\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 05, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2005/05\hfil Multiplicity of symmetric solutions]
{Multiplicity of symmetric solutions for a nonlinear eigenvalue
  problem in $\mathbb{R}^n$}

\author[Daniela Visetti\hfil EJDE-2005/05\hfilneg]
{Daniela Visetti} 

\address{Dipartimento di Matematica Applicata
    ``U.~Dini'', Universit\`a degli studi di Pisa,
    via Bonanno Pisano 25/B, 56126 Pisa, Italy}
\email{visetti@mail.dm.unipi.it}

\date{}
\thanks{Submitted October 22, 2004. Published January 2, 2005.}
\thanks{Supported  by M.U.R.S.T., project ``Metodi
  variazionali e topologici nello studio di  \hfill\break\indent
  fenomeni non lineari''.}
\subjclass[2000]{35Q55, 45C05}
\keywords{Nonlinear Schr\"odinger equations; nonlinear eigenvalue problems}

\begin{abstract}
  In this paper, we study the nonlinear  eigenvalue field equation
  $$
  -\Delta u+V(|x|)u+\varepsilon(-\Delta_p u+W'(u))=\mu u
  $$
  where $u$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$
  with $n\geq 3$,  $\varepsilon$ is a positive parameter and
  $p>n$.   We find a multiplicity of solutions, symmetric with 
  respect to an action of the orthogonal group $O(n)$:
  For any $q\in\mathbb{Z}$ we prove the existence of finitely many
  pairs $(u,\mu)$ solutions for $\varepsilon$ sufficiently small,
  where $u$ is   symmetric and has topological charge $q$.
  The multiplicity of our solutions can be as large as desired,
  provided that the singular point   of $W$ and $\varepsilon$ are
  chosen accordingly.
\end{abstract}

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

\section{Introduction}

In this paper, we find infinitely many solutions of the nonlinear
eigenvalue field equation
\begin{equation} \label{Pe}
 -\Delta u+V(|x|)u+\varepsilon(-\Delta_p u+W'(u))=\mu u\, ,
\end{equation}
where $u$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$
with $n\geq 3$, $\varepsilon$ is a positive parameter and
$p\in\mathbb{N}$ with $p>n$.

The choice of the nonlinear operator $-\Delta_p+W'$ is very important.
The presence of the $p$-Laplacian comes from a conjecture by
Derrick (see \cite{D}). He was looking for a model for elementary
particles, which extended the features of the sine-Gordon equation in
higher dimension; he showed that equation
$$
- \Delta u + W'(u) = 0
$$
has no nontrivial stable localized solutions for any $W\in C^1$ on
$\mathbb{R}^n$ with $n\geq 2$.  He proposed then to consider a
higher power of the derivatives in the Lagrangian function and
this has been done for the first time in \cite{BFP}.   So the
$p$-Laplacian is responsible for the existence of nontrivial
solutions.  As concerns $W'$, it denotes the gradient of a
function $W$, which is singular in a point:  this fact constitutes
a sort of topological constraint and permits to characterize the
solutions of \eqref{Pe} by a topological invariant, called
topological charge (see \cite{BFP}).

The free problem
$$
- \Delta u - \varepsilon\Delta_6 u + W'(u) = 0
$$
has been studied in \cite{BFP}, while the concentration of the
solutions has been considered in \cite{BBD}.   In \cite{BMV} and
\cite{BMV2} the authors have studied problem \eqref{Pe}
respectively in a bounded domain and in $\mathbb{R}^n$.   In
\cite{BDFP} the authors have proved the existence of infinitely
many solutions of the free problem, which are symmetric with
respect to the action of the orthogonal group $O(n)$.

In this paper, we find a  multiplicity of solutions,
symmetric with respect to the action considered in \cite{BDFP}, of
problem \eqref{Pe} in $\mathbb{R}^n$:  For any
$q\in\mathbb{Z}$ we prove the existence of finitely many pairs
$(u,\mu)$ solutions of problem \eqref{Pe} for $\varepsilon$
sufficiently small, where $u$ is symmetric and has topological
charge $q$.  The multiplicity of the solutions can be as large as
one wants, provided that the singular point $\xi_\star =(\xi_0,0)$
($\xi_0\in\mathbb{R}$, $0\in\mathbb{R}^n$) of $W$ and
$\varepsilon$ are chosen accordingly.

The basic idea is to consider problem \eqref{Pe} as a perturbation
of the linear problem when $\epsilon=0$.   In terms of the associated energy
functionals, one passes from the non-symmetric functional $J_\epsilon$
(defined in (\ref{funzionale})) to the symmetric functional $J_0$.
Non-symmetric perturbations of a symmetric problem, in order to preserve
critical values, have been studied by several authors.   We recall only
\cite{BB}, which seems to be the first work on the subject, and the papers
\cite{B} and \cite{BG}.

In fact, the existence result is a result of preservation for the
functional $J_\epsilon$ of some critical values of the functional
$J_0$, constrained on the unitary sphere of
$L^2(\mathbb{R}^n,\mathbb{R}^{n+1})$.

Since the topological charge divides the domain $\Lambda$ of the
energy functional $J_\epsilon$ into connected components
$\Lambda_q$ with $q\in\mathbb{Z}$, the solutions are found in each
connected component and in two different ways:  as minima  and as
min-max critical points of the energy functional constrained on
the unitary sphere of $L^2(\mathbb{R}^n,\mathbb{R}^{n+1})$.
More precisely we can state:

\medskip

\textit{Given $q\in\mathbb{Z}$, for any $\xi_\star =(\xi_0,0)$
(with $\xi_0>0$ and $0\in\mathbb{R}^n$) and for any
$\varepsilon>0$, there exist $\mu_1(\varepsilon)$ and
$u_1(\varepsilon)$ respectively eigenvalue and eigenfunction of
the problem \eqref{Pe}, such that the topological charge of
$u_1(\varepsilon)$ is $q$.}


\textit{Moreover, given $q\in\mathbb{Z}\setminus\{ 0\}$ and
$k\in\mathbb{N}$, we consider $\xi_\star =(\xi_0,0)$ with $\xi_0$
large enough and $0\in\mathbb{R}^n$.   Let $\lambda_j$ be the
eigenvalues of the linear problem \eqref{Pe} with $\epsilon=0$.   Then for
$\varepsilon$ sufficiently small and for any $j\leq k$ with
$\lambda_{j-1}<\lambda_j$, there exist $\mu_j(\varepsilon)$ and
$u_j(\varepsilon)$ respectively eigenvalue and eigenfunction of
the problem \eqref{Pe}, such that the topological charge of
$u_j(\varepsilon)$ is $q$.}

\section{Functional setting}

\subsection*{Statement of the problem}

We consider from now on the field equation
\begin{equation}  \label{Pe1}
 -\Delta u+V(|x|)u+\varepsilon^r(-\Delta_p u+W'(u))=\mu u\, ,
\end{equation}
where $u$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$
with $n\geq 3$, $\epsilon$ is a positive parameter and $p,\,r\in\mathbb{N}$
with $p>n$ and $r>p-n$ (for technical reasons we
prefer to re-scale the parameter $\epsilon$).  The function $V$ is
real and we denote with $W'$ the gradient of a function
$W:\mathbb{R}^{n+1}\setminus\{\xi_\star \}\to\mathbb{R}$, where
$\xi_\star$ is a point of $\mathbb{R}^{n+1}$, different from the
origin, which for simplicity we choose on the first component:
\begin{equation}
\xi_\star =(\xi_0,0)\, ,
\label{xistar}
\end{equation}
%
with $\xi_0\in\mathbb{R}$, $\xi_0>0$ and $0\in\mathbb{R}^n$.

Throughout the paper, we  assume the following hypotheses on the
function $V: [0,+\infty)\to\mathbb{R}$:
\begin{itemize}
\item[(V1)] $\displaystyle\lim_{r\to +\infty}V(r)=+\infty$
\item[(V2)] $V(|x|)e^{-|x|}\in L^p(\mathbb{R}^n,\mathbb{R})$
\item[(V3)] $\mathop{\rm ess\,inf}_{r\in[0,+\infty)} V(r)>0$
\end{itemize}

The assumptions on the function
$W:\mathbb{R}^{n+1}\setminus\{\xi_\star\}\to\mathbb{R}$ are as
follows:

\begin{itemize}
\item[(W1)] $W\in C^1(\mathbb{R}^{n+1}\setminus\{ \xi_\star \},\mathbb{R})$
\item[(W2)] $W(\xi)\geq 0$ for all $\xi\in\mathbb{R}^{n+1}\setminus
  \{\xi_\star\}$ and $W(0)=0$
\item[(W3)] There exist two constants $c_1,\, c_2>0$ such that
  $$
  \xi\in\mathbb{R}^{n+1},\; 0<|\xi|<c_1 \Longrightarrow W(\xi_\star +\xi)
    \geq \frac{c_2}{|\xi|^{\frac{np}{p-n}}}
  $$
\item[(W4)] There exist two constants $c_3,\, c_4>0$ such that
  $$
  \xi\in\mathbb{R}^{n+1},\; 0\leq |\xi|<c_3 \Longrightarrow |W'(\xi)|\leq
    c_4|\xi|\, .
  $$
\item[(W5)] For all $\xi\in\mathbb{R}^{n+1}\setminus\{\xi_\star\}$,
  $\xi=(\xi^1,\tilde\xi)$ with $\xi^1\in\mathbb{R}$, $\tilde\xi\in\mathbb{R}^n$ and
  for all $g\in O(n)$, there holds
  $$
  W(\xi^1,g\tilde\xi) = W(\xi^1,\tilde\xi)\, .
  $$
\end{itemize}


\subsection*{The space $E$}
We define the following functional spaces:\\
$\Gamma(\mathbb{R}^n,\mathbb{R})$ is  the completion of
$C_0^\infty(\mathbb{R}^n,\mathbb{R})$  with respect to the norm
\begin{equation}
\| z\|^2_{\Gamma(\mathbb{R}^n,\mathbb{R})} = \int_{\mathbb{R}^n}
[ V(|x|)\, |z(x)|^2 + |\nabla z(x)|^2 ] dx
\label{Gamma1}
\end{equation}
$\Gamma(\mathbb{R}^n,\mathbb{R}^{n+1})$ is the completion of
$C_0^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})$ with respect to the norm
\begin{equation}
\| u\|^2_{\Gamma(\mathbb{R}^n,\mathbb{R}^{n+1})} =
\int_{\mathbb{R}^n} [ V(|x|)\, |u(x)|^2 + |\nabla u(x)|^2 ] dx\,.
\label{Gamma2}
\end{equation}
For $s\geq 1$, we set
\begin{equation}
\|\nabla u\|^s_{L^s} = \int_{\mathbb{R}^n} |\nabla u|^s dx =
  \sum_{i=1}^{n+1} \|\nabla u^i\|^s_{L^s(\mathbb{R}^n,\mathbb{R}^n)}
\label{grad(u)Ls3}
\end{equation}
with $u=(u^1,u^2,\dots,u^{n+1})$.

The spaces $\Gamma(\mathbb{R}^n,\mathbb{R})$ and
$\Gamma(\mathbb{R}^n,\mathbb{R}^{n+1})$ are Hilbert spaces, with
scalar products
\begin{gather}
(z_1,z_2)_{\Gamma(\mathbb{R}^n,\mathbb{R})}
   =  \int_{\mathbb{R}^n} \left[ V(|x|)\, z_1 z_2 + \nabla z_1\cdot
        \nabla z_2 \right] dx\, , \label{()Gamma1} \\
(u_1,u_2)_{\Gamma(\mathbb{R}^n,\mathbb{R}^{n+1})}
   =  \int_{\mathbb{R}^n} \left[ V(|x|)\, u_1\cdot u_2 + \nabla u_1\cdot
        \nabla u_2 \right] dx\, . \label{()Gamma2}
\end{gather}
We recall a compact embedding theorem (see for example \cite{BF}) into
$L^2$.

\begin{theorem} \label{trm-L2}
  The embedding of the space $\Gamma(\mathbb{R}^n,\mathbb{R})$ into the space $L^2(\mathbb{R}^n,
  \mathbb{R})$ is compact.
\end{theorem}

We define the Banach space $E$ as the completion of the space
$C_0^\infty (\mathbb{R}^n,\mathbb{R}^{n+1})$ with respect to the
norm
\begin{equation}
\| u\|_{E} = \| u\|_{\Gamma(\mathbb{R}^n,\mathbb{R}^{n+1})} +
\|\nabla u\|_{L^p}\, . \label{normaE}
\end{equation}
The space $E$ satisfies some useful properties which are listed in the next
proposition.   They follow from Sobolev embedding theorem and from
 \cite[Proposition 8]{BDFP}.

\begin{proposition} \label{prp-E}
    The Banach space $E$ has the following properties:
  \begin{enumerate}
  \item It is continuously embedded into $L^s(\mathbb{R}^n,\mathbb{R}^{n+1})$ for
    $2\leq s\leq +\infty$;
  \item It is continuously embedded into $W^{1,p}(\mathbb{R}^n,\mathbb{R}^{n+1})$;
  \item There exist two constants $C_0$, $C_1>0$ such that
    for every $u\in E$
    %
    \begin{gather*}
    \| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}
       \leq  C_0 \| u\|_E\, ,                                   \\
    |u(x)-u(y)|
       \leq  C_1 |x-y|^{1-\frac{n}{p}} \| u\|_{W^{1,p}(\mathbb{R}^n,
        \mathbb{R}^{n+1})}\, ;
    \end{gather*}
    %
  \item If $u\in E$ then
    $\lim_{|x|\to\infty} u(x) = 0$.
  \end{enumerate}
\end{proposition}


\subsection*{The energy functional $J_\epsilon$}

In the space $E$, by Proposition \ref{prp-E}, it is possible to consider the
open subset
%
\begin{equation}
\Lambda = \{ u\in E : \xi_\star\not\in u(\mathbb{R}^n)
\}\, . \label{Lambda}
\end{equation}
On $\Lambda$ we consider the functional
\begin{equation}
J_\epsilon(u) = \int_{\mathbb{R}^n} \big[ \frac{1}{2}|\nabla u|^2
+   \frac{1}{2}V(|x|)|u|^2 + \frac{\epsilon^r}{p}|\nabla u|^p +
   \epsilon^r W(u) \big] \, dx\, ,
\label{funzionale}
\end{equation}
which is the energy functional associated to the problem \eqref{Pe1}.

It is easy to verify the following lemma (see Lemma 2.3 of \cite{BMV2}).

\begin{lemma} \label{lmm-C^1}
  The functional $J_\epsilon$ is of class $C^1$ on the open set
  $\Lambda$ of $E$.
\end{lemma}

\subsection*{The topological charge}

On the open set $\Lambda$ a topological invariant can be defined.
Let $\Sigma$ be the sphere of center $\xi_\star$ and radius
$\xi_0$ in $\mathbb{R}^{n+1}$.  Let $P$ be the projection of
$\mathbb{R}^{n+1}\setminus\{\xi_\star\}$ onto $\Sigma$:
\begin{equation}
P(\xi) = \xi_\star+\frac{\xi-\xi_\star}{|\xi-\xi_\star|}\, .
\label{P}
\end{equation}

\begin{definition} \label{def-ch} \rm
  For any $u\in\Lambda$, $u=(u^1,\dots,u^{n+1})$ the open and bounded set
  $$
  K_u = \{ x\in\mathbb{R}^n : u^1(x)>\xi_0 \}
  $$
  is called support of $u$.   Then the topological charge of
  $u$ is the number
  $$
  \mathop{\rm ch}(u) = \deg(P\circ u,K_u,2\xi_\star)\, .
  $$
\end{definition}

To use some properties of the topological charge, we need to
recall the following result, whose proof can be found in  \cite{BFP}.

\begin{proposition}
  If a sequence $\{u_m\}\subset\Lambda$ converges to $u\in\Lambda$ uniformly
  on $A\subset\mathbb{R}^n$, then also $P\circ u_m$ converges to $P\circ u$
  uniformly on $A$.
\end{proposition}

This proposition permits to prove the continuity of the charge
with respect to the uniform convergence:

\begin{theorem} \label{trm-chcontinua}
  For every $u\in\Lambda$ there exists $r=r(u)>0$ such that, for every
  $v\in\Lambda$
  $$
  \| v-u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}\leq r    \Longrightarrow
     \mathop{\rm ch}(v)=\mathop{\rm ch}(u)\, .
  $$
\end{theorem}


\subsection*{The connected components of $\Lambda$}

The topological charge divides the open set $\Lambda$ into the
following sets, each of them associated to an integer number
$q\in\mathbb{Z}$:
%
\begin{equation}
\Lambda_q = \{ u\in\Lambda : \mathop{\rm ch}(u)=q\}\, .
\label{Lambdaq}
\end{equation}
%
By Theorem \ref{trm-chcontinua}, we can conclude that the sets
$\Lambda_q$ are open in $E$.  Moreover it is easy to see that
\[
 \Lambda = \bigcup_{q\in\mathbb{Z}} \Lambda_q\, , \quad
\Lambda_p\cap\Lambda_q = \emptyset \mbox{ if } p\neq q
\]
%
and each $\Lambda_q$ is a connected component of $\Lambda$.


\section{Symmetry and compactness properties}

\subsection*{Action of $O(n)$}

We consider the following action of the orthogonal group $O(n)$ on
the space of the continuous functions
$C(\mathbb{R}^n,\mathbb{R}^{n+1})$:
%
\begin{equation}
\begin{array}{rccc}
T: & O(n) \times C(\mathbb{R}^n,\mathbb{R}^{n+1}) & \longrightarrow & C(\mathbb{R}^n,\mathbb{R}^{n+1}) \\
   & (g,u)                          & \longmapsto     & T_gu
\end{array}
\label{azione}
\end{equation}
%
where
%
\begin{equation}
T_gu(x) = (u^1(gx),g^{-1}\tilde u(gx))\, ,
\label{azione2}
\end{equation}
%
with
%
\begin{equation}
u(x) = (u^1(x),\tilde u(x)) = (u^1(x),u^2(x),\dots,u^{n+1}(x))\, .
\label{u1tildeu}
\end{equation}
%
In particular $O(n)$ acts on the space $E$ and so one can prove the following
result.

\begin{lemma}
  The open subset $\Lambda\subset E$ and the energy functional $J_\epsilon$
  are invariant with respect to the action (\ref{azione}-\ref{u1tildeu}).
\end{lemma}


\begin{remark} \label{rmk1} \rm
  More precisely every connected component $\Lambda_q$ of $\Lambda$ is
  invariant with respect to the action (\ref{azione}-\ref{u1tildeu}) of
  the orthogonal group $O(n)$.   Moreover for any $u\in E$ and for any
  $g\in O(n)$
  $$
  \| T_gu\|_E = \| u\|_E\, .
  $$
\end{remark}

Let $F$ denote the subspace of the fixed points with respect to the action
(\ref{azione}-\ref{u1tildeu}) of $O(n)$ on $E$:
%
\begin{equation}
F = \{ u\in E : \forall g\in O(n)\; T_gu=u \}\, .
\label{F}
\end{equation}
%

\begin{remark} \label{rmk-Fchiuso} \rm
  The set $F$ is a closed subspace.
\end{remark}

The set
$$
\Lambda^F = \Lambda\cap F
$$
is a natural constraint for the energy functional $J_\epsilon$.  In fact,
if $u\in\Lambda^F$ is a critical point for $J_\epsilon
\big|_{\Lambda^F}$, it is a global critical point (see \cite{BDFP}):

\begin{lemma} \label{lem3.2}
  For every $u\in\Lambda^F$ and $v\in E$, we have
  $$
  J_\epsilon '(u)(v) = J_\epsilon '(u)(Pv)\, ,
  $$
  being $P$ the projection of $E$ onto $F$.
\end{lemma}

We denote by $\Lambda^F_q$ the subset of the invariant functions of
topological charge $q$:
$$
\Lambda^F_q = \Lambda_q\cap F\, .
$$


\subsection*{Results of compactness}

Next proposition provides a compact embedding for the subspace of
the invariant functions of $E$ into
$L^s(\mathbb{R}^n,\mathbb{R}^{n+1})$:

\begin{proposition}
\label{prp-imm.comp.}
  The space $F$ equipped with the norm $\|\cdot\|_E$ is compactly embedded
  into $L^s(\mathbb{R}^n,\mathbb{R}^{n+1})$ for every 
  $s\in[2,\frac{2n}{n-2})$.
\end{proposition}

The proof is a consequence  of \cite[Proposition 4]{BDFP} and of Theorem
\ref{trm-L2}.

We set
\begin{equation}
S = \{ u\in E: \|
u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}=1\}\, . \label{S}
\end{equation}
%

To get some critical points of the functional $J_\epsilon$ on the
$C^2$ manifold $\Lambda\cap S$ we use the following version of
Palais-Smale condition.   For $J_\epsilon\in
C^1(\Lambda,\mathbb{R})$, the norm of the derivative at $u\in S$
of the restriction $\hat J_\epsilon = \left. J_\epsilon
\right|_{\Lambda\cap S}$ is defined by
$$
\| \hat{J}_\epsilon '(u)\|_\star = \min_{t\in\mathbb{R}}
   \| J_\epsilon '(u)-tg'(u)\|_{E^*}\, ,
$$
where $g:E\to\mathbb{R}$ is the function defined by
$g(u)=\int_{\mathbb{R}^n} |u(x)|^2 dx$.

\begin{definition} \rm
  The functional $J_\epsilon$ is said to satisfy the Palais-Smale condition
  in $c\in\mathbb{R}$ on $\Lambda\cap S$ (on $\Lambda_q\cap S$, for $q\in\mathbb{Z}$) if,
  for any sequence $\{ u_i\}_{i\in\mathbb{N}}\subset\Lambda\cap S$ ($\{ u_i\}_
  {i\in\mathbb{N}}\subset\Lambda_q\cap S$) such that $J_\epsilon(u_i)\to c$ and
  $\|\hat{J}_\epsilon '(u_i)\|_\star\to 0$, there exists a subsequence
  which converges to $u\in\Lambda\cap S$ ($u\in\Lambda_q\cap S$).
\end{definition}

To obtain the Palais-Smale condition, we need a few technical lemmas
(see \cite{BMV2} and \cite{BFP}).

\begin{lemma} \label{lmm1}
  Let $\{ u_i\}_{i\in\mathbb{N}}$ be a sequence in $\Lambda_q$ (with $q\in\mathbb{Z}$) such
  that the sequence $\{ J_\epsilon(u_i)\}_{i\in\mathbb{N}}$ is bounded.   We consider
  the open bounded sets
  %
  \begin{equation}
  Z_i = \{ x\in\mathbb{R}^n : |u_i(x)|>c_3\}\, .
  \label{Zi}
  \end{equation}
  %
  Then the set $\cup_{i\in\mathbb{N}} Z_i\subset\mathbb{R}^n$ is bounded.
\end{lemma}

\begin{lemma} \label{lmm2}
  Let $\{ u_i\}_{i\in\mathbb{N}}\subset\Lambda$ be a sequence weakly converging
  to $u$ and such that $\{ J_\epsilon(u_i)\}_{i\in\mathbb{N}}\subset\mathbb{R}$ is
  bounded, then $u\in\Lambda$.
\end{lemma}

\begin{lemma} \label{lmm3}
  For any $a>0$, there exists $d>0$ such that for every $u\in\Lambda$
  $$
  J_\epsilon(u) \leq a \quad\Rightarrow\quad \inf_{x\in\mathbb{R}^n}
    |u(x)-\xi_\star| \geq d\, .
  $$
\end{lemma}

Now it is possible to prove (see \cite{BMV2}) that the functional
$J_\epsilon$ satisfies the Palais-Smale condition on $\Lambda\cap
S$ for any $c\in\mathbb{R}$ and $0<\epsilon\leq 1$.  As a
consequence the following proposition holds:

\begin{proposition} \label{prp-PSF}
  The functional $J_\epsilon$ satisfies the Palais-Smale condition on
  $\Lambda^F\cap S$ (on $\Lambda^F_q\cap S$ for $q\in\mathbb{Z}$) for any $c\in\mathbb{R}$
  and $0<\epsilon\leq 1$.

\begin{proof}
  Given a Palais-Smale sequence $\{ u_m\}_{m\in\mathbb{N}}$ for $J_\epsilon$ on
  $\Lambda^F\cap S\subset\Lambda\cap S$, it strongly converges to a
  function $u\in\Lambda\cap S$ by Proposition 2.1 of \cite{BMV2}.
  As the subspace $F$ is closed (see Remark \ref{rmk-Fchiuso}), $u\in
  \Lambda^F$.
\end{proof}
\end{proposition}


\section{Eigenvalues of the Schr\"odinger operator}

\subsection*{Existence of the eigenvalues}

We define the following subspace of invariant functions with respect to
the action of $O(n)$ (see (\ref{azione}-\ref{u1tildeu})):
%
\begin{equation}
\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1}) = \{
u\in\Gamma(\mathbb{R}^n,\mathbb{R}^{n+1}) : \forall g\in O(n)\; T_gu=u \}\, .
\label{GammaF}
\end{equation}
%

By Proposition \ref{prp-imm.comp.} the identical embedding of
$\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$ into
$L^2(\mathbb{R}^n,\allowbreak\mathbb{R}^{n+1})$ is continuous and
compact.
Then there exists a monotone increasing sequence
$\{\tilde\lambda_m\}_{m \in\mathbb{N}}$ of eigenvalues
$$
0<\tilde\lambda_1\leq\tilde\lambda_2\leq\dots\leq\tilde\lambda_m
  \stackrel{m\to\infty}{\longrightarrow} +\infty
$$
with
$$
\tilde\lambda_m = \inf_{E_m\in\mathcal{E}_m} \max_{v\in E_m,\, v\neq 0}
  \frac{\| v\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| v\|^2_{L^2(\mathbb{R}^n,
  \mathbb{R}^{n+1})}}\,,
$$
where $\mathcal{E}_m$ is the family of all $m$-dimensional
subspaces $E_m$ of $\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$. Also
there exists a  sequence
$\{\varphi_m\}_{m\in\mathbb{N}}\subset\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$
of eigenfunctions, orthonormal in
$L^2(\mathbb{R}^n,\mathbb{R}^{n+1})$, such that
$$
(\varphi_m,v)_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})} =
\tilde\lambda_m
  (\varphi_m,v)_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}\, ,\quad
  \forall v\in \Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})\, ,\quad
  \forall m\in\mathbb{N}\, .
$$


\subsection*{Regularity of the eigenfunctions}

The eigenfunctions $\varphi_m$ have
been found in the space
$\Gamma_F(\mathbb{R}^n,
\mathbb{R}^{n+1})$. Nevertheless they possess some more regularity
properties, as it can be shown using the following theorem:

\begin{theorem} \label{trm-decadimento}
  If $V(x)\to +\infty$ as $|x|\to\infty$, then for any $z\in H^1(\mathbb{R}^n,\mathbb{R})$
  such that
  $$
  -\Delta z + V(x)z = \lambda z
  $$
  the following estimate holds:
  %
  \begin{equation}
  |z(x)| \leq C_a e^{-a|x|}\, ,
  \label{exp}
  \end{equation}
  %
  where $a>0$ is arbitrary and $C_a>0$ depends on $a$.
\end{theorem}

For the proof of this theorem, see \cite[p.~169]{BS}.

\begin{proposition}
\label{prp-regolarita'}
  The eigenfunctions $\varphi_m\in\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$ of the
  Schr\"odinger operator $-\Delta +V(|x|)$ belong to the Banach space $E$.
\end{proposition}

\begin{proof}
  We prove the result for the real-valued eigenfunctions $e_m$ so that the
  statement of the proposition follows immediately.   By the regularity
  result of Agmon-Douglis-Nirenberg, if $z\in\Gamma_F(\mathbb{R}^n,\mathbb{R})$ is such
  that $-\Delta z-\lambda z=-Vz$ and if $Vz\in L^2(\mathbb{R}^n,\mathbb{R})\cap
  L^p(\mathbb{R}^n,\mathbb{R})$, then $z\in W^{2,p}(\mathbb{R}^n,\mathbb{R})$.

  So we only have to verify that $Vz\in L^2(\mathbb{R}^n,\mathbb{R})\cap L^p(\mathbb{R}^n,
  \mathbb{R})$.   By Theorem \ref{trm-decadimento} and $\big(V_2\big)$
  we get
  $$
  \int_{\mathbb{R}^n} |V(|x|)z(x)|^p dx \leq
    C \left\| V(|x|)e^{-|x|} \right\|^p_{L^p(\mathbb{R}^n,\mathbb{R})}
    < +\infty\, .
  $$
  Moreover, if $R>0$ is such that for $x\in\mathbb{R}^n\setminus B_{\mathbb{R}^n}(0,R)$
  $V(|x|)>1$, we have
  %
  \begin{align*}
  &\int_{\mathbb{R}^n} |V(|x|)z(x)|^2 dx\\
    & <  C \Big( \int_{B_{\mathbb{R}^n}(0,R)} |V(|x|)|^2 e^{-p|x|} dx 
    +  \int_{\mathbb{R}^n\setminus B_{\mathbb{R}^n}(0,R)}
          |V(|x|)|^p e^{-p|x|} dx \Big)  
    < + \infty\, .
  \end{align*}
\end{proof}


\subsection*{Useful properties}

We give here another variational characterization of the eigenvalues (see
for example \cite{Courant-Hilbert} and \cite{Micheletti}) and we introduce
the subspaces spanned by the eigenfunctions.

\begin{definition} \rm
  For $m\in\mathbb{N}$ we consider the following subspaces of
$\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$:
  %
  \begin{align}
  F_m       & =  \mathop{\rm span}[\varphi_1,\ldots,\varphi_m]\, ,\label{Fm3}\\
  F_m^\perp & =  \{ u\in\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1}): (u,\varphi_i)_
                  {L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}=0 \mbox{ for } 1\leq i\leq m\}\, .
                  \label{Fmperp3}
  \end{align}
\end{definition}

\begin{lemma}
  The following properties hold:
  %
  \begin{equation}
  \tilde\lambda_m = \min_{{u\in\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1}),\; u\neq 0 \atop
    (u,\varphi_i)_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}=0} \atop \forall i=1,\dots,m-1}
    \frac{\| u\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| u\|^2_{L^2
    (\mathbb{R}^n,\mathbb{R}^{n+1})}}
  \label{tildelambdam}
  \end{equation}
  %
  and
  %
  \begin{equation}
  (\varphi_i,\varphi_j)_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})} =
    \tilde\lambda_i\delta_{ij} \quad \forall i,j\in\mathbb{N}\, .
  \label{(fii,fij)GammaF}
  \end{equation}
  %
  Moreover,
  %
  \begin{eqnarray}
  u\in F_m\, , u \neq 0
    & \Longrightarrow & \tilde\lambda_1 \leq \frac{\| u\|^2_{\Gamma_F
                        (\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| u\|^2_{L^2(\mathbb{R}^n,
                        \mathbb{R}^{n+1})}} \leq \tilde\lambda_m\, ,
                        \label{disvar1F}                              \\
  u\in F_m^\perp\, , u \neq 0
    & \Longrightarrow & \frac{\| u\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}}
                        {\| u\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}
                        \geq \tilde\lambda_{m+1}\, . \label{disvar2F}
  \end{eqnarray}
  %
\end{lemma}



\section{Min-max values}


\subsection*{The functions $\Phi^q_\epsilon$}

We introduce here a particular class of functions in $E$, which
are invariant with respect to the action of the orthogonal group
$O(n)$. Let us consider the functions
$\varphi:\mathbb{R}^n\to\mathbb{R}^{n+1}$ defined in the following
way (see \cite{CF}):
%
\begin{equation}
\varphi(x) = \begin{cases}
    \begin{pmatrix}
    \varphi_1(|x|) \\ \varphi_2(|x|)\frac{x}{|x|} \end{pmatrix}
     &\mbox{for }x\neq 0 \\
    \begin{pmatrix}
    \varphi_1(0) \\ 0
    \end{pmatrix}   &
    \mbox{for }x=0
  \end{cases}
\label{varphi}
\end{equation}
%
where $\varphi_i:[0,+\infty)\to\mathbb{R}$ for $i=1,2$.   In fact
for any $g\in O(n)$ and $x\in\mathbb{R}^n$
$$
T_g\varphi(x) = \varphi(x)\, .
$$

By Proposition \ref{prp-regolarita'}, the set $F_m$ defined in
(\ref{Fm3}) is a subset of $E$.   Then, for any $m\in\mathbb{N}$,
let $S(m)$ denote the $m$-dimensional sphere:
%
\begin{equation}
S(m)=F_m\cap S\, , \label{S(m)3}
\end{equation}
%
where $S$ has been defined in (\ref{S}).

Fixed an integer $k\in\mathbb{N}$, we introduce the number
%
\begin{equation}
M_k=\sup_{u\in S(k)} \|
u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}\, . \label{Mk3}
\end{equation}
%
Then we choose the first coordinate $\xi_0$ of the point $\xi_\star=
(\xi_0,0)$ in such a way that
%
\begin{equation}
\xi_0>2M_k\, .
\label{xi0>2Mk3}
\end{equation}
%
We can now introduce for any $q\in\mathbb{Z}\setminus\{ 0\}$ the
 functions $\Phi^q_a:\mathbb{R}^n\to\mathbb{R}^{n+1}$ of
type (\ref{varphi}):
%
\begin{equation}
\Phi^q_a(x) = \begin{cases}
   \begin{pmatrix} \Phi^q_{a,1}(|x|) \\
   \Phi^q_{a,2}(|x|)\frac{x}{|x|} \end{pmatrix} &   \mbox{for }x\neq 0 \\
  \begin{pmatrix}
  \Phi^q_{a,1}(0) \\ 0
  \end{pmatrix} &  \mbox{for }x=0
\end{cases} \label{Phiqa}
\end{equation}
where
\begin{equation}
\begin{gathered}
\Phi^q_{a,1}(|x|) = \begin{cases}
    2\xi_0[\cos(\pi|x|)+1] & \mbox{for }R_1\leq |x|\leq R_2 \\
    0 & \mbox{for } 0\leq |x|\leq R_1 \mbox{ or } |x|\geq R_2
    \end{cases} \\
\Phi^q_{a,2}(|x|) = a|x|e^{-|x|}\sin(\pm\pi|x|)
\end{gathered}
\label{Phiqa12}
\end{equation}
%
with
\begin{itemize}
\item[(i)] $a>0$
\item[(ii)] The sign in the argument of the sine in $\Phi^q_{a,2}$ is
  equal to the sign of $q$,
\item[(iii)] $R_1$ is a constant depending on the parity of $q$:
  $$
  R_1 = R_1(q) = \begin{cases}
      0 & \mbox{if $q$ is odd,} \\
      1 & \mbox{if $q$ is even,}
    \end{cases}
  $$
\item[(iv)] $R_2$ is a positive constant depending on $q$:
  $$
  R_2 = R_2(q) = \begin{cases}
      |q|   & \mbox{if $q$ is odd,} \\
      |q|+1 & \mbox{if $q$ is even.}
    \end{cases}
  $$
\end{itemize}

Next lemma computes the topological charge of the functions just defined
(see \cite{BDFP}).

\begin{lemma} \label{lmm-PhiqainLambdaq}
  For any $q\in\mathbb{Z}\setminus\{ 0\}$, the functions $\Phi^q_a$ defined in
  (\ref{Phiqa}), (\ref{Phiqa12}), with the hypotheses $(i)$-$(iv)$, belong
  to $E$ and have topological charge
  $$
  \mathop{\rm ch}\left(\Phi^q_a\right) = q\, .
  $$
\end{lemma}

\begin{proof}
The functions $\Phi^q_a$ belong to the space $E$.  If we consider
  the components
  \begin{gather*}
  f_1(x^1,x^2,\dots,x^n)  =  \Phi^q_{a,1}(|x|)\, ,                \\
  f_i(x^1,x^2,\dots,x^n)  =  \Phi^q_{a,2}(|x|)\frac{x^i}{|x|}\, ,
  \end{gather*}
  %
  where $2\leq i\leq n+1$, we have
  %
  \begin{gather*}
  |\nabla_x f_1|^2  =     |{\Phi^q_{a,1}}'(|x|)|^2            \\
  |\nabla_x f_i|^2  \leq  C \left( |{\Phi^q_{a,2}}'(|x|)|^2 +
    \frac{|\Phi^q_{a,2}(|x|)|^2}{|x|^2} \right)
  \end{gather*}
  %
  and consequently
  %
  \begin{align*}
  \sum_{i=1}^{n+1} \|\nabla f_i\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^n)}
    & \leq  C\int_0^\infty \left( |{\Phi^q_{a,1}}'(r)|^2 +
             |{\Phi^q_{a,2}}'(r)|^2 + \frac{|\Phi^q_{a,2}(r)|^2}{r^2}
             \right)r^{n-1}\, dr                                     \\
    & <     +\infty                                                 \\
  \sum_{i=1}^{n+1} \|\nabla f_i\|^p_{L^p(\mathbb{R}^n,\mathbb{R}^n)}
    & \leq  C\int_0^\infty \left( |{\Phi^q_{a,1}}'(r)|^p +
             |{\Phi^q_{a,2}}'(r)|^p + \frac{|\Phi^q_{a,2}(r)|^p}{r^p}
             \right) r^{n-1}\, dr                                    \\
    & <     +\infty
  \end{align*}
  %
  Moreover the following inequalities hold:
  \begin{align*}
  &\int_{\mathbb{R}^n} V(|x|)\sum_{i=0}^n |f_i(x)|^2 dx\\
    & \leq  C \int_{R_1}^{R_2} V(r)(\Phi^q_{a,1}(r))^2 r^{n-1} dr
     + \int_{\mathbb{R}^n} V(|x|) (\Phi^q_{a,2}(|x|))^2 dx \\
    & \leq  C' + \| V(|x|)e^{-|x|}\|_{L^p(\mathbb{R}^n,\mathbb{R})}
      \| a^2|x|^2e^{-|x|}\|_{L^q(\mathbb{R}^n,\mathbb{R})}
     <  + \infty\, ,
  \end{align*}
  %
  where $q=\frac{p}{p-1}$.

  The functions $\Phi^q_a$ belong to the space $\Lambda$.   In fact,
  if $\Phi^q_{a,2}(|x|)=0$, then $|x|\in\mathbb{N}\cup\{ 0\}$ and hence
  $\Phi^q_{a,1}(|x|)\in\{ 0,4\xi_0\}$, so that $\Phi^q_a(\mathbb{R}^n)\not\ni
  \xi_\star$.

  The functions $\Phi^q_a$ have topological charge $q$.
  Let $P$ be the projection introduced in (\ref{P}) of $\mathbb{R}^{n+1}$ onto
  the sphere $\Sigma$ of center $\xi_\star$ and radius $\xi_0$ in
  $\mathbb{R}^{n+1}$; then
  $$
  P\circ\Phi^q_a(x) = \begin{pmatrix}
    \frac{\Phi^q_{a,1}(|x|)-\xi_0}{\sqrt{(\Phi^q_{a,1}(|x|)-\xi_0)^2+
    (\Phi^q_{a,2}(|x|))^2}} + \xi_0 \\[3pt]
   \frac{\Phi^q_{a,2}(|x|)}{\sqrt{(\Phi^q_{a,1}(|x|)-\xi_0)^2+
    (\Phi^q_{a,2}(|x|))^2}} \frac{x}{|x|}
    \end{pmatrix}
  $$
  If $K_{\Phi^q_a}$ is the support of $\Phi^q_a$, we can consider on it the
  local coordinates obtained by the stereographic projection of the sphere
  $\Sigma$ from the origin onto the plane $\Pi=\{\xi^1=2\xi_0\}$:
  $$
  \begin{array}{ccccc}
  p & : & \Sigma                        & \longrightarrow & \Pi         \\
    &   & (\xi^1,\xi^2,\dots,\xi^{n+1}) & \longmapsto     & 2\xi_0
      \left( \frac{\xi^2}{\xi^1},\frac{\xi^3}{\xi^1},\dots,
      \frac{\xi^{n+1}}{\xi^1} \right).
  \end{array}
  $$
  Then the function $\Phi^q_a$ in the new coordinates becomes
  $$
  \overline\Phi^q_a(x) = p\circ P\circ\Phi^q_a(x) =
    f^q_a(|x|)\frac{x}{|x|}\, ,
  $$
  where
  %
  \begin{equation}
  f^q_a(|x|) = \frac{\Phi^q_{a,2}(|x|)}{\Phi^q_{a,1}(|x|)-\xi_0+
    \xi_0\sqrt{(\Phi^q_{a,1}(|x|)-\xi_0)^2+(\Phi^q_{a,2}(|x|))^2}}\, .
  \label{fqa}
  \end{equation}
  %
  The topological charge is therefore
  $$
  \mathop{\rm ch}(\Phi^q_a) = \deg\big(\overline\Phi^q_a,K_{\Phi^q_a},
    0\big)\, .
  $$
Let $\delta$ be a positive parameter, $\delta<\frac{3}{4}$ and let
  $i_1,\, i_2\in\mathbb{N}\cup\{ 0\}$, with
  %
  \[
  i_1  =  R_1\, , \quad
  i_2  =  \max\{ i\in\mathbb{N}\cup\{0\} : 2i+1\leq R_2\}\, .
  \]
  %
  Then the sets
  $$
  K_i = \{ x\in\mathbb{R}^n : 2i-\delta<|x|<2i+\delta\}\, ,
  $$
  for $i\in\mathbb{N}\cup\{ 0\}$, $i_1\leq i\leq i_2$, are disjoint and their
  union satisfies the inclusion:
  $$
  \bigcup_{i=i_1}^{i_2} K_i \subset K_{\Phi^q_a}\, .
  $$
  Moreover this subset of $K_{\Phi^q_a}$ contains all the zeros of the
  function $\overline\Phi^q_a$, that is:
  $$
  \left\{ x\in K_{\Phi^q_a} : \overline\Phi^q_a=0 \right\} \subset
    \bigcup_{i=i_1}^{i_2} K_i\, .
  $$
  By the excision and the additive properties of the topological degree
  we can write
  $$
  \deg (\overline\Phi^q_a,K_{\Phi^q_a},0) =
    \sum_{i=i_1}^{i_2} \deg(\overline\Phi^q_a,K_i,0)\, .
  $$
 To conclude we want to prove that
  $$
  \deg(\overline\Phi^q_a,K_i,0) = \begin{cases}
      \mathop{\rm sign}(q)  & \mbox{for } i=0\, ,     \\
      2\mathop{\rm sign}(q) & \mbox{for } i\in\mathbb{N}\, . \\
    \end{cases}
  $$
  In fact consider the function
  $$
  v_0(x) = \frac{f^q_a(\delta)}{\delta}x\, ,
  $$
  where $f^q_a(|x|)$ is defined in (\ref{fqa}).  Since $v_0$ coincides with
  $\overline\Phi^q_a$ on the boundary of $K_0$, i.e. for any $x\in
  \partial K_0$
  $$
  \overline\Phi^q_a(x) =
    f^q_a(|x|)\frac{x}{|x|} = v_0(x)\, ,
  $$
  the degrees of the two functions coincide too, so
  $$
  \deg(\overline\Phi^q_a,K_0,0) = \deg(v_0,K_0,0) =
    \mathop{\rm sign}(q)\, .
  $$
  Finally, for $1\leq i\leq i_2$, set
  %
  \[
  K_i^+  =  \{ x\in\mathbb{R}^n : |x|<2i+\delta \}\, , \quad
  K_i^-  =  \{ x\in\mathbb{R}^n : |x|<2i-\delta \}\, ;
  \]
  %
  then the degrees satisfy
  $$
  \deg\big(\overline\Phi^q_a,K_i,0\big) =
    \deg\big(\overline\Phi^q_a,K_i^+,0\big) -
    \deg\big(\overline\Phi^q_a,K_i^-,0\big)\, .
  $$
  Analogously to the previous argument, we introduce the
  functions:
  %
  \[
  v_i^+(x)  =  \frac{f^q_a(2i+\delta)}{2i+\delta} x\, , \quad
  v_i^-(x)  =  \frac{f^q_a(2i-\delta)}{2i-\delta} x\, .
  \]
  %
  As $v_i^\pm$ coincides with $\overline\Phi^q_a$ on the boundary of
  $K_i^\pm$, we conclude that
  %
  \begin{gather*}
  \deg\big(\overline\Phi^q_a,K_i^+,0\big)
     =  \deg(v_i^+,K_i^+,0) = \mathop{\rm sign}(q)\, , \\
  \deg\big(\overline\Phi^q_a,K_i^-,0\big)
     =  \deg(v_i^-,K_i^-,0) = -\mathop{\rm sign}(q)\,.
  \end{gather*}
  %
This completes the proof.
\end{proof}

The following corollary is now immediate.

\begin{corollary}
  For all $q\in\mathbb{Z}$ the connected component $\Lambda^F_q$ is not empty.
\end{corollary}

\begin{lemma} \label{lmm-Phiq}
  Fixed $q\in\mathbb{Z}\setminus\{0\}$, there exists $\hat a_q>0$ such that for
  every $a\geq\hat a_q$ the functions $\Phi^q_a$ have the following
  properties:
  \begin{itemize}
  \item[$(i)$] The distance of $\Phi^q_a$ from the point $\xi_\star$ is
    $\xi_0$, i.e.
    $$
    d(\Phi^q_a,\xi_\star) = \inf_{x\in\mathbb{R}^n} |\Phi^q_a(x)-\xi_\star| =
       \xi_0\, .
    $$
  \item[$(ii)$] If we expand $\Phi^q_a$ of a factor $t\geq 1$, $t\Phi^q_a
    \in\Lambda^F$ and
    $$
    d(t\Phi^q_a,\xi_\star) = \inf_{x\in\mathbb{R}^n} |t\Phi^q_a(x)-\xi_\star| =
       \xi_0\, .
    $$
  \end{itemize}
\end{lemma}

\begin{proof}
 $(i)$ We prove that there exists $a$ sufficiently large such
  that
  $$
  |\Phi^q_a(x)-\xi_\star| \geq \xi_0
  $$
  for all $x\in\mathbb{R}^n$.   For $x\in\mathbb{R}^n$ with $0\leq |x|\leq R_1$ or
  $|x|\geq R_2$, it is immediate that
  $$
  |\Phi^q_a(x)-\xi_\star|^2 = a^2|x|^2e^{-2|x|}\sin^2(\pi |x|) + \xi_0^2
    \geq \xi_0^2\, .
  $$
  As for $x\in\mathbb{R}^n$ with $R_1\leq |x|\leq R_2$ there holds:
  %
  \begin{align*}
  |\Phi^q_a(x)-\xi_\star|^2
    & =  \xi_0^2[2\cos(\pi|x|)+1]^2 + a^2|x|^2e^{-2|x|}\sin^2(\pi|x|) \\
    & =  \big( 4\xi_0^2-a^2|x|^2e^{-2|x|} \big) \cos^2(\pi|x|) +
          4\xi_0^2\cos(\pi|x|)
     + \xi_0^2 + a^2|x|^2e^{-2|x|}\, .
  \end{align*}
  %
  Let $f_a:[0,+\infty)\to\mathbb{R}$ be the  function
  $$
  f_a(r) = \left( 4\xi_0^2-a^2r^2e^{-2r} \right) \cos^2(\pi r) +
    4\xi_0^2\cos(\pi r) + a^2r^2e^{-2r}\, .
  $$
  We consider the polynomial
  $$
  P(y) = P_\alpha(y) = \left( 4\xi_0^2-\alpha^2 \right) y^2 + 4\xi_0^2 y +
    \alpha^2\, ,
  $$
  where
  $  \alpha = \alpha_a(r) = are^{-r}$,
  on the interval $[-1,+1]$.

  Now, if $\alpha^2=4\xi_0^2$, the only zero of $P(y)$ is $y=-1$ and
  therefore on $[-1,1]$ $P(y)$ is nonnegative.   On the contrary, if
  $\alpha^2\neq 4\xi_0^2$, the zeros of $P(y)$ are:
  $$
  y_{1,2} = \frac{-2\xi_0^2\pm\left(\alpha^2-2\xi_0^2\right)}
    {4\xi_0^2-\alpha^2} = \begin{cases}
      -1 \\ \frac{\alpha^2}{\alpha^2-4\xi_0^2}
    \end{cases}
  $$
  For $\alpha^2>4\xi_0^2$ we have $y_1=-1$ and $y_2>1$, so $P(y)\geq 0$
  on $[-1,1]$.   For $2\xi_0^2\leq\alpha^2<4\xi_0^2$, we have $y_2\leq -1$,
  so $P(y)\geq 0$ and consequently
  $$
  a^2r^2e^{-2r}\geq 2\xi_0^2 \Longrightarrow f_a(r)\geq 0\, .
  $$
  If we consider
  %
  \begin{equation}
  a \geq \frac{\sqrt{2}\xi_0}{R_2e^{-R_2}}   \label{cond.a}
  \end{equation}
  %
  and $R_1=1$ (i.e. $q$ even), there always holds $\alpha^2\geq 2\xi_0^2$.

  If on the contrary $q$ is odd and so $R_1=0$, for $a$ as in
  (\ref{cond.a}) $(\alpha_a(r))^2<2\xi_0^2$ for $0\leq r<r_1$, where
  $r_1$ is such that
  %
  \begin{equation}
  (\alpha_a(r_1))^2 = 2\xi_0^2\, .
  \label{r1}
  \end{equation}
  %
  We choose $a$ sufficiently large to have $r_1\leq\frac{1}{2}$:
  then $\cos(\pi r)\in (0,1]$ for any $r\in[0,r_1)$ and so
  $$
  \min_{r\in[0,r_1)} f_a(r) \geq 0\, .
  $$

  \noindent $(ii)$ For any $x\in\mathbb{R}^n$ with $0\leq |x|\leq R_1$ or $|x|
  \geq R_2$, it is immediate that
  $$
  \left|t\Phi^q_a(x)-\xi_\star\right|^2 =
    t^2a^2|x|^2e^{-2|x|}\sin^2(\pi |x|) + \xi_0^2 \geq \xi_0^2\, .
  $$
  On the contrary for $x\in\mathbb{R}^n$, $R_1\leq |x|\leq R_2$, there holds:
  %
  \begin{align*}
  \left|t\Phi^q_a(x)-\xi_\star\right|^2
    & =     \xi_0^2[2t\cos(\pi |x|)+2t-1]^2 +
             t^2a^2|x|^2e^{-2|x|}\sin^2(\pi |x|)           \\
    & =     t^2 \left( 4\xi_0^2-a^2|x|^2e^{-2|x|} \right)
             \cos^2(\pi|x|) + 4t(2t-1)\xi_0^2\cos(\pi |x|) \\
    &\quad + \xi_0^2(2t-1)^2 + t^2a^2|x|^2e^{-2|x|}\, .
  \end{align*}
  %
  As before we consider $\widetilde f_a:[0,+\infty)\to\mathbb{R}$ defined by
  %
  \begin{align*}
  &\widetilde f_a(r) \\
  &=  t^2 \left( 4\xi_0^2-a^2r^2e^{-2r} \right) \cos^2(\pi r) +
          4t(2t-1)\xi_0^2\cos(\pi r) + 4t(t-1)\xi_0^2
     + t^2a^2r^2e^{-2r}\, .
  \end{align*}
  %
  The polynomial $P(y)$ becomes
  $$
  \widetilde P(y) = \widetilde P_\alpha(y) = t^2 ( 4\xi_0^2-\alpha^2
  ) y^2 + 4t(2t-1)\xi_0^2 y + 4t(t-1)\xi_0^2 + t^2\alpha^2\, .
  $$
  If $\alpha^2=4\xi_0^2$, the only zero of $\widetilde P(y)$ is
  $y=-1$ and so on $[-1,1]$ $\widetilde P(y)$ is nonnegative.   If
  $\alpha^2\neq 4\xi_0^2$, the zeros of $\widetilde P(y)$ are
  $$
  y_{1,2} = \frac{(2-4t)\xi_0^2\pm\left(2\xi_0^2-t\alpha^2\right)}
    {t(4\xi_0^2-\alpha^2)}
  =  \begin{cases}
      -1 \\ \frac{4(1-t)\xi_0^2-t\alpha^2}{t(4\xi_0^2-\alpha^2)}
    \end{cases}
  $$
  For $\alpha^2>4\xi_0^2$ we have $y_1=-1$ and $y_2>1$, then
  $\widetilde P(y)\geq 0$ in $[-1,1]$.
  For $2\xi_0^2\leq\alpha^2<4\xi_0^2$, there holds $y_2\leq -1$, so
  $\widetilde P(y)\geq 0$ and consequently
  $$
  a^2r^2e^{-2r}\geq 2\xi_0^2 \Longrightarrow \widetilde f_a(r)\geq 0\, .
  $$
  Now, with the choice of $a$ done in $(i)$ and $R_1=1$ ($q$ even),
  $\alpha^2\geq 2\xi_0^2$.

  Finally, if $R_1=0$ and $a$ is as in $(i)$, $\alpha^2<2\xi_0^2$ for
  $0\leq r<r_1\leq\frac{1}{2}$ (where $r_1$ is as in (\ref{r1})),
  $$
  \min_{r\in[0,r_1)} \widetilde f_a(r) \geq 0\,.
  $$
  This completes the proof.
\end{proof}

\begin{definition} \rm
  For any $q\in\mathbb{Z}\setminus\{ 0\}$ and for $\hat a_q$ as in Lemma
  \ref{lmm-Phiq}, we define the  function
  %
  \begin{equation}
  \Phi^q = \Phi^q_{\hat a_q}\, .
  \label{Phiq}
  \end{equation}
  %
  Evidently for $i=1,2$ we pose $(\Phi^q)_i=\Phi^q_{\hat a_q,i}\,$.

  Moreover we introduce the rescaled functions $\Phi^q_\epsilon$,
  with $q\in\mathbb{Z}\setminus\{ 0\}$ and $0<\epsilon\leq 1$:
  %
  \begin{equation}
  \Phi^q_\epsilon(x) = \Phi^q\left(\frac{x}{\epsilon}\right) .
  \end{equation}
  %
\end{definition}

\begin{remark} \label{rmk-Phiqepsilon} \rm
  \begin{enumerate}
  \item The functions $\Phi^q_\epsilon$ belong to $\Lambda^F_q$.
  \item By definition of $\Phi^q_\epsilon$ and by Lemma \ref{lmm-Phiq}
    the image of $\Phi^q_\epsilon$ does not intersect the point
    $\xi_\star$ and the distance of the image from the point is
    $\xi_0$.
  \item Even if we expand the functions $\Phi^q_\epsilon$ ($0<\epsilon
    \leq 1$) of a factor $t\geq 1$, their image is such that they do not
    meet the point $\xi_\star$ and the distance is still $\xi_0$.   Hence
    $t\Phi^q_\epsilon\in\Lambda^F_q$ for all $t\geq 1$ and $\epsilon\in
    (0,1]$.
  \end{enumerate}
\end{remark}

\begin{remark} \label{rmk-norme3} \rm
  The norms of the functions $\Phi^q_\epsilon$ satisfy the following
  equalities depending on the parameter $\epsilon$:
  %
  \begin{gather}
   \|\Phi^q_\epsilon\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}^2 =
      \epsilon^n\|\Phi^q\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}^2\, , \label{L2-3}    \\
   \|\nabla\Phi^q_\epsilon\|_{L^2}^2 =
      \epsilon^{n-2}\|\nabla\Phi^q\|_{L^2}^2\, , \label{H12-3}          \\
   \|\nabla\Phi^q_\epsilon\|_{L^p}^p =
      \frac{1}{\epsilon^{p-n}}\|\nabla\Phi^q\|_{L^p}^p\, . \label{H1p-3}
  \end{gather}
\end{remark}


The functions $\Phi^q_\epsilon$ own some fundamental properties, which are
presented in the following lemma.

\begin{lemma} \label{lmm-Phiqepsilon}
  Given $q\in\mathbb{Z}\setminus\{ 0\}$ and $k\in\mathbb{N}$, we consider
  $\xi_\star =  (\xi_0,0)$ with $\xi_0>2M_k$, where
  $$
  M_k=\sup_{u\in S(k)} \| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}\, ,
  $$
  and $0\in\mathbb{R}^n$.
  There exist $\hat\rho_q>0$ and $\overline\epsilon_q$, with $0<
  \overline\epsilon_q\leq 1$, such that for all $0<\epsilon\leq
  \overline\epsilon_q$ we have
  \begin{itemize}
  \item[(i)] $\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}
    \leq 1$ for all $u\in S(k)$,
  \item[(ii)] $\displaystyle\inf_{\epsilon\in (0,\overline\epsilon_q] \atop
    u\in S(k)} \|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}>0$,
  \item[(iii)] $\displaystyle\inf_{{x\in\mathbb{R}^n \atop \epsilon\in (0,
    \overline\epsilon_q]} \atop u\in S(k)} \left| \frac{\Phi^q_\epsilon(x) +
    \hat\rho_q u(x)}{\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,
    \mathbb{R}^{n+1})}} -\xi_\star \right| >\frac{\xi_0}{2}$,
  \item[(iv)] $\displaystyle\frac{\Phi^q_\epsilon +\hat\rho_q u}
    {\|\Phi^q_\epsilon + \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}\in\Lambda_q
    \cap S$ for all $u\in S(k)$.
  \end{itemize}
\end{lemma}

\begin{proof}
  \noindent (i) For any $\rho>0$ and $0<\epsilon\leq 1$ we have
  $$
  \|\Phi^q_\epsilon +\rho u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}\leq
    \epsilon^{\frac{n}{2}}\|\Phi^q\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}+\rho\, .
  $$
  Let $\overline\epsilon_q$ be such that
  %
  \begin{equation}
  \begin{gathered}
  \overline\epsilon_q < \left( \frac{1}{\|\Phi^q\|_{L^2(\mathbb{R}^n,
    \mathbb{R}^{n+1})}} \right)^\frac{2}{n}\, , \\
  \overline\epsilon_q \leq 1\, .
  \end{gathered}
  \label{overlineepsilonq}
  \end{equation}
  %
  Then there exists $\hat\rho_q>0$ such that $\|\Phi^q_{\overline\epsilon_q}
  \|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}+\hat\rho_q\leq 1$.

  \noindent (ii) As $\|\Phi^q_\epsilon+\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}
  \geq\hat\rho_q-\|\Phi^q_\epsilon\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}$, reducing if
  necessary $\overline\epsilon_q$, we get $\|\Phi^q_\epsilon +\hat\rho_q
  u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}>0$.


  \noindent (iii) By (ii) of Lemma \ref{lmm-Phiq} we deduce that for all
  $u\in S(k)$
  $$
  \inf_{x\in\mathbb{R}^n \atop \epsilon\in (0,\overline\epsilon_q]}
    \left| \frac{\Phi^q_\epsilon(x)}{\|\Phi^q_\epsilon +\hat\rho_q u\|_
    {L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} -\xi_\star \right| = \xi_0\, .
  $$
  To get (iii) it is sufficient to prove that, reducing if necessary
  $\overline\epsilon_q$, for all $\epsilon\leq\overline\epsilon_q$
  $$
  \sup_{u\in S(k)} \frac{\hat\rho_q\| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}}
    {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} <
    \frac{\xi_0}{2}\, .
  $$
  We observe that
  %
  \begin{align*}
  \sup_{u\in S(k)} \frac{\hat\rho_q\| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}}
             {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}
    & \leq  \frac{\hat\rho_q M_k}{\displaystyle\inf_{u\in S(k)}
             \|\Phi^q_\epsilon + \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \\
    & \leq  \frac{M_k}{1-\frac{\epsilon^\frac{n}{2}}{\hat\rho_q}
             \|\Phi^q\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}\, .
  \end{align*}
  %
  Since $M_k<\frac{\xi_0}{2}$, for $\overline\epsilon_q$ sufficiently
  small we have (iii).

  \noindent (iv) follows immediately from (iii).
\end{proof}


\subsection*{The values $c^q_{\epsilon,j}$}

Using the properties of the functions $\Phi^q_\epsilon$ seen in Lemma
\ref{lmm-Phiqepsilon}, it is possible to introduce the following subsets
of $\Lambda^F\cap S$:

\begin{definition} \label{def-M} \rm
  Fixed $k\in\mathbb{N}$, $q\in\mathbb{Z}\setminus\{ 0\}$ and $0<\epsilon\leq
  \overline\epsilon_q$, where $\overline\epsilon_q$ is defined in Lemma
  \ref{lmm-Phiqepsilon}, we set
  %
  \begin{equation}
  \mathcal{M}^q_{\epsilon,j} = \big\{ \frac{\Phi^q_\epsilon +\hat\rho_q u}
    {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}
    :     u\in S(j) \big\}
  \label{M3}
  \end{equation}
  %
  with $j\leq k$ and $\hat\rho_q$ defined in Lemma \ref{lmm-Phiqepsilon}.
  We pose by convention $\mathcal{M}^q_{\epsilon,0}=\emptyset$.
\end{definition}

\begin{remark} \label{rmk-M3} \rm
  We outline the following properties of the sets $\mathcal{M}^q_{\epsilon,j}$:
  \begin{itemize}
  \item[(i)] $\mathcal{M}^q_{\epsilon,j-1}\subset\mathcal{M}^q_{\epsilon,j}$;
  \item[(ii)] $\mathcal{M}^q_{\epsilon,j}\subset \Lambda^F_q\cap S$;
  \item[(iii)] $\mathcal{M}^q_{\epsilon,j}$ is a compact set;
  \item[(iv)] $\mathcal{M}^q_{\epsilon,j}$ is a sub-manifold of
    $\Lambda^F_q$ for $0<\epsilon\leq\overline\epsilon_q$ (see Lemma
    \ref{lmm-Phiqepsilon}).
  \end{itemize}
\end{remark}

Next definition introduces the min-max values $c^q_{\epsilon,j}$.

\begin{definition} \rm
  Fixed $k\in\mathbb{N}$, for all $q\in\mathbb{Z}\setminus\{ 0\}$, $j\leq k$ and
  $0<\epsilon\leq\overline{\epsilon}_q$ ($\overline\epsilon_q$ is defined
  in Lemma \ref{lmm-Phiqepsilon}), we define the following values:
  %
  \begin{equation}
  c^q_{\epsilon,j} = \inf_{h\in\mathcal{H}^q_{\epsilon,j}}
    \sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_\epsilon \big( h(v) \big)\, ,
  \label{cqepsilonj3}
  \end{equation}
  %
  where $\mathcal{H}^q_{\epsilon,j}$ are the following sets of continuous
  transformations:
  $$
  \mathcal{H}^q_{\epsilon,j} = \left\{ h:\Lambda^F_q\cap S\to\Lambda^F_q\cap S
    : h\,\mbox{continuous},\,
     h\big|_{\mathcal{M}^q_{\epsilon,j-1}}=
    \mathrm{id}_{\mathcal{M}^q_{\epsilon,j-1}} \right\}\, .
  $$
\end{definition}

We observe that
$\mathcal{H}^q_{\epsilon,j+1} \subset \mathcal{H}^q_{\epsilon,j}$.

\begin{lemma}
  Fixed $k\in\mathbb{N}$, for all $q\in\mathbb{Z}\setminus\{ 0\}$, $j<k$ and
  $0<\epsilon\leq\overline{\epsilon}_q$, we have
  \begin{itemize}
  \item[(i)] $c^q_{\epsilon,j}\in\mathbb{R}$,
  \item[(ii)] $c^q_{\epsilon,j}\leq c^q_{\epsilon,j+1}$.
  \end{itemize}
\end{lemma}


\section{Main results}

\subsection*{Minima}

We recall now the Deformation Lemma:

\begin{lemma}[Deformation Lemma] \label{lmm-def}
  Let $J$ be a $C^1$-functional defined on a $C^2$-Finsler manifold $E$.
  Let $c$ be a regular value for $J$.   We assume that:
  \begin{itemize}
  \item[(i)] $J$ satisfies the Palais-Smale condition in $c$ on $M$,
  \item[(ii)] there exists $k>0$ such that the sublevel $J^{c+k}$ is
    complete.
  \end{itemize}
  Then there exist $\delta>0$ and a deformation $\eta:[0,1]\times E
  \longrightarrow E$ such that:
  \begin{itemize}
  \item[(a)] $\eta (0,u)=u$ for all $u\in E$,
  \item[(b)] $\eta (t,u)=u$ for all $t\in [0,1]$ and $u$ such that
    $|J(u)-c|\geq 2\delta$,
  \item[(c)] $J(\eta (t,u))$ is non-increasing in $t$ for any $u\in E$,
  \item[(d)] $\eta (1,J^{c+\delta})\subset J^{c-\delta}$.
  \end{itemize}
\end{lemma}

To apply Lemma \ref{lmm-def} on each connected component
$\Lambda^F_q$, with $q\in\mathbb{Z}\setminus\{ 0\}$, intersected
with the unitary sphere $S$ we need the completeness of the
sub-levels of the functional $J_\epsilon$. It is simple to verify
next:

\begin{lemma} \label{lmm-compl}
  For any $q\in\mathbb{Z}$, $\epsilon\in (0,1]$ and $c\in\mathbb{R}$, the subset
  $\Lambda^F_q\cap S\cap J_\epsilon ^c$ of the Banach space $E$ is complete.
\end{lemma}

Now we  get easily the minimum values of the functional $J_\epsilon$
on each set $\Lambda^F_q\cap S$:

\begin{theorem} \label{trm-minimo}
  Given $q\in\mathbb{Z}$, for any $\xi_\star =(\xi_0,0)$ with $\xi_0>0$ and
  $0\in\mathbb{R}^n$ and for any $\epsilon>0$, there exists a minimum for the
  functional $J_\epsilon$ on the subset $\Lambda^F_q\cap S$ of $\Lambda
  \cap S$.
\end{theorem}

\begin{proof}
  For any $t\geq 1$ we have that $t\Phi^q\in\Lambda^F_q$ (see (iii) of
  Remark \ref{rmk-Phiqepsilon}) and in particular the function
  $\frac{\Phi^q}{\|\Phi^q\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}$ is in $\Lambda^F_q
  \cap S$.   This means that $\Lambda^F_q\cap S$ is not empty for all
  $q\in\mathbb{Z}$, since it is obvious that $\Lambda^F_0\cap S\neq\emptyset$.

  The claim follows by the fact that $\Lambda^F_q\cap S$ is not empty,
  the functional $J_\epsilon$ is bounded from below and satisfies the
  Palais-Smale condition on $\Lambda^F_q\cap S$ (see Proposition
  \ref{prp-PSF}).
\end{proof}

\begin{remark} \rm
  We point out that to have this result there is no need to require that
  the first coordinate $\xi_0$ of the point $\xi_\star$ is sufficiently
  large (see (\ref{xi0>2Mk3})).  In fact this assumption is necessary to
  have properties $(iii)$ and $(iv)$ of Lemma \ref{lmm-Phiqepsilon},
  while here we only have to show that $\Lambda^F_q\cap S$ is not empty
  for all $q\in\mathbb{Z}$.
\end{remark}


\subsection*{Critical values}
The next theorem is an existence and multiplicity result of solutions for the
problem $\big(P_\epsilon\big)$.

\begin{theorem} \label{trm-main}
  Given $q\in\mathbb{Z}\setminus\{ 0\}$ and $k\in\mathbb{N}$,
we consider $\xi_\star =  (\xi_0,0)$ with $\xi_0>2M_k$, where
  $$
  M_k=\sup_{u\in S(k)} \| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}\, ,
  $$
and $0\in\mathbb{R}^n$.

  Then there exists $\hat\epsilon_q\in(0,1]$ such that for any $\epsilon\in
  (0,\hat\epsilon_q]$ and for any $2\leq j\leq k$ with $\tilde\lambda_{j-1}<
  \tilde\lambda_j$, we get that $c^q_{\epsilon,j}$ is a critical value for
  the functional $J_\epsilon$ restricted to the manifold $\Lambda^F_q\cap
  S$.   Moreover $c^q_{\epsilon,j-1}<c^q_{\epsilon,j}$.
\end{theorem}

The proof of this theorem is similar to the proof of Theorem 3.1 in
\cite{BMV2}, but for the convenience of the reader we summarize
it here.

\begin{proof}
  We begin with some notation:   if $u\in F$ we define the projections
  %
  \begin{equation}
  P_{F_j}u = \sum_{i=1}^j (u,\varphi_i)_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}
              \varphi_i \,,\quad
  Q_{F_j}u = u-P_{F_j}u\,.  \label{PQ}
  \end{equation}
  %
  It is immediate that
  %
  \begin{equation}
  (Q_{F_j}u,\varphi_i)_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})} = \tilde\lambda_i
    (Q_{F_j}u,\varphi_i)_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})} = 0 \quad
    \forall i=1,\dots,j\, .
  \label{(Q,P)=0}
  \end{equation}
  %
We divide the argument into five steps.

  \noindent\textbf{Step 1}
  \emph{For any $h\in\mathcal{H}^q_{\epsilon,j}$ the
  intersection of the set $h(\mathcal{M}^q_{\epsilon,j})$ with the set
  $\{ u\in F: (u,\varphi_i)_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}=0\,
  \forall i=1,\dots,j-1\}$ is not empty: in fact there exists $v\in
  \mathcal{M}^q_{\epsilon,j}$ such that $P_{F_{j-1}}h(v)=0$.}

  This is obtained by an argument of degree theory (for the proof see
  \cite{BMV}).

  \noindent\textbf{Step 2}
\emph{We prove that
  %
  \begin{gather}
  \sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_\epsilon(v)
     \leq  \tilde\lambda_j + \sigma(\epsilon) \label{dis1} \\
  c^q_{\epsilon,j}
     \leq  \tilde\lambda_j + \sigma(\epsilon) \label{dis2}
  \end{gather}
  where $\lim_{\epsilon\rightarrow 0}\sigma(\epsilon)=0$.}
 First of all we verify that
  %
  \begin{equation}
  \sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_0(v) \leq \tilde\lambda_j
    + \sup_{u\in S(j)} \frac{\| Q_{F_j}\Phi^q_\epsilon\|^2_{\Gamma_F
    (\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| P_{F_j}\Phi^q_\epsilon +\hat\rho_q u\|^2_{L^2
    (\mathbb{R}^n,\mathbb{R}^{n+1})}+\| Q_{F_j}\Phi^q_\epsilon\|^2_{L^2(\mathbb{R}^n,
    \mathbb{R}^{n+1})}}\, .
  \label{dis3}
  \end{equation}
  %
  In fact by Definition \ref{def-M}, (\ref{PQ}) and (\ref{(Q,P)=0})
  we have:
  %
  \begin{align*}
  \sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_0(v)
    & =  \sup_{u\in S(j)} \left\|\frac{\Phi^q_\epsilon+\hat\rho_q u}
             {\|\Phi^q_\epsilon+\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}
             \right\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}                     \\
    & = \sup_{u\in S(j)} \frac{\| P_{F_j}\Phi^q_\epsilon+
             \hat\rho_q u\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})} +
             \| Q_{F_j}\Phi^q_\epsilon\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}}
             {\| P_{F_j}\Phi^q_\epsilon+\hat\rho_q u\|^2_{L^2(\mathbb{R}^n,
             \mathbb{R}^{n+1})}+\| Q_{F_j}\Phi^q_\epsilon\|^2_{L^2(\mathbb{R}^n,
             \mathbb{R}^{n+1})}}                                               \\
    & \leq \sup_{u\in S(j)} \Big( \frac{\| P_{F_j}\Phi^q_\epsilon +
             \hat\rho_q u\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| P_{F_j}
             \Phi^q_\epsilon +\hat\rho_q u\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}
                                                  \\
    & \quad + \frac{\| Q_{F_j}\Phi^q_\epsilon\|^2_{\Gamma_F
             (\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| P_{F_j}\Phi^q_\epsilon +\hat\rho_q u
             \|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}+\| Q_{F_j}\Phi^q_\epsilon
             \|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \Big)                    \\
    & \leq  \tilde\lambda_j + \sup_{u\in S(j)}\frac{\| Q_{F_j}
             \Phi^q_\epsilon\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}}
             {\| P_{F_j}\Phi^q_\epsilon +\hat\rho_q u\|^2_{L^2(\mathbb{R}^n,
             \mathbb{R}^{n+1})}+\| Q_{F_j}\Phi^q_\epsilon\|^2_{L^2(\mathbb{R}^n,
             \mathbb{R}^{n+1})}}\, .
  \end{align*}
  %
  Using the definition of $J_\epsilon$ and  (\ref{dis3}), we prove the
  following inequalities:
  %
  \begin{equation}
  \begin{aligned}
  c^q_{\epsilon,j}
    &=    \inf_{h\in\mathcal{H}^q_{\epsilon,j}}
          \sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_\epsilon (h(v))         \\
    &\leq \sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_\epsilon(v)             \\
    &\leq \sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_0(v) +
          \epsilon^r \sup_{v\in\mathcal{M}^q_{\epsilon,j}}
          \int_{\mathbb{R}^n} \left( \frac{1}{p} |\nabla v|^p+W(v) \right)\, dx\\
    &\leq \tilde\lambda_j + \sup_{u\in S(j)}
          \frac{\| Q_{F_j}\Phi^q_\epsilon\|^2_{\Gamma_F(\mathbb{R}^n,
          \mathbb{R}^{n+1})}}{\| P_{F_j}\Phi^q_\epsilon +\hat\rho_q u\|^2_{L^2
          (\mathbb{R}^n,\mathbb{R}^{n+1})}+ \| Q_{F_j}\Phi^q_\epsilon\|^2_{L^2(\mathbb{R}^n,
          \mathbb{R}^{n+1})}}                                                  \\
    &\quad  + \frac{\epsilon^r}{p} \sup_{u\in S(j)}
          \frac{\|\nabla(\Phi^q_\epsilon +\hat\rho_q u)\|^p_{L^p}}
          {\|\Phi^q_\epsilon +\hat\rho_q u\|^p_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}  \\
    &\quad  + \epsilon^r \sup_{u\in S(j)} \int_{\mathbb{R}^n} W \left(
          \frac{\Phi^q_\epsilon +\hat\rho_q u}{\|\Phi^q_\epsilon +
          \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right) dx\, .
  \end{aligned}
  \label{dis4}
  \end{equation}
  %
  At this point we note that $\lim_{\epsilon\to 0}\| Q_{F_j}
  \Phi^q_\epsilon\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}=0$, in fact
  %
  \begin{align*}
  \| Q_{F_j}\Phi^q_\epsilon\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}
    & \leq  \|\Phi^q_\epsilon\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})} \\
    & =     \int_{\mathbb{R}^n} |\nabla\Phi^q_\epsilon(x)|^2 dx +
             \int_{\mathbb{R}^n} V(|x|)|\Phi^q_\epsilon(x)|^2 dx\, ,
  \end{align*}
  %
  where the right-hand side tends to zero for $\epsilon\to 0$, because
  (\ref{H12-3}) holds and
  \begin{align*}
  \int_{\mathbb{R}^n} V(|x|)|\Phi^q_\epsilon(x)|^2 dx
    &=     \int_{\mathbb{R}^n} V(|x|) \left[ \left| (\Phi^q)_1 \left(
           \frac{|x|}{\epsilon} \right) \right|^2 + \left|
           (\Phi^q)_2 \left( \frac{|x|}{\epsilon} \right)
           \right|^2 \right] dx                                    \\
    &\leq  c\int_{\epsilon R_1}^{\epsilon R_2} V(r) \left|
           (\Phi^q)_1 \left( \frac{r}{\epsilon} \right) \right|^2
           r^{n-1} dr                                              \\
    &\quad+ \| V(|x|)e^{-|x|}\|_{L^p(\mathbb{R}^n,\mathbb{R})}
           \left\| e^{|x|} \left| (\Phi^q)_2 \left( \frac{|x|}
           {\epsilon} \right) \right|^2 \right\|_{L^q(\mathbb{R}^n,\mathbb{R})}  \\
    &\leq  \Big( c \max_{r\in[0,R_2]} V(r) R_2^{n-1}(R_2-R_1)
           \Big) \epsilon^n                                      \\
    &\quad + \Big( \big\| V(|x|)e^{-|x|} \big\|_{L^p(\mathbb{R}^n,
             \mathbb{R})} \| |x|^2e^{-|x|}\|_{L^q(\mathbb{R}^n,\mathbb{R})} \Big)
             \epsilon^\frac{n}{q}
  \end{align*}
where $q$ denotes the dual exponent of $p$.

  Moreover by (ii) of Lemma \ref{lmm-Phiqepsilon} we obtain
  $$
  \sup_{0<\epsilon\leq\overline\epsilon} \; \sup_{u\in S(j)}
    \frac{1}{\| P_{F_j}\Phi^q_\epsilon +\hat\rho_q u\|^2_{L^2(\mathbb{R}^n,
    \mathbb{R}^{n+1})} + \| Q_{F_j}\Phi^q_\epsilon\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}
    < +\infty\, ,
  $$
  in fact
  %
  \begin{gather*}
  \| P_{F_j}\Phi^q_\epsilon\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}
   \leq  \epsilon^n\|\Phi^q\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}\, , \\
  \| Q_{F_j}\Phi^q_\epsilon\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}
   \leq  \epsilon^n\|\Phi^q\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}\, .
  \end{gather*}
  %
  Therefore the second term of the last inequality of (\ref{dis4})
  goes to zero when $\epsilon$ goes to zero.

  Now we observe that the following inequality holds:
  $$
  \epsilon^r \|\nabla\Phi^q_\epsilon +\hat\rho_q u\|^p_{L^p} \leq
    \Big( \epsilon^\frac{r-(p-n)}{p} \|\nabla\Phi^q\|_{L^p} +
    \epsilon^\frac{r}{p}\hat\rho_q \|\nabla u\|_{L^p} \Big)^p\, .
  $$
  Then by this inequality and $(ii)$ of Lemma \ref{lmm-Phiqepsilon} (we
  recall that $r>p-n$), we have that the third term of the last
  inequality of (\ref{dis4}) tends to zero when $\epsilon$ tends to zero.

  Regarding the last term, we verify that
  $$
  \int_{\mathbb{R}^n} W \left( \frac{\Phi^q_\epsilon +\hat\rho_q u}
    {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right) dx
  $$
  is bounded uniformly with respect to $\epsilon\in (0,
  \overline{\epsilon}]$ and $u\in S(k)$.   In fact by definition of
  $\Phi^q_\epsilon$ and by the exponential decay of the eigenfunctions
  (see Theorem \ref{trm-decadimento}) there exists a ball $B_{\mathbb{R}^n}(0,R)$
  such that, if we write $u=\sum_{m=1}^j a_m\varphi_m$ with $\sum_{m=1}^j
  a_m^2=1$, for all $x\in\mathbb{R}^n\setminus B_{\mathbb{R}^n}(0,R)$ the following
  inequalities hold
  %
  \begin{align*}
  \left| \frac{\Phi^q_\epsilon(x)+\hat\rho_q u(x)}{\|\Phi^q_\epsilon +
             \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right|
    & =     \frac{\hat\rho_q |u(x)|}{\|\Phi^q_\epsilon +
             \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}                    \\
    & \leq  \frac{C\,\hat\rho_q \Big( \sum_{m=1}^j |a_m| \Big)
             e^{-|x|}}{\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,
             \mathbb{R}^{n+1})}}                                              \\
    & \leq  M e^{-|x|} < c_3
  \end{align*}
  %
  where the constant $M$ does not depend on $u\in S(j)$ nor on
  $\epsilon$ for $\epsilon$ small enough (see the point $(ii)$ of Lemma
  \ref{lmm-Phiqepsilon}).   By $\big(W_4\big)$ we get
  $$
  \left| W \left( \frac{\Phi^q_\epsilon(x) +\hat\rho_q u(x)}
    {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right)
    \right| \leq c_4 \frac{|\Phi^q_\epsilon(x) +\hat\rho_q u(x)|^2}
    {\|\Phi^q_\epsilon +\hat\rho_q u\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}
  $$
  for any $x\in\mathbb{R}^n\setminus B_{\mathbb{R}^n}(0,R)$.   Concluding we have
  %
  \begin{eqnarray*}
  \lefteqn{\left| \int_{\mathbb{R}^n} W \left( \frac{\Phi^q_\epsilon
             +\hat\rho_q u}{\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2
             (\mathbb{R}^n,\mathbb{R}^{n+1})}} \right) dx \right|}                  \\
    & \leq & c_4 + \int_{B_{\mathbb{R}^n}(0,R)} \left| W \left(
             \frac{\Phi^q_\epsilon + \hat\rho_q u}{\|\Phi^q_\epsilon +
             \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right) \right| dx
  \end{eqnarray*}
  %
  where the integral on the right hand side is bounded by $(iii)$ of Lemma
  \ref{lmm-Phiqepsilon}.   So we have the claim.


  \noindent\textbf{Step 3}
  \emph{We prove that $c^q_{\epsilon,j}\geq\tilde\lambda_j$.}
By Step 1 and by the positivity of $W$ we get
  %
  \begin{align*}
  c^q_{\epsilon,j}
    & \geq  \inf_{h\in\mathcal{H}^q_{\epsilon,j}}
             \sup_{v\in\mathcal{M}^q_{\epsilon,j}}
             \| h(v)\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}                     \\
    & \geq  \inf_{h\in\mathcal{H}^q_{\epsilon,j}}
             \sup_{{v\in\mathcal{M}^q_{\epsilon,j}} \atop
             {P_{F_{j-1}}h(v)=0}} \| h(v)\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}\\
    & \geq  \tilde\lambda_j
  \end{align*}
  %
  In fact by Step 1 for all $h\in\mathcal{H}^q_{\epsilon,j}$ we have
  that the set $h(\mathcal{M}^q_{\epsilon,j})$ intersects the set
  $\{ u\in F: (u,\varphi_i)=0\,\forall i=1,\dots,j-1\}$ and so
  from (\ref{disvar2F}) we get the claim.


  \noindent\textbf{Step 4}
    \emph{If $\tilde\lambda_{j-1}<\tilde\lambda_j$, then for
  $\epsilon$ small enough we have:
  %
  \begin{eqnarray}
  c^q_{\epsilon,j-1}
    & < & c^q_{\epsilon,j}\, , \label{dis5}           \\
  \sup_{v\in\mathcal{M}^q_{\epsilon,j-1}} J_\epsilon(v)
    & < & c^q_{\epsilon,j}\, . \label{dis6}
  \end{eqnarray}
  %
  }
  By Step 2 and 3 we obtain for $\epsilon$ small enough
  %
  \begin{gather*}
  c^q_{\epsilon,j-1}
     \leq  \tilde\lambda_{j-1} + \sigma(\epsilon) <
             \tilde\lambda_j \leq c^q_{\epsilon,j}\, , \\
  \sup_{v\in\mathcal{M}^q_{\epsilon,j-1}} J_\epsilon(v)
     \leq  \tilde\lambda_{j-1} + \sigma(\epsilon) <
             \tilde\lambda_j \leq c^q_{\epsilon,j}\, .
  \end{gather*}
  %


  \noindent\textbf{Step 5}
  \emph{If $\tilde\lambda_{j-1}<\tilde\lambda_j$, then
  $c^q_{\epsilon,j}$ is a critical value for the functional
  $J_\epsilon$ on the manifold $\Lambda_q^F\cap S$.}

  By contradiction we suppose that $c^q_{\epsilon,j}$ is a regular value
  for $J_\epsilon$ on $\Lambda_q^F\cap S$.   By Proposition \ref{prp-PSF}
  and Lemmas \ref{lmm-def} and \ref{lmm-compl}, there exist $\delta>0$
  and a deformation $\eta:[0,1]\times\Lambda_q^F\cap S\rightarrow
  \Lambda_q^F\cap S$ such that
  \begin{gather*}
    \eta (0,u)=u \quad \forall u\in \Lambda_q^F\cap S\, ,   \\
    \eta (t,u)=u \quad \forall t\in [0,1],\, \forall u\in
      J_\epsilon^{c^q_{\epsilon,j}-2\delta}\, ,              \\
    \eta (1,J_\epsilon^{c^q_{\epsilon,j}+\delta})\subset
      J_\epsilon^{c^q_{\epsilon,j}-\delta}\, .
  \end{gather*}
  By (\ref{dis6}) we can suppose
  %
  \begin{equation}
  \sup_{v\in\mathcal{M}^q_{\epsilon,j-1}} J_\epsilon(v) < c^q_{\epsilon,j}
    -2\delta\, .
  \label{dis7}
  \end{equation}
  %
  Moreover by definition of $c^q_{\epsilon,j}$ there exists a
  transformation $\hat h\in\mathcal{H}^q_{\epsilon,j}$ such that
  $\sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_\epsilon \big( \hat h(v) \big)<
  c^q_{\epsilon,j}+\delta$.   Now by the properties of the deformation
  $\eta$ and by (\ref{dis7}) we get $\eta \big( 1,\hat h( .) \big)
  \in\mathcal{H}^q_{\epsilon,j}$ and $\sup_{v\in\mathcal{M}^q_{\epsilon,j}}
  J_\epsilon \big( \eta(1,\hat h(v)) \big) <c^q_{\epsilon,j}-\delta$
  and this is a contradiction.
\end{proof}

\begin{remark}   \begin{enumerate} \rm
  \item In the assumptions of Theorem \ref{trm-main}
    $$
    \min_{u\in\Lambda^F_q\cap S} J_\epsilon(u) = c^q_{\epsilon,1}\, .
    $$
    Nevertheless the critical point corresponding to the minimum, found in
    Theorem \ref{trm-minimo}, is not attained in the framework of Theorem
    \ref{trm-main}.  In fact, to conclude that a value $c^q_{\epsilon,j}$
    is critical, $j$ must be strictly greater than one.
  \item Provided that we choose suitably $\xi_\star$ and $\epsilon$, it
    is possible to find as many solutions of $\big(P_\epsilon\big)$ as we
    want.   In fact, let us suppose that we want $K\in\mathbb{N}$ solutions, then,
    since $\tilde\lambda_j\to\infty$, there exists $k\in\mathbb{N}$, $k>K$, such
    that among the first $k$ eigenvalues $\tilde\lambda$ there are $K$
    ``jumps'' $\tilde\lambda_j<\tilde\lambda_{j+1}$, so that Theorem
    \ref{trm-main} gives $K$ critical values.
  \item For all $q\in\mathbb{Z}\setminus\{ 0\}$, $\epsilon\in (0,1]$ the critical
    values $c^q_{\epsilon,j}$ tend to the eigenvalues $\tilde\lambda_j$
    when $\epsilon$ tends to zero.
  \end{enumerate}
\end{remark}


\subsection*{Acknowledgements}
This paper arises from the Ph.D. thesis (see \cite{tesi}) of the author,
who heartily thanks her supervisors V.~Benci and A.~M.~Micheletti.


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\end{document}