\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 112, 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 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/112\hfil Singular semilinear elliptic problems] {Multiple solutions for a singular semilinear elliptic problems with critical exponent and symmetries} \author[A. Cano, S. Hern\'andez-Linares, E. Hern\'andez-Mart\'inez \hfil EJDE-2010/112\hfilneg] {Alfredo Cano, Sergio Hern\'andez-Linares, Eric Hern\'andez-Mart\'inez} % in alphabetical order \address{Alfredo Cano Rodr\'iguez \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\'{e}xico, M\'exico} \email{calfredo420@gmail.com} \address{Sergio Hern\'andez-Linares \newline Universidad Aut\'onoma Metropolitana, Cuajimalpa \\ Departamento de Matem\'aticas Aplicadas y Sistemas \\ Artificios No. 40, Col. Hidalgo \\ Del. \'Alvaro Obreg\'on, C.P. 01120\\ M\'exico D.F., M\'exico} \email{slinares@correo.cua.uam.mx} \address{Eric Hern\'andez-Mart\'inez \newline Universidad Aut\'onoma de la Ciudad de M\'exico \\ Colegio de Ciencia y Tecnolog\'ia. \newline Academia de Matem\'aticas, Calle Prolongaci\'{o}n San Isidro No. 151, Col. San Lorenzo Tezonco\\ Del. Iztapalapa, C.P. 09790 \\ M\'exico D.F., M\'{e}xico} \email{ebric2001@hotmail.com} \thanks{Submitted November 15, 2009. Published August 16, 2010.} \thanks{This work was presented in the Poster Sessions at the III CLAM Congreso Latino \hfill\newline\indent Americano de Matem\'aticos, 2009, Santiago, Chile} \subjclass[2000]{35J20, 35J25, 49J52, 58E35,74G35} \keywords{Critical points; critical Sobolev exponent; multiplicity of solutions; \hfill\newline\indent invariant under the action of a orthogonal group; Palais-Smale condition; \hfill\newline\indent singular semilinear elliptic problem; relative category} \begin{abstract} We consider the singular semilinear elliptic equation $-\Delta u-\frac{\mu }{| x| ^2}u-\lambda u=f(x)| u| ^{2^{\ast }-1}$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain, in $\mathbb{R}^N$, $N\geq 4$, $2^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $f:\mathbb{R} ^N\to \mathbb{R}$ is a continuous function, $0<\lambda <\lambda _1$, where $\lambda _1$ is the first Dirichlet eigenvalue of $-\Delta -\frac{\mu }{| x| ^2}$ in $\Omega $ and $0<\mu < \overline{\mu }:=(\frac{N-2}{2})^2$. We show that if $\Omega $ and $f$ are invariant under a subgroup of $O(N)$, the effect of the equivariant topology of $\Omega $ will give many symmetric nodal solutions, which extends previous results of Guo and Niu \cite{gn}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Much attention has been paid to the singular semilinear elliptic problem \begin{equation} \label{wp-lambda-mu-f} \begin{gathered} -\Delta u-\mu \frac{u}{| x| ^2}-\lambda u=f(x)| u| ^{2^{\ast }-2}u\quad\text{in }\Omega , \\ u=0\quad\quad \text{on }\partial \Omega , \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N$ $(N\geq 4)$ is a smooth bounded domain, $0\in \Omega $, $0\leq \mu <\overline{\mu }:=((N-2)/2)^2$, $\lambda \in (0,\lambda _1)$, where $\lambda _1$ is the first Dirichlet eigenvalue of $-\Delta-\frac{\mu}{|x|^2} $ on $\Omega $ and $2^{\ast }:=2N/(N-2)$ is the critical Sobolev exponent, and $f$ is a continuous function. We state some related work here about this problem. Brezis and Nirenberg \cite{bn} proved the existence of one positive solution for \eqref{wp-lambda-mu-f} with $\mu=0$ and $f=1$, with $\lambda \in (0,\lambda _1)$, where $\lambda _1$ is the first Dirichlet eigenvalue of $-\Delta $ on $ \Omega $ and $N\geq 4$. Rey \cite{r} and Lazzo \cite{la} established a close relationship between the number of positive solutions for \eqref{wp-lambda-mu-f} with $\mu=0$ and $f=1$ and the domain topology if $\lambda $ is positive and sufficiently small. Cerami, Solimini, and Struwe \cite{css} proved that \eqref{wp-lambda-mu-f} with $\mu=0$ and $f=1$ has one solution changing sign exactly once for $N\geq 6$ and $\lambda \in (0,\lambda _1)$. In \cite{cc} Castro and Clapp proved that there is an effect of the domain topology on the number of minimal nodal solutions changing sign just once of \eqref{wp-lambda-mu-f} with $\mu=0$ and $f=1$, with $\lambda $ positive sufficiently small. Recently Cano and Clapp \cite{CnC} proved the multiplicity of sign changing solutions for \eqref{wp-lambda-mu-f} with $\lambda=a$ and $\mu=0$, where $a$ and $f$ are continuous functions. The existence of non trivial positive solution for \eqref{wp-lambda-mu-f} with $f=1$ and $\mu \in [0,\overline{\mu }-1]$ and $\lambda \in (0,\lambda _1)$ where $\lambda _1$ is the first Dirichlet eigenvalue of $-\Delta -\frac{\mu }{| x| ^2}$ on $\Omega $, was proved by Janelli \cite{jan}. Cao and Peng \cite{caopeng} proved the existence of a pair of sign changing solutions for \eqref{wp-lambda-mu-f} with $f=1$, $N\geq 7$, $\mu \in [ 0,\overline{\mu }-4]$, $\lambda \in (0,\lambda _1)$. Han and Liu \cite{hl} proved the existence of one non trivial solution for \eqref{wp-lambda-mu-f} with $\lambda >0$, $f(x)>0$ and some additional assumptions. Chen \cite{chen} proved the existence of one positive solution for \eqref{wp-lambda-mu-f} with $\lambda \in (0,\lambda _1)$ and $f$ not necessarily positive but satisfying additional hypothesis. Guo and Niu \cite{gn} proved the existence of a symmetric nodal solution and a positive solution for $0<\lambda <\lambda _1$, where $\lambda _1$ is the first Dirichlet eigenvalue of $-\Delta -\frac{\mu }{| x| ^2}$ on $\Omega $, with $\Omega $ and $f$ invariant under a subgroup of $O(N)$. \section{Statement of results} Let $\Gamma $ be a closed subgroup of the orthogonal transformations $O(N)$. We consider the problem \begin{equation} \label{wp-lambda-mu-f-Gamma} \begin{gathered} -\Delta u-\mu \frac{u}{| x| ^2}-\lambda 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} \end{equation} where $\Omega$ is a smooth bounded domain, $\Gamma$-invariant in $\mathbb{R}^N$, $N\geq 4$, $2^{\ast}:=(2N)/(N-2)$ is the critical Sobolev exponent, $f:\mathbb{R}^N\to \mathbb{R}$ is a $\Gamma$-invariant continuous function, $0<\lambda <\lambda _1$, where $\lambda _1$ is the first Dirichlet eigenvalue of $-\Delta-\frac{\mu }{|x| ^2}$ on $\Omega $ and $0<\mu <\overline{\mu }:=((N-2)/2)^2$. Note that a subset $X$ of $\mathbb{R}^N$ is $\Gamma $-invariant if $\gamma x\in X$ for all $x\in X$ and $\gamma \in \Gamma $. A function $h:X\to \mathbb{R}$ is $\Gamma $-invariant if $h(\gamma x)=h(x)$ for all $x\in X$ and $\gamma \in \Gamma $. 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. Let $X/\Gamma :=\{ \Gamma x:x\in X\} $ denote 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} \label{wp-0-0-1-infty} \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} \end{equation} are the instantons \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 $C(N)=(N(N-2))^{(N-2)/2}$ (see \cite{a}, \cite{t}). If the domain is not $\mathbb{R} ^N$, there is no minimal energy solutions. These solutions minimize \[ S_0:=\min_{u\in D^{1,2}(\mathbb{R}^N)\backslash \{0\}} \frac{\int_{\mathbb{R}^N}| \nabla u|^2dx} {\big(\int_{\mathbb{R}^N}| u| ^{2^{\ast }}dx\big)^{2/2^{\ast}}}, \] where $D^{1,2}(\mathbb{R}^N)$ is the completion of $C_{c}^{\infty }(\mathbb{R}^N)$ with respect to the norm \[ \|u\|^2:=\int_{\mathbb{R}^N}| \nabla u| ^2dx. \] Also, for $0<\mu <\overline{\mu }$ it is well known that the positive solutions to \begin{equation} \label{wp-0-mu-1-infty} \begin{gathered} -\Delta u-\mu \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} \end{equation} are \[ U_{\mu }(x):=C_{\mu }(N)\Big(\frac{\varepsilon }{\varepsilon ^2| x| ^{(\sqrt{\overline{\mu }}-\sqrt{\overline{ \mu }-\mu })/\sqrt{\overline{\mu }}}+| x| ^{( \sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu })/\sqrt{\overline{ \mu }}}}\Big)^{(N-2)/2}, \] where $\varepsilon >0$ and $C_{\mu }(N)=(\frac{4N(\overline{\mu }-\mu )}{N-2})^{(N-2)/4}$ (see \cite{te}). These solutions minimize \[ S_{\mu }:=\min_{u\in D^{1,2}(\mathbb{R}^N)\backslash \{0\}} \frac{ \int_{\mathbb{R}^N}\big(| \nabla u| ^2-\mu \frac{u^2}{| x| ^2}\big)dx} {\big(\int_{\mathbb{R}^N}| u| ^{2^{\ast }}dx\big)^{2/2^{\ast }}}. \] We denote \[ M:=\big\{ y\in \overline{\Omega }:\frac{\#\Gamma y}{f(y) ^{(N-2/2}}=\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{(N-2)/2}}\big\} . \] We shall assume that $f$ satisfies: \begin{itemize} \item[(F1)] $f(x)>0$ for all $x\in \overline{\Omega }$. \item[(F2)] $f$ is \emph{locally flat} at $M$, that is, there exist $r>0$, $\nu >N$ and $A>0$ such that \[ | f(x)-f(y)| \leq A| x-y| ^{\nu }\quad \text{if }y\in M\text{ and }| x-y| 0$ let \begin{equation} M_{\delta }^{-}:=\{ y\in M:\operatorname{dist}(y,\partial \Omega )\geq \delta \}, \; B_{\delta }(M) :=\{ z\in \mathbb{R}^N:\operatorname{dist}(z,M)\leq \delta \} . \label{Mdelta} \end{equation} \begin{theorem}\label{thm1} Let $N\geq 4$, $\Omega $ and $f$ be $\Gamma$-invariant, and {\rm (F1), (F2), (A1)} and $\ell_{f}^{\Gamma }\leq S_{\mu }^{N/2}$ hold. For each $\delta ,\delta '>0$ there exist $\lambda ^{\ast }\in (0,\lambda _1)$, $\mu ^{\ast }\in (0,\overline{\mu })$ such that for all $\lambda \in (0,\lambda ^{\ast })$, $\mu \in (0,\mu ^{\ast })$ the problem \eqref{wp-lambda-mu-f-Gamma} has at least \[ \operatorname{cat}_{B_{\delta }(M)/\Gamma }(M_{\delta }^{-}/\Gamma ) \] positive solutions which satisfy \[ \ell _{f}^{\Gamma }-\delta '\leq \|u\| _{\lambda ,\mu }^2<\ell _{f}^{\Gamma }. \] \end{theorem} \subsection{Multiplicity of nodal solutions} We assume that $\Gamma $ is the kernel of an epimorphism $\tau :G\to\mathbb{Z}/2:=\{ -1,1\} $, where $G$ is a closed subgroup of $O(N) $ for which, $\Omega $ is $G$-invariant and $f:\mathbb{R}^N\to \mathbb{R}$ is a $G$-invariant function. A real valued function $u$ defined in $\Omega $ will be called $\tau $-equivariant if \[ u(gx)=\tau (g)u(x)\quad \forall x\in \Omega ,\; g\in G. \] In this section we study the problem \begin{equation} \label{wp-lambda-mu-f-tau} \begin{gathered} -\Delta u-\mu \frac{u}{| x| ^2}-\lambda 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} \end{equation} Note that all $\tau $-equivariant functions $u$ are $\Gamma$-invariant; i.e., $u(gx)=u(x)$ for all $x\in \Omega $, $g\in \Gamma $. If $u$ is a $\tau $-equivariant function then $u(gx)=-u(x)$ for all $x\in \Omega $ and $g\in \tau ^{-1}(-1)$. Thus all non trivial $\tau $-equivariant solution of \eqref{wp-lambda-mu-f-Gamma} change sign. \begin{definition} \label{def2.2} \rm We call a $\Gamma $-invariant subset $X$ of $\mathbb{R}^N$ $\Gamma $-connected 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 \[ \{ x\in \Omega :u(x)>0\} \quad \text{and}\quad \{ x\in \Omega :u(x)<0\} \] are nonempty and $\Gamma $-connected. \end{definition} For each $G$-invariant subset $X$ of $\mathbb{R}^N$, we define \[ X^{\tau }:=\{ x\in X:Gx=\Gamma x\} . \] Let $\delta >0$, define \[ M_{\tau ,\delta }^{-}:=\{ y\in M:\operatorname{dist} (y,\partial \Omega \cap \Omega ^{\tau })\geq \delta \} , \] and $B_{\delta }(M)$ as in \eqref{Mdelta}. The next theorem is a multiplicity result for $\tau$-equivariant $(\Gamma,2)$-nodal solutions for the problem \eqref{wp-lambda-mu-f-Gamma}. \begin{theorem} \label{thm2} Let $N\geq 4$, and {\rm (F1), (F2), (A1)} and $\ell _{f}^{\Gamma }\leq S_{\mu }^{N/2}$ 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 $f$ are $G$-invariant. Given $\delta ,\delta '>0$ there exists $\lambda ^{\ast }\in (0,\lambda _1)$, $\mu ^{\ast }\in (0,\overline{\mu })$ such that for all $\lambda \in (0,\lambda ^{\ast })$, $\mu \in (0,\mu ^{\ast })$ the problem \eqref{wp-lambda-mu-f-Gamma} has at least \[ \operatorname{cat}_{(B_{\delta }(M)\backslash B_{\delta }( M)^{\tau })/G}(M_{\tau ,\delta }^{-}/G) \] pairs $\pm u$ of $\tau $-equivariants $(\Gamma ,2)$-nodal solutions which satisfy \[ 2\ell _{f}^{\Gamma }-\delta '\leq \|u\| _{\lambda ,\mu }^2<2\ell _{f}^{\Gamma }. \] \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{thm3} Let $N\geq 4$ and assume that $f$ satisfies {\rm (F1), (F2), (A1)} and $\ell _{f}^{\Gamma }\leq S_{\mu }^{N/2}$. Let $\widetilde{\Gamma }$ be a closed subgroup of $O(N)$ containing $\Gamma$, for which $\Omega$ and $f$ are $\widetilde{\Gamma }$-invariant and \[ \min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{\frac{N-2 }{2}}}<\min_{x\in \overline{\Omega }}\frac{\#\widetilde{\Gamma }x}{f(x)^{(N-2)/2}}. \] Given $\delta ,\delta '>0$ there exist $\lambda^{\ast }\in(0,\lambda _1)$, $\mu ^{\ast }\in (0,\overline{\mu })$ such that for all $\lambda \in (0,\lambda ^{\ast})$, $\mu \in (0,\mu ^{\ast })$ the problem \eqref{wp-lambda-mu-f-Gamma} has at least \[ \operatorname{cat}_{B_{\delta }(M)/\Gamma }(M_{\delta }^{-}/\Gamma ) \] positive solutions which are not $\widetilde{\Gamma }$-invariant and satisfy \[ 2\ell _{f}^{\Gamma }-\delta '\leq \|u\| _{\lambda ,\mu }^2<2\ell _{f}^{\Gamma }. \] \end{theorem} \section{The variational problem} Let $\tau :G\to \mathbb{Z}/2$ be a homomorphism defined on a closed subgroup $G$ of $O(N)$, and $\Gamma :=\ker \tau $. Consider the problem \begin{equation} \label{wp-lambda-mu-f-taub} \begin{gathered} -\Delta u-\mu \frac{u}{| x| ^2}-\lambda 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} \end{equation} where $\Omega $ is a $G$-invariant bounded smooth subset of $\mathbb{R}^N$, and $f:\mathbb{R}^N\to \mathbb{R}$ is a $G$-invariant continuous function which satisfies (F1). If $\tau \equiv 1$ then the problems \eqref{wp-lambda-mu-f-tau} and \eqref{wp-lambda-mu-f-Gamma} coincide. If $\tau $ is an epimorphism then a solution of \eqref{wp-lambda-mu-f-tau} is a solution of \eqref{wp-lambda-mu-f-Gamma} with the additional property $u(gx)=-u(x)$ for all $x\in \Omega $ and $g\in \tau ^{-1}(-1)$. So every non trivial solution of \eqref{wp-lambda-mu-f-tau} is a sign changing solution for \eqref{wp-lambda-mu-f-Gamma}. The homomorphism $\tau $ induces the action of $G$ on $H_0^1(\Omega )$ given by \[ (gu)(x):=\tau (g)u(g^{-1}x). \] The fixed point space of the action is given by \begin{align*} H_0^1(\Omega )^{\tau } &:= \{ u\in H_0^1(\Omega ):gu=u\quad \forall g\in G\} \\ &=\{ u\in H_0^1(\Omega ):u(gx) =\tau (g)u(x)\quad \forall g\in G,\quad \forall x\in \Omega \} , \end{align*} is the space of $\tau $-equivariant functions. The fixed point space of the restriction of this action to $\Gamma $ \[ H_0^1(\Omega )^{\Gamma }=\{ u\in H_0^1(\Omega ):u(gx)=\tau (g)u(x), \forall g\in \Gamma ,\; \forall x\in \Omega \} \] are the $\Gamma $-invariant functions of $H_0^1(\Omega)$. The norms $\|\cdot \|_{\lambda ,\mu}$, $\|\cdot \|$ 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_{\lambda ,\mu ,f}(u) &:=\frac{1}{2}\int_{\Omega }(| \nabla u| ^2-\mu \frac{u^2}{| x| ^2} -\lambda | u| ^2)dx-\frac{1}{2^{\ast }} \int_{\Omega }f(x)| u| ^{2^{\ast }}dx \\ &= \frac{1}{2}\|u\|_{\lambda ,\mu }^2-\frac{1}{2^{\ast } }| u| _{f,2^{\ast }}^{2^{\ast }} \end{align*} is $G$-invariant, with derivative \[ DE_{\lambda ,\mu ,f}(u)v=\int_{\Omega }\Big(\nabla u\cdot \nabla v-\mu \frac{uv}{| x| ^2}-\lambda uv\Big) dx-\int_{\Omega }f(x)| u| ^{2^{\ast }-2}uvdx. \] 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{wp-lambda-mu-f-tau}, and all non trivial solutions lie on the Nehari manifold \begin{align*} \mathcal{N}_{\lambda ,\mu ,f}^{\tau } &:= \{ u\in H_0^1(\Omega )^{\tau }:u\neq 0,DE_{\lambda ,\mu ,f}(u)u=0\} \\ &= \{u\in H_0^1(\Omega )^{\tau }:u\neq 0,\|u\| _{\lambda ,\mu }^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 \[ \pi _{\lambda ,\mu ,f}:H_0^1(\Omega )^{\tau }\setminus \{ 0\} \to \mathcal{N}_{\lambda ,\mu ,f}^{\tau }\quad \pi _{\lambda ,\mu ,f}(u):=\Big(\frac{\|u\|_{\lambda ,\mu }^2}{ | u| _{f,2^{\ast }}^{2^{\ast }}}\Big)^{(N-2)/4}u. \] Therefore, the nontrivial solutions of \eqref{wp-lambda-mu-f-tau} are precisely the critical points of the restriction of $E_{\lambda ,\mu ,f}$ to $\mathcal{N}_{\lambda ,\mu ,f}^{\tau }$. If $\tau \equiv 1$ we write $\mathcal{N}_{\lambda ,\mu ,f}^{\Gamma }$ and if $G$ is a trivial group $\mathcal{N}_{\lambda ,\mu ,f}$. Note that \begin{equation} E_{\lambda ,\mu ,f}(u)=\frac{1}{N}\|u\|_{\lambda ,\mu }^2=\frac{1}{N}| u| _{f,2^{\ast }}^{2^{\ast }}\quad \forall u\in \mathcal{N}_{\lambda ,\mu ,f}^{\tau }. \label{enerneh} \end{equation} and \[ E_{\lambda ,\mu ,f}(\pi _{\lambda ,\mu ,f}(u)) =\frac{1}{N} \Big(\frac{\|u\|_{\lambda ,\mu }^2}{| u| _{f,2^{\ast }}^2}\Big) ^{N/2}\quad \forall u\in H_0^1(\Omega )^{\tau }\backslash \{0\}. \] We define \begin{align*} m(\lambda ,\mu ,f) &:= \inf_{\mathcal{N}_{\lambda ,\mu ,f}}E_{\lambda ,\mu ,f}(u) =\inf_{\mathcal{N}_{\lambda ,\mu ,f}}\frac{1}{N}\| u\|_{\lambda ,\mu }^2 \\ &= \inf_{u\in H_0^1(\Omega )\setminus \{0\}}\frac{1}{N} \Big(\frac{\|u\|_{\lambda ,\mu }^2}{| u| _{f,2^{\ast }}^2}\Big)^{N/2}. \end{align*} In particular, $E_{\lambda ,\mu ,f}$ are bounded below on $\mathcal{N}_{\lambda ,\mu ,f}$. We denote by \[ m^{\Gamma }(\lambda ,\mu ,f):=\inf_{\mathcal{N}_{\lambda ,\mu ,f}^{\Gamma }}E_{\lambda ,\mu ,f},\quad m^{\tau }(\lambda ,\mu ,f):=\inf_{ \mathcal{N}_{\lambda ,\mu ,f}^{\tau }}E_{\lambda ,\mu ,f}. \] \subsection{Estimates for the infimum} \begin{proposition} \label{prop3.1} $m^{\Gamma }(\lambda ,\mu ,f)>0$. \end{proposition} \begin{proof} Assume that $m^{\Gamma }(\lambda ,\mu ,f)=0$. Then there exist a sequence $(u_{n})$ on $\mathcal{N}_{\lambda ,\mu ,f}^{\Gamma }$ such that \[ E_{\lambda ,\mu ,f}(u_{n})\to m^{\Gamma }(\lambda ,\mu ,f)=0. \] So $E_{\lambda ,\mu ,f}(u_{n})=\frac{1}{N}\| u_{n}\|_{\lambda ,\mu }^2$. Since $\|\cdot \|_{\lambda ,\mu }$ and $\|\cdot \|$ are equivalent norms of $ H_0^1(\Omega )$\ we have that $u_{n}\to 0$ strongly in $ H_0^1(\Omega )$; but $\mathcal{N}_{\lambda ,\mu ,f}^{\Gamma }$ is closed in $H_0^1(\Omega )$ then $0\in \mathcal{N}_{\lambda ,\mu ,f}^{\Gamma }$ which is a contradiction. \end{proof} \begin{proposition}\label{propinfimo} Let $0<\lambda \leq \lambda '<\lambda _1$, $0<\mu \leq \mu '<\overline{\mu }$ and $f:\mathbb{R}^N\to \mathbb{R}$ a continuous function $\Sigma $-invariant, such that $f$ satisfies (F1), and $\Sigma $ is a closed subgroup of $O(N)$. Then $\|u\|_{\lambda ',\mu '}^2\leq \|u\|_{\lambda ,\mu }^2$, \[ m(\lambda ',\mu ',f)\leq m(\lambda ,\mu ,f)\text{ \ and \ } m^{\Sigma }(\lambda ',\mu ',f)\leq m^{\Sigma }(\lambda ,\mu ,f). \] \end{proposition} \begin{proof} By definition of $\|\cdot \|_{\lambda ,\mu }$ we obtain the first inequality. Let $u\in H_0^1(\Omega )\setminus \{0\}$, then \begin{align*} m(\lambda ',\mu ',f) &\leq E_{\lambda ',\mu ',f}(\pi _{\lambda ',\mu ',f}(u)) \\ &= \frac{1}{N}\Big(\frac{\|u\|_{\lambda',\mu '}^2}{| u| _{f,2^{\ast }}^2}\Big)^{N/2} \\ &\leq \frac{1}{N}\Big(\frac{\|u\|_{\lambda ,\mu }^2}{ | u| _{f,2^{\ast }}^2}\Big)^{N/2} \\ &= E_{\lambda ,\mu ,f}(\pi _{\lambda ,\mu ,f}(u)). \end{align*} From this inequality there proof follows. \end{proof} We denote by $\lambda _1$ the first Dirichlet eigenvalue of $-\Delta -\frac{\mu }{| x| ^2}$ in $H_0^1(\Omega )$. \begin{lemma}\label{aproxenergia} For all $\lambda \in (0,\lambda _1)$, $\mu \in (0,\overline{\mu })$, $u\in H_0^1(\Omega)^{\tau }$, it follows that \[ E_{0,0,f}(\pi _{0,0,f}(u))\leq (\frac{\bar{ \mu}}{\bar{\mu}-\mu })^{\tfrac{N}{2}}\big(\frac{\lambda _1}{\lambda _1-\lambda }\big)^{\tfrac{N}{2}}E_{\lambda ,\mu ,f}(\pi _{\lambda ,\mu ,f}(u)). \] \end{lemma} \begin{proof} Since \[ E_{\lambda ,\mu ,f}(\pi _{\lambda ,\mu ,f}(u)) =\frac{1}{N}\Big(\frac{\|u\|_{\lambda ,\mu }^2}{ | u| _{f,2^{\ast }}^2}\Big)^{N/2} = \frac{1}{N}\Big(\frac{\|u\|_{\lambda ,\mu }^N}{ | u| _{f,2^{\ast }}^N}\Big), \] and by \eqref{NEquivalentes1} \[ (1-\frac{\mu }{\bar{\mu}})(1-\frac{\lambda }{\lambda _1})\|u\|^2\leq \| u\|_{\lambda ,\mu }^2, \] then \begin{gather*} (1-\frac{\mu }{\bar{\mu}})^{\tfrac{N}{2}}(1-\frac{ \lambda }{\lambda _1})^{\tfrac{N}{2}}\| u\|^N \leq \|u\|_{\lambda ,\mu }^N \\ (1-\frac{\mu }{\bar{\mu}})^{\tfrac{N}{2}}(1-\frac{ \lambda }{\lambda _1}) ^{\tfrac{N}{2}}\frac{1}{N}\frac{\|u\| ^N}{| u| _{f,2^{\ast }}^N} \leq E_{\lambda ,\mu ,f}(\pi _{\lambda ,\mu ,f}(u)) \end{gather*} so \[ E_{0,0,f}(\pi _{0,0,f}(u)) \leq \big(\frac{\bar{\mu}}{\bar{\mu}-\mu }\big)^{\tfrac{N}{2}} \big(\frac{\lambda _1}{\lambda _1-\lambda }\big)^{\tfrac{N}{2}}E_{\lambda ,\mu ,f}(\pi _{\lambda ,\mu ,f}(u)), \] which concludes the proof. \end{proof} As a immediately consequence we have the following result. \begin{corollary}\label{aproxinfimos} \[ m^{\tau }(0,0,f)\leq (\frac{\bar{\mu}}{\bar{\mu} -\mu })^{\tfrac{N}{2}}\big(\frac{\lambda _1}{\lambda _1-\lambda }\big)^{\tfrac{N}{2}}m^{\tau }(\lambda ,\mu ,f). \] \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 \begin{itemize} \item[(a)] $m^{\Gamma }(0,0,f)\leq \frac{1}{N}\ell _{f}^{\Gamma }$. \item[(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{itemize} \end{lemma} \subsection{A compactness result} \begin{definition} \label{def3.6} \rm A sequence $\{u_{n}\}\subset H_0^1(\Omega )$ satisfying \[ E_{\lambda ,\mu ,f}(u_{n})\to c\quad\text{and}\quad \nabla E_{\lambda ,\mu ,f}(u_{n})\to 0. \] is called a Palais-Smale sequence for $E_{\lambda ,\mu ,f}$ at $c$. We say that $E_{\lambda ,\mu ,f}$ satisfies the Palais-Smale condition $(PS)_{c}$ if every Palais-Smale sequence for $E_{\lambda ,\mu ,f}$ 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 $E_{\lambda ,\mu ,f}$ satisfies the $\tau$-equivariant Palais-Smale condition, $(PS)_{c}^{\tau }$. If $\tau \equiv 1$ $\{u_{n}\}$ is a $\Gamma $-invariant Palais-Smale sequence and $E_{\lambda ,\mu ,f}$ satisfies the $\Gamma $-invariant Palais-Smale condition $(PS)_{c}^{\Gamma }$. \end{definition} The next theorem, proved by Guo-Niu \cite{gn}, describes the $\tau $-equivariant Palais-Smale sequence for $E_{\lambda ,\mu ,f}$. \begin{theorem} \label{thm3.7} Let $(u_{n})$ be a Palais-Smale in $H_0^1(\Omega )^{\tau }$, for $E_{\lambda ,\mu ,f}$ at $c\geq 0$. Then there exist a solution $u$ of \eqref{wp-lambda-mu-f-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{wp-0-0-1-infty}, for $i=1,\dots,m$; and $\{R_{n}^{j}\}\subset \mathbb{R}^{+}$, a solution $\widehat{u}_{\mu}^{j}$ of \eqref{wp-0-mu-1-infty} for $j=1,\dots,l$. Such that \begin{itemize} \item[(i)] $G_{y_{n}^{i}}=G^{i}$ \item[(ii)] $(r_{n}^{i})^{-1}dist(y_{n}^{i},\partial \Omega )\to \infty$, $y_{n}^{i}\to y^{i}$, 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)] $\widehat{u}_0^{i}(gx)=\tau (g)\widehat{u}_0^{i}(x)$ $\forall z\in \mathbb{R}^N$ and $g\in G^{i}$, \item[(v)] $\widehat{u}_{\mu }^{j}(gx)=\tau (g)\widehat{u}_{\mu }^{j}(x)$ $\forall z\in \mathbb{R}^N$ 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})^{\frac{2-N}{2}}f(y^{i}) ^{\frac{2-N}{4}}\tau (g)\widehat{u }_0^{i}(g^{-1}(\frac{x-gy_{n}^{i}}{r_{n}^{i}}))\\ &\quad +\sum_{j=1}^{l}(R_{n}^{j})^{\frac{2-N}{2}}\widehat{u}_{\mu }^{i}(\frac{x}{ R_{n}^{j}})+o(1), \end{align*} \item[(vii)] $E_{\lambda ,\mu ,f}(u_{n})\to E_{\lambda ,\mu ,f}(u)+\sum_{i=1}^{m}(\frac{\#(G/G^{i})}{f(y^{i})^{\frac{N-2}{2 }}})E_{0,0,1}^{\infty }(\widehat{u}_0^{i})+\sum _{j=1}^{l}E_{0,\mu ,1}^{\infty }(\widehat{u}_{\mu }^{j})$, as $ n\to \infty $ \end{itemize} \end{theorem} \begin{corollary}\label{existsoluc} $E_{\lambda ,\mu ,f}$ satisfies $(PS)_{c}^{\tau }$ at every \[ c<\min \big\{ \#(G/\Gamma )\frac{\ell _{f}^{\Gamma }}{N}, \frac{\#(G/\Gamma )}{N}S_{\mu }^{N/2}\big\} . \] \end{corollary} \section{The bariorbit map} We will assume the 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} Corollary \ref{existsoluc} and Lemma \ref{lema9} imply \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} if (NE) is assumed. It is well known that $(NE)$ holds, if $\Gamma=\{1\}$ and $f$ is constant (see \cite[Cap. III, Teorema 1.2]{s}). Set \[ M:=\big\{ y\in \overline{\Omega }:\frac{\#\Gamma y}{f(y)^{(N-2)/2}} =\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{(N-2)/2}} \big\} . \] 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 $, $i=1,\dots,m$, one in each conjugacy class of an isotropy subgroup of $M$. Set \[ V^{i}:=\big\{z\in V:\gamma z=z\text{ \ }\forall \gamma \in \Gamma _{i} \big\} \] the fixed point space of $V\subset \mathbb{R}^N$ under the action of $\Gamma _{i}$. Set \begin{gather*} M^{i} := \{y\in M:\Gamma _{y}=\Gamma _{i}\}, \\ \Gamma M^{i} := \{\gamma y:\gamma \in \Gamma ,\text{ }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 $\Gamma M^{i}$. Set \[ f_{i}:=f(\Gamma M^{i})\in \mathbb{R}. \] 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\delta _0\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 $M_{\delta _0}^{i}:=\{z\in V^{i}:$ dist$(z,M^{i})\leq \delta _0\}$ is precisely $ \Gamma _{i}$. Define \[ W_{\varepsilon ,z}:=\sum_{[g]\in \Gamma /\Gamma _{i}}f_{i}^{\frac{2-N }{4}}U_{\varepsilon ,gz}\quad\text{if }z\in M_{\delta _0}^{i}, \] where $U_{\varepsilon ,y}:=U_0^{\varepsilon ,y}$ as in \eqref{instanton AT}. For each $\delta \in (0,\delta _0)$ define \begin{gather*} M_{\delta } := M_{\delta }^1\cup \cdots \cup M_{\delta}^{m}, \\ B_{\delta } := \{(\varepsilon ,z):\varepsilon \in (0,\delta ),\;z\in M_{\delta }\}, \\ \Theta _{\delta } := \{\pm W_{\varepsilon ,z}:(\varepsilon ,z)\in B_{\delta }\},\quad \Theta _0:=\Theta _{\delta _0}. \end{gather*} For the proof of next proposition see \cite{CnC}. \begin{proposition}\label{teo de ckr} Let $\delta \in (0,\delta _0)$, and assume that $(NE)$ 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 \[ \inf_{W\in \Theta _0}\|u-W\|<\sqrt{\frac{1}{2} Nm^{\Gamma }(0,0,f)}, \] and there exist precisely one $\nu \in \{-1,1\}$, one $\varepsilon \in (0,\delta _0)$ and one $\Gamma $-orbit $\Gamma z\in M_{\delta _0}$ such that \[ \|u-\nu W_{\varepsilon ,z}\|=\inf_{W\in \Theta _0}\|u-W\|. \] 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{teo de ckr}. 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:\text{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{teo de ckr} we can define \begin{definition} \label{defbeta} \rm The bariorbit map \[ \beta ^{\Gamma }:\mathcal{N}_{0,0,f}^{\Gamma }\cap E_{0,0,f}^{\eta }\to B_{\delta }(M)/\Gamma , \] is defined by \[ \beta ^{\Gamma }(u)=\Gamma y\overset{def}{\Longleftrightarrow } \|u\pm W_{\varepsilon ,y}\|=\min_{W\in \Theta _0}\|u-W\|. \] \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 }\Longrightarrow u^{\pm }\in \mathcal{N} _{0,0,f}^{\Gamma }\quad\text{and}\quad E_{0,0,f}(u)=2E_{0,0,f}(u^{\pm }). \label{relneh} \end{equation} \begin{lemma}\label{dobinf} If $E_{0,0,f}$ does not achieve its infimum at $\mathcal{N}_{0,0,f}^{\tau }$, then \[ m^{\tau }(0,0,f):=\inf_{\mathcal{N}_{0,0,f}^{\tau }}E_{0,0,f}=\Big(\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{ f(x)^{(N-2)/2}}\Big)\frac{2}{N}S^{N/2}=2m^{\Gamma}(0,0,f). \] \end{lemma} \begin{proof} By contradiction. 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 \[ m^{\tau }(0,0,f)\leq \Big(\min_{x\in \overline{\Omega }}\frac{ \#\Gamma x}{f(x)^{(N-2)/2}}\Big)\frac{2}{N}S^{N/2}. \] Hence \[ m^{\Gamma }(0,0,f)\leq E_{0,0,f}(u^{+})=\frac{1}{2}m^{\tau }(0,0,f)\leq \Big(\min_{x\in \overline{\Omega }}\frac{\#\Gamma x}{f(x)^{\frac{N-2 }{2}}}\Big)\frac{1}{N}S^{N/2}=m^{\Gamma }(0,0,f). \] 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 \[ m^{\tau }(0,0,f)=\Big(\min_{x\in \overline{\Omega }}\frac{\#\Gamma x }{f(x)^{(N-2)/2}}\Big)\frac{2}{N}S^{N/2}. \] \end{proof} Then property \eqref{relneh} implies \[ u^{\pm }\in \mathcal{N}_{0,0,f}^{\Gamma }\cap E_{0,0,f}^{\eta }\quad \forall u\in \mathcal{N}_{0,0,f}^{\tau }\cap E_{0,0,f}^{2\eta }, \] 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 \[ \beta ^{\tau }:\mathcal{N}_{0,0,f}^{\tau }\cap E_{0,0,f}^{2\eta }\to (B_{\delta }(M)\setminus B_{\delta }(M)^{\tau })/\Gamma , \quad\text{}\beta ^{\tau }(u):=\beta ^{\Gamma }(u^{+}), \] is well defined, continuous and $\mathbb{Z}/2$-equivariant; i.e., \[ \beta ^{\tau }(-u)=\Gamma (g_{\tau }y)\Longleftrightarrow \beta ^{\tau }(u)=\Gamma y. \] \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 $\varrho _{X}\circ \varrho _{X}=id_{X}$. Given an involution we can define 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 $\mathcal{N}_{\lambda ,\mu ,f}^{\tau }$. 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 $\Theta :X\times [ 0,1] \to Y$ 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 [ 0,1] $. A subset $A$ of $X$ is $\mathbb{Z}/2$-equivariant if $\varrho _{X}a\in A$ for every $a\in A$. \begin{definition} \label{def} \rm The $\mathbb{Z}/2$-category of a $\mathbb{Z}/2$-map $f:X\to Y$ is the smallest integer $k:=\mathbb{Z}/2$-$\operatorname{cat}(f)$ with following properties \begin{itemize} \item[(i)] There exists a cover of $X=X_1\cup \dots \cup X_{k}$ 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 $\mathbb{Z}/2$-$\operatorname{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 $$ \mathbb{Z}/2\text{-}cat_{X}(A):=\mathbb{Z}/2\medskip \text{-} \operatorname{cat}(\iota ), \quad \mathbb{Z}/2\medskip \text{-}cat_{X}(X):=\mathbb{Z}/2\text{-}\operatorname{cat}(X). $$ Note that if $\varrho _{x}=id_{X}$ then \[ \mathbb{Z}/2\text{-}cat_{X}(A):=cat_{X}(A), \quad \mathbb{Z}/2\text{-}\operatorname{cat}(X):=\operatorname{cat}(X), \] are the usual Lusternik-Schnirelmann category (see \cite[definition 5.4]{w}). \begin{theorem} \label{thm5.2} 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 $\mathbb{Z}/2$-$\operatorname{cat}(\phi ^{d})$ pairs critical points such that $\phi (u)\leq d$. \end{theorem} \subsection{Proof of Theorems} We prove Theorem \ref{thm2} only; the proof of Theorem \ref{thm1} 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{thm2}] By Corollary \ref{existsoluc}, $E_{\lambda ,\mu ,f}$ satisfies $(PS)_{\theta }^{\tau }$ for \[ \theta <\min \{ \#(G/\Gamma )\frac{\ell _{f}^{\Gamma }}{N} ,\frac{\#(G/\Gamma )}{N}S_{\mu }^{N/2}\} . \] By Lusternik-Schnirelmann theory $E_{\lambda ,\mu ,f}$ has at least $\mathbb{Z}/2$-$\operatorname{cat}(\mathcal{N} _{\lambda ,\mu ,f}^{\tau }\cap E_{\lambda ,\mu ,f}^{\theta })$ pairs $\pm u$ of critical points in $\mathcal{N}_{\lambda ,\mu ,f}^{\tau }\cap E_{\lambda ,\mu ,f}^{\theta }$. 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)$ such that \[ (\frac{\bar{\mu}}{\bar{\mu}-\mu ^{\ast }})^{N/2}(\frac{ \lambda _1}{\lambda _1-\lambda ^{\ast }})^{N/2}=\min \{ 2, \frac{N\eta }{\#(G/\Gamma )\ell _{f}^{\Gamma }},\frac{\ell _{f}^{\Gamma }}{\ell _{f}^{\Gamma }-\delta '}\} . \] By Lemma \ref{aproxenergia}, if $u\in \mathcal{N}_{\lambda ,\mu ,f}^{\tau }\cap E_{\lambda ,\mu ,f}^{\theta }$, $\mu \in ( 0,\mu ^{\ast })$, $\lambda \in (0,\lambda ^{\ast })$ we have \begin{align*} E_{0,0,f}(\pi _{0,0,f}(u))&\leq (\frac{ \bar{\mu}}{\bar{\mu}-\mu })^{\tfrac{N}{2}}\big(\frac{\lambda _1}{\lambda _1-\lambda }\big)^{\tfrac{N}{2}}E_{\lambda ,\mu ,f}( u)\\ &< \big(\frac{\bar{\mu}}{\bar{\mu}-\mu }\big) ^{\tfrac{N}{2}}\big(\frac{\lambda _1}{\lambda _1-\lambda }\big)^{\tfrac{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 \[ \beta ^{\tau }\circ \pi _{0,0,f}:\mathcal{N}_{\lambda ,\mu ,f}^{\tau }\cap E_{\lambda ,\mu ,f}^{\theta }\to (B_{\delta }(M)\setminus B_{\delta }(M)^{\tau })/\Gamma , \] is a well defined $\mathbb{Z}/2$-invariant continuous function. By the \cite[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 \[ E_{\lambda ,\mu ,f}(\pi _{\lambda ,\mu ,f}( w_{\varepsilon ,y}^{\tau }))\leq \theta <\#(G/\Gamma )\frac{ \ell _{f}^{\Gamma }}{N},\quad \forall \text{ }y\in M_{\delta }^{-}, \] where $w_{\varepsilon ,y}^{\tau }=w_{\varepsilon ,y}^{\Gamma }-w_{\varepsilon ,g_{\tau }y}^{\Gamma }$, $\tau (g_{\tau })=-1$, and \[ w_{\varepsilon ,y}^{\Gamma }(x)=\sum_{[\gamma ]\in \Gamma /\Gamma _{y}}f(y)^{(2-N)/4}U_{\varepsilon ,\gamma y}(x) \varphi _{\gamma y}(x). \] Thus the map \begin{gather*} \alpha _{\delta }^{\tau } :M_{\tau ,\delta }^{-}/\Gamma \to \mathcal{N}_{\lambda ,\mu ,f}^{\tau } \cap E_{\lambda ,\mu ,f}^{\theta }, \\ \alpha _{\delta }^{\tau }(\Gamma y):= \pi _{\lambda ,\mu ,f}(w_{\varepsilon ,y}^{\tau }), \end{gather*} 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, \[ \mathbb{Z}/2\text{-}\operatorname{cat}(\mathcal{N}_{\lambda , \mu ,f}^{\tau }\cap E_{\lambda ,\mu ,f}^{\theta }) \geq \operatorname{cat}_{(( B_{\delta }(M)\setminus B_{\delta } (M)^{\tau })/\Gamma )}(M_{\tau ,\delta }^{-}/\Gamma ). \] So \eqref{wp-lambda-mu-f-tau} has at least \[ \operatorname{cat}_{((B_{\delta }(M)\setminus B_{\delta }(M)^{\tau })/G)}(M_{\tau ,\delta }^{-}/G) \] pairs $\pm u$ solution which satisfy \[ E_{\lambda ,\mu ,f}(u)<\#(G/\Gamma ) \frac{\ell _{f}^{\Gamma }}{N}. \] By the choice of $\lambda ^{\ast }$ and $\mu ^{\ast }$ we have \[ (\frac{\bar{\mu}}{\bar{\mu}-\mu ^{\ast }})^{N/2}(\frac{ \lambda _1}{\lambda _1-\lambda ^{\ast }})^{N/2}\leq \frac{\ell _{f}^{\Gamma }}{\ell _{f}^{\Gamma }-\delta '}. \] Then \begin{align*} \#(G/\Gamma )\frac{\ell _{f}^{\Gamma }-\delta '}{N} &\leq (\frac{\bar{\mu}-\mu }{\bar{\mu}})^{N/2}(\frac{ \lambda _1-\lambda }{\lambda _1})^{N/2}\#(G/\Gamma ) \frac{\ell _{f}^{\Gamma }}{N} \\ &\leq m^{\tau }(\lambda ,\mu ,f)\leq E_{\lambda ,\mu ,f}(u)\\ &= \frac{1}{N}\|u\|_{\lambda ,\mu }^2<\#( G/\Gamma )\frac{\ell _{f}^{\Gamma }}{N} \end{align*} therefore \[ \#(G/\Gamma )\ell _{f}^{\Gamma }-\delta ^{\prime \prime }\leq \|u\|_{\lambda ,\mu }^2<\#(G/\Gamma )\ell _{f}^{\Gamma }. \] \end{proof} \begin{proof}[Proof of Theorem \ref{thm3}] By Theorem \ref{thm1} there exist $\lambda $ and $\mu $ sufficiently close to zero such that the problem \eqref{wp-lambda-mu-f-Gamma} has at least $\operatorname{cat}_{B_{\delta }(M)/\Gamma }(M_{\delta }^{-}/\Gamma )$ positive solutions such that $E_{\lambda ,\mu ,f}(u)<\frac{\ell _{f}^{\Gamma }}{N}$. We will prove that $\frac{\ell _{f}^{\Gamma }}{N}\frac{\ell _{f}^{\Gamma }}{N}$. If $m^{\widetilde{\Gamma }}(0,0,f)$ is achieved there exists $u\in \mathcal{N}_{0,0,f}^{\widetilde{\Gamma }}\subset \mathcal{N}_{0,0,f}^{\Gamma }$ and \[ \frac{\ell _{f}^{\Gamma }}{N}=m^{\Gamma }(0,0,f)