ \frac{2_{\ast }(2-q)}{2(2_{\ast }-2)}}\big( \tfrac{2-q}{(2_{\ast
}-q) |k^{+}|_{\infty }}
\big)^{\frac{2-q}{2_{\ast }-2}}\\
&\quad -\lambda \tfrac{2_{\ast }-q}{2_{\ast }-2}(S_{a,b,\mu
})^{-q/2}C_1|h^{+}|_{\infty }\Big).
\end{align*}
\end{proof}
As in \cite[Proposition 9]{W}, we have the following result.


\begin{lemma} \label{lem4}
\begin{itemize}
\item[(i)] If $\lambda \in (0,\Lambda _1)$, then there exists a
$(PS)_{c_{\lambda }}$ sequence $( u_{n})\subset
\mathcal{N}_{\lambda }$ for $I_{\lambda ,\mu }$.

\item[(ii)] If $\lambda \in (0,\frac{q}{2}\Lambda _1)$, then there
exists a $(PS)_{c_{\lambda }^{-}}$ sequence $(u_{n})\subset
\mathcal{N}_{\lambda }^{-}$ for $I_{\lambda ,\mu }$.
\end{itemize}
\end{lemma}

We define
\begin{gather*}
K^{+}:=\big\{u\in \mathcal{N}_{\lambda }:\int_{\Omega }k(
x)\frac{|u|^{2_{\ast }}}{|x|^{2_{\ast }b}}dx>0\big\},
\quad
K_{0}^{-}:=\big\{u\in \mathcal{N} _{\lambda }:\int_{\Omega
}k(x)\frac{|u|^{2_{\ast }}}{|x|^{2_{\ast }b}}dx\leq 0\big\},
\\
H^{+}:=\{u\in \mathcal{N}_{\lambda }:\int_{\Omega }h(
x)\frac{|u|^{q}}{|x|^{c}} dx>0\},
\quad
H_{0}^{-}:=\{u\in \mathcal{N}_{\lambda }:
\int_{\Omega }h(x)\frac{|u|^{q}}{ |x|^{c}}dx\leq 0\},
\end{gather*}
and
\[
t_{\rm max}=t_{\rm max}(u):=\big(\frac{2-q}{2_{\ast }-2}\big)^{1/( 2_{\ast
}-2)}\| u\|_{\mu ,a}^{2/( 2_{\ast }-2) }
\Big(\int_{\Omega
}k(x)\frac{|u|^{2_{\ast }}}{| x| ^{2_{\ast }b}}dx
\Big)^{-1/(2_{\ast}-2)},
\]
for $u\in K^{+}$.
Then we have the following result.

\begin{proposition} \label{prop2}
 For $\lambda \in (0,\Lambda _1)$ we have
\begin{itemize}
\item[(1)]  If $u\in K^{+}\cap H_{0}^{-}$ then there exists unique
$t^{+}>t_{\rm max}$ such that $t^{+}u\in \mathcal{N}_{\lambda
}^{-}$and
\[
I_{\lambda ,\mu }(t^{+}u)\geq I_{\lambda ,\mu }( tu) \quad
\text{for } t\geq t_{\rm max};
\]

\item[(2)] If $u\in K^{+}\cap H^{+}$, then there exist unique $t^{-}$,
$t^{+}$ such that $0<t^{-}<t_{\rm max}<t^{+}$,
$t^{-}u\in \mathcal{N} _{\lambda }^{+}$,
$t^{+}u\in \mathcal{N}_{\lambda }^{-}$ and
\[
I_{\lambda ,\mu }(t^{+}u)\geq I_{\lambda ,\mu }( tu)\text{ for
}t\geq t^{-}\text{ and }I_{\lambda ,\mu }( t^{-}u)\leq I_{\lambda
,\mu }(tu)\text{ for }t\in \lbrack 0,t^{+}].
\]
\item[(3)] If $u\in K^{-}\cap H^{-}$, then does not exist $t>0$ such
that $tu\in \mathcal{N}_{\lambda }$.

\item[(4)] If $u\in K_{0}^{-}\cap H^{+}$, then there exists unique
$0<t^{-}<+\infty $ such that $t^{-}u\in \mathcal{N}_{\lambda
}^{+}$ and
\[
I_{\lambda ,\mu }(t^{-}u)=\inf_{t\geq 0}I_{\lambda ,\mu }(tu).
\]
\end{itemize}
\end{proposition}

\begin{proof}
For $u\in H_{\mu }$, we have
\[
\Psi _{u}(t)=I_{\lambda ,\mu }(tu)=\frac{t^{2}}{2} \| u\| _{\mu
,a}^{2}-\lambda \frac{t^{q}}{q}\int_{\Omega
}h(x)\frac{|u|^{q}}{|x|^{c}}dx-\frac{t^{2_{\ast }}}{2_{\ast
}}\int_{\Omega }k( x) \frac{|u|^{2_{\ast }}}{|x| ^{2_{\ast }b}}dx
\]
and
\[
\Psi _{u}'(t)=t^{q-1}\Big(\varphi _{u}(t)-\lambda
\int_{\Omega }h(x)\frac{|u|^{q}}{| x|^{c}}\Big),
\]
where
\[
\varphi _{u}(t)=t^{2-q}\| u\|_{\mu ,a}^{2}-t^{2_{\ast
}-q}\int_{\Omega }k(x)\frac{| u| ^{2_{\ast }}}{|x|^{2_{\ast }b}}.
\]
Easy computations show that $\varphi _{u}$ is concave and achieves
its
maximum at
\[
t_{\rm max}:=\big(\frac{2-q}{2_{\ast }-2}\big)^{1/(2_{\ast }-2) }
\| u\|_{\mu,a}^{2/(2_{\ast }-2)}
\Big( \int_{\Omega }k(x) \frac{|u|^{2_{\ast
}}}{|x|^{2_{\ast }b}}dx\Big)^{-1/( 2_{\ast }-2)}\]
 for $u\in K^{+}$;
that is,
\[
\Psi (t_{\rm max})= C_{a,b,q,N}\| u\|_{\mu ,a}^{(2_{\ast
}-q)/(2_{\ast }-2)}\Big(\int_{\Omega }k(x) \frac{|u|^{2_{\ast }}}{
|x|^{2_{\ast }b}}dx\Big)^{( q-2)/(2_{\ast }-2) },
\]
where
\[
C_{a,b,q,N} = \frac{2_{\ast }+q-4}{2_{\ast
}-2}(\frac{ 2-q}{2_{\ast }-2})^{(2-q)/(2_{\ast }-2)}.
\]
Then we can get the conclusion of our proposition easily.
\end{proof}

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

\subsection*{Existence of a local minimum for
$I_{\lambda ,\mu }$ on $\mathcal{N}_{\lambda }^{+}$}
We want to prove that $I_{\lambda ,\mu }$ can achieve a local minimizer
on $\mathcal{N}_{\lambda }^{+}$.

\begin{proposition} \label{prop3}
Let $\lambda \in (0,\Lambda _1)$, then $I_{\lambda ,\mu }$ has
a minimizer $u_{\lambda }$ in $\mathcal{N}_{\lambda }^{+}$ such
that
\[
I_{\lambda ,\mu }(u_{\lambda })=c_{\lambda }^{+}<0.
\]
\end{proposition}

\begin{proof}
By Lemma \ref{lem4}, there exists a minimizing sequence
 $(u_{n}) \subset\mathcal{N}_{\lambda }$ such that
\[
I_{\lambda ,\mu }(u_{n})\to c_{\lambda }\text{ \ and \ }
I_{\lambda ,\mu }'(u_{n})\to 0\text{ \ in } H_{\mu
}^{-1}\text{ (dual of }H_{\mu }\text{).}
\]
Since
\[
I_{\lambda ,\mu }(u_{n})=\big( \frac{1}{2}-\frac{1}{2_{\ast }} \big)
\|u_{n}\|_{\mu ,a}^{2}-\lambda (\frac{1}{q}-\frac{ 1}{2_{\ast
}})\int_{\Omega }h(x) \frac{|u_{n}|^{q}}{|x|^{c}},
\]
by Caffarelli-Kohn-Nirenberg inequality, we have
\[
c_{\lambda }+\circ _{n}(1)\geq \big(\frac{1}{2}-\frac{1}{ 2_{\ast
}}\big)\| u_{n}\|_{\mu ,a}^{2}-\lambda \frac{ 2_{\ast }-q}{2_{\ast
}q}(S_{a,b,\mu }) ^{-q/2}C_1| h^{+}|_{\infty }\| u_{n}\|_{\mu
,a}^{q},
\]
where $\circ _{n}(1)$ denotes that $\circ _{n}( 1) \to 0$ $as$
$n\to \infty .$Thus $( u_{n})$ is bounded in $H_{\mu }$, then
passing to a subsequence if necessary, we have the following
convergence:
\begin{gather*}
u_{n} \rightharpoonup u_{\lambda }\quad \text{in } H_{\mu }, \\
u_{n} \rightharpoonup u_{\lambda }\quad  \text{in }L_{2_{\ast
}}(\Omega ,|x|^{-2_{\ast }b}), \\
u_{n} \to u_{\lambda }\quad  \text{in } L_{q}(\Omega,|x|^{-c}), \\
u_{n} \to u_{\lambda }\quad \text{a.e. in }\Omega .
\end{gather*}
Thus $u_{\lambda }\in \mathcal{N}_{\lambda }$ is a weak solution
of \eqref{p-lambda-mu}. As $c_{\lambda }<0$ and
$I_{\lambda ,\mu }(0)=0$, then $u_{\lambda }\not\equiv 0$. Now we
show that $u_{n}\to u_{\lambda }$ in $H_{\mu }$. Suppose
otherwise, then
$\| u_{\lambda }\|_{\mu }<\liminf_{n\to \infty }
\| u_{n}\|_{\mu }$, and we obtain
\begin{align*}
c_{\lambda }
&\leq  I_{\lambda ,\mu }(u_{\lambda })\\
&=\big(\frac{1}{2}-\frac{1}{2_{\ast }}\big)
 \| u_{\lambda }\|_{\mu,a}^{2}-\lambda \frac{2_{\ast }-q}{2_{\ast }q}
 \int_{\Omega}h(x)\frac{|u_{\lambda }|^{q}}{|x|^{c}} \\
&< \liminf_{n\to \to \infty } \Big(
(\frac{1}{2}-\frac{1}{2_{\ast }})\| u_{n}\|_{\mu ,a}^{2}-\lambda
\frac{2_{\ast }-q}{2_{\ast }q}\int_{\Omega
}h(x)\frac{|u_{n}|^{q}}{|x|^{c}}\Big)\\
&= c_{\lambda }.
\end{align*}
We obtain a contradiction. Consequently $u_{n}\to u_{\lambda }$
strongly in $H_{\mu }$. Moreover, we have $u_{\lambda }\in
\mathcal{N} _{\lambda }^{+}$. If  not $u_{\lambda }\in
\mathcal{N}_{\lambda }^{-}$, thus $\Psi _{u}'(1)=0$ and
$\Psi _{u}''( 1) <0$, which implies that
$I_{\lambda ,\mu }( u_{\lambda })
>0$, contradiction.
\end{proof}

\subsection*{Existence of a local minimum for $I_{\lambda ,
\mu }$ on $\mathcal{N}_{\lambda }^{-}$}
To prove the existence of a second nonnegative solution we need the
following results.

\begin{lemma} \label{lem5}
Let $(u_{n})$ is a $(PS)_l$ sequence with
$u_{n}\rightharpoonup u$ in $H_{\mu }$. Then there exists positive
constant $\tilde{C}:=C(a,b,N,q,|h^{+}|_{\infty },S_{a,b,\mu })$
such that
\[
I_{\lambda ,\mu }'(u)=0\quad \text{and}\quad
I_{\lambda ,\mu }(u)\geq -\tilde{C}\lambda ^{2/(2-q)}.
\]
\end{lemma}

\begin{proof}
It is easy to prove that $I_{\lambda ,\mu }'(u) =0$,
which implies that $\langle I_{\lambda ,\mu }^{'}(u),u\rangle =0$,
and
\[
I_{\lambda ,\mu }(u)-\frac{1}{2_{\ast }}\langle I_{\lambda
,\mu }^{'}(u),u\rangle
=( \frac{1}{2}-\frac{1}{2_{\ast }})\| u\|_{\mu ,a}^{2}
-\lambda (\frac{1}{q}-\frac{1}{2_{\ast }})\int_{\Omega }h(x)
\frac{|u|^{q}}{|x|^{c}}dx.
\]
By Caffarelli-Kohn-Nirenberg, H\"{o}lder and Young inequalities
we find that
\[
I_{\lambda ,\mu }(u)\geq ( \frac{1}{2}-\frac{1}{2_{\ast }} ) \|
u\|_{\mu ,a}^{2}-\lambda \frac{2_{\ast }-q}{ 2_{\ast
}q}(S_{a,b,\mu })^{-q/2}C_1|h^{+}|_{\infty }\| u\| _{\mu
,a}^{q}.
\]
There exists $\tilde{C}$ such that
\[
(\frac{1}{2}-\frac{1}{2_{\ast }})t^{2}-\lambda \frac{2_{\ast }-q
}{2_{\ast }q}(S_{a,b,\mu })^{-q/2}C_1|h^{+}|_{\infty }t^{q}\geq
-\tilde{C}\text{ }\lambda ^{2/(2-q)} \quad  \text{for all
}t\geq 0.
\]
Then we conclude that
$I_{\lambda ,\mu }(u)\geq -\tilde{C}\text{ }\lambda ^{2/(2-q)}$.
\end{proof}


\begin{lemma} \label{lem6}
Let $(u_{n})$ in $H_{\mu }$ be such that
\begin{gather}
I_{\lambda ,\mu }(u_{n})\to l<l^{\ast }:=( \frac{1
}{2}-\frac{1}{2_{\ast }})|k^{+}|_{\infty }(S_{a,b,\mu })^{2_{\ast
}/(2_{\ast }-2)}-\tilde{C}\lambda ^{2/(2-q)}, \label{e1}
\\
I_{\lambda ,\mu }'(u_{n})\to 0\text{\ in } H_{\mu }^{-1}.
\label{e2}
\end{gather}
Then there exists a subsequence strongly convergent.
\end{lemma}

\begin{proof}
 From \eqref{e1} and \eqref{e2} we deduce that $(u_{n})$ is
bounded. Thus up a subsequence, we have the following convergence:
\begin{gather*}
u_{n} \rightharpoonup u\quad \text{in } H_{\mu }, \\
u_{n} \rightharpoonup u\quad  \text{in }L_{2_{\ast}}(\Omega
,|x|^{-2_{\ast }b}), \\
u_{n} \to u\quad \text{in } L_{q}(\Omega ,|x|^{-c}), \\
u_{n} \to u\quad \text{a.e. in }\Omega .
\end{gather*}
Then $u$ is a weak solution of problem \eqref{p-lambda-mu}.

Denote $v_{n}=u_{n}-u$. As $k$ is continuous on $\Omega $,
then the Br\'{e}zis - Lieb \cite{BL} leads to
\begin{equation}
\int_{\Omega }k(x)\frac{|u_{n}|^{2_{\ast }}}{|x| ^{2_{\ast
}b}}dx=\int_{\Omega }k(x)\frac{|v_{n}|^{2_{\ast }}}{| x|^{2_{\ast
}b}}dx+\int_{\mathbb{R}^{N}}k(x)
\frac{|u|^{2_{\ast }}}{ | x|^{2_{\ast }b}}dx, \label{a}
\end{equation}
and
\begin{equation}
\| u_{n}\|_{\mu ,a}^{2}=\| v_{n}\|_{\mu ,a}^{2}+\| u\| _{\mu
,a}^{2}+\circ _{n}(1). \label{c}
\end{equation}
Using the Lebesgue theorem, it follows that
\begin{equation}
\lim_{n\to \infty }\int_{\Omega }h(x) \frac{|
u_{n}|^{q}}{|x|^{c}}dx=\int_{\Omega }h(x)\frac{|u| ^{q}}{|x|^{c}}
dx. \label{d}
\end{equation}
 From \eqref{a}, \eqref{c} and \eqref{d}, we deduce that
\[
I_{\lambda ,\mu }(u_{n})=I_{\lambda ,\mu }(u)+ \frac{1}{2}\|
v_{n}\|_{\mu ,a}^{2}-\frac{1}{2_{\ast }} \int_{\Omega }k(x)
\frac{| v_{n}|^{2_{\ast }}}{|x|^{2_{\ast }b}}dx+\circ _{n}(1),
\]
and
\[
\langle I_{\lambda ,\mu }'(u_{n})
,u_{n}\rangle
=\langle I_{\lambda ,\mu }'(u),u\rangle
+\| v_{n}\|_{\mu ,a}^{2}-\int_{\Omega }k(x)\frac{|v_{n}|^{2_{\ast }}}
{ |x|^{2_{\ast }b}}dx+\circ_{n}(1),
\]
using the fact that $v_{n}\rightharpoonup 0$ in $H_{\mu }$, we
can assume that
\[
\| v_{n}\|_{\mu ,a}^{2}\to \theta \quad\text{and}\quad
 \int_{\Omega }k(x)\frac{|v_{n}|^{2_{\ast }}}{|x| ^{2_{\ast }b}}dx
\to \theta \geq 0.
\]
By the definition of $S_{a,b,\mu }$ we have
\[
\| v_{n}\|_{\mu ,a}^{2}\geq S_{a,b,\mu }\Big( \int_{\Omega
}\frac{|v_{n}|^{2_{\ast }}}{| x|^{2_{\ast }b}}dx\Big)^{2/2_{\ast }},
\]
and so $\theta \geq |k^{+}|_{\infty }S_{a,b,\mu }\theta^{2/2_{\ast }}$.

Assume $\theta \neq 0$, then $\theta \geq | k^{+}| _{\infty
}(S_{a,b,\mu })^{2_{\ast }/( 2_{\ast }-2)} $, and we get by
Lemma \ref{lem6} that
\begin{align*}
l &= I_{\lambda ,\mu }(u)+( \frac{1}{2}-\frac{1}{2_{\ast }}
)\theta \\
&\geq -\tilde{C}\lambda ^{2/(2-q)}+( \frac{1}{2}-\frac{1}{ 2_{\ast
}})|k^{+}|_{\infty }( S_{a,b,\mu
})^{2_{\ast }/(2_{\ast }-2)}
= l^{\ast }
\end{align*}
which is a contradiction. So $l=0$; i.e., $u_{n}\to u$ in
$\ H_{\mu}$.
\end{proof}

In the following, we shall give some estimates for the extremal
functions defined in \eqref{e2.1}.
Let $\Psi (x)\in C_{0}^{\infty}(\Omega )$ such that
$0\leq \Psi (x)\leq 1$, $\Psi (x)=1$ for
$|x|\leq \rho _{0}$, $\Psi (x)=0$ for $| x|\geq 2\rho _{0}$, where
$\rho _{0}$ is a small positive number.
Set
\[
\tilde{u}(x)=\Big(|x|^{\frac{2_{\ast }-2}{2}(\sqrt{\bar{\mu
}_{a}}-\sqrt{\bar{\mu }_{a}-\mu })}+|x| ^{\frac{2_{\ast
}-2}{2}( \sqrt{\bar{\mu } _{a}}+\sqrt{\bar{\mu }_{a}-\mu
}) }\Big)^{-\frac{2}{2_{\ast }-2 }}
\]
\[
\tilde{v}_{\varepsilon }(x)=
\begin{cases}
\Psi (x)\Big(\varepsilon ^{\frac{2\sqrt{\bar{\mu }_{a}-\mu
}}{\sqrt{ \bar{\mu }_{a}-\mu }-b}}|x| ^{\frac{2_{\ast }-2}{2}
(\sqrt{\bar{\mu }_{a}}-\sqrt{\bar{\mu }_{a}-\mu
})}+|x|^{\frac{2_{\ast }-2}{2}(\sqrt{\bar{\mu }
_{a}}+\sqrt{\bar{\mu }_{a}-\mu })}\Big)^{-\frac{2}{2_{\ast }-2
}}\\
\quad \text{if (A1) holds,} \\[2pt]
\Psi (x)\Big(\varepsilon ^{2}|x|^{\frac{2_{\ast }-2}{2
}(\sqrt{\bar{\mu }_{a}}-\sqrt{\bar{\mu }_{a}-\mu})
}+|x|^{\frac{2_{\ast }-2}{2}( \sqrt{\bar{\mu }
_{a}}+\sqrt{\bar{\mu }_{a}-\mu }) }\Big)^{-\frac{2}{2_{\ast }-2
}} \\
\quad \text{if (A2) holds}.
\end{cases}
\]
By a straightforward computation, one finds
\[
\int_{\Omega }k(x)\frac{|\tilde{v}_{\varepsilon }| ^{2_{\ast
}}}{|x|^{2_{\ast }b}} dx=\varepsilon
^{-\frac{N-2(a+1-b)}{2(a+1-b)} }|k^{+}|_{\infty }\int_{\Omega
}\frac{|\tilde{u}|^{2_{\ast }}}{|x|^{2_{\ast }b}}
dx+O(\varepsilon ),
\]
where $O(\varepsilon ^{\zeta })$ denotes
$|O(\varepsilon ^{\zeta})|/\epsilon ^{\zeta }\leq C$,
\begin{gather*}
\| \tilde{v}_{\varepsilon }\|_{\mu ,a}^{2}=\varepsilon ^{-
\frac{N-2(a+1-b)}{2( a+1-b) }}\| \tilde{u} \|_{\mu ,a}^{2}+O(1),
\\
\frac{\| \tilde{v}_{\varepsilon }\|_{\mu ,a}^{2}}{ \int_{\Omega
}k(x)\frac{| \tilde{v}_{\varepsilon }|^{2_{\ast }}}{|x|^{2_{\ast
}b}}dx} =O(\varepsilon ^{\frac{N-2( a+1-b)}{2(a+1-b)}}).
\end{gather*}

\begin{lemma} \label{lem7}
Let $l^{\ast }$be defined in Lemma \ref{lem6}, then there exists $\Lambda
_{4}>0$ such that for all $\lambda \in (0,\Lambda _{4})$ we have
$l^{\ast }>0$ and $\underset{t\geq 0}{\text{ }\sup }I_{\lambda
,\mu }(t\tilde{v}_{\varepsilon })<l^{\ast }$.
\end{lemma}

\begin{proof}
We consider the following two functions
\[
f(t)=I_{\lambda ,\mu }(t\tilde{v}_{\varepsilon
})=\frac{t^{2}}{2}\| \tilde{v}_{\varepsilon }\|_{\mu
,a}^{2}-\frac{t^{2_{\ast }}}{ 2_{\ast }}\int_{
\mathbb{R}^{N}}k(x)\frac{|\tilde{v}_{\varepsilon }|^{2_{\ast
}}}{|x|^{2_{\ast }b}}dx-\lambda \frac{t^{q}}{q }
\int_{\mathbb{R}
^{N}}h(x)\frac{|\tilde{v}_{\varepsilon }| ^{q}}{|x| ^{c}}dx,
\]
and
\[
\tilde{f}(t)=\frac{t^{2}}{2}\| \tilde{v}_{\varepsilon }\| _{\mu
,a}^{2}-\frac{t^{2_{\ast }}}{2_{\ast }}|k^{+}|_{\infty }\int_{
\mathbb{R} ^{N}}
\frac{|\tilde{v}_{\varepsilon }|^{2_{\ast }}}{ |x| ^{2_{\ast
}b}}dx.
\]
Let $\Lambda _{2}>0$ be such that
\[
(\frac{1}{2}-\frac{1}{2_{\ast }})|k^{+}|_{\infty }(S_{a,b,\mu
})^{2_{\ast }/( 2_{\ast }-2)}- \tilde{C}\lambda ^{2/(2-q)}>0\quad
\text{for all }\lambda \in (0,\Lambda _{2}).
\]
Then
\[
f(0)=0<(\frac{1}{2}-\frac{1}{2_{\ast }})| k^{+}| _{\infty
}(S_{a,b,\mu })^{2_{\ast }/( 2_{\ast }-2)}-\tilde{C}\lambda
^{2/(2-q)}\quad \text{for all } \lambda \in (0,\Lambda _{2}).
\]
By the continuity of $f(t)$, there exists $t_1>0$ small enough such that
\[
f(t)<(\frac{1}{2}-\frac{1}{2_{\ast }})|k^{+}|_{\infty }(S_{a,b,\mu
})^{2_{\ast }/( 2_{\ast }-2) }-\tilde{C}\lambda ^{2/(2-q)}\quad
\text{for all }t\in (0,t_1).
\]
On the other hand,
\[
\max_{t\geq 0}\tilde{f}(t)=(\frac{1}{2}-\frac{1}{2_{\ast }})
|k^{+}|_{\infty }(S_{a,b,\mu }) ^{2_{\ast }/(2_{\ast
}-2)}+O(\varepsilon ^{\frac{N-2( a+1-b)}{ 2(a+1-b)}}).
\]
Then
\begin{align*}
\sup_{t\geq 0} I_{\lambda ,\mu }(t\tilde{v}_{\varepsilon })
&< (\frac{1}{2}-\frac{1}{2_{\ast
}})|k^{+}|_{\infty }(S_{a,b,\mu }) ^{2_{\ast }/(2_{\ast
}-2)}+O(\varepsilon ^{\frac{N-2(a+1-b)}{2(a+1-b)}})\\
&\quad -\lambda \frac{t_1^{q}}{q}h_{0}\int_{B(0,\rho _{0})
}\frac{ |\tilde{v}_{\varepsilon }|^{q}}{|x|^{c}}dx.
\end{align*}
Let $0<\varepsilon <\rho _{0}^{(2_{\ast }-2) \sqrt{\bar{\mu
}_{a}-\mu }}$ then
\begin{align*}
&\int_{B(0,\rho _{0})}\frac{|\tilde{v}_{\varepsilon }|
^{q}}{|x|^{c}}dx \\
&= \int_{B(0,\rho _{0})}| x| ^{-c}(\varepsilon
^{\frac{2\sqrt{ \bar{\mu }_{a}-\mu }}{\sqrt{\bar{\mu
}_{a}-\mu }-b}}|x|^{\frac{2_{\ast }-2}{2}(\sqrt{\bar{\mu
}_{a}}-\sqrt{ \bar{\mu }_{a}-\mu })}+|x| ^{\frac{2_{\ast
}-2}{2}(\sqrt{\bar{\mu }_{a}}+\sqrt{\bar{\mu }_{a}-\mu }
)})^{-\frac{2q}{2_{\ast }-2}}dx \\
&\geq C_{2}.
\end{align*}
Now, taking $\varepsilon =\lambda ^{\frac{2(2_{\ast }-2)}{ 2_{\ast
}-q}}$ we get $\lambda <\rho _{0}^{(2-q)\sqrt{\bar{\mu}_{a}-\mu }}$ and
\[
\sup_{t\geq 0} I_{\lambda ,\mu
}(t\tilde{v}_{\varepsilon })<(\frac{1}{2}-\frac{1}{2_{\ast
}})|k^{+}|_{\infty }(S_{a,b,\mu }) ^{2_{\ast }/(2_{\ast
}-2)}+O(\lambda ^{2/( 2-q)})-\lambda
\frac{t_1^{q}}{q}h_{0}C_{2}.
\]
Choosing $\Lambda _{3}>0$ such that
\[
O(\lambda ^{2/(2-q)})-\lambda \frac{t_1^{q}}{q}h_{0}C_{2}<-
\tilde{C}\lambda ^{2/( 2-q)}\quad \text{for all }\lambda \in (0,\Lambda
_{3}).
\]
Then if we take $\Lambda _{4}=\min \{\Lambda _{2},\Lambda
_{3},\rho _{0}^{(2-q)\sqrt{\bar{\mu }_{a}-\mu }}\}$ we deduce
that
\[
\sup_{t\geq 0} J_{\lambda }(t\tilde{v}_{\varepsilon
})<l^{\ast } \quad \text{for all }\lambda \in (0,\Lambda _{4}).
\]
\end{proof}

Now, we prove that $I_{\lambda ,\mu }$ can achieve a local
minimizer on $\mathcal{N}_{\lambda }^{-}$.

\begin{proposition} \label{prop4}
Let $\Lambda ^{\ast }=\min \{q\Lambda _1/2,\Lambda _{4}\} $.
Then for all $\lambda \in (0,\Lambda ^{\ast })$,
 $I_{\lambda ,\mu }$ has a minimizer $v_{\lambda }$ in
$\mathcal{N}_{\lambda }^{-}$
such that $I_{\lambda ,\mu }( v_{\lambda })=c_{\lambda }^{-}$.
\end{proposition}

\begin{proof}
By Lemma \ref{lem4}, there exists a minimizing sequence\ $(u_{n}) \subset
\mathcal{N}_{\lambda }^{-}$ for all $\lambda \in ( 0,q\Lambda
_1/2)$ such that $I_{\lambda ,\mu }( u_{n})\to c_{\lambda }^{-}$
\ and \ $I_{\lambda ,\mu }'(u_{n})\to 0$ \ in $H_{\mu
}^{-1}$. Since $I_{\lambda ,\mu }$ is coercive on
$\mathcal{N}_{\lambda }^{-}$ thus $(u_{n})$ bounded. Then, passing
to a subsequence if necessary, we have the following convergence:
\begin{gather*}
u_{n} \rightharpoonup v_{\lambda }\quad \text{in } H_{\mu }, \\
u_{n} \rightharpoonup v_{\lambda }\quad  \text{in }L_{2_{\ast
}}(\Omega ,|x|^{-2_{\ast }b}), \\
u_{n} \to v_{\lambda }\quad \text{in } L_{q}(\Omega,|x|^{-c}), \\
u_{n} \to  v_{\lambda }\quad \text{a.e. in }\Omega .
\end{gather*}
By Lemma \ref{lem7}, $c_{\lambda }^{-}<l^{\ast }$, thus from
Lemma \ref{lem6}
we deduce that $u_{n}\to v_{\lambda }$ in
$\ H_{\mu}$. Then we conclude that $I_{\lambda ,\mu }(v_{\lambda })
=c_{\lambda }^{-}>0$. Similarly as the proof of Proposition
\ref{prop3}, we conclude that $I_{\lambda ,\mu }$ has a
minimizer $v_{\lambda }$
in $\mathcal{N}_{\lambda }^{-}$ for all $\lambda \in ( 0,\Lambda
^{\ast })$ such that $I_{\lambda ,\mu }(v_{\lambda }) =c_{\lambda
}^{-}>0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
By Propositions \ref{prop2} and \ref{prop4},
there exists $\Lambda ^{\ast }>0$
such that \eqref{p-lambda-mu} has two
nonnegative solutions $u_{\lambda }\in \mathcal{N}_{\lambda }^{+}$
and $v_{\lambda }\in \mathcal{ N}_{\lambda }^{-}$ since
$\mathcal{N}_{\lambda }^{+}\cap \mathcal{N} _{\lambda
}^{-}=\emptyset$.
\end{proof}

\begin{thebibliography}{00}

\bibitem{AB} A. Ambrosetti, H. Br\'{e}zis, G. Cerami:
Combined effects of
concave and convex nonlinearities in some elliptic problems, J. Funct. Anal.
122 (1994), 519--543.

\bibitem{BA} M. Bouchekif, A. Matallah: Multiple positive solutions for
elliptic equations involving a concave term and critical Sobolev-Hardy
exponent, Appl. Math. Lett. 22 (2009), 268-275.

\bibitem{BA2} M. Bouchekif, A. Matallah: On singular nonhomogeneous elliptic
equations involving critical Caffarelli-Kohn-Nirenberg exponent, Ricerche
math. doi 10. 1007/s 11587-009-0056-y.

\bibitem{BL} H. Br\'{e}zis, E. Lieb: A relation between pointwise
convergence of functions and convergence of functionals, Proc. Amer. Math.
Soc. 88 (1983), 486-490.

\bibitem{BZ} K.J. Brown, Y. Zhang: The Nehari manifold for a semilinear
elliptic equation with a sign-changing weight function, J. Differential
Equations 193 (2003), 481-499.

\bibitem{CK} L. Caffarelli, R. Kohn, L. Nirenberg: First order interpolation
inequality with weights, Compos. Math. 53 (1984), 259--275.

\bibitem{CW} F. Catrina, Z. Wang: On the Caffarelli-Kohn-Nirenberg
inequalities: sharp constants, existence (and nonexistence), and symmetry of
extremal functions, Comm. Pure Appl. Math. 54 (2001), 229-257.

\bibitem{C} J. Chen: Multiple positive solutions for a class of nonlinear
elliptic equations, J. Math. Anal. Appl. 295 (2004), 341-354.

\bibitem{CC} K.S. Chou, C.W. Chu: On the best constant for a weighted
Sobolev-Hardy Inequality, J. London Math. Soc. 2 (1993), 137-151.

\bibitem{E} L.C. Evans: Partial differential equations, in: graduate studies
in mathematics 19, Amer. Math. Soc. Providence, Rhode Island, (1998).

\bibitem{FG} A. Ferrero, F. Gazzola: Existence of solutions for singular
critical growth semilinear elliptic equations. J. Differential Equations 177
(2001), 494-522.

\bibitem{HH} T.S. Hsu, H.L. Lin: Multiple positive solutions for singular
elliptic equations with concave-convex nonlinearities and sign-changing
weights. Boundary Value Problems, doi: 10 1155/2009/584203.

\bibitem{K} D. Kang, G. Li, S. Peng: Positive solutions and critical
dimensions for the elliptic problem involving the Caffarelli-Kohn-Nirenberg
inequalities, Preprint.

\bibitem{Lin} M. Lin: Some further results for a class of weighted nonlinear
elliptic equations, J. Math. Anal. Appl. 337(2008), 537-546.

\bibitem{X.} B. J. Xuan: The solvability of quasilinear 
Br\'{e}zis-Nirenberg-type problems with singular weights, Nonlinear Anal.
62 (4) (2005), 703-725.

\bibitem{X} B. J. Xuan, S. Su, Y. Yan: Existence results for 
Br\'{e}zis-Nirenberg problems with Hardy potential and singular
coefficients, Nonlinear Anal. 67 (2007), 2091-2106.

\bibitem{T} G. Tarantello: On nonhomogeneous elliptic equations involving
critical Sobolev exponent, Ann. Inst. H. Poincar\'{e} Anal. Non.
Lin\'{e}aire 9 (1992), 281-304.

\bibitem{W} T. F. Wu: On semilinear elliptic equations involving
concave-convex nonlinearities and sign-changing weight function, J. Math.
Anal. Appl. 318 (2006), 253-270.
\end{thebibliography}

\end{document}