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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 122, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/122\hfil Multiple symmetric solutions]
{Multiple symmetric solutions for a singular semilinear
elliptic problem with critical exponent}

\author[A. Cano, E. Hern\'andez-Mart\'inez\hfil EJDE-2012/122\hfilneg]
{Alfredo Cano, Eric Hern\'andez-Mart\'inez}  % in alphabetical order

\address{Alfredo Cano \newline
Universidad Aut\'onoma del Estado de M\'exico,
Facultad de Ciencias, Departamento de Matem\'aticas, Campus El Cerrillo
Piedras Blancas, Carretera Toluca-Ixtlahuaca, Km 15.5, Toluca, Estado de
 M\'exico, M\'exico}
\email{calfredo420@gmail.com}

\address{Eric Hern\'andez-Mart\'{\i}nez \newline
Universidad Aut\'onoma de la Ciudad de M\'exico,
Colegio de Ciencia y Tecnolog\'{\i}a, Acade\-mia de Matem\'aticas,
 Calle Prolongaci\'on San Isidro No. 151, Col. San Lorenzo
Tezonco, Del. Iztapalapa, C.P. 09790, M\'exico D.F., M\'exico}
\email{ebric2001@hotmail.com}

\thanks{Submitted March 27, 2012. Published July 20, 2012.}
\thanks{This work was presented in the II Joint meeting RSME-SMM, 2012,
 M\'alaga, Espa\~na}
\subjclass[2000]{35J75, 35J57, 35J60}
\keywords{Critical exponent; singular problem; symmetric solutions}

\begin{abstract}
 Let be $\Gamma$ a closed subgroup of $O(N)$.
 We consider the semilinear elliptic problem
 \begin{gather*}
 -\Delta u-\frac{b(x)}{| x|^2}u-a(x)u=f(x)| u|^{2^{\ast }-2}u\quad
 \text{in }\Omega ,\\
  u=0 \quad\text{on } \partial \Omega ,
 \end{gather*}
 where $\Omega \subset \mathbb{R}^{N}$ is a smooth
 bounded domain, $N\geq 4$.
 We establish the multiplicity of symmetric positive solutions,
 nodal solutions, and solutions which are $\Gamma$ invariant but
 are not $\widetilde{\Gamma }$ invariant, where
 $\Gamma \subset \widetilde{\Gamma}\subset O(N)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

We consider the singular semilinear elliptic problem with critical
nonlinearity,
\begin{equation}
\begin{gathered}
-\Delta u-b(x)\frac{u}{| x|^2}-a(x)u=f(x)|
u|^{2^{\ast }-2}u \quad \text{in }  \Omega , \\
u=0 \quad \text{on }  \partial \Omega ,
\end{gathered}  \label{problem-abf}
\end{equation}
where $\Omega \subset \mathbb{R}^{N}$ $(N\geq 4) $ is a smooth
bounded domain, $0\in \Omega $, $2^{\ast }:=\frac{2N}{N-2}$ is the critical
Sobolev exponent, and $f$, $a$, $b$ are continuous real function defined on
 $\mathbb{R}^{N}$, $f>0$ on $\overline{\Omega }$,
$0<b(x)<\overline{\mu }:=(\frac{N-2}{2})^2$ for all $x\in \overline{\Omega }$,
 and $ 0<\max_{\overline{\Omega }}a(x)<\lambda _{1,b}$ where $\lambda _{1,b}$ is
the first Dirichlet eigenvalue of $-\Delta -\frac{b_0}{|x|^2}$ on
 $\Omega $ with $b_0:=\max_{\overline{\Omega }}b(x)$.

Some previous works about this problem, are as follows:

When $a(x)=\lambda $, $b(x)=0$ and $f(x)=1$,  problem \eqref{problem-abf}
has been studied by many authors  \cite{bn,r,la,css,cc}.
In \cite{CnC} the authors proved for $b(x)=0$ a multiplicity
sign changing result where $a$ and $f$ are continuous functions.
Jannelli \cite{jan}  investigate the problem with
$b(x)=\mu \in [ 0,\overline{\mu }-1]$, $f(x)=1$ and
$a(x)=\lambda \in (0,\lambda _1)$ where
 $ \lambda _1$ is the first Dirichlet eigenvalue of
$-\Delta -\frac{\mu }{| x|^2}$ on $\Omega $ and got the existence of
nontrivial positive solution. Cao and Peng \cite{caopeng} proved the
existence of a pair of sign changing solutions for $N\geq 7$,
 $b(x)=\mu \in [ 0,\overline{\mu }-4]$, $a(x)=\lambda \in (0,\lambda _1)$, and
$f(x)=1$. For $a(x)=\lambda $ and $b(x)=\mu $, Han and Liu \cite{hl} proved
the existence of one non trivial solution.

Guo and Niu \cite{gn} proved the existence of a symmetric nodal solution and
a positive solution for $a(x)=\lambda \in (0,\lambda _1)$, where
$\lambda_1$ is the first Dirichlet eigenvalue of
$-\Delta -\frac{\mu }{|x|^2}$ on $\Omega $, with $b(x)=\mu $, $\Omega $ and $f$
invariant under a subgroup of $O(N) $, this result was
generalized by Guo, Niu, Cui \cite{gnc} changing the term $a(x)u$ by a
function depend on $x$ and $u$, both proofs was based on previous work by
Smets \cite{S}.

\section{Statement of results}

We write again the partial differential equations to consider
\begin{equation}
\begin{gathered}
-\Delta u-b(x)\frac{u}{| x|^2}-a(x)u=f(x)|
u|^{2^{\ast }-2}u \quad \text{in }  \Omega \\
u=0\quad  \text{on }  \partial \Omega \\
u(\gamma x) =u(x) \quad \forall x\in \Omega ,\;\gamma \in \Gamma .
\end{gathered}  \label{problem abf GAMMA}
\end{equation}

In this problem the symmetries are given by $\Gamma $ a closed subgroup of
orthogonal transformation $O(N)$. We suppose $\Omega $ a $\Gamma $-invariant
smooth bounded domain in $\mathbb{R}^{N}$ such that $0\in \Omega $, and
$N\geq 4$. The critical Sobolev exponent is given by
 $2^{\ast }:=\frac{2N}{N-2}$. The functions $a$, $b$ and $f$ are
$\Gamma $-invariant continuous real valued defined on $\mathbb{R}^{N}$,
 with the following additional hypothesis,
 $0<a(x) <\lambda _{1,b}$, where $\lambda _{1,b}$ is
the first Dirichlet eigenvalue of
$-\Delta -\frac{b_0}{| x|^2}$, where $b_0=\max_{\bar{\Omega}}b(x) $ and
$0<b(x) <\overline{\mu }:=(\frac{N-2}{2})^2$. We
note that $\lambda _{1,b}$ depends of the domain of
$-\Delta -\frac{b_0}{| x|^2}$.

Let $\Gamma x:=\{ \gamma x:\gamma \in \Gamma \} $ be the
$\Gamma $-orbit of a point $x\in \mathbb{R}^{N}$, and
$\#\Gamma x$ its cardinality,
and denote by $X/\Gamma :=\{ \Gamma x:x\in X\} $ the
$\Gamma $-orbit space of $X\subset $ $\mathbb{R}^{N}$ with the quotient topology.

Let us recall that the least energy solutions of
\begin{equation}
\begin{gathered}
-\Delta u=| u|^{2^{\ast }-2}u \quad \text{in }  \mathbb{R}^{N} \\
u\to 0\quad  \text{as }  | x| \to \infty
\end{gathered}  \label{criticalproblemUNO}
\end{equation}
are the instantons given by Aubin and Talenti (see \cite{a,t}.)
\begin{equation}
U_0^{\varepsilon ,y}(x):=C(N)\Big(\frac{\varepsilon }{\varepsilon
^2+| x-y|^2}\Big)^{(N-2)/2},
\label{instanton AT}
\end{equation}
where $C(N)=(N(N-2))^{(N-2)/4}$. Is well known that if the domain is
not $\mathbb{R}^{N}$, there is no minimal energy solutions of
\eqref{criticalproblemUNO}. These solutions are minimizers for
\begin{equation*}
S:=\min_{u\in D^{1,2}(\mathbb{R}^{N})\backslash \{0\}}\frac{\int_{
\mathbb{R}^{N}}| \nabla u|^2dx}
{\big(\int_{\mathbb{R}^{N}}| u|^{2^{\ast }}dx\big)^{2/2^*}},
\end{equation*}
where $D^{1,2}(\mathbb{R}^{N}) $ is the completion of $
C_{c}^{\infty }(\mathbb{R}^{N}) $ with respect to the norm
\begin{equation*}
\| u\|^2:=\int_{\mathbb{R}^{N}}| \nabla u|^2dx.
\end{equation*}

Similarly, for $0<b(0) <\overline{\mu }$, the critical problem
\begin{equation}
\begin{gathered}
-\Delta u-b(0) \frac{u}{| x|^2}=|u|^{2^{\ast }-2}u \quad \text{in } \mathbb{R}^{N} \\
u\to 0\quad  \text{as }  | x|\to \infty ,
\end{gathered}  \label{critical problem DOS}
\end{equation}
was studied by Terracini \cite{te} and gives the solutions
\begin{equation*}
U_{b(0) }(x):=C_{b(0) }(N)\Big(\frac{\varepsilon }{
\varepsilon^2| x|^{(\sqrt{\overline{\mu }}-
\sqrt{\overline{\mu }-b(0) }) /\sqrt{\overline{\mu }}
}+| x|^{(\sqrt{\overline{\mu }}+\sqrt{\overline{
\mu }-b(0) }) /\sqrt{\overline{\mu }}}}\Big)^{(N-2)/2},
\end{equation*}
where $\varepsilon >0$ and
$C_{b(0) }(N)=(\frac{4N(\overline{\mu }-b(0) )}{N-2})^{(N-2)/4}$. In this
case the solutions are minimizers for
\begin{equation*}
S_{b(0) }:=\min_{u\in D^{1,2}(\mathbb{R}^{N})\backslash \{0\}}
\frac{\int_{\mathbb{R}^{N}}(| \nabla u|
^2-b(0) \frac{u^2}{| x|^2}) dx}{
\big(\int_{\mathbb{R}^{N}}| u|^{2^{\ast}}dx\big)^{2/2^*}}.
\end{equation*}

In the following we denote by
\begin{equation*}
M:=\{ y\in \overline{\Omega }:\frac{\#\Gamma y}{f(y)^{(N-2)/2}}
=\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{(N-2)/2}}\} .
\end{equation*}

We shall assume that $f$, $a$, and $b$ satisfy:

\begin{itemize}
\item[(F1)] $f(x) >0$ for all $x\in \overline{\Omega }$ and $f(0)=1$.

\item[(F2)] $f$ is locally flat at $M$; that is,
there exist $r>0$, $\nu >N$ and $A>0$ such that
\begin{equation*}
| f(x) -f(y) | \leq A|x-y|^{\nu }\quad \text{if }y\in M\text{ and }|x-y| <r.
\end{equation*}

\item[(B1)] $0<b(x)<\overline{\mu }$ for all
$x\in \overline{ \Omega }$, We denote by $b_0:=\max_{\overline{\Omega }}b(x)$.

\item[(A1)] If $a_0:=\max_{\overline{\Omega }}a(x)$
it must hold $0<a_0<\lambda _{1,b}$, where $\lambda _{1,b}$ denote the
first eigenvalue of $-\Delta -\frac{b_0}{| x|^2}$.

\item[(A2)] $a(x)>0$ for all $x\in M$.
\end{itemize}

With the above conditions we define
\begin{equation*}
\langle u,v\rangle _{a,b}:=\int_{\Omega }\Big(\nabla
u\cdot \nabla v-b(x)\frac{uv}{| x|^2}-a(x)uv\Big) dx
\end{equation*}
which is an inner product in $H_0^1(\Omega ) $ and its
induced norm is
\begin{equation*}
\| u\| _{a,b}:=\sqrt{\langle u,u\rangle _{a,b}}
=\Big(\int_{\Omega }(| \nabla u|^2-b(x)
\frac{u^2}{| x|^2}-a(x)u^2) dx\Big)^{1/2}.
\end{equation*}
Using the Hardy inequality,
\begin{equation}
\int_{\Omega }\frac{u^2}{| x|^2}dx\leq \frac{1
}{\overline{\mu }}\int_{\Omega }| \nabla u|
^2dx,\quad \forall u\in H_0^1(\Omega ) ,  \label{DHardy}
\end{equation}
we will prove the equivalence of the norms $\| u\| _{a,b}$
and $\| u\| :=\| u\| _{0,0}$ in $H_0^1(\Omega ) $.
Since $\lambda _{1,b}$ is the first
eigenvalue of $-\Delta -\frac{b_0}{| x|^2}$ on $H_0^1(\Omega ) $,
\begin{equation}
\int_{\Omega }a_0| u|^2dx\leq \frac{a_0}{
\lambda _{1,b}}\int_{\Omega }\Big(| \nabla u|
^2-b_0\frac{u^2}{| x|^2}\Big) dx.
\label{DvalorPropio}
\end{equation}
Therefore,
\begin{equation} \label{NEquivalentes1}
\begin{aligned}
\| u\| _{a,b}^2 &:=\int_{\Omega }\Big(
| \nabla u|^2-b(x)\frac{u^2}{| x|^2}-a(x)| u|^2|Big) dx   \\
&\geq \int_{\Omega }\Big(| \nabla u|^2-b_0
\frac{u^2}{| x|^2}\Big) dx
-\frac{a_0}{\lambda_{1,b}}\int_{\Omega }(| \nabla u|^2-b_0\frac{u^2}{| x|^2})   \\
&\geq (1-\frac{a_0}{\lambda _{1,b}}) \int_{\Omega}
\Big(| \nabla u|^2-b_0\frac{u^2}{|
x|^2}\Big) dx,\quad \text{and by \eqref{DHardy}}   \\
&\geq (1-\frac{a_0}{\lambda _{1,b}}) (1-\frac{b_0}{
\overline{\mu }}) \int_{\Omega }| \nabla u|^2dx   \\
&= (1-\frac{a_0}{\lambda _{1,b}}) (1-\frac{b_0}{\overline{\mu }}) \| u\|^2.
\end{aligned}
\end{equation}
The other inequality holds since $0<a_0<\lambda _{1,b}$, implies
$a_1=\min_{\bar{\Omega}}a(x) \leq a_0<\lambda _{1,b}<\lambda_1$ where
 $\lambda _1$ denote the first eigenvalue of $-\Delta $ on
$H_0^1(\Omega )$; therefore,
\begin{align*}
\| u\| _{a,b}^2
&\leq \int_{\Omega }( | \nabla u|^2-a(x)| u|^2)
dx \\
&\leq \int_{\Omega }| \nabla u|^2dx
 -\frac{a_1}{\lambda _1}\int_{\Omega }| \nabla u|^2dx,
 \\
&\leq (1-\frac{a_1}{\lambda _1}) \int_{\Omega}| \nabla u|^2dx\,.
\end{align*}

If $f\in C(\overline{\Omega }) $ and  (F1) is satisfied, then the norms
\begin{equation*}
| u| _{2^{\ast }}:=(\int_{\Omega }|
u|^{2^{\ast }}dx)^{1/2^{\ast}},\quad\text{and}\quad
|u| _{f,2^{\ast }}:=(\int_{\Omega }f(x)| u|^{2^{\ast }}dx)^{1/2^{\ast}}
\end{equation*}
are equivalent.
We denote
\begin{equation*}
\ell _f^{\Gamma }:=\Big(\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{
f(x)^{(N-2)/2}}\Big) S.
\end{equation*}
We will use the following non existence assumption.
\begin{itemize}
\item[(A3)] The problem
\begin{equation}
\begin{gathered}
-\Delta u=f(x)| u|^{2^{\ast }-2}u \quad \text{in }  \Omega\\
u=0\quad   \text{on }  \partial \Omega \\
u(\gamma x) =u(x) \quad \forall x\in \Omega ,\;
\gamma \in \Gamma
\end{gathered}  \label{nonexistenceproblemUNO}
\end{equation}
does not have a positive solution $u$ which satisfies
$\|u\|^2\leq \ell _f^{\Gamma }$.
\end{itemize}

If $\Omega $ is a smooth starshaped domain is well known that
(A3) is satisfied \cite{Sp}.

\subsection{Multiplicity of positive solutions}

Our next result generalizes the work of Guo and  Niu \cite{gn} for
problem \eqref{problem abf GAMMA} and establishes a relationship between the
topology of the domain and the multiplicity of positive solutions. For $
\delta >0$ let
\begin{equation}
M_{\delta }^{-}:=\{ y\in M:\mathsf{dist}(y,\partial \Omega
) \geq \delta \} ,\;B_{\delta }(M) :=\{ z\in
\mathbb{R}^{N}:\mathsf{dist}(z,M) \leq \delta \} .
\label{Mdelta}
\end{equation}

\begin{theorem}\label{teoremaUNO}
Let $N\geq 4$, {\rm (A1), (A2), (B1), (F1), (F2), (A3)} and 
$\ell _f^{\Gamma }\leq S_{b(0) }^{N/2}$ hold. 
Given $\delta ,\delta'>0 $ there exist $\lambda^{\ast }\in (0,\lambda _{1,b}) $, $
\mu^{\ast }\in (0,\overline{\mu }) $ such that for all $a(x)\in
(0,\lambda^{\ast }) $, $b(x)\in (0,\mu^{\ast }) $ $
\forall x\in \Omega $ the problem \eqref{problem abf GAMMA} has at least
\begin{equation*}
\operatorname{cat}{}_{B_{\delta }(M) /\Gamma }(M_{\delta }^{-}/\Gamma)
\end{equation*}
positive solutions which satisfy
\begin{equation*}
\ell _f^{\Gamma }-\delta'\leq \| u\|_{a,b}^2<\ell _f^{\Gamma }.
\end{equation*}
\end{theorem}

\subsection{Multiplicity of nodal solutions}

Let $G$ be a closed subgroup of $O(N)$ for which $\Omega $ and $f:\mathbb{R}
^{N}\to \mathbb{R}$ are $G$-invariant. We denote by $\Gamma $ the
kernel of an epimorphism $\tau :G\to \mathbb{Z}/2:=\{-1,1\} $.

A real valued function $u$ defined in $\Omega $ will be called $\tau $-equivariant
if
\begin{equation*}
u(gx)=\tau (g)u(x)\quad \forall x\in \Omega ,\text{ }g\in G.
\end{equation*}
In this section we study the problem
\begin{equation}
\begin{gathered}
-\Delta u-b(x)\frac{u}{| x|^2}-a(x)u=f(x)|u|^{2^{\ast }-2}u \quad \text{in } \Omega \\
u=0\quad \text{on }  \partial \Omega \\
u(gx) =\tau (g)u(x) \quad \forall x\in \Omega ,\;g\in G
\end{gathered} \label{problem abf TAU}
\end{equation}

If $g\in \Gamma $ then all $\tau $-equivariant functions $u$ satisfy
$u(gx)=u(x)$ for all $x\in \Omega $; i.e., are $\Gamma $-invariant. If $u$ is a
$\tau $-equivariant function and $g\in \tau^{-1}(-1)$ then $u(gx)=-u(x)$
for all $x\in \Omega $. Thus all non trivial $\tau $-equivariant solution of
\eqref{problem abf TAU} change sign.

\begin{definition} \label{def1} \rm
A subset $X$ of $\mathbb{R}^{N}$ is $\Gamma $-connected if it is a
$\Gamma $-invariant subset $X$ of $\mathbb{R}^{N}$ and if cannot be written as the
union of two disjoint open $\Gamma $-invariant subsets. A real valued
function $u:\Omega \to \mathbb{R}$ is $(\Gamma ,2) $-nodal if the sets
\begin{equation*}
\{ x\in \Omega :u(x) >0\} \quad \text{and}\quad
\{ x\in \Omega :u(x) <0\}
\end{equation*}
are nonempty and $\Gamma $-connected.
\end{definition}

For each $G$-invariant subset $X$ of $\mathbb{R}^{N}$, we define
\begin{equation*}
X^{\tau }:=\{ x\in X:Gx=\Gamma x\} .
\end{equation*}
Let $\delta >0$, define
\begin{equation*}
M_{\tau ,\delta }^{-}:=\{ y\in M:\operatorname{dist}(y,\partial \Omega \cup
\Omega^{\tau }) \geq \delta \} ,
\end{equation*}
and $B_{\delta }(M) $ as in \eqref{Mdelta}.

The next theorem is a multiplicity result for
 $\tau $-equivariant $(\Gamma,2)$-nodal solutions for  \eqref{problem abf GAMMA}.

\begin{theorem}\label{teoremaDOS}
Let $N\geq 4$, {\rm (A1), (A2), (B1), (F1), (F2), (A3)},
 and $\ell _f^{\Gamma }\leq S_{b(0) }^{N/2}$ hold.
If $\Gamma $ is the kernel of an epimorphism $\tau :G\to \mathbb{Z}/2$
defined on a closed subgroup $G$ of $O(N) $ for which $\Omega $
and the functions $a$, $b$, $f$ are $G$-invariant.
Given $\delta ,\delta '>0$ there exists
$\lambda^{\ast }\in (0,\lambda_{1,b}) $, $\mu^{\ast }\in (0,\overline{\mu }) $ such
that for all $a(x)\in (0,\lambda^{\ast }) $,
$b(x)\in (0,\mu^{\ast }) $ for all $x\in \Omega $  problem
\eqref{problem abf GAMMA} has at least
\begin{equation*}
\operatorname{cat}{}_{(B_{\delta }(M) \backslash B_{\delta }(M)
^{\tau }) /G}(M_{\tau ,\delta }^{-}/G)
\end{equation*}
pairs $\pm u$ of $\tau $-equivariants $(\Gamma ,2) $-nodal
solutions which satisfy
\begin{equation*}
2\ell _f^{\Gamma }-\delta'\leq \| u\|_{a,b}^2<2\ell _f^{\Gamma }.
\end{equation*}
\end{theorem}

\subsection{Non symmetric properties for solutions}

Let $\Gamma\subset\widetilde{\Gamma}\subset O (N)$. Next we give
sufficient conditions for the existence of many solutions which are
$\Gamma$-invariant but are not $\widetilde{\Gamma}$-invariant.

\begin{theorem}\label{teorema TRES}
Let $N\geq 4$, {\rm (A1), (A2), (B1), (F1), (F2), (A3)}, and
$\ell _f^{\Gamma }\leq S_{b(0) }^{N/2}$ hold.
Let $ \widetilde{\Gamma }$ be a closed subgroup of $O(N) $ containing
$\Gamma $, for which $\Omega $ and the functions $a$, $b$, $f$ are $
\widetilde{\Gamma }$-invariant and
\begin{equation*}
\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{\frac{N-2
}{2}}}<\min_{x\in \overline{\Omega }}\frac{\#\widetilde{\Gamma }x}{f(
x)^{(N-2)/2}}.
\end{equation*}
Given $\delta ,\delta'>0$ there exist
$\lambda^{\ast }\in (0,\lambda _{1,b}) $,
$\mu^{\ast }\in (0,\overline{\mu }) $
such that for all $a(x)\in (0,\lambda^{\ast }) $,
$b(x)\in(0,\mu^{\ast }) $ for all $x\in \Omega $
 problem \eqref{problem abf GAMMA} has at least
\begin{equation*}
\operatorname{cat}{}_{B_{\delta }(M) /\Gamma }(M_{\delta }^{-}/\Gamma)
\end{equation*}
positive solutions which are not $\widetilde{\Gamma }$-invariant and satisfy
\begin{equation*}
2\ell _f^{\Gamma }-\delta'\leq \| u\|_{a,b}^2<2\ell _f^{\Gamma }.
\end{equation*}
\end{theorem}

\section{The variational problem}

To generalize the notation we introduce a homomorphism
$\tau:G\to \mathbb{Z}/2$ defined on a closed subgroup $G$ of
$O(N) $. Recall the problem \eqref{problem abf TAU},
\begin{gather*}
-\Delta u-b(x)\frac{u}{| x|^2}-a(x)u=f(x)|
u|^{2^{\ast }-2}u \quad \text{in }  \Omega \\
u=0\quad \text{on }  \partial \Omega \\
u(gx) =\tau (g) u(x) \quad \forall x\in \Omega ,\; g\in G,
\end{gather*}
where $\Omega $ is a $G$-invariant bounded smooth subset of $\mathbb{R}^{N}$,
and $a$, $b$, and $f$ are a $G$-invariant continuous functions which satisfy
(A1), (A2), (B1), (F1) and (F2).

Let $\Gamma :=\ker \tau $. If $\tau $ is not an epimorphism then the
problems \eqref{problem abf TAU} and \eqref{problem abf GAMMA} coincide. In
the other case we obtain solutions for the problem \eqref{problem abf TAU}
and in particular are sign changing solutions of \eqref{problem abf GAMMA}.

The homomorphism $\tau $ induces the natural action of $G$ on
$H_0^1(\Omega ) $ given by
\begin{equation*}
(gu) (x) :=\tau (g) u(g^{-1}x) .
\end{equation*}
Due the symmetries, the solutions are in the fixed point space of the action
or the space of $\tau $-equivariant functions
\begin{align*}
H_0^1(\Omega )^{\tau }&:= \{ u\in H_0^1(\Omega ) :gu=u\; \forall g\in G\} \\
&= \{ u\in H_0^1(\Omega ) :u(gx) =\tau(g) u(x)  \; g\in G,\; \forall x\in
\Omega \} .
\end{align*}
The fixed point space of the restriction of this action to $\Gamma $
\begin{equation*}
H_0^1(\Omega )^{\Gamma }=\{ u\in H_0^1(\Omega ) :u(gx) =u(x) \; \forall g\in
\Gamma ,\; \forall x\in \Omega \}
\end{equation*}
are the $\Gamma $-invariant functions of $H_0^1(\Omega ) $.
The norms $\| \cdot \| _{a,b}$,
$\| \cdot\| $ on $H_0^1(\Omega ) $ and $| \cdot | _{2^{\ast }}$,
 $| \cdot | _{f,2^{\ast }}$ on $L^{2^{\ast }}(\Omega ) $ are $G$-invariant
with respect to the action induced by $\tau$; therefore the functional
\begin{align*}
E_{a,b,f}(u) &:=\frac{1}{2}\int_{\Omega }\Big(| \nabla
u|^2-a(x)\frac{u^2}{| x|^2}
-b(x)| u|^2\Big) dx-\frac{1}{2^{\ast }}
\int_{\Omega }f(x)| u|^{2^{\ast }}dx \\
&= \frac{1}{2}\| u\| _{a,b}^2-\frac{1}{2^{\ast }}
| u| _{f,2^{\ast }}^{2^{\ast }}
\end{align*}
is $G$-invariant, with derivative
\begin{equation*}
DE_{a,b,f}(u)v=\int_{\Omega }\Big(\nabla u\cdot \nabla v-b(x)\frac{uv}{
| x|^2}-a(x)uv\Big) dx-\int_{\Omega }f(x)|u|^{2^{\ast }-2}uv\,dx.
\end{equation*}
By the principle of symmetric criticality \cite{p}, the critical points of
its restriction to $H_0^1(\Omega )^{\tau }$ are the
solutions of \eqref{problem abf TAU}, and all non trivial solutions lie on
the Nehari manifold
\begin{align*}
\mathcal{N}_{a,b,f}^{\tau } &:=\{ u\in H_0^1(\Omega )^{\tau
}:u\neq 0,DE_{a,b,f}(u)u=0\} \\
&= \{u\in H_0^1(\Omega )^{\tau }:u\neq 0,\| u\|
_{a,b}^2=| u| _{f,2^{\ast }}^{2^{\ast }}\}.
\end{align*}
which is of class $C^2$ and radially diffeomorphic to the unit sphere in
$H_0^1(\Omega )^{\tau }$ by the radial projection
\begin{equation*}
\pi _{a,b,f}:H_0^1(\Omega )^{\tau }\setminus \{ 0\}
\to \mathcal{N}_{a,b,f}^{\tau }\quad
\pi _{a,b,f}(u):=(\frac{\| u\| _{a,b}^2}{| u| _{f,2^{\ast
}}^{2^{\ast }}})^{(N-2)/4}u.
\end{equation*}
Therefore, the nontrivial solutions of \eqref{problem abf TAU} are precisely
the critical points of the restriction of $E_{a,b,f}$ to
$\mathcal{N}_{a,b,f}^{\tau }$. If $\tau \equiv 1$ we write
 $\mathcal{N}_{a,b,f}^{\Gamma} $.

An easy computation gives
\begin{equation}
E_{a,b,f}(u)=\frac{1}{N}\| u\| _{a,b}^2=\frac{1}{N}
| u| _{f,2^{\ast }}^{2^{\ast }}\quad \forall u\in
\mathcal{N}_{a,b,f}^{\tau }  \label{enerneh}
\end{equation}
and
\begin{equation*}
E_{a,b,f}(\pi _{a,b,f}(u)) =\frac{1}{N}(\frac{\|
u\| _{a,b}^2}{| u| _{f,2^{\ast }}^2})
^{N/2}\quad \forall u\in H_0^1(\Omega )^{\tau }\backslash
\{0\}.
\end{equation*}
We define
\begin{align*}
m(a,b,f) &:= \inf_{\mathcal{N}_{a,b,f}}E_{a,b,f}(u)=\inf_{\mathcal{N}
_{a,b,f}}\frac{1}{N}\| u\| _{a,b}^2 \\
&= \inf_{u\in H_0^1(\Omega )\setminus \{0\}}\frac{1}{N}(\frac{
\| u\| _{a,b}^2}{| u| _{f,2^{\ast
}}^2})^{N/2}.
\end{align*}
In the restrictions for the Nehari manifolds we denote by
\begin{equation*}
m^{\Gamma }(a,b,f):=\inf_{\mathcal{N}_{a,b,f}^{\Gamma }}E_{a,b,f},\quad
m^{\tau }(a,b,f):=\inf_{\mathcal{N}_{a,b,f}^{\tau }}E_{a,b,f}.
\end{equation*}

\subsection{Estimates for the infimum}

From the definition of Nehari Manifold and \eqref{enerneh} we obtain that
$m^{\Gamma }(a,b,f)>0$.

\begin{proposition}
Let  $a(x)\leq a'(x)<\lambda _{1,b}$, $b(x) \leq
b'(x) <\bar{\mu}$, for all $x\in \bar{\Omega}$, and
 $f: \mathbb{R}^{N}\to \mathbb{R}$, with the conditions above. Then
\begin{equation*}
m(a',b',f) \leq m(a,b,f), \quad
m^{\Sigma }(a',b',f) \leq m^{\Sigma}(a,b,f) ,
\end{equation*}
with $\Sigma =\Gamma $ or $\Sigma =\tau $.
\end{proposition}

\begin{proof}
By definition of $\| \cdot \| _{a,b}$ we obtain
$\|u\| _{a',b'}^2\leq \| u\|_{a,b}^2$. Let $u\in H_0^1(\Omega )\setminus \{0\}$,
then
\begin{align*}
m(a',b',f) 
&\leq E_{a',b',f}(\pi_{a',b',f}(u)) \\
&= \frac{1}{N}\Big(\frac{\| u\| _{a',b'}^2}{| u| _{f,2^{\ast }}^2}\Big)^{N/2} \\
&\leq \frac{1}{N}\Big(\frac{\| u\| _{a,b}^2}{
| u| _{f,2^{\ast }}^2}\Big)^{N/2}
= E_{a,b,f}(\pi _{a,b,f}(u)),
\end{align*}
and from this inequality, the conclusion follows.
\end{proof}

We denote by $\lambda _{1,b}$ the first Dirichlet eigenvalue of
$-\Delta -\frac{b_0}{| x|^2}$ in $H_0^1(\Omega )$.

\begin{lemma}\label{comparacionE}
With the conditions $(a_1)$ and $(b)$, for $u\in
H_0^1(\Omega )^{\tau }$, we obtain
\begin{equation*}
E_{0,0,f}(\pi _{0,0,f}(u) ) \leq \Big(\frac{\bar{
\mu}}{\bar{\mu}-b_0}\Big)^{N/2}(\frac{\lambda _{1,b}}{
\lambda _{1,b}-a_0})^{N/2}E_{a,b,f}(\pi_{a,b,f}(u) ) .
\end{equation*}
\end{lemma}

\begin{proof}
Since
\[
E_{a,b,f}(\pi _{a,b,f}(u) )
= \frac{1}{N}\Big(\frac{\| u\| _{a,b}^2}{| u|_{f,2^{\ast }}^2}\Big)^{N/2}
= \frac{1}{N}\Big(\frac{\| u\| _{a,b}^{N}}{|u| _{f,2^{\ast }}^{N}}\Big) ,
\]
 by \eqref{NEquivalentes1} we have
\begin{equation*}
\big(1-\frac{a_0}{\lambda _{1,b}}\big)^{N/2}\big(1-\frac{
b_0}{\overline{\mu }}\big)^{N/2}\| u\|^{N}\leq \| u\| _{a,b}^{N}
\end{equation*}
then
\begin{equation*}
\big(1-\frac{a_0}{\lambda _{1,b}}\big)^{N/2}\big(1-\frac{
b_0}{\overline{\mu }}\big)^{N/2}\frac{1}{N}\frac{\|
u\|^{N}}{| u| _{f,2^{\ast }}^{N}}\leq
E_{a,b,f}(\pi _{a,b,f}(u) )
\end{equation*}
so
\begin{equation*}
E_{0,0,f}(\pi _{0,0,f}(u) ) \leq \big(\frac{\bar{
\mu}}{\bar{\mu}-b_0}\big)^{N/2}\big(\frac{\lambda _{1,b}}{
\lambda _{1,b}-a_0}\big)^{N/2}E_{a,b,f}(\pi_{a,b,f}(u) ) ,
\end{equation*}
which completes the proof.
\end{proof}

\begin{corollary}\label{corolario7}
$m^{\tau }(0,0,f) \leq (\frac{\bar{\mu}
}{\bar{\mu}-b_0})^{N/2}(\frac{\lambda _{1,b}}{
\lambda _{1,b}-a_0})^{N/2}m^{\tau }(a,b,f) $.
\end{corollary}

For the proof of the next lemma we refer the reader to \cite{CnC}.

\begin{lemma}\label{lema9}
If $\Omega \cap M\neq \emptyset $ then:
(a) $m^{\Gamma }(0,0,f) \leq \frac{1}{N}\ell _f^{\Gamma }$.
(b) If there exists $y\in \Omega \cap M$ with $\Gamma x\neq Gy$, then $
m^{\tau }(0,0,f) \leq \frac{2}{N}\ell _f^{\Gamma }$.
\end{lemma}

\subsection{A compactness result}

\begin{definition}\label{def2}
A sequence $\{u_n\}\subset H_0^1(\Omega ) $ satisfying
\begin{equation*}
E_{a,b,f}(u_n)\to c\quad \text{and}\quad
\nabla E_{a,b,f}(u_n)\to 0.
\end{equation*}
is called a Palais-Smale sequence for $E_{a,b,f}$ at $c$. We say
that $E_{a,b,f}$ satisfies the Palais-Smale condition
$(PS) _{c}$ if every Palais-Smale sequence for $E_{a,b,f}$ at
 $c $ has a convergent subsequence.
If $\{u_n\}\subset H_0^1(\Omega )^{\tau }$ then $\{u_n\}$ is a
$\tau $-equivariant Palais-Smale sequence and $E_{a,b,f}$ satisfies
the $\tau $-equivariant Palais-Smale condition,
$(PS) _{c}^{\tau }$.
If $\tau \equiv 1$ $\{u_n\}$ is a $\Gamma $-invariant
Palais-Smale sequence and $E_{a,b,f}$ satisfies the $\Gamma $-invariant Palais-Smale
condition $(PS) _{c}^{\Gamma }$.
\end{definition}

To describe the $\tau $-equivariant Palais-Smale sequence
for $E_{a,b,f}$ we use the next theorem proved by Guo and Niu \cite{gn}.
which is based on results of Struwe \cite{s}.

\begin{theorem} \label{thm4}
Let $(u_n)$ be a $\tau $-equivariant Palais-Smale sequence in
 $ H_0^1(\Omega )^{\tau }$ for $E_{a,b,f}$ at $c\geq 0$. Then there exist a
solution $u$ of \eqref{problem abf TAU}, $m$, $l\in \mathbb{N}$; a closed
subgroup $G^i$ of finite index in $G$, sequences
$\{y_n^i\}\subset \Omega $, $\{r_n^i\}\subset (0,\infty )$, a
solution $\widehat{u}_0^i $ of \eqref{criticalproblemUNO} for
$i=1,\dots,m$; and $\{R_n^{j}\}\subset (0,\infty )$, a solution
$\widehat{u}_{b}^{j}$ of \eqref{critical problem DOS} for $j=1,\dots,l$.
Such that
\begin{itemize}
\item[(i)] $G_{y_n^i}=G^i$,

\item[(ii)] $(r_n^i)^{-1}\operatorname{dist}(y_n^i,\partial \Omega )\to \infty $,
 $y_n^i\to y^i$, if $n\to \infty $, for $i=1,\dots,m$,

\item[(iii)] $(r_n^i)^{-1}| gy_n^i-g'y_n^i|\to \infty $,
 if $n\to \infty $, and $[g]\neq [g']\in G/G^i$ for $i=1,\dots,m$,

\item[(iv)] $\widehat{u}_0^i(gx)=\tau (g)\widehat{u}_0^i(x)$, $\forall x\in
\mathbb{R}^{N}$ and $g\in G^i$,

\item[(v)] $\widehat{u}_{b}^{j}(gx)=\tau (g)\widehat{u}_{b}^{j}(x)$, for all
$x\in \mathbb{R}^{N}$ and $g\in G$, $R_n^{j}\to 0$ for $j=1,\dots,l$,

\item[(vi)] 
\begin{align*}
u_n(x)&=u(x)+\sum_{i=1}^{m}\sum_{[g]\in G/G^i}(
r_n^i)^{(2-N)/2}f(y^i)^{(2-N)/4}\\
&\quad\times \tau (g)\widehat{u}_0^i(g^{-1}(\frac{x-gy_n^i}{r_n^i}) )
+\sum_{j=1}^{l}(R_n^{j})^{\frac{2-N}{2}}\widehat{u}_{b}^{j}
(\frac{x}{R_n^{j}})+o(1)
\end{align*}

\item[(vii)] $E_{a,b,f}(u_n)\to E_{a,b,f}(u)+\sum_{i=1}^{m}(\frac{
\#(G/G^i)}{f(y^i)^{(N-2)/2}}) E_{0,0,1}^{\infty }(\widehat{u}
_0^i)+\sum_{j=1}^{l}E_{0,b(0),1}^{\infty }(\widehat{u}_{b}^{j})$, as
$n\to \infty $
\end{itemize}
\end{theorem}

\begin{corollary}\label{existsoluc}
 $E_{a,b,f}$ satisfies $(PS)_{c}^{\tau }$ at every value
\begin{equation*}
c<\min \big\{ \#(G/\Gamma ) \frac{\ell _f^{\Gamma }}{N},
\frac{\#(G/\Gamma ) }{N}S_{b(0) }^{N/2}\big\} .
\end{equation*}
\end{corollary}

\begin{proof}
From the inequality of the value $c$ and the part (vii) of the
theorem, we obtain that $m$ and $l$ are equal to zero. The convergence
follows from (vi).
\end{proof}

\section{The bariorbit map}

In the following we suppose the condition
 $\ell _f^{\Gamma }\leq S_{b(0)}^{N/2}$ hold and we will assume the next nonexistence
condition.
\begin{itemize}
\item[(NE)] The infimum of $E_{0,0,f}$ is not
achieved in $\mathcal{N}_{0,0,f}^{\Gamma }$.
\end{itemize}

With these conditions, Corollary \ref{existsoluc} and Lemma
\ref{lema9} imply that
\begin{equation}
m^{\Gamma }(0,0,f):=\inf_{\mathcal{N}_{0,0,f}^{\Gamma }}E_{0,0,f}
=\Big(\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{(N-2)/2}}
\Big) \frac{1}{N}S^{N/2}.  \label{inf0}
\end{equation}
Let
\begin{equation*}
M:=\{ y\in \overline{\Omega }:\frac{\#\Gamma y}{f(y)^{(N-2)/2}}
=\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{(N-2)/2}}
\}.
\end{equation*}
For every $y\in \mathbb{R}^{N}$, $\gamma \in \Gamma $, the isotropy
subgroups satisfy $\Gamma _{\gamma y}=\gamma \Gamma _{y}\gamma^{-1}$.
Therefore the set of isotropy subgroups of $\Gamma $-invariant subsets
consists of complete conjugacy classes. We choose $\Gamma _i\subset \Gamma
$, $i=1,..,m,$\ one in each conjugacy class of an isotropy subgroup of $M$.
Set
\begin{gather*}
M^i := \{y\in M:\Gamma _{y}=\Gamma _i\}
=\{y\in M:\gamma y=y\; \forall \gamma \in \Gamma _i\}, \\
\Gamma M^i := \{\gamma y:\gamma \in \Gamma ,\; y\in M^i\}
=\{y\in M:(\Gamma _{y})=(\Gamma _i)\}.
\end{gather*}
By definition of $M$ it follows that $f$ is constant on each $\Gamma M^i$,
then we can define
\begin{equation*}
f_i:=f(\Gamma M^i)\in \mathbb{R}.
\end{equation*}
The compactness of $M$ allows us to fix $\delta _0>0$ such that
\begin{equation}
\begin{gathered}
| y-\gamma y| \geq 3\delta _0\quad \forall y\in M,
\; \gamma \in \Gamma \text{ if }\gamma y\neq y, \\
\operatorname{dist}(\Gamma M^i,\Gamma M^{j})\geq 3\quad \forall
i,j=1,\dots,m\text{ if }i\neq j,
\end{gathered}\label{desig2}
\end{equation}
and such that the isotropy subgroup of each point in 
$M_{\delta_0}^i:=\{z\in \mathbb{R}^{N}:\gamma z=z \forall \gamma \in \Gamma
_i, \operatorname{dist}(z,M^i)\leq \delta _0\}$ is precisely $\Gamma _i$. Define
\begin{equation*}
W_{\varepsilon ,z}:=\sum_{[g]\in \Gamma /\Gamma _i}f_i^{\frac{2-N
}{4}}U_{\varepsilon ,gz}\quad \text{if }z\in M_{\delta _0}^i,
\end{equation*}
where $U_{\varepsilon ,y}:=U_0^{\varepsilon ,y}$ is defined by
\eqref{instanton AT}. For each $\delta \in (0,\delta _0)$ define
\begin{gather*}
M_{\delta } := M_{\delta }^1\cup \cdot \cdot \cdot \cup M_{\delta }^{m},\\
B_{\delta } := \{(\varepsilon ,z):\varepsilon \in (0,\delta ),\text{ }z\in
M_{\delta }\}, \\
\Theta _{\delta } := \{\pm W_{\varepsilon ,z}:(\varepsilon ,z)\in B_{\delta
}\},\text{ \ \ \ \ }\Theta _0:=\Theta _{\delta _0}.
\end{gather*}
We mention the next result proved in \cite{CnC} about the construction of
bariorbit maps.

\begin{proposition}\label{teodeckr}
Let $\delta \in (0,\delta _0)$, and assume that {\rm (NE)}
holds. There exists $\eta >m^{\Gamma }(0,0,f)$ with following properties: 
For each $u\in \mathcal{N}_{0,0,f}^{\Gamma }$ such that 
$E_{0,0,f}(u)\leq \eta $ we have
\begin{equation*}
\inf_{W\in \Theta _0}\| u-W\| <\sqrt{\frac{1}{2} Nm^{\Gamma }(0,0,f)},
\end{equation*}
and there exist precisely one $\nu \in \{-1,1\}$, one $\varepsilon \in
(0,\delta _0)$ and one $\Gamma $-orbit $\Gamma z\subset M_{\delta _0}$
such that
\begin{equation*}
\| u-\nu W_{\varepsilon ,z}\| =\inf_{W\in \Theta_0}\| u-W\| .
\end{equation*}
Moreover $(\varepsilon ,z)\in B_{\delta }$
\end{proposition}

\subsection{Definition of the bariorbit map}

Fix $\delta \in (0,\delta _0)$ and choose $\eta >m^{\Gamma }(0,0,f)$ as in
Proposition \ref{teodeckr}. Define
\begin{gather*}
E_{0,0,f}^{\eta } := \{u\in H_0^1(\Omega ):E_{0,0,f}(u)\leq \eta \}, \\
B_{\delta }(M) := \{z\in \mathbb{R}^{N}:\operatorname{dist}(z,M)\leq \delta \},
\end{gather*}
and the space of $\Gamma $-orbits of $B_{\delta }(M)$ by $B_{\delta}(M)/\Gamma $.
From Proposition \ref{teodeckr} we have the following definition.

\begin{definition}\label{defbeta}\rm
The \emph{bariorbit map}
$\beta^{\Gamma }:\mathcal{N}_{0,0,f}^{\Gamma }\cap E_{0,0,f}^{\eta
}\to B_{\delta }(M)/\Gamma$
is defined by
\begin{equation*}
\beta^{\Gamma }(u)=\Gamma y\overset{\emph{def}}{\Longleftrightarrow }
\| u\pm W_{\varepsilon ,y}\| =\min_{W\in \Theta_0}\| u-W\| .
\end{equation*}
\end{definition}

This map is continuous and $\mathbb{Z}/2$-invariant by the compactness of 
$M_{\delta }$.
If $\Gamma $ is the kernel of an epimorphism
 $\tau :G\to \mathbb{Z} /2$, choose $g_{\tau }\in \tau^{-1}(-1)$. 
Let $u\in \mathcal{N}_{0,0,f}^{\tau }$ then $u$ changes sign and 
$u^{-}(x)=-u^{+}(g_{\tau}^{-1}x). $ Therefore, 
$\| u^{+}\|^2=\|u^{-}\|^2$ and $| u^{+}| _{f,2^{\ast
}}^{2^{\ast }}=| u^{-}| _{f,2^{\ast }}^{2^{\ast }}$. So
\begin{equation}
u\in \mathcal{N}_{0,0,f}^{\tau }\Rightarrow u^{\pm }\in \mathcal{N}
_{0,0,f}^{\Gamma }\text{  and  }E_{0,0,f}(u)=2E_{0,0,f}(u^{\pm }).
\label{relneh}
\end{equation}

\begin{lemma}\label{dobinf} 
$E_{0,0,f}$ does not achieve its infimum at $\mathcal{N}_{0,0,f}^{\tau }$, moreover
\begin{equation*}
m^{\tau }(0,0,f) :=\inf_{\mathcal{N}_{0,0,f}^{\tau
}}E_{0,0,f}=\Big(\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{
f(x)^{(N-2)/2}}\Big) \frac{2}{N}S^{N/2}=2m^{\Gamma }(0,0,f).
\end{equation*}
\end{lemma}

\begin{proof}
Suppose that there exists $u\in \mathcal{N}_{0,0,f}^{\tau }$ such that 
$E_{0,0,f}(u)=m^{\tau }(0,0,f)$. Then $u^{+}\in \mathcal{N}_{0,0,f}^{\Gamma }$
and by Lemma \ref{lema9},
\begin{equation*}
m^{\tau }(0,0,f)\leq \Big(\min_{x\in \overline{\Omega }}\frac{
\#\Gamma x}{f(x)^{(N-2)/2}}\Big) \frac{2}{N}S^{N/2}.
\end{equation*}
Hence
\begin{equation*}
m^{\Gamma }(0,0,f)\leq E_{0,0,f}(u^{+})=\frac{1}{2}m^{\tau }(0,0,f)\leq
\Big(\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{\frac{N-2
}{2}}}\Big) \frac{1}{N}S^{N/2}=m^{\Gamma }(0,0,f).
\end{equation*}
Thus $u^{+}$ is a minimum of $E_{0,0,f}$ on $\mathcal{N}_{0,0,f}^{\Gamma }$,
which contradicts (NE). The corollary \ref{existsoluc} implies 
\begin{equation*}
m^{\tau }(0,0,f)=\Big(\min_{x\in \overline{\Omega }}\frac{\#\Gamma x
}{f(x)^{(N-2)/2}}\Big) \frac{2}{N}S^{N/2}.
\end{equation*}
\end{proof}

 The property \eqref{relneh} implies
$u^{\pm }\in \mathcal{N}_{0,0,f}^{\Gamma }\cap E_{0,0,f}^{\eta }$
for all $u\in \mathcal{N}_{0,0,f}^{\tau }\cap E_{0,0,f}^{2\eta }$,
so
\begin{equation}
\| u^{+}-\nu W_{\varepsilon ,y}\| =\min_{W\in \Theta
_0}\| u^{+}-W\|\Leftrightarrow \| u^{-}+\nu
W_{\varepsilon ,g_{\tau }y}\| =\min_{W\in \Theta _0}\|
u^{-}-W\| .  \label{sim}
\end{equation}
Therefore,
\begin{equation}
\beta^{\Gamma }(u^{+})=\Gamma y\Longleftrightarrow \beta^{\Gamma
}(u^{-})=\Gamma (g_{\tau }y),  \label{simbeta}
\end{equation}
and
\begin{equation}
\beta^{\Gamma }(u^{+})\neq \beta^{\Gamma }(u^{-})\quad \forall
u\in \mathcal{N}_{0,0,f}^{\tau }\cap E_{0,0,f}^{2\eta }.
\label{difbariorbita}
\end{equation}
Set
$B_{\delta }(M)^{\tau }:=\{z\in B_{\delta }(M):Gz=\Gamma z\}$.

\begin{proposition}\label{barorbequiv}
The map
\begin{equation*}
\beta^{\tau }:\mathcal{N}_{0,0,f}^{\tau }\cap E_{0,0,f}^{2\eta }\to
(B_{\delta }(M)\setminus B_{\delta }(M)^{\tau }) /\Gamma ,\quad
 \beta^{\tau }(u):=\beta^{\Gamma }(u^{+}),
\end{equation*}
is well defined, continuous and $\mathbb{Z}/2$-equivariant; i.e.,
$\beta^{\tau }(-u)=\Gamma (g_{\tau }y)$ if and only if 
$\beta^{\tau }(u)=\Gamma y$.
\end{proposition}

\begin{proof}
If $u\in \mathcal{N}_{0,0,f}^{\tau }\cap E_{0,0,f}^{2\eta }$ and
 $\beta^{\tau }(u)=\Gamma y\in B_{\delta }(M)^{\tau }/\Gamma $ then
 $\beta^{\Gamma}(u^{+})=\Gamma y=\Gamma (g_{\tau }y)=\beta^{\Gamma }(u^{-})$, 
this is a contradiction to \eqref{difbariorbita}. We conclude that 
$\beta^{\tau}(u)\not\in B_{\delta }(M)^{\tau }/\Gamma $. The continuity and 
$\mathbb{Z}/2 $-equivariant properties follows by $\beta^{\Gamma }$ ones.
\end{proof}

\section{Multiplicity of solutions}

\subsection{Lusternik-Schnirelmann theory}

An involution on a topological space $X$ is a map $\varrho _{X}:X\to
X$, such that $\varrho _{X}\circ \varrho _{X}=id_{X}$. Providing $X$ with an
involution amounts to defining an action of $\mathbb{Z}/2$ on $X$ and
viceversa. The trivial action is given by the identity $\varrho _{X}=id_{X}$,
the action of $G/\Gamma \simeq \mathbb{Z}/2$ on the orbit space $\mathbb{R}
^{N}/\Gamma $ where $G\subset O(N) $ and $\Gamma $ is the kernel
of an epimorphism $\tau :G\to \mathbb{Z}/2$, and the antipodal
action $\varrho (u) =-u$ on $\mathcal{N}_{a,b,f}^{\tau }$. A map
$f:X\to Y$ is called $\mathbb{Z}/2$-equivariant $(\text{or a }
\mathbb{Z}/2\text{-map}) $ if $\varrho _{Y}\circ f=f\circ \varrho
_{X}, $ and two $\mathbb{Z}/2$-maps, $f_0,f_1:X\to Y$, are said
to be $\mathbb{Z}/2$-homotopic if there exists a homotopy $\Theta :X\times
\left[ 0,1\right] \to Y$ such that $\Theta (x,0)
=f_0(x) $, $\Theta (x,1) =f_1(x) $
and $\Theta (\varrho _{X}x,t) =\varrho _{Y}\Theta (
x,t) $ for every $x\in X$, $t\in \left[ 0,1\right] $. A subset $A$ of $
X$ is $\mathbb{Z}/2$-equivariant if $\varrho _{X}a\in A$ for every $a\in A$.

\begin{definition} \label{def4} \rm
The $\mathbb{Z}/2$-category of a $\mathbb{Z}/2$-map $f:X\to Y$ is
the smallest integer $k:=\mathbb{Z}/2-{\rm cat}(f) $ with following
properties
\begin{itemize}
\item[(i)] There exists a cover of $X=X_1\cup \ldots \cup X_{k}$
by $k$ open $\mathbb{Z}/2$-invariant subsets,

\item[(ii)] The restriction $f\mid _{X_i}:X_i\to Y$ is 
$\mathbb{Z}/2$-homotopic to the composition $\kappa _i\circ \alpha _i$ of
a $\mathbb{Z}/2$-map 
$\alpha _i:X_i\to \{ y_i,\varrho_{Y}y_i\} $, $y_i\in Y$, and the inclusion
$\kappa _i:\{y_i,\varrho _{Y}y_i\} \hookrightarrow Y$.
\end{itemize}
If not such covering exists, we define $\mathbb{Z}/2-{\rm cat}(f):=\infty $.
\end{definition}

If $A$ is a $\mathbb{Z}/2$-invariant subset of $X$ and 
$\iota :A\hookrightarrow X$ is the inclusion we write
\begin{equation*}
\mathbb{Z}/2-\operatorname{cat}{}_{X}(A) :=\mathbb{Z}/2-{\rm cat}(\iota ) ,\quad
\mathbb{Z}/2-\operatorname{cat}{}_{X}(X) :=\mathbb{Z}/2-{\rm cat}(X) .
\end{equation*}
Note that if $\varrho _{x}={\rm id}_{X}$ then
\begin{equation*}
\mathbb{Z}/2-\operatorname{cat}{}_{X}(A) :=\operatorname{cat}{}_{X}(A) ,\quad
\mathbb{Z}/2-{\rm cat}(X) :={\rm cat}(X) ,
\end{equation*}
are the usual Lusternik-Schnirelmann category (see \cite[definition 5.4]{w}).

\begin{theorem} \label{thm5}
Let $\phi :M\to \mathbb{R}$ be an even functional of class $C^1$,
and $M$ a submanifold of a Hilbert space of class $C^2$, symmetric with
respect to the origin. If $\phi $ is bounded below and satisfies $(PS)_{c}$
for each $c\leq d$, then $\phi $ has at least $\mathbb{Z}/2$-cat$(\phi^{d})$
pairs critical points such that $\phi (u)\leq d$ (see \cite{cp}).
\end{theorem}

\subsection{Proof of Theorems}

We prove Theorem \ref{teoremaDOS}; the proof of Theorem \ref{teoremaUNO}
is analogous. Recall that if $\tau $ is the identity or an epimorphism then 
$\#(G/\Gamma ) $ is $1$ or $2$.

\begin{proof}[Proof of Theorem \ref{teoremaDOS}]
By Corollary \ref{existsoluc}, $E_{a,b,f}$ satisfies $(PS)_{\theta }^{\tau }$ for
\begin{equation*}
\theta <\min \big\{ \#(G/\Gamma ) \frac{\ell _f^{\Gamma }}{N}
,\frac{\#(G/\Gamma ) }{N}S_{b(0) }^{N/2}\big\} .
\end{equation*}
By Lusternik-Schnirelmann theory $E_{a,b,f}$ has at least $\mathbb{Z}/2$-cat$
(\mathcal{N}_{a,b,f}^{\tau }\cap E_{a,b,f}^{\theta }) $ pairs $
\pm u$ of critical points in $\mathcal{N}_{a,b,f}^{\tau }\cap
E_{a,b,f}^{\theta }$. We are going to estimate this category for an
appropriate value of $\theta $.

Without lost of generality we can assume that 
$\delta \in (0,\delta_0) $, with $\delta _0$ as in \eqref{desig2}. Let 
$\eta >\frac{\ell _f^{\Gamma }}{N}$, $\mu^{\ast }\in (0,\overline{\mu }) $
and $\lambda^{\ast }\in (0,\lambda _{1,b}) $ such that
\begin{equation*}
(\frac{\bar{\mu}}{\bar{\mu}-\mu^{\ast }})^{N/2}(\frac{
\lambda _{1,b}}{\lambda _{1,b}-\lambda^{\ast }})^{N/2}
=\min \{2,\frac{N\eta }{\#(G/\Gamma ) \ell _f^{\Gamma }},\frac{\ell
_f^{\Gamma }}{\ell _f^{\Gamma }-\delta'}\} .
\end{equation*}
By Lemma \ref{comparacionE}, if $u\in \mathcal{N}_{a,b,f}^{\tau }\cap
E_{a,b,f}^{\theta }$, $b_0\in (0,\mu^{\ast }) $, $a_0\in
(0,\lambda^{\ast }) $ we have
\begin{align*}
E_{0,0,f}(\pi _{0,0,f}(u) ) 
&\leq (\frac{\bar{\mu}}{\bar{\mu}-b_0})^{N/2}(\frac{\lambda
_{1,b}}{\lambda _{1,b}-a_0})^{N/2}E_{a,b,f}(u)
\\
&< (\frac{\bar{\mu}}{\bar{\mu}-b_0})^{N/2}(
\frac{\lambda _{1,b}}{\lambda _{1,b}-a_0})^{N/2}\#(
G/\Gamma ) \frac{\ell _f^{\Gamma }}{N} \\
&\leq \#(G/\Gamma ) \eta .
\end{align*}

Let $\beta^{\tau }$ be the $\tau$-bariorbit function, defined in
Proposition \ref{barorbequiv}. Hence the composition map
\begin{equation*}
\beta^{\tau }\circ \pi _{0,0,f}:\mathcal{N}_{a,b,f}^{\tau }\cap
E_{a,b,f}^{\theta }\to (B_{\delta }(M)\setminus B_{\delta
}(M)^{\tau }) /\Gamma ,
\end{equation*}
is a well defined $\mathbb{Z}/2$-invariant continuous function.

Since $N\geq 4$, by \cite[Lemma 3 and Proposition 3]{CnC},
 using (F2) we can choose $\varepsilon >0$ small enough and 
$\theta :=\theta_{\varepsilon }<\#(G/\Gamma ) \frac{\ell _f^{\Gamma }}{N}$
such that
\begin{equation*}
E_{a,b,f}(\pi _{a,b,f}(w_{\varepsilon ,y}^{\tau }) )
\leq \theta <\#(G/\Gamma ) \frac{\ell _f^{\Gamma }}{N},\quad
\forall y\in M_{\delta }^{-},
\end{equation*}
where $w_{\varepsilon ,y}^{\tau }=w_{\varepsilon ,y}^{\Gamma
}-w_{\varepsilon ,g_{\tau }y}^{\Gamma }$, $\tau (g_{\tau })=-1$, and
\begin{equation*}
w_{\varepsilon ,y}^{\Gamma }(x)=\sum_{[\gamma ]\in \Gamma /\Gamma _{y}}f(y)^{
\frac{2-N}{4}}U_{\varepsilon ,\gamma y}(x)\varphi _{\gamma y}(x).
\end{equation*}
Thus the map
$\alpha _{\delta }^{\tau }: M_{\tau ,\delta }^{-}/\Gamma \to
\mathcal{N}_{a,b,f}^{\tau }\cap E_{a,b,f}^{\theta }$, defined by 
\[
\alpha _{\delta }^{\tau }(\Gamma y) := \pi _{a,b,f}(
w_{\varepsilon ,y}^{\tau })
\]
is a well defined $\mathbb{Z}/2$-invariant continuous function.
 Moreover $\beta^{\tau }(\pi _{0,0,f}(\alpha _{\delta }^{\tau }(
\Gamma y) ) ) =\Gamma y$ for all $y\in M_{\tau ,\delta
}^{-}$. Therefore,
\begin{equation*}
\mathbb{Z}/2-{\rm cat}(\mathcal{N}_{a,b,f}^{\tau }\cap
E_{a,b,f}^{\theta }) \geq \operatorname{cat}{}_{((B_{\delta }(M)\setminus
B_{\delta }(M)^{\tau }) /\Gamma ) }(M_{\tau ,\delta
}^{-}/\Gamma ) .
\end{equation*}
So \eqref{problem abf TAU} has at least
\begin{equation*}
{\rm ca}t_{((B_{\delta }(M)\setminus B_{\delta }(M)^{\tau })
/G) }(M_{\tau ,\delta }^{-}/G)
\end{equation*}
pairs $\pm u$ solution which satisfy
\begin{equation*}
E_{a,b,f}(u) <\#(G/\Gamma ) \frac{\ell _f^{\Gamma }}{N}.
\end{equation*}
By the choice of $\lambda^{\ast }$ and $\mu^{\ast }$ we have that
\begin{equation*}
(\frac{\bar{\mu}}{\bar{\mu}-\mu^{\ast }})^{N/2}(\frac{
\lambda _1}{\lambda _1-\lambda^{\ast }})^{N/2}\leq \frac{\ell
_f^{\Gamma }}{\ell _f^{\Gamma }-\delta'},
\end{equation*}
then
\begin{align*}
\#(G/\Gamma ) \frac{\ell _f^{\Gamma }-\delta'}{N}
&\leq (\frac{\bar{\mu}-b_0}{\bar{\mu}})^{N/2}(\frac{
\lambda _1-a_0}{\lambda _1})^{N/2}\#(G/\Gamma )
\frac{\ell _f^{\Gamma }}{N} \\
&\leq m^{\tau }(a,b,f)\leq E_{a,b,f}(u) \\
&= \frac{1}{N}\| u\| _{a,b}^2<\#(G/\Gamma )
\frac{\ell _f^{\Gamma }}{N}
\end{align*}
therefore,
$\#(G/\Gamma ) \ell _f^{\Gamma }-\delta^{\prime \prime }\leq
\| u\| _{a,b}^2<\#(G/\Gamma ) \ell
_f^{\Gamma }$.
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{teorema TRES}]
By Theorem \ref{teoremaUNO} there exist $\lambda $ and $\mu $ sufficiently
close to zero such that \eqref{problem abf GAMMA} has at least 
$\operatorname{cat}{}_{B_{\delta }(M) /\Gamma }(M_{\delta }^{-}/\Gamma) $ positive
solutions with $E_{a,b,f}(u)<\frac{\ell _f^{\Gamma }}{N}$.

Observe that $\frac{\ell _f^{\Gamma }}{N}<m^{\widetilde{\Gamma }}(0,0,f)$. 
Indeed, if $m^{\widetilde{\Gamma }}(0,0,f)$ is not achieved then by the
hypothesis
 $m^{\widetilde{\Gamma }}(0,0,f)=\frac{\ell _f^{\widetilde{
\Gamma }}}{N}>\frac{\ell _f^{\Gamma }}{N}$.
 On the other hand if 
$u\in\mathcal{N}_{0,0,f}^{\widetilde{\Gamma }}\subset \mathcal{N}_{0,0,f}^{\Gamma}$ 
satisfies $E_{0,0,f}(u)=m^{\widetilde{\Gamma }}(0,0,f)$ we obtain 
\begin{equation*}
\frac{\ell _f^{\Gamma }}{N}=m^{\Gamma }(0,0,f)<m^{\widetilde{\Gamma }
}(0,0,f)=E_{0,0,f}(u).
\end{equation*}
By Corollary \ref{corolario7}, there exist 
$\widehat{\lambda }\in (0,\lambda _1) $ and$\ \widehat{\mu }\in (0,\bar{\mu}) $
such that for each $\lambda \in (0,\widehat{\lambda })$ and
$\mu \in (0,\widehat{\mu })$ such that
\begin{equation*}
\frac{\ell _f^{\Gamma }}{N}<m^{\tilde{\Gamma}}(0,0,f)\leq (\frac{
\lambda _1}{\lambda _1-\lambda })^{N/2}(\frac{
\overline{\mu }}{\overline{\mu }-\mu })^{N/2}m^{\tilde{\Gamma}
}(a,b,f).
\end{equation*}
Then
\begin{equation*}
E_{a,b,f}(u)<\frac{\ell _f^{\Gamma }}{N}<m^{\tilde{\Gamma}}(a,b,f).
\end{equation*}
Therefore $u$ is not $\tilde{\Gamma}$-invariant solution.
\end{proof}



\subsection*{Acknowledgements}
The authors wish to thank the Instituto de Matem\'aticas of the
Universidad Nacional Aut\'onoma de M\'exico for hospitality during 2012
which contributed essentially to this research.

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