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\markboth{\hfil Behaviour of symmetric solutions \hfil EJDE--2000/69}
{EJDE--2000/69\hfil Teresa D'Aprile \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~69, pp.~1--40. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
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Behaviour of symmetric solutions of a nonlinear
elliptic field equation in the semi-classical limit:
Concentration around a circle
%
\thanks{ {\em Mathematics Subject Classifications:} 35J20 35J60.
\hfil\break\indent
{\em Key words:} nonlinear Schr\"odinger equations, topological charge,
existence, concentration.
\hfil\break\indent
\copyright 2000 Southwest Texas State University. \hfil\break\indent
Submitted May 15, 2000. Published November 16, 2000. \hfil\break\indent
Supported by M.U.R.S.T., ``Equazioni Differenziali e Calcolo delle
Variazioni''.} }
\date{}
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\author{Teresa D'Aprile}
\maketitle
\begin{abstract}
In this paper we study the existence of concentrated solutions of
the nonlinear field equation
$$ -h^{2}\Delta v+V(x)v-h^{p}\Delta_{p}v+ W'(v)=0\,,
$$
where $v:{\mathbb R}^{N}\to{\mathbb R}^{N+1}$, $N\geq 3$, $p>N$, the
potential $V$ is positive and radial, and $W$ is an appropriate singular
function satisfying a suitable symmetric property.
Provided that $h$ is sufficiently small, we are able to find solutions
with a certain spherical symmetry which exhibit a
concentration behaviour near a circle centered at zero as
$h\to 0^{+}$. Such solutions are obtained as critical points
for the associated energy functional; the proofs of the results are
variational and the arguments rely on topological tools.
Furthermore a penalization-type method is developed for the identification
of the desired solutions.
\end{abstract}
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\section{Introduction }
This paper carries on the study started in
\cite{BBD,BBD1,D}, which considers the nonlinear elliptic
equation
\begin{equation} \label{equ1}
-h^{2}\Delta v+V( x)v-h^{p}\Delta_{p}v+ W'(v)=0\,,
\end{equation}
where $h>0$, $v:{\mathbb R}^{N}\to {\mathbb R}^{N+1}$,
$N\geq 3$, $p>N$, $V:{\mathbb R}^{N}\to {\mathbb R}$,
$W:\Omega\to {\mathbb R}$ with $\Omega \subset
{\mathbb R}^{N+1}$ an open set, denoting $W'$ the gradient of $W.$
Here $\Delta v=(\Delta v_1,\ldots,\Delta
v_{N+1})$, being $\Delta $ the classical Laplacian operator, while
$\Delta _{p}v$ denotes the ($N$+1)-vector whose {\it j}-th component
is given by
\[
\mathop{\rm div}(|\nabla v|^{p-2}\nabla v_{j}).\]
By making the change of
variables $x\to h x$, (\ref{equ1}) can be rewritten as
\begin{equation} \label{equ2}
-\Delta u + V_h(x)u-\Delta_{p}u+ W'(u)=0
\end{equation}
where $V_h(x)=V(h x)$ and $u(x)=v(h x)$.
The interest in studying this equation lies in its relationship with
the particle physics and the relativistic quantum field theory.
More precisely, equations like (\ref{equ1}) or (\ref{equ2}) have been
introduced in a set of recent papers (see \cite{B}-\cite{BFP});
in such works the authors look for soliton-like solutions, i.e.
solutions whose energy is finite and which preserve their shape after
interactions; in this respect the solitons resemble as closely as
possible classical particles and their dynamics is studied in order to
provide some examples of classical models which exhibit a quantistic
behaviour. We refer to \cite{Benci-Abb.,Sol.rel.,Sol.part.}
for a more precise description of such developments.
The presence of a small diffusion parameter $h$ in equation (\ref{equ1})
leads to the problem of finding bound states (i.e. solutions with finite
energy) at least for small $h$. Once one has obtained existence results,
other natural questions arise: do these bound states exhibit some notable
behaviour in the semi-classical limit, i.e. as $h\to 0^{+}$? If so,
is it possible to locate their concentration points?
In \cite{BBD,BBD1,D} the authors have given a partial positive answer.
Let us recall the results obtained in the quoted papers.
In \cite{BBD}, under the assumption
$\liminf_{|x|\to +\infty}V(x)>\inf_{x\in{\mathbb R}^{N}}V(x)>0$,
it was proved that if $h$ is small enough (\ref{equ1}) possesses at
least a solution obtained as a minimum for the associated energy
functional; furthermore this solution concentrates around an absolute
minimum of $V$ as $h\to 0^{+}$, in the sense that its shape is a sharp
peak near that point, while it vanishes everywhere else. In \cite{BBD1}
the authors removed any global assumption on $V$ except for
$\inf_{x\in{\mathbb R}^{N}}V(x)>0$ and constructed solutions with
multiple peaks which concentrate at any prescribed finite set of local
minimum points of $V$ in the semi-classical limit. Again such solutions
are captured as minima for the energy functional and the technique is
based on the analysis of the behaviour of sequences with bounded energy,
in the spirit of the concentration-compactness principle (\cite{P.L.}).
Finally \cite{D}, under the assumption
$\lim_{|x|\to +\infty}V(x)=+\infty$, deals with the existence of critical
points, instead of minima, for small $h$; variational methods based on
variants of the Mountain Pass Theorem are used to obtain a critical
value for the energy functional characterized by a mini-max argument;
moreover the associated solutions to equation (\ref{equ1}) vanish
uniformly outside a bounded set in ${\mathbb R}^{N}$.
This paper intends to go further in the analysis begun
in \cite{BBD,BBD1,D}: in particular we deal with the existence of
solutions which satisfy some symmetry properties and which are not
necessarily with least energy; we are also interested in finding a
sufficient condition for the concentration of such ``radial'' bound
states. What will derive is the appearance of a new phenomenon, i.e.
the concentration near a circle; this provides one of the first
example of solutions concentrating around a curve and no more around
a finite set of points as for most of the literature concerning with
concentration phenomena. In what follows we recall the results contained
in some of a large number of works which has been devoted in studying
single and multiple spike solutions.
The existence of concentrated solutions and related problems has
attracted considerable attention in recent years, particularly in
relation with the study of standing waves for the nonlinear scalar
Schr\"odinger equation:
\begin{equation} \label{equ3} ih\frac{\partial\psi}{\partial
t}=-\frac{h^{2}}{2m}\Delta \psi+V(x)\psi-\gamma
|\psi|^{p-1}\psi ,
\end{equation}
where $\gamma>0$, $p>1$ and
$\psi:{\mathbb R}^{N}\to{\mathbb C}.$
Looking for standing waves of (\ref{equ3}), i.e. solutions of the form
$\psi(x,t)=\exp(-iEt/h)v(x)$, the equation for $v$ becomes
\begin{equation} \label{real4}-h^{2}\Delta v+V(x)v-|v|^{p-1}v=0
\end{equation}
where we have assumed $\gamma=2m=1$ and the parameter $E$ has been
absorbed by $V$. The first result in this line, at our knowledge,
is due to Floer and Weinstein (\cite{FW}).
These authors considered the one-dimensional case and
constructed for small $h>0$ such a concentrating family via a
Lyapunov-Schmidt reduction around any non-degenerate critical
point of the potential $V$, under the condition that $V$ is
bounded and $p=3$. In \cite{Oh1} and \cite{Oh2} Oh generalized this
result to higher
dimensions when $1
\inf_{x\in{\mathbb R}^{N}}V(x)$
in \cite{Wang} Wang established that these mountain-pass solutions
concentrate at global minimum points of $V$ as $h\to
0^{+}$; moreover a point at which a sequence of solutions
concentrates must be critical for $V$. This line of research has
been extensively pursued in a set of papers by Del Pino and Felmer
(\cite{gen2}-\cite{Del2}). We also recall the nonlinear finite
dimensional reduction used in \cite{ABC} and a recent paper by
Grossi (\cite{G}). The most complete and general results for this
kind of problems seem due to Del Pino and Felmer (\cite{Del1}) and
Li (\cite{Li}). As for radial positive bound states,
in \cite{Wang} Wang has worked on equation (\ref{real4}) in case when
$V$ is radial: he proved the existence of a family $\{v_h\}$ of
positive radial solutions with least energy among all nontrivial radial
solutions; such a family must concentrate at the origin as $h\to 0^{+}$.
In most of the above examples (but with some exceptions such as
\cite{gen2}, \cite{Del1}, \cite{Del2}), the method employed, local in
nature, seems to use in an essential way the splitting of the functional
space into a direct sum of good invariant subspaces of the linearized
operator; in such a linearization process the non-degeneracy of the
concentration points plays a basic role, even though this assumption
can be somewhat relaxed. However the finite dimensional reduction does
not seem possible in the study of equation (\ref{equ1}) because of the
presence of the p-Laplacian operator. Instead the direct use of
variational methods, relying on topological tools, permits to obtain
good results under relatively minimal assumptions and this is exactly
the direction we will follow. Indeed $W$ is chosen to be a suitable
singular function so that the presence of the term $W'(u)$ in
(\ref{equ2}) implies that the solutions have to be searched among the
maps which take value in a certain open set
$\Omega \subset{\mathbb R}^{N+1}$ (see hypotheses a)-g) below).
So the nontrivial topological properties of $\Omega$ allow us to give a
topological classification of such maps. This classification is carried
out by means of a topological invariant, the topological charge,
which is an integer number depending only on the behaviour of the
function on a bounded set (see definition 3.1).
Let us see more precisely the class of the nonlinearity $W$ and of the
potential $V$ we deal with.
First we introduce the following notation: for every
$\xi\in{\mathbb R}^{N+1}$ we write
$$\xi=(\xi_0,\xi_1),\;\;\xi_0\in{\mathbb R},\;\;\xi_1
\in{\mathbb R}^{N}.
$$
Throughout this paper we always make the following assumptions:
\begin{enumerate}
\item[\textbf{a)}] $V\in C^{1}({\mathbb R}^{N},{\mathbb R})$,
$V_0\equiv\inf_{x\in {\mathbb R}^{N}} V(x)>0$ and $V$ is a radially
symmetric function;
\item[\textbf{b)}] $W\in C^{1}(\Omega,\,{\mathbb R})$ where
$\Omega={\mathbb R}^{N+1}\setminus\{\overline{\xi}\}$ with
$\overline{\xi}=(1,0)$;
\item[\textbf{c)}] $W(\xi)\geq W(0)=0$ for all $\xi\in\Omega$;
$W $ is two times differentiable in $0$;
\item[\textbf{d)}] there exist $c$, $r>0$ such that
$$|\xi|c|\xi|^{-q} $$
where
$$\frac{1}{q}=\frac{1}{N}-\frac{1}{p},\quad N\geq 3,\quad p>N;
$$
\item[\textbf{e)}] for every $\xi=(\xi_0,\xi_1)\in\Omega$
and for every $g\in O(N)$ there results
$$W(\xi_0,g\xi_1)=W(\xi_0,\xi_1).$$
\end{enumerate}
With obvious notation by $O(N)$ we have denoted the group of the
rotation matrices which acts on ${\mathbb R}^{N}$. Hence in
assumption a) by radially symmetric function we mean that for all
$g\in O(N)$ and $x\in{\mathbb R}^{N}$ it holds: $V(gx)=V(x)$.
In most of this paper equations (\ref{equ1}) and (\ref{equ2}) will be
considered for a more restricted class of $W$'s; more precisely we will
study (\ref{equ2}) when $W$ satisfies the further assumptions:
\begin{enumerate}
\item[\textbf{f)}] $|W''(0)|<\frac{V_0}{(2(N+1))^{1/2}+2}$;
\item[\textbf{g)}] there exists $\overline{\varepsilon}\in (0,1)$
such that for every
$\xi\in\Omega$ with $|\xi|\leq\overline{\varepsilon}:$
$$W(\lambda \xi)\leq W(\xi)\quad \forall \lambda\in[0,1].$$
\end{enumerate}
We notice that no restriction on the global behaviour of $V$ and $W$ is
required other than a). In particular they are not required to be
bounded or to satisfy some assumption at infinity.
A simple function $W$ satisfying hypotheses b)-g) is the following
$$W(\xi)=\frac{|\xi|^{4}}{|\xi-\overline{\xi}|^{q}}.$$
Under the regularity assumptions on $V$ and $W$ it is standard to check
that the weak solutions of (\ref{equ2}) correspond to the critical
points for the associated energy functional:
\begin{equation} \label{funct}
E_h(u)=\frac{1}{2}\int_{{\mathbb R}^{N}}\left(|\nabla u|^{2}
+V_h(x)|u|^{2}\right)\,dx+\frac{1}{p}\int_{{\mathbb R}^{N}}|\nabla u|^{p}
\,dx+\int_{{\mathbb R}^{N}}W(u)\,dx.
\end{equation}
In order to sum up our main arguments we observe that, since we are
interested in maps $u$ which lie in some open subset of
$W^{1,2}({\mathbb R}^{N},{\mathbb R}^{N+1})\cap W^{1,p}({\mathbb R}^{N},
{\mathbb R}^{N+1}) $, it makes sense to adopt the following convention
\begin{equation} \label{nota}
u=(u_0,u_1),\quad\mbox{with}\quad u_0:{\mathbb R}^{N}\to {\mathbb R},
\quad u_1:{\mathbb R}^{N}\to {\mathbb R}^{N}.
\end{equation}
Following the idea used in \cite{BFPD}, we look for solutions $u$ of
(\ref{equ2}) which satisfy the following symmetry property:
\begin{equation} \label{prope}u_0(gx)=u_0(x),\;\;g^{-1}u_1(gx)=u_1(x)\quad
\forall g\in O(N),\;\;\forall x\in{\mathbb R}^{N}.
\end{equation}
In \cite{BFPD} the authors showed that under assumptions b)-e) equation
like (\ref{equ1}), but without the potential term $V(x)v$, admits
infinitely many solutions. Obviously equation (\ref{equ1}) deprived of
the potential term exhibits an invariance under the group of the rotations
and translations in ${\mathbb R}^{N}$, while the presence of $V$ leads
to a loss of such invariance and then the arguments in \cite{BFPD}
partially fall. However observe that if
$u\in W^{1,2}({\mathbb R}^{N}, {\mathbb R}^{N+1})\cap
W^{1,p}({\mathbb R}^{N}, {\mathbb R}^{N+1}) $ verifies (\ref{prope}),
then $|u|$ is a radially symmetric real valued function; hence, by using
the compactness properties of the radial functions, we are able to
overcome the difficulty due to the addition of $V$ and under the only
hypotheses a)-e) we prove (in section 5) the existence of infinitely many
solutions for arbitrary $h>0$; furthermore such solutions concentrate
near the origin as $h\to 0^{+}$ (see theorem 5.1).
Most of this paper is devoted to a problem which raises naturally from
the symmetry properties of equation (\ref{equ1}), i.e. that of seeking
solutions satisfying (\ref{prope}) and concentrating near a circle
centered in $0$ in the semi-classical limit. We summerize the result
achieved in section 8 as follows: assuming that there exists $r_0>0$
such that $V(x)$ is sufficiently big for $|x|= r_0$, then at least
for small $h$ we can find a solution $v_h$ to equation (\ref{equ1})
with the following property: for every sequence $h_n\to 0^{+}$ there
exists a subsequence, still denoted by $h_n$, such that $|v_{h_n}|$
has a circle of local maximum points
$\{x\in{\mathbb R}^{N}:|x|=r_{h_n}\}$ with
$r_{h_n}\to \overline{r}>r_0$ as $n\to +\infty$, while $v_{h_n}$
vanishes to zero away from the circle
$\{x\in{\mathbb R}^{N}:|x|=\overline{r}\}$.
We also provide an implicit formula which permits us to locate exactly
the concentration set of the solutions and simultaneously gives us an
estimate of how big must be $V(x)$ for $|x|=r_0$ to make the above
result available, even though we cannot express explicitly the radius
$\overline{r}$ in terms of the potential $V$. Our solutions are obtained
by using variational methods and the method employed requires some
restrictions of a technical nature on the nonlinear term which are
specified in assumptions f)-g). For the exact statement of the above
result see theorem 8.1 which provides the main result of this paper
about existence and concentration of solutions.
We now briefly outline the organization of the contents of this paper.
In section 2 we introduce the abstract setting, i.e. the functional set
in which it is convenient to study the functional $E_h$ and we point
out some of its compactness properties. Section 3 is devoted to the
definition of a topological device, the ``topological charge'',
which we will need in order to give a classification of the maps
we deal with. In section 4 we study the energy functional $E_h$
and provide an important invariance property under a certain class of
transformations. The first existence and concentration result for
equation (\ref{equ1}) is given in section 5, where we find a family
$\{v_h\}$ of solutions concentrating near zero as $h\to 0^{+}$.
Section 6 and 7 pave the way for the achievement, in last section,
of the second result of the paper, i.e. the existence of solutions
exhibiting a concentration behaviour near a circle.
\paragraph{Notation.}
For the rest of this paper we use the following notation.
\begin{itemize}
\item $x\,y$ is the standard scalar product between $x$,
$y\in{\mathbb R}^{N}$. $|x|$ is the Euclidean norm of
$x\in{\mathbb R}^{N}$. Analogously $|M|$ is the Euclidean norm of a
$m\times n$ real matrix $M$.
\item $W^{1,2}({\mathbb R}^{N},{\mathbb R}^{N+1})$ and
$W^{1,p}({\mathbb R}^{N},{\mathbb R}^{N+1})$ are the standard Sobolev
spaces.
\item For $u:{\mathbb R}^{N}\to {\mathbb R}^{N+1}$, $\nabla
u$ is the $(N+1)\times N$ real matrix whose rows are given by
the gradient of each component function $u_{j}$.
\item For any $U\subset {\mathbb R}^{N}$, $int (U)$ is its internal
part, $\overline{U}$ its closure and $\partial U$ its boundary.
Furthermore $\chi_{U}$ denotes the characteristic function of $U$,
while by $meas\,(U)$ we intend the Lebesgue measure of $U$. Finally
$\mathop{\rm dist}(x,U)$ is the Euclidean distance between a point
$x\in{\mathbb R}^{N}$ and $U$, i.e.
$\mathop{\rm dist}(x,U)=\inf_{y\in U}|x-y|$.
\item $\omega_{N}$ is the surface measure of the unit sphere
$S^{N-1}$ of ${\mathbb R}^{N}$.
\item If $x\in{\mathbb R}^{N}$ and $r>0$, then $B_{r}(x)$ or $B(x,r)$
is the open ball with center in $x$ and radius $r$.
\item Given $R_{2}>R_1>0$, by $C(R_1,R_{2})$ we denote the ring
in ${\mathbb R}^{N}$ centered in $0$ and with internal radius $R_1$
and external radius $R_{2}$, i.e.
$C(R_1,R_{2})=\{x\in{\mathbb R}^{N}:R_1<|x| 0$ let $H_h$ denote the
subspace of $W^{1,2}({\mathbb R}^{N},{\mathbb R}^{N+1})$
consisting of functions $u$ such that
\begin{equation} \label{norm}\|u\|_{H_h}
\equiv\bigg(\int_{{\mathbb R}^{N}}\Big(|\nabla
u|^{2}+V_h(x)|u|^{2}\Big)\,dx\bigg)^{1/2}+\bigg(\int_{{\mathbb R}^{N}}|\nabla
u|^{p}\,dx\bigg)^{1/p}<+\infty.
\end{equation} The space
$H_h$ can also be defined as the closure of
$C_0^{\infty}({\mathbb R}^{N},{\mathbb R}^{N+1})$ with respect to the
norm (\ref{norm}). The main properties of $H_h$ are
summarized in the following lemma.
\begin{theorem} \label{space}
For every $h> 0$ the following statements
hold: \begin{description}
\item{i)} $H_h$ is continuously embedded in
$W^{1,2}({\mathbb R}^{N},{\mathbb R}^{N+1})$ and
$W^{1,p}({\mathbb R}^{N},{\mathbb R}^{N+1})$.
\item{ii)} There exist two constants $C_0$, $C_1>0$ such that,
for every $u\in H_h$,\\
$\|u\|_{L^{\infty}}\leq C_0\|u\|_{H_h}$
and \begin{equation} \label{equi}|u(x)-u(y)|\leq
C_1\,|x-y|^{(p-N)/p}\|\nabla u\|_{L^{p}}\quad\forall
\,x,y\in{\mathbb R}^{N}.
\end{equation}
\item{iii)} For every $u\in
H_h$\begin{equation} \label{inf}\lim_{|x|\to+\infty}u(x)=0\,.
\end{equation}
\item{iv)} If $\{u_n\}$ converges weakly in $H_h$ to some
function $u$, then it converges uniformly on every compact set in
${\mathbb R}^{N}$.
\end{description}
\end{theorem}
The proof is a direct consequence of the Sobolev embedding
theorems (see\cite{D} for the proof of the continuous immersion
$H_h\subset W^{1,p}({\mathbb R}^{N},{\mathbb R}^{N+1})$).
Note that from (\ref{equi}) we derive the following property we
are going to use several times in the proofs of our results: given
$\{u_{\alpha}\}\subset H_h$ a family of functions verifying
$\|\nabla u_{\alpha}\|_{L^{p}}\leq M$ for some $M\geq 0$, then
there results: for every $\varepsilon >0$ there exists $\delta>0$
such that $$|x-y|\leq \delta \Rightarrow
|u_{\alpha}(x)-u_{\alpha}(y)|\leq \varepsilon\quad \forall\,
\alpha.$$ We refer to the above property as to the ``equi-uniform
continuity" of the family
$\{u_{\alpha}\}$.
Now let $H_{h,r}$ denote the subset of $H_h$ given by $$H_{h,r}=\{u\in H_h:|u| \;\;\mbox{is a radially symmetric function}\}.$$
By $iv)$ of lemma \ref{space} it follows that $H_{h,r}$ is weakly closed in $H_h$. We next prove an easy useful radial lemma concerning a compactness property of the functions in $H_{h,r}$.
\begin{theorem}\label{compactinj} For every $h>0$ $H_{h,r}$ is relatively
compact in $L^{s}({\mathbb R}^{N},{\mathbb R}^{N+1})$ for every
$s\in (2,+\infty]$. \end{theorem}
\paragraph{Proof.} Let $\{u_n\}\subset H_{h,r}$ be a sequence such
that $\|u_n\|_{H_h}$ is bounded. Then, up to a subsequence,
it is
$$u_n\rightharpoonup u\quad \mbox{weakly in}\quad H_h
$$
for some $u\in H_{h,r}$. Fix $\gamma\in(0,1)$ arbitrarily and consider
$R>0$ such that, according to (\ref{inf}),
$$\forall x\in{\mathbb R}^{N}\setminus \overline{B}_{R}(0):\;\;|u(x)|
<\frac{\gamma}{2}.
$$
The objective is to prove that, for $n$ sufficiently large,
\begin{equation} \label{object}\forall x\in{\mathbb R}^{N}\setminus
\overline{B}_{R}(0):\;\;|u_n(x)|<\frac{\gamma}{2}.
\end{equation}
Assume by the contrary that, up to a subsequence,
$$\forall n\in{\mathbb N}\;\;\exists x_n\in{\mathbb R}^{N}\setminus
\overline{B}_{R}(0)\;\;\mbox{s.t.}\;\; |u_n(x_n)|\geq
\frac{\gamma}{2}.
$$
First we'll prove that the sequence $\{x_n\}$
is bounded in ${\mathbb R}^{N}$. Indeed, arguing by contradiction, we
suppose that, up to a subsequence,
\begin{equation} \label{abs}
|x_n|\to+\infty\quad\mbox{as}\quad
n\to+\infty\,.
\end{equation}
By (\ref{equi}), since the sequence $\{\|\nabla u_n\|_{p}\}$
is bounded, there exists $\delta>0$ such that
\begin{equation} \label{equi1} \forall\,n\in{\mathbb N} , \quad \forall\,x,\,y\in
{\mathbb R}^{N}: \quad |x-y|<\delta \quad \Rightarrow \quad
|u_n(x)-u_n(y)| <\frac{\gamma}{4} .
\end{equation}
>From (\ref{equi1}) we immediately get
$$ B_{\delta}(x_n)\subset \left\{
x\in {\mathbb R}^{N}:|u_n(x)|>\frac{\gamma}{4}\right\}\quad \forall
n\in{\mathbb N}.
$$
Because of (\ref{abs}) we may assume $|x_n|-\delta>0$ for every
$n\in{\mathbb N}$. Now, taking into account that $\{|u_n|\}$ is a
sequence of radial functions, putting
$C_n\equiv C(|x_n|-\delta,|x_n|+\delta)$ (see the notations at
the end of the introduction), it holds
$$|u_n(x)|>\frac{\gamma}{4}\quad \forall x\in C_n\quad \forall
n\in{\mathbb N}.
$$
Then we can write
$$\int_{{\mathbb R}^{N}}V_h(x)|u_n|^{2}\,dx\geq V_0
\int_{C_n}|u_n|^{2}\,dx\geq V_0\frac{\gamma^{2}}{16}
\mathop{\rm meas} (C_n)\to +\infty\quad \mbox{as}\quad n\to +\infty,
$$
but this contradicts the fact that $\{u_n\}$ is bounded in $H_h$.
Therefore, the sequence $\{x_n\}$ turns out to be bounded in
${\mathbb R}^{N};$ up to a subsequence we obtain
$$x_n\to x\in{\mathbb R}^{N}\quad\mbox{as}\quad n\to +\infty.
$$
Now we have
$$|u_n(x_n)- u(x)|\leq
|u_n(x_n)-u_n(x)|+|u_n(x)-u(x)|.
$$
For $n$ large enough
both terms on the right side are arbitrary small, the first
because of the equi-uniform continuity of $\{u_n\}$ and the
second because of $iv)$ of lemma \ref{space}.
This implies $u_n(x_n)\to u(x)$ as $n\to+\infty $ and then
$|u(x)|\geq\frac{\gamma}{2}$. The choice of $\gamma$ yields $|x|0$ let $F_h$ denotes the subspace of fixed points:
$$F_h=\{u\in H_h:T_{g}(u) =u\;\;\forall g\in O(N)\}.
$$
An easy computation shows that $F_h\subset H_{h,r}$. Furthermore
$F_h$ is weakly closed in $H_h$, i.e. for every
$\{u_n\}\subset F_h$ and $u\in H_h$ it is
\begin{equation} \label{wc}
u_n\rightharpoonup u\;\;\mbox{weakly in}\;\;H_h\Rightarrow u\in F_h.
\end{equation}
Then define the set $\Lambda_{F}(h)=F_h\cap\Lambda_h$.
Part i) of lemma \ref{space} assures that for every $h>0$ the space
$H_h$ is continuously embedded in
$H\equiv W^{1,2}({\mathbb R}^{N},{\mathbb R}^{N+1})\cap
W^{1,p}({\mathbb R}^{N},{\mathbb R}^{N+1})$;
setting $\Lambda=\{u\in H:u(x)\neq \overline{\xi}\equiv(1,0)\;\;
\forall x\in{\mathbb R}^{N}\}$, in the course of the paper
we will often make use of the following two sets
$$F=\{u\in H:T_{g}(u) =u\;\;\forall g\in O(N)\},\quad
\Lambda_{F}=F\cap \Lambda.$$
Now we want to give a topological classification of the maps
$u\in\Lambda$. More precisely we introduce a topological
invariant with suitable ``localization'' properties in the sense
that, roughly speaking, it depends on the compact region where $u$
is concentrated. This invariant consists of an integer number
called ``topological charge'' and it will be defined by means of
the topological degree.
\section{Topological Charge}
In this section we take from \cite{BFPD}some crucial results.
With the help of the notation introduced in (\ref{nota}), we give
the following definition.
\paragraph{Definition} Given $u=(u_0,u_1)\in \Lambda$ and
$U\subset{\mathbb R}^{N}$ an open set such that $u_1(x)\neq 0$
if $x\in\partial U$, then we define the (topological) charge of
$u$ in $U$ as the following integer number
$$
ch(u,U)= \deg\left(u_1,\,K(u)\cap U,\,0\right)
$$
where the open set
$K(u)$ is defined as follows:
$$K(u)=\left\{x\in{\mathbb R}^{N}:u_0(x)>1\right\}.
$$
We recall the convention
$\deg\left(u_1,\,\emptyset,\,0\right)=0.$ Furthermore given
$u\in\Lambda$ we define the (topological) charge of $u$ as the
integer number
$$ch(u)= \deg\left(u_1,\,K(u),\,0\right).$$
We note that the topological charge is well defined thanks to (\ref{inf}) and the definition of $\Lambda$.
>From well known properties of the topological degree we get other
useful properties of the topological charge. For example notice
that if $U\subset{\mathbb R}^{N}$ is open and such that $0\not\in u_1(\partial U)$ and if $U$ consists of $m$ connected components
$U_1,\ldots,U_{m}$, i.e. $u$ has the energy concentrated in
different regions of the space, then by the additivity property
of the degree we get
\begin{equation} \label{summ}
ch(u,U)=\sum_{j=1}^{m}ch\left(u,\,U_{j}\right).
\end{equation}
The next lemma shows how the topological charge exhibits a sort of
invariance under a class of transformations in $\Lambda$.
\begin{theorem} \label{novi} Let $w\in \Lambda$ compactly supported
with $w\equiv 0$ in a neighbourhood of the origin and for $r>0$
arbitrarily fixed define
$\widetilde{w}:{\mathbb R}^{N}\to {\mathbb R}^{N+1}$ by setting
$$\widetilde{w}(x)=\cases{
0& if $|x|\leq r$, \cr
w\left(x-r\frac {x}{|x|}\right) & if $|x|>r$. \cr}.$$
Then there results: $ch(w)=ch(\widetilde{w})$.
\end{theorem}
\paragraph{Proof.} First observe that by definition $\widetilde{w}$
belongs to $\Lambda$ too. It is immediate to show that if
$K(w)=\emptyset$ it follows $K(\widetilde{w})=\emptyset$ and then
$ch(w)=ch(\widetilde{w})=0$. Now assume $K(w)\neq\emptyset$;
it makes sense to choose $m\geq1$ and $01\quad \forall x\in \cup_{i=1}^{m-1}C(R_{i},R_{i+1}),\cr
w_0(x)\leq1\quad \forall x\in {\mathbb R}^{N}\setminus
\cup_{i=1}^{m-1}C(R_{i},R_{i+1}).
}$$
Remember that for $a1\}$. By
construction we infer
$$\widetilde{w}_0(x)>1\quad \forall x\in
\cup_{i=1}^{m}C\left(R_{i}+r,R_{i+1}+r\right).
$$
To simplify notation, set $$C_{i}=C(R_{i},R_{i+1}),\quad
\widetilde{C}_{i}=C\left(R_{i}+r,R_{i+1}+r\right)\quad
\mbox{for}\;\;i=1,\ldots, m-1.
$$
According to (\ref{summ}),
$$ch(w)=\sum_{i=1}^{m-1}\deg(w_1,C_{i},0),\quad
ch(\widetilde{w})=\sum_{i=1}^{m-1}
\deg(\widetilde{w}_1,\widetilde{C}_{i},0).
$$
Hence, in order to conclude, it will be sufficient to show that
$$\deg(w_1,C_{i},0)=\deg(\widetilde{w}_1,\widetilde{C}_{i},0)\quad
\forall i=1,\ldots,m-1.
$$
Take for example $i=1$ and consider
the functions $G,\;\widetilde{G}:C\left(R_1,r+R_{2}\right)\to
{\mathbb R}^{N+1}$ defined by setting
$$G(x)= \cases{ w_1(x)& if $R_1\leq|x|\leq R_{2}$, \cr
w_1\left(\frac{x}{|x|}R_{2}\right)
& if $R_{2}\leq|x|\leq r+R_{2}$. \cr},
$$
$$
\widetilde{G}(x)= \cases{ \widetilde{w}_1(x)&
if $r+R_1\leq|x|\leq r+R_{2}$, \cr
\widetilde{w}_1\left(\frac{x}{|x|}(r+R_1)\right) & if
$R_1\leq|x|\leq r+R_1$. \cr}
$$
It is obvious that $G$ and $\widetilde{G}$ are continuous and moreover
$G(x)=\widetilde{G}(x)$ for $ x\in\partial C\left(R_1,r+R_{2}\right)$.
Furthermore, since by construction
$w_0(x)=1$ for $|x|=R_{2}$ and $\widetilde{w}_0(x)=1$ for
$|x|=r+R_1$, the definition of $\Lambda$ implies $w_1(x)\neq
0$ for $|x|=R_{2}$ and $\widetilde{w}_1(x)\neq 0$ for
$|x|=r+R_1$ and then
$$G(x)\neq 0\quad \forall R_{2}\leq|x|\leq
r+R_{2}, \quad \widetilde{G}(x)\neq 0 \quad \forall
R_1\leq|x|\leq r+R_1.
$$
Then, because of the excision
property and the homotopy invariance of the topological degree,
we conclude
\begin{eqnarray*}
\deg(w_1,C_1,0)&=&\deg\left(G,C\left(R_1,r+R_{2}\right),0\right)\\
&=&\deg\left(\widetilde{G},C\left(R_1,r+R_{2}\right),0\right) \\
&=&\deg(\widetilde{w}_1,\widetilde{C}_1,0),
\end{eqnarray*}
and hence the proof is complete.\hfill $\diamondsuit$
Another consequence is that the topological charge is stable under
uniform convergence as the following lemma states.
\begin{theorem} \label{cont} Let $\{u_n\}\subset\Lambda$,
$u\in\Lambda$ and $U\subset{\mathbb R}^{N}$ an open set such that
$u_n\to u$ uniformly in $U$ and $$(u_n)_1(x)\neq
0,\;\;u_1(x)\neq0\quad \forall x\in\partial U, \;\;\forall
n\in{\mathbb N}.
$$
Then, for $n$ large enough, $ch(u,U)=ch(u_n,U)$. \end{theorem}
\begin{corollary}
For every $u\in\Lambda_h$ there exists $\varrho=\varrho(u)>0$
such that, for every $v\in\Lambda_h$,
$$\|u-v\|_{L^{\infty}}\leq \varrho\Rightarrow ch(u)=ch(v).$$
\end{corollary}
By definition we easily deduce that for every $u\in \Lambda$ and
$U\subset{\mathbb R}^{N}$ an open set with $u_1(x)\neq0$ for
$x\in \partial U$,
\begin{equation} \label{prop}ch(u,U)\neq0\Rightarrow \|u\|_{L^{\infty}(U)}>1.
\end{equation}
Finally we define the set
$$\Lambda_{F}^{\ast}(h)=\{u\in\Lambda_F(h):ch(u)\neq 0\}.
$$
In \cite{BFPD} the authors proved the existence a function
in $\Lambda_{F}$ with $ch(u)\neq 0$. Now multiply such $u$ for a
cut-off radially symmetric function $\tau:{\mathbb R}^{N}\to {\mathbb R}$
verifying
$$
\tau\in C^{\infty}({\mathbb R}^{N},{\mathbb R}),\quad \tau=1\;\;
\mbox{in}\;\;B_{R}(0), \quad \tau=0\;\;\mbox{in}\;\;{\mathbb R}^{N}
\setminus B_{R+1}(0),\quad 0\leq\tau\leq1
$$
with $R$ sufficiently large such that $K(u)\subset B_{R}(0);$ we
immediately obtain that $\tau u\in\Lambda_{F}^{\ast}(h)$ for all $h>0$.
In particular we get $\Lambda_{F}^{\ast}(h)\neq \emptyset$ for every
$h> 0$.
Our aim is to find critical points of the functional $E_h$ defined
by (\ref{funct}) in the class $\Lambda_{F}^{\ast}(h)$ to obtain the
existence of solutions for (\ref{equ2}) in the set of the fields
$u$ with nontrivial charge. This is what we'll do in the next sections.
\section{The Energy Functional}
Now we are going to study the properties of the functional
$E_h$; first recall that under the hypotheses a), b) and c) in
section 4 of \cite{BBD} the authors proved that $E_h$ is well defined
in the space $\Lambda_h$, i.e. for every $u\in
\Lambda_h$ we have $ E_h(u)<+\infty$.
Obviously $E_h$ is bounded from below and is coercive in the
$H_h$-norm:
\begin{equation} \label{coercive0}
\lim_{\|u\|_{H_h}\to+\infty}E_h(u)=+\infty\,.
\end{equation}
Moreover in \cite{BBD} it is proved that the energy functional $E_h$
belongs to the class $C^{1}(\Lambda_h,{\mathbb R})$ under the
assumptions a)-d). An immediate corollary is that the critical points
$u\in\Lambda_h$ (particularly the minima) for the functional $E_h$
are weak solutions of equation (\ref{equ2}).
The next two propositions deal with some other
properties of the functional $E_h$; we omit the proofs because
they are the same as in \cite{BFP}, provided that we substitute
$E_h$ for $E$ and
$\Lambda_h$ for $\Lambda$. The first deals with the
behaviour of $E_h$ when $u$ approaches the boundary of
$\Lambda_h$.
\begin{proposition}[\cite{BFP}, Lemma 3.7, p. 326]
Let $\{u_n\}\subset\Lambda_h$ be bounded in the $H_h$-norm
and weakly converging to
$u\in\partial\Lambda_h$, then
$$\int_{{\mathbb R}^{N}}W(u_n)\,dx\to+\infty\quad\mbox{as}\quad
n\to +\infty.
$$
As a consequence, if $\{u_n\}\subset\Lambda_h$ is weakly converging
to $u$ and such that $E_h(u_n)$ is bounded,
then $u\in\Lambda_h$.
\end{proposition}
The second proposition states the weakly lower semi-continuity of the
energy functional $E_h$.
\begin{proposition}[\cite{BFP}, Prop. 3.10, p. 328]
For every $u\in\Lambda_h$ and for every
sequence $\{u_n\}
\subset \Lambda_h$, if $\{u_n\}$ weakly converges to $u$,
then
$$\liminf_{n\to+\infty}E_h(u_n)\geq
E_h(u).$$
\end{proposition}
Taking into account of the radial assumptions a) and e), an easy
calculation shows that the open set $\Lambda_h$ and the functional
$E_h$ are invariant under the action (\ref{action}), i.e.
for every $u\in\Lambda_h$ and $g\in O(N)$,
$$T_{g}(u)\in\Lambda_h,\quad E_h(T_{g}(u))=E_h(u).
$$
This fact suggests the idea that the set $\Lambda_{F}(h)$ is a natural
constrain to obtain critical points of $E_h$: more precisely the
following important result holds:
\begin{theorem} \label{action1}Let $f:\Lambda_h\to {\mathbb R}$ a
$C^{1}$ functional verifying $f(T_{g}(u))=f(u)$ for every
$u\in\Lambda_h$. Then for every $u\in\Lambda_h$ and $v\in H_h$ it is
$$\langle f'(u),v\rangle=\langle f'(u),P_hv\rangle,
$$
where $P_h$ is the projection of $H_h$ onto $F_h$. As a corollary,
if $u\in\Lambda_{F}(h)$ is such that, for any $v\in F_h$ ,
$\langle f'(u),v\rangle=0$, then $f'(u)=0$, i.e. $u$ is a critical point
for $f$.\end{theorem}
The proof can be found in \cite{BFPD}.
By applying lemma \ref{action1} to our functional $E_h$ we deduce that
every local minimum point of $E_h$ in $\Lambda_{F}(h)$ is also a
critical point of $E_h$. This fact suggests a useful technique to
obtain solutions for equation (\ref{equ2}), i.e. that of minimizing the
functional $E_h$ in some open subset of $\Lambda_{F}(h)$; this is
exactly what we'll do in the next sections to achieve our existence and
concentration results.
\section{First Existence Result: Concentration Near the Origin}
In this section we will show the existence of a family of solutions
$\{v_h\}$ to equation (\ref{equ1}) which concentrate near the origin
as $h\to 0^{+}$. First observe that $\Lambda_{F}^{\ast}(h)$ is open
in $\Lambda_{F}(h)$: indeed
$\Lambda_{F}^{\ast}(h)=\Lambda_{F}(h)\cap \{u\in\Lambda_h: ch(u)
\neq 0\}$, and the set $\{u\in\Lambda_h: ch(u)\neq 0\}$ is open in
$\Lambda_h$ because of $ii)$ of lemma \ref{space} and corollary 3.1.
Then, according to last section, our solutions $v_h$ will be obtained
by minimizing the functional $E_h$ in the set $\Lambda_{F}^{\ast}(h)$.
To this aim put
$$E_h^{\ast}=\inf_{u\in\Lambda_{F}^{\ast}(h)}E_h(u).$$
In what follows we will consider the functional $E_0$ defined on
$\Lambda$ by setting
$$E_0(u)=\frac{1}{2}\int_{{\mathbb R}^{N}}\left(|\nabla u|^{2}+V(0)
|u|^{2}\right)\,dx+\frac{1}{p}\int_{{\mathbb R}^{N}}|\nabla u|^{p}\,dx
+\int_{{\mathbb R}^{N}}W(u)\,dx.$$
$E_0$ is the functional associated with equation (\ref{equ2}) replacing
$V_h$ by $V(0)$. Now we are able to provide the first existence and
concentration result of this paper.
\begin{theorem} Assume that a)$-$e) hold. Then for every $h>0$ the
minimum $E_h^{\ast}$ is attained in the set $\Lambda_{F}^{\ast}(h)$.
Furthermore if we put
\begin{equation} \label{related}v_h(x)=u_h\left(\frac{x}{h}\right)
\end{equation}
where $u_h\in \Lambda_{F}^{\ast}(h)$ is the minimizing function for
$E_h^{\ast}$, then $v_h$ is a solution of (\ref{equ1}) and there
exist at least a points $x_h\in{\mathbb R}^{N}$ such that
$|v_h(x_h)|>1$. Finally the family $\{v_h\}$ concentrates at
the origin in the following sense: for every $\delta>0$, it holds:
$$v_h\to 0\quad as \quad h\to 0^{+}\quad uniformly\;\; in\;\; the\;\;
set\;\; \quad \{|x|\geq\delta\}.
$$ In particular this implies $x_h\to 0$ as $h\to 0^{+}$.
\end{theorem}
\paragraph{Proof.} Fix $h>0$ and consider
$\{u_n^{h}\}$ a minimizing sequence in $\Lambda_{F}^{\ast}(h)$. Remember that $\Lambda_{F}^{\ast}(h)\neq \emptyset$ for every $h>0$.
Applying lemma \ref{compactinj} we
have, up to a subsequence:$$u_n^{h}\rightharpoonup u_h\quad
\mbox{weakly in}\quad H_h\quad\mbox{as}\quad n\to+\infty$$
$$u_n^{h}\to u_h\quad\mbox{uniformly
in}\quad{\mathbb R}^{N}\quad\mbox{as}\quad n\to+\infty,$$ for
some $u_h\in F_h$. Proposition 4.2 implies $u_h\in \Lambda_{F}(h)$.
The continuity of the topological
degree with respect to the uniform convergence implies that the
sequence $\{ch(u_n^{h})\}$ is constant for large $n$'s, and then
$$ch(u_h)=\lim_{n\to+\infty}ch(u_n^{h})\neq 0 ,$$ hence $u_h\in\Lambda_{F}^{\ast}(h)$. By the weakly lower semi-continuity of the energy functional $E_h$,
$$\quad E_h^{\ast}\leq
E_h(u_h)\leq\liminf_{n\to+\infty}
E_h(u_n^{h})=E_h^{\ast},$$ i.e. $E_h(u_h)= E_h^{\ast}$.
Hence we have proved that each $u_h$ is a minimum point of $E_h$ in $\Lambda_{F}^{\ast}(h)$ and, by applying lemma \ref{action1}, it is a critical point of $E_h$ and hence a solution to equation (\ref{equ2}); by rescaling, we immediately obtain that the family $\{v_h\}$ related to $\{u_h\}$ by (\ref{related}) provides solutions to equation (\ref{equ1}). We recall that for every $x\in K(u_h)$ with $ (u_h)_1(x)=0$ it is $|u_h(x)|>1$; then the definition of $\Lambda_{F}^{\ast}(h)$ implies the existence of at least a point $x_h$ verifying $|v_h(x_h)|>1$. It remains to prove that $\{v_h\}$ satisfies the concentration property announced in the theorem.
To this aim we first establish some important facts in the following
two steps.
\paragraph{Step 1.} $\limsup_{h\to 0^{+}}E_h^{\ast}<+\infty$.
Take $u\in\Lambda_{F}$ compactly supported such that $ch(u)\neq 0$.
In the last section we have pointed out that such a function exists.
Furthermore we obviously have $u\in\Lambda_{F}^{\ast}(h)$ for all $h>0$.
Now observe: \begin{equation} \label{int} E_h^{\ast}
\leq
E_h(u)=E_0(u)+\frac{1}{2}\int_{{\mathbb R}^{N}}\left(V_h(x)-V(0)
\right)|u|^{2}\,dx.
\end{equation}
Since $V_h$ converges to $V(0)$ uniformly on compact sets as
$h\to 0^{+}$, then, if we take $h$ sufficiently
small, the integral in (\ref{int}) is close to zero. This implies
$$\limsup_{h\to 0^{+}}E_h^{\ast}\leq E_0(u)
$$
and the conclusion follows.
\paragraph{Step 2.} Let $h_n\to 0^{+}$ an arbitrary sequence.
Then for every $\gamma>0$ there exists $R_{\gamma}>0$ such that
for $n$ sufficiently large:
$$|u_{h_n}(x)|< \gamma\quad \forall x\in {\mathbb R}^{N}\setminus
B_{R_{\gamma}}(0).
$$
Fix $\gamma>0$. By Step 1 $E_h^{\ast}$ is bounded for $h>0$ small.
In particular, according to (\ref{equi}), the sequence $\{u_{h_n}\}$
is equi-uniformly continuous; then there exists $\delta>0$ such that
$$ \forall\,n\in{\mathbb N} , \quad \forall\,x,\,y\in
{\mathbb R}^{N}: \quad |x-y|<\delta \quad \Rightarrow \quad
|u_{h_n}(x)-u_{h_n}(y)| <\frac{\gamma}{2} .
$$
Assume by absurd that, passing to a subsequence, we can find a sequence
$\{z_{h_n}\}\subset{\mathbb R}^{N}$ such that
$$|u_{h_n}(z_{h_n})|\geq\gamma,\quad |z_{h_n}|\to +\infty
\quad \mbox{as}\quad n\to +\infty;
$$
we immediately get $$ B_{\delta}(z_{h_n})\subset \left\{
x\in {\mathbb R}^{N}:|u_{h_n}(x)|>\frac{\gamma}{2}\right\}\quad
\forall n\in{\mathbb N}.$$ Without loss of generality we may assume
$|z_{h_n}|-\delta>0$ for every $n\in{\mathbb N}$.
Now, taking into account that $\{|u_{h_n}|\}$ is a sequence of radial
functions, putting $C_{h_n}\equiv C(|z_{h_n}|-\delta,|z_{h_n}|
+\delta)$ (see the notations at the end of the introduction), it holds
$$|u_{h_n}(x)|>\frac{\gamma}{2}\quad \forall x\in C_{h_n}\quad
\forall n\in{\mathbb N}.
$$
Then we can write
$$\int_{{\mathbb R}^{N}}V_{h_n}(x)|u_{h_n}|^{2}\,dx\geq V_0
\int_{C_{h_n}}|u_{h_n}|^{2}\,dx\geq\frac{\gamma^{2}}{4}
\mathop{\rm meas} (C_{h_n})\to +\infty
$$
as $n\to +\infty$, but this contradicts the fact that $\{E_h^{\ast}\}$
is bounded for small $h>0$. The proof of Step 2 is complete.
>From Step 2 it directly follows that, taken an arbitrary sequence
$h_n\to 0^{+}$, $\{u_{h_n}\}$ has uniform behaviour at
infinity, i.e. $\lim_{|x|\to +\infty}|u_{h_n}(x)|=0$ uniformly
with respect to $n$. By re-scaling it is easy to prove that
$\{v_{h_n}\}$ decays uniformly to zero for $x$ outside every
fixed neighbourhood of the origin as $n\to +\infty$. Finally the
fact that $h_n$ is arbitrary allows us to conclude. \hfill
$\diamondsuit$
\section{Second Existence Result: Preliminaries}
This section is devoted to establish some preliminary results
concerning the study of the behaviour of sequences
$\{u_{h_n}\}$, with $u_{h_n}\in\Lambda_{F}(h_n)$, whose
energy grows like $1/h_n^{N-1}$. First we put
\begin{equation}
\label{notation}{\mathcal{V}}(|x|)=V(x)\quad \forall x\in{\mathbb
R}^{N}.
\end{equation}
We notice that by hypothesis $V$ is a radially
symmetric function, so the above definition is well posed.
Toward our aims the following lemma constitutes a crucial step;
it uses in an essential way the radial properties of the functions
in $\Lambda_{F}(h)$.
\begin{theorem} \label{preliminary0}
Let $h_n\to 0^{+}$ an arbitrary sequence and
$u_{h_n}\in\Lambda_{F}(h_n)$ such that
$$
\limsup_{n\to +\infty}h_n^{N-1}\int_{{\mathbb R}^{N}}
\left(|\nabla u_{h_n}|^{2}+|u_{h_n}|^{2}\right)\,dx<+\infty.
$$
For every $n\in{\mathbb N}$ and $x\in{\mathbb R}^{N}$ set
$\varphi_n(|x|)\equiv |u_{h_n}(x)|$. Then for all $t>0$ it holds
\begin{equation} \label{ht}
\limsup_{n\to +\infty}\int_{t/(2h_n)}^{+\infty}\left(|\varphi_n'|^{2}
+|\varphi_n|^{2}\right)\,dr<+\infty\,.
\end{equation}
Furthermore the sequence $\{\varphi_n\}$ is equi-uniformly continuous
in $\left[\frac{t}{2h_n},+\infty\right)$ in the following sense:
for every $\eta>0$ there exists $\delta>0$ such that :
$$
\forall n\in{\mathbb N},\quad\forall s,s'\in\Big[\frac{t}{2h_n},
+\infty\Big):\quad |s-s'|\leq\delta\Rightarrow
|\varphi_n(s)-\varphi_n(s')|\leq\eta\,.
$$ \end{theorem}
\paragraph{Proof.} An immediate computation leads to
\begin{eqnarray}
\lefteqn{\limsup_{n\to +\infty}h_n^{N-1}\int_{{\mathbb R}^{N}}
\left(|\nabla u_{h_n}|^{2}+|u_{h_n}|^{2}\right)\,dx }\nonumber\\
&\geq&\limsup_{n\to +\infty}h_n^{N-1}\int_{{\mathbb R}^{N}}
\left(|\nabla |u_{h_n}||^{2}+|u_{h_n}|^{2}\right)\,dx\nonumber\\
&=&\limsup_{n\to +\infty}\omega_{N}\int_0^{+\infty}(h_nr)^{N-1}
\left(|\varphi_n'|^{2}+|\varphi_n|^{2}\right)\,dr \label{ine11} \\
&\geq& \omega_{N}\limsup_{n\to +\infty}\int_{\frac{t}{2h_n}}^{+\infty}
(h_nr)^{N-1}\left(|\varphi_n'|^{2}+|\varphi_n|^{2}\right)\,dx \nonumber\\
&\geq& \omega_{N} \frac{t^{N-1}}{2^{N-1}} \limsup_{n\to +\infty}
\int_{\frac{t}{2h_n}}^{+\infty}\left(|\varphi_n'|^{2}
+|\varphi_n|^{2}\right)\,dx. \nonumber
\end{eqnarray}
It is obvious that each $\varphi_n$ belongs to the class
$W^{1,2}\left(\left(\frac{t}{2h_n},+\infty\right),{\mathbb R}\right)$.
Then the theory of the one-dimensional Sobolev spaces leads to
\begin{equation} \label{equi11}|\varphi_n(s)-\varphi_n(s')|\leq
\left(\int_{t/(2h_n)}^{+\infty}|\varphi_n'(r)|^{2}\,dr\right)^{1/2}\;
|s-s'|^{1/2}\quad \forall s,s'\in \left(\frac{t}{2h_n},+\infty\right).
\end{equation}
Using (\ref{ine11}) it immediately follows that the sequence
$\int_{t/(2h_n)}^{+\infty}|\varphi_n'|^{2}\,dr$ is bounded, hence
from (\ref{equi11}) we easily deduce the equi-uniform continuity of
$\{\varphi_n\}$. \hfill $\diamondsuit$
A second preliminary result we will need is given in the following
lemma inspired by the ``splitting lemma'' of \cite{BFP} in the spirit
of the concentration compactness principle of \cite{P.L.}.
\begin{theorem} \label{preliminary}
Let $h_n\to 0^{+}$ an arbitrary sequence and
$u_{h_n}\in\Lambda_{F}(h_n)$ such that
\begin{equation} \label{13}
\limsup_{n\to +\infty}h_n^{N-1}\int_{{\mathbb R}^{N}}
\left(|\nabla u_{h_n}|^{2}+|u_{h_n}|^{2}\right)\,dx<+\infty,
\end{equation}
\begin{equation} \label{131}\sup_{|x|\geq\frac{t}{h_n}}|u_{h_n}(x)|>
\frac{\varepsilon}{2}\quad \forall n\in{\mathbb N}
\end{equation}
and \begin{equation} \label{14}|u_{h_n}(x)|\leq \frac{\varepsilon}{2}\quad
\forall x\in C\left(\frac{t}{2h_n},\frac{t}{h_n}\right)
\quad\forall n\in{\mathbb N},
\end{equation}
for some $t,\varepsilon>0$. Then, setting $\varphi_{h_n}(|x|)\equiv
|u_{h_n}(x)|$, there exist $\ell\in{\mathbb N}$,
$R_1,\ldots,R_{\ell}>0$ and $\ell$ sequences of positive numbers
$\{r_{h_n}^{1}\},\ldots,\{r_{h_n}^{\ell}\}$,
with $r_{h_n}^{i}>\frac{t}{h_n}$, such that, up to subsequence
\begin{equation} \label{dim}
\{h_nr_{h_n}^{i}\}\quad\mbox{is bounded for }i=1,\ldots,\ell ;
\end{equation}
\begin{equation} \label{mag}
\varphi_{h_n}(r_{h_n}^{i})>\frac{\varepsilon}{2}\quad\forall\,
n\in{\mathbb N},\quad \forall i=1,\ldots,\ell ;
\end{equation}
$r_{h_n}^{i}$ is maximal point for $\varphi_{h_n}$ in
\begin{equation} \label{max}
\big[\frac{t}{2h_n},+\infty\big)
\setminus\cup_{j0\}$
for a fixed $x\neq 0$. For sake of simplicity we will use the
following abbreviations:
$$
\theta\equiv(\theta_1,\ldots ,\theta_{N-1}),\;\psi(x)
\equiv(\psi_1(x),\ldots,\psi_{N-1}(x)),\;
f(\theta)\equiv (f_1(\theta),\ldots,f_{N}(\theta)).
$$
For every $u\in\Lambda_{F}$ put
\begin{equation} \label{convention}
\Phi_{u}(\rho,\theta)\equiv u(\rho f(\theta))
\end{equation}
and define
$$\frac{\partial u}{\partial\rho}(x)\equiv
\frac{\partial \Phi_{u}}{\partial\rho}(|x|,\psi(x))\quad \forall x\neq 0.
$$
Next consider the set $$\Upsilon_{F}\equiv\{u\in\Lambda_{F}:u
\equiv 0\;\;\mbox{in a neighbourhood of 0},\;\;u\;\;
\mbox{compactly supported}\};
$$
then for $r>0$ and $u\in\Upsilon_{F}$ the auxiliary functional
${\mathcal{E}}_{r}$ is defined as
$${\mathcal{E}}_{r}(u)=\int_{{\mathbb R}^{N}}\frac{1}{|x|^{N-1}}
\Big(\frac{1}{2}\Big({\mathcal{V}}(r)|u|^{2}+\Big|
\frac{\partial u}{\partial\rho}\Big|^{2}\Big)+\frac{1}{p}
\Big|\frac{\partial u}{\partial\rho}\Big|^{p}\,dx+W(u)\Big)\,dx
$$
where $\mathcal{V}$ has been defined in (\ref{notation}).
We notice that ${\mathcal{E}}_{r}$ is well defined in $\Upsilon_{F}$.
Using the Cauchy-Schwartz inequality and the relation
$\sum_{i=1}^{N}|f_{i}(\psi(x))|^{2}\equiv1$ for all $x\neq 0$, we obtain
$$\Big|\frac{\partial u}{\partial\rho}(x)\Big|=\Big|\frac{\partial
\Phi_{u}}{\partial \rho}(|x|,\psi(x))\Big|
=\Big|\sum_{i=1}^{N}\frac{\partial u}{\partial x_{i}}(x)f_{i}(\psi(x))
\Big|\leq |\nabla u(x)|\quad \mbox{a.e. in}\;\;{\mathbb R}^{N}.
$$
Hence, taking $\gamma=2,p$, we obtain that for $u\in \Lambda_{F}$
\begin{equation} \label{uti}\int_{{\mathbb R}^{N}}
\frac{1}{|x|^{N-1}}\left|\frac{\partial u}{\partial \rho}
\right|^{\gamma}\equiv\int_{{\mathbb R}^{N}}\frac{1}{|x|^{N-1}}\left|
\frac{\partial \Phi_{u}}{\partial \rho}(|x|,\psi(x))\right|^{\gamma}
\leq \int_{{\mathbb R}^{N}}\frac{1}{|x|^{N-1}}|\nabla u|^{\gamma}\,dx.
\end{equation}
To study the properties of the functional ${\mathcal{E}}_{r}$ we will
need the elementary results given by the following two lemmas.
\begin{lemma} \label{change} Let $r,R,\tau>0$ with $r-R>0$ and
$f\in L^{1}(C(r-R, r+R),{\mathbb R})$. Then there results
\begin{eqnarray*}
\lefteqn{ \int_{C(\tau,\tau +2R)}f\Big(x+(r-R-\tau)\frac{x }{|x|}\Big)
\,dx }\\
&=& \int_{C(r-R, r+R)}\Big(\frac{|x|-r+R+\tau}{|x|}\Big)^{N-1}f(x)\,dx
\end{eqnarray*}
and
\begin{eqnarray*}
\lefteqn{\int_{C(\tau,\tau+2R)}\Big(\frac{|x|+r-R-\tau}{|x|}\Big)^{N-1}
f\big(x+(r-R-\tau)\frac{x }{|x|}\big)\,dx }\\
&=&\int_{C(r-R, r+R)}f(x)\,dx\,.\hspace{4cm}
\end{eqnarray*}
\end{lemma}
\begin{theorem} \label{change1} Let $A\subset {\mathbb R}^{N}$ a
measurable set, $\{f_n\}\subset C(A,{\mathbb R})$ and
$g_n:A\to {\mathbb R}$ two sequences of functions such that
$g_n$ measurable, $g_n(x)\geq 0$ a.e. in ${\mathbb R}^{N}$,
and $f_n\to f$ uniformly in $A$ as $n\to+\infty$
for some $f\in C(A,{\mathbb R})$ with $\inf_{A}|f(x)|=\nu>0$.
Then
$$\displaylines{
\limsup_{n\to+\infty}\int_{A}f_ng_n\,dx=\limsup_{n\to+\infty}
\int_{A}f\,g_n\,dx,\cr
\liminf_{n\to+\infty}\int_{A}f_ng_n\,dx
=\liminf_{n\to+\infty}\int_{A}f\,g_n\,dx.
}$$ \end{theorem}
The usefulness of having introduced the auxiliary functional
${\mathcal{E}}_{r}$ will be clear in the next lemma which establish a
relation between the behaviour of the two functional $E_h$ and
${\mathcal{E}}_{r}$ with respect to suitable sequences of functions in
the set $\Upsilon_{F}$.
\begin{theorem} \label{impo} The following two statements hold:
\begin{enumerate}
\item Let $h_n\to 0$ be an arbitrary sequence and consider
$\{u_{h_n}\}\subset\Upsilon_{F}$ and
$\{r_{h_n}\}\subset (0,+\infty)$ such that
\begin{equation} \label{hyp1}
u_{h_n}\equiv 0\quad in\quad{\mathbb R}^{N}\setminus
C(r_{h_n}-R,r_{h_n}+R
)\end{equation}
and
\begin{equation} \label{hyp2}
h_nr_{h_n}\to r\quad as\quad n\to +\infty
\end{equation}
for some $R,r>0$. For $n\in{\mathbb N}$ define
$w_{h_n}(x)\equiv u_{h_n}\left(x+\left(r_{h_n}-2R\right)
\frac{x}{|x|}\right)$, in $C(R,3R)$ and $w_{h_n}\equiv 0$
everywhere else. Then there results
$$\liminf_{n\to +\infty}h_n^{N-1}E_{h_n}(u_{h_n})\geq r^{N-1}
\liminf_{n\to +\infty}{\mathcal{E}}_{r}(w_{h_n}).$$
\item Let $w\in\Upsilon_{F}$ and $r>0$ and for every $h>0$ set
$ v_h(x)\equiv w\left(x-\frac{r}{h}\frac{x}{|x|}\right)$ for
$|x|>\frac{r}{h}$ and $ v_h\equiv 0$ everywhere else. Then it is
$$\limsup_{h\to 0^{+}}h^{N-1}E_h(v_h)\leq r^{N-1}
{\mathcal{E}}_{r}(w).$$
\end{enumerate}
\end{theorem}
\paragraph{Proof.}
{\bf Part 1.} To avoid triviality, assume $\liminf_{n\to
+\infty}h_n^{N-1}E_{h_n}(u_{h_n})<+\infty$. According to
(\ref{hyp2}) it is $r_{h_n}\to +\infty$ as $n\to +\infty$ and
then it makes sense to assume $r_{h_n}-R>0$ for every
$n\in{\mathbb N}$. Then by lemma \ref{change},
\begin{eqnarray*}
\lefteqn{\liminf_{n\to +\infty}\int_{C(R,3R)}|w_{h_n}|^{2}\,dx}\\
&=&\liminf_{n\to +\infty}\int_{C(r_{h_n}-R,r_{h_n}+R)}
\Big(\frac{|x|-r_{h_n}+2R}{|x|}\Big)^{N-1}
|u_{h_n}|^{2}\,dx\\
&\leq&
\liminf_{n\to +\infty}h_n^{N-1}\Big(\frac{3R}{h_n(r_{h_n}-R)}
\Big)^{N-1}\int_{{\mathbb R}^{N}}|u_{h_n}|^{2}\,dx \\
&=&\frac{2}{V_0}\big(\frac{3R}{r}\big)^{N-1}\liminf_{n\to
+\infty}h_n^{N-1}E_{h_n}(u_{h_n})<+\infty\,.
\end{eqnarray*}
Observe that each $w_{h_n}$ belongs to $\Upsilon_{F}$, so that
${\mathcal{E}}_{r}(w_{h_n})$ is well defined. Using
lemma \ref{change} again, there follows
\begin{eqnarray}
\lefteqn{\int_{C(r_{h_n}-R,
r_{h_n}+R)}\Big(\frac{1}{2}{\mathcal{V}}_{h_n}(|x|)
|u_{h_n}|^{2}+W(u_{h_n})\Big)\,dx } \nonumber\\
&=&
\int_{C(R,3R)}\Big(\frac{|x|+r_{h_n}-2R}{|x|}\Big)^{N-1}\label{there}\\
&&\times \Big(\frac{1}{2}{\mathcal{V}}_{h_n}(|x|+r_{h_n}-2R)
|w_{h_n}|^{2}+W(w_{h_n})\Big)\,dx.\nonumber
\end{eqnarray}
Using the notation in (\ref{convention}), we
infer that $\Phi_{w_{h_n}}(|x|,\psi(x))=\Phi_{u_{h_n}}(|x|
+r_{h_n}-2R,\psi(x))$ for all $x\neq 0$. Then, by applying
again lemma \ref{change} and using (\ref{uti}), for $\gamma=2,p$,
we can write
\begin{eqnarray}
\lefteqn{ \int_{C(r_{h_n}-R,
r_{h_n}+R)}\frac{1}{\gamma}|\nabla u_{h_n}|^{\gamma}\,dx}\nonumber\\
&\geq& \int_{C(r_{h_n}-R,
r_{h_n}+R)}\frac{1}{\gamma}\left|\frac{\partial
u_{h_n}}{\partial \rho}\right|^{\gamma}\,dx \label{there1}\\
&=&\int_{C(R,3R)}\frac{1}{\gamma}\left(\frac{|x|+r_{h_n}-2R}{|x|}
\right)^{N-1}\left|\frac{\partial
w_{h_n}}{\partial \rho}\right|^{\gamma}\,dx.\nonumber
\end{eqnarray}
Note
that we have used the equality
$\psi\left(x+(r_{h_n}-2R)\frac{x}{|x|}\right)=\psi(x)$. Now
combine (\ref{there}) with (\ref{there1}) and obtain
\begin{eqnarray*}
\lefteqn{\liminf_{n\to +\infty}h_n^{N-1}E_{h_n}(u_{h_n})}\\
&\geq& \liminf_{n\to
+\infty}\Big[\int_{C(R,3R)}\big(\frac{h_n(|x|+r_{h_n}-2R)}{|x|}
\big)^{N-1}\frac{1}{2}\big({\mathcal{V}}(r)
|w_{h_n}|^{2}+\big|\frac{\partial w_{h_n}}{\partial
\rho}\big|^{2}\big)\,dx\\
&&+\int_{C(R,3R)}\big(\frac{h_n(|x|+r_{h_n}-2R)}{|x|}\big)^{N-1}
\big(\frac{1}{p}\big|\frac{\partial w_{h_n}}{\partial
\rho}\big|^{p}+W(w_{h_n})\big)\,dx\Big]\\
&&+\frac{1}{2}\liminf_{n\to+\infty}\int_{C(R,3R)}
\big(\frac{h_n(|x|+r_{h_n}-2R)}{|x|}\big)^{N-1}\\
&&\times \big({\mathcal{V}}_{h_n}(|x|+r_{h_n}-2R)-{\mathcal{V}}(r)\big)
|w_{h_n}|^{2}\,dx\,.
\end{eqnarray*}
It's immediate to prove that
$\left(\frac{h_n(|x|+r_{h_n}-2R)}{|x|}\right)^{N-1}\to
\frac{1}{|x|^{N-1}}r^{N-1}$ uniformly in $C(R,3R)$; hence lemma
\ref{change1} applies in both integral and gives
\begin{eqnarray*}
\lefteqn{ \liminf_{n\to
+\infty}h_n^{N-1}E_{h_n}(u_{h_n})}\\
&\geq& r^{N-1}\liminf_{n\to +\infty}{\mathcal{E}}_{r}(w_{h_n})\\
&&+ \frac{r^{N-1}}{2}\liminf_{n\to
+\infty}\int_{C(R,3R)}\frac{1}{|x|^{N-1}}\left({\mathcal{V}}_{h_n}(|x|
+r_{h_n}-2R)-{\mathcal{V}}(r)\right)
|w_{h_n}|^{2}\,dx\,.
\end{eqnarray*}
On the other hand
${\mathcal{V}}_{h_n}(|x|+r_{h_n}-2R)\to {\mathcal{V}}(r)$
uniformly in $C(R,3R)$, while in the first part of the proof we
have obtained $\liminf_{n\to
+\infty}\int_{C(R,3R)}|w_{h_n}|^{2}\,dx<+\infty$; then we easily
deduce $$\liminf_{n\to
+\infty}\int_{C(R,3R)}\frac{1}{|x|^{N-1}}\left({\mathcal{V}}_{h_n}(|x|+r_{h_n}-2R)-{\mathcal{V}}(r)\right)
|w_{h_n}|^{2}\,dx=0$$ and the desired conclusion follows.
\noindent{\bf Part 2.} Since $w$ is compactly supported, for every $h>0$
it is $v_h\in\Lambda_{F}(h)$. Then we can write
\begin{equation} \label{gng}
E_h(v_h)=\frac{1}{2}\int_{{\mathbb R}^{N}}
\left({\mathcal{V}}_h(|x|)|v_h|^{2}+|\nabla v_h|^{2}\right)\,dx
+\frac{1}{p}\int_{{\mathbb R}^{N}}|\nabla v_h|^{p}\,dx
+\int_{{\mathbb R}^{N}}W(v_h)\,dx.
\end{equation}
We begin by analyzing the behaviour of the terms which do not involve
the gradient. The definition of $\Upsilon_{F}$ assure the existence
of $\tau>0$ and $ R>0$ such that $w\equiv 0$ in
${\mathbb R}^{N}\setminus C(\tau, R)$. Then by applying
lemma \ref{change} we deduce
\begin{eqnarray*}
\lefteqn{\int_{C\left(\frac{r}{h}+\tau,\frac{r}{h}+R\right)}
\left(\frac{1}{2}{\mathcal{V}}_h(|x|)|v_h|^{2}+W(v_h)\right)\,dx}\\
&=&\frac{1}{2}\int_{C(\tau,R)}\left(\frac{|x|+\frac{r}{h}}{|x|}\right)
^{N-1}{\mathcal{V}}_h\left(|x|+\frac{r}{h}\right)|w|^{2}\,dx \\
&&+\int_{C(\tau,R)}\left(\frac{|x|+\frac{r}{h}}{|x|}\right)^{N-1}W(w)\,dx.
\end{eqnarray*}
Since ${\mathcal{V}}\left(h|x|+r\right)$ and
$h^{N-1}\left(\frac{|x|+\frac{r}{h}}{|x|}\right)^{N-1}$
converge respectively to ${\mathcal{V}}(r)$ and to
$r^{N-1}/|x|^{N-1}$ uniformly on $C(\tau,R)$ as $h\to 0^{+}$,
\begin{eqnarray}
\lefteqn{\lim_{h\to 0}h^{N-1}\int_{C(\frac{r}{h}+\tau,\frac{r}{h}+R)}
\left(\frac{1}{2}{\mathcal{V}}_h(|x|)|v_h|^{2}+W(v_h))\right)\,dx
}\label{gng1}\\
&=& r^{N-1}\int_{C(\tau,R)}\frac{1}{|x|^{N-1}}\left(\frac{1}{2}
{\mathcal{V}}(r)|w|^{2}+W(w)\right)\,dx\,. \nonumber
\end{eqnarray}
Now we estimate of the integrals in (\ref{gng}) containing gradient terms.
According to the notation in (\ref{convention}), it is
immediate to show that
$ \Phi_{v_h}(|x|,\psi(x))\equiv \Phi_{w}\left(|x|-\frac{r}{h},\psi(x)
\right)$. Then compute
\begin{eqnarray}
\lefteqn{ \int_{C(\frac{r}{h}+\tau,\frac{r}{h}+R)}|\nabla
v_h|^{2}\,dx} \nonumber\\
&=&\int_{C(\frac{r}{h}+\tau,\frac{r}{h}+R)}\sum_{\alpha=1}^{N}
\big|\frac{\partial{ v_h}}{\partial x_{\alpha}}\big|^{2}\,dx
\label{inse}\\
&=&\int_{C(\frac{r}{h}+\tau,\frac{r}{h}+R)}\sum_{\alpha=1}^{N}
\big|\frac{\partial \Phi_{v_h}}{\partial \rho}(|x|,\psi(x))
\frac{x_{\alpha}}{|x|}+\sum_{i=1}^{N-1}\frac{\partial
\Phi_{v_h}}{\partial \theta_{i}}(|x|,\psi(x))\frac{\partial
\psi_{i}(x)}{\partial x_{\alpha}}\big|^{2}\,dx \nonumber\\
&=& \int_{C(\tau,R)}\big(\frac{|x|+\frac{r}{h}}{|x|}\big)^{N-1}
\sum_{\alpha=1}^{N}\big|\frac{\partial \Phi_{w}}{\partial \rho}(|x|,
\psi(x))\frac{x_{\alpha}}{|x|} \nonumber\\
&&+\sum_{i=1}^{N-1}\frac{\partial
\Phi_{w}}{\partial \theta_{i}}(|x|,\psi(x))
\frac{\partial \psi_{i}}{\partial x_{\alpha}}\big(x+\frac{r}{h}
\frac{x}{|x|}\big)\big|^{2}dx\,.\nonumber
\end{eqnarray}
In the above equalities we have used the fact that
$\psi(x)\equiv\psi(\lambda x) $ for every $x\neq 0$ and $\lambda>0
$. Observe that another consequence of the invariance of the
functions $\psi_{i}$ over the rays is the following
$$\left(1+\frac{r}{h|x|}\right)\frac{\partial \psi_{i}}{\partial
x_{\alpha}}\left(x+\frac{r}{h}\frac{x}{|x|}\right)\equiv
\frac{\partial \psi_{i}}{\partial x_{\alpha}}(x)\quad \forall
x\neq 0.
$$
By inserting this equality in (\ref{inse}) one has
\begin{eqnarray*}
\lefteqn{\int_{C(\frac{r}{h}+\tau,\frac{r}{h}+R)}\left|\nabla
v_h\right|^{2}\,dx}\\
&=&\int_{C(\tau,R)}\left(\frac{|x|+\frac{r}{h}}{|x|}\right)^{N-1}\\
&&\times\sum_{\alpha=1}^{N}\Big|\frac{\partial\Phi_{w}}{\partial
\rho}(|x|,\psi(x))\frac{x_{\alpha}}{|x|}+\frac{h|x|}{h|x|+r}\sum_{i=1}^{N-1}\frac{\partial
\Phi_{w}}{\partial \theta_{i}}(|x|,\psi(x))\frac{\partial
\psi_{i}}{\partial x_{\alpha}}(x)\Big|^{2}dx.
\end{eqnarray*}
Then, since
$\left(\frac{h|x|+r}{|x|}\right)^{N-1}\to
\frac{1}{|x|^{N-1}}r^{N-1}$ uniformly in $C(\tau,R)$, by taking
the limit as $h\to 0^{+}$ and using lemma \ref{change1}, we get
\begin{eqnarray}
\lefteqn{\limsup_{h\to 0^{+}}h^{N-1}\int_{C(\frac{r}{h}
+\tau,\frac{r}{h}+R)}\left|\nabla v_h\right|^{2}\,dx}\nonumber\\
&=& \limsup_{h\to
0^{+}}\int_{C(\tau,R)}\frac{r^{N-1}}{|x|^{N-1}} \label{inse1}\\
&&\times \sum_{\alpha=1}^{N}\left|\frac{\partial \Phi_{w}}{\partial
\rho}(|x|,\psi(x))\frac{x_{\alpha}}{|x|}+\frac{h|x|}{h|x|+r}
\sum_{i=1}^{N-1}\frac{\partial
\Phi_{w}}{\partial \theta_{i}}(|x|,\psi(x))\frac{\partial
\psi_{i}}{\partial x_{\alpha}}(x)\right|^{2}dx\,.\nonumber
\end{eqnarray}
>From (\ref{uti}) we deduce $\frac{\partial \Phi_{w}}{\partial
\rho}(|x|,\psi(x))\in L^{2}(C(\tau,R),{\mathbb R}^{N+1})$ and, as
a consequence, $\frac{\partial \Phi_{w}}{\partial
\rho}(|x|,\psi(x))\frac{x_{\alpha}}{|x|}\in
L^{2}(C(\tau,R),{\mathbb R}^{N+1})$. On the other hand an
elementary calculus shows that $$
\sum_{i=1}^{N-1}\frac{\partial
\Phi_{w}}{\partial \theta_{i}}(|x|,\psi(x))\frac{\partial
\psi_{i}}{\partial x_{\alpha}}(x)=\frac{\partial{ w}}{\partial
x_{\alpha}}(x)-\frac{\partial \Phi_{w}}{\partial
\rho}(|x|,\psi(x))\frac{x_{\alpha}}{|x|}
$$
by which
$\sum_{i=1}^{N-1}\frac{\partial \Phi_{w}}{\partial
\theta_{i}}(|x|,\psi(x))\frac{\partial \psi_{i}}{\partial
x_{\alpha}}(x)\in L^{2}(C(\tau,R),{\mathbb R}^{N+1})$. Thus we
derive \begin{equation}
\label{inse2}\frac{h|x|}{h|x|+r}\sum_{i=1}^{N-1}\frac{\partial
\Phi_{w}}{\partial \theta_{i}}(|x|,\psi_{j}(x))\frac{\partial
\psi_{i}}{\partial x_{\alpha}}(x)\to 0\quad \mbox{in}\quad
L^{2}(C(\tau,R),{\mathbb R}^{N+1}).
\end{equation}
Combining (\ref{inse1}) together with (\ref{inse2}) we obtain
\begin{eqnarray*}
\lefteqn{\limsup_{h\to
0^{+}}h^{N-1}\int_{C(\frac{r}{h}+\tau,\frac{r}{h}+R)}\left|\nabla
v_h\right|^{2}\,dx }\\
&=& r^{N-1}\int_{C(\tau,R)}\frac{1}{|x|^{N-1}}
\sum_{\alpha=1}^{N}\left|\frac{\partial \Phi_{w}}{\partial
\rho}(|x|,\psi(x))\frac{x_{\alpha}}{|x|}\right|^{2}\,dx\\
&\leq& r^{N-1}\int_{C(\tau,R)}\frac{1}{|x|^{N-1}}\left|\frac{\partial
w}{\partial \rho}\right|^{2}\sum_{\alpha=1}^{N}
\frac{x_{\alpha}^{2}}{|x|^{2}}\,dx\\
&=&r^{N-1}\int_{C(\tau,R)}\frac{1}{|x|^{N-1}}\left|\frac{\partial
w}{\partial \rho}\right|^{2}\,dx\,.
\end{eqnarray*}
In the same way one obtains
$$\limsup_{h\to 0^{+}}h^{N-1}\int_{C(\frac{r}{h}
+\tau,\frac{r}{h}+R)}\left|\nabla
v_h\right|^{\gamma}\,dx\leq
r^{N-1}\int_{C(\tau,R)}\frac{1}{|x|^{N-1}}\left|\frac{\partial
w}{\partial \rho}\right|^{\gamma}\,dx.$$ The thesis follows from
(\ref{gng1}) and the above inequalities.\hfill $\diamondsuit$
Before going on with the statements of our results for every $r>0$
set
$$\widetilde{{\mathcal{E}}}_{r}^{\ast}=\inf_{u\in\Upsilon_{F}^{\ast}}{\mathcal{E}}_{r}(u),$$
where, with obvious notation, we have put
$\Upsilon_{F}^{\ast}=\{u\in \Upsilon_{F}: ch(u)\neq 0\}$. We
point out that in \cite{BFPD} the authors exhibited some examples
of functions $u\in \Lambda_{F}$ with $ch(u)\neq 0$ and such that
$K(u)\subset C(R_1,R_{2})$ for some $00\longmapsto
\widetilde{{\mathcal{E}}}_{r}^{\ast}$ is continuous.
\item [ii)] For every $M>0$ there exists $\sigma=\sigma(M)>0$ such that
for every $r>0$,
$$\widetilde{{\mathcal{E}}}_{r}^{\ast}0}\widetilde{{\mathcal{E}}}_{r}^{\ast}
>0\,.\end{equation}
\item [iii)] If $\{r_n\}\subset (0,+\infty)$ is a sequence verifying
${\mathcal{V}}(r_n)\to +\infty$ as $n\to +\infty$, then there results
$\limsup_{n\to +\infty}\widetilde{{\mathcal{E}}}_{r_n}^{\ast}=+\infty$.
\end{enumerate}
\end{theorem}
\paragraph{Proof.} i) Choose $b>a>0$. From the continuity of
${\mathcal{V}}$ we deduce that the function
$r\in[a,b]\longmapsto \widetilde{{\mathcal{E}}}_{r}^{\ast}$ is
bounded; then choose $M>0$ such that
$\widetilde{{\mathcal{E}}}_{r}^{\ast}< M$ for all $r\in[a,b]$.
Hence the definition of ${\mathcal{E}}_{r}$ implies that each
$\widetilde{{\mathcal{E}}}_{r}^{\ast}$ can be also written in the
form
$$\widetilde{{\mathcal{E}}}_{r}^{\ast}\equiv\inf
\left\{{\mathcal{E}}_{r}(u):u\in\Upsilon_{F}^{\ast},\;\int_{{\mathbb
R}^{N}} \frac{1}{|x|^{N-1}}|u|^{2}\,ds\leq \frac{2M}{
V_0}\right\} .$$ Thus for every $u\in \Upsilon_{F}^{\ast}$ with
$\int_{{\mathbb R}^{N}}|u|^{2}\,ds\leq \frac{2M}{V_0}$ and for
every $s,s'\in[a,b]$ it is
$$
\widetilde{{\mathcal{E}}}_{s}^{\ast}\leq{\mathcal{E}}_{s}(u)
\leq{\mathcal{E}}_{s'}(u)+|{\mathcal{V}}(s)-{\mathcal{V}}(s')|
\frac{M}{V_0}.
$$
By taking the infimum on the right side we get $
\widetilde{{\mathcal{E}}}_{s}^{\ast}\leq
\widetilde{{\mathcal{E}}}_{s'}^{\ast}+|{\mathcal{V}}(s)-{\mathcal{V}}(s')|\frac{M}{V_0}$.
By changing the role of $s$ and $s'$ we obtain the symmetric
inequality, by which, taking into account of the continuity of
the potential $V$, the part i) of the thesis.
ii) Fix $M>0$ arbitrarily. By the definition of ${\mathcal{E}}_{r}$
there follows that for every $r>0$ with $\widetilde{{\mathcal{E}}}_{r }
^{\ast}s'>0$,
by which it follows that the set of functions
$\{\varphi_{[u]}:u\in\Gamma^{\ast}\}$ is equi-uniformly continuous
in $(0,+\infty)$. This fact provides the existence of $\sigma>0$ such
that
\begin{equation} \label{euc}
\forall u\in\Gamma^{\ast},\quad\forall s,s'\in(0,+\infty),\;\;|s-s'|
<\sigma\Rightarrow |\varphi_{[u]}(s)- \varphi_{[u]}(s')|\leq\frac{1}{2}.
\end{equation}
On the other hand, taking into account that if
$u\in\Upsilon_{F}^{\ast}$ it is $ch(u)\neq 0$, by (\ref{prop}) we deduce
that for every $u\in\Gamma^{\ast}$ there is $s_{u}\in (0,+\infty)$
verifying $\varphi_{[u]}(s_{u})>1$. From (\ref{euc}) there follows
$$|\varphi_{[u]}(s)|>\frac{1}{2}\quad \forall s\in(s_{u}-\sigma,s_{u}
+\sigma).
$$
By using (\ref{euc0}), an elementary calculus shows that for every $r>0$
with $\widetilde{{\mathcal{E}}}_{r}^{\ast}0}\widetilde{{\mathcal{E}}}_{r}^{\ast}=0$
and take $\{r_n\}\subset (0,+\infty)$ such that
$\widetilde{{\mathcal{E}}}_{r_n}^{\ast}\to 0$. Then, since the sequence
$\{\widetilde{{\mathcal{E}}}_{r_n}^{\ast}\}$ is bounded, we can apply
the first part of ii) and obtain for every $n\in{\mathbb N}$ and
for some $\sigma>0$: $\widetilde{{\mathcal{E}}}_{r_n}^{\ast}\geq
{\mathcal{V}}(r_n)\sigma \geq V_0\sigma$, which is a contradiction.
iii) Assume by absurd that $\limsup_{n\to
+\infty}\widetilde{{\mathcal{E}}}_{r_n}^{\ast}<+\infty$ and fix
$M>0$ such that $\widetilde{{\mathcal{E}}}_{r_n}^{\ast}0$ such that
$$\widetilde{{\mathcal{E}}}_{r_n}^{\ast}\geq
{\mathcal{V}}(r_n)\sigma\to +\infty\;\;\mbox{as}\;\; n\to
+\infty $$ and we achieve the desired contradiction. \hfill
$\diamondsuit$
Now we have in our hands all the instruments to prove in the next
section the second existence and concentration result of this
work.
\section{Concentration Around a Circle}
In this section we solve the problem of finding a family of
solutions $\{v_h\}$ to equation (\ref{equ1}) exhibiting a
concentration behaviour around a circle $\{x\in{\mathbb
R}^{N}:|x|=\overline{r}\}$. Toward this end suppose
assumptions a)-g) hold and assume $r_0>0$ such that
\begin{equation}
\label{assures}r_0^{N-1}\widetilde{{\mathcal{E}}}_{r_0}^{\ast}>
\inf_{r\geq r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}
.\end{equation}
To achieve our results we need a suitable modification
of the nonlinear term $W$: because of assumption f) choose
$\overline{c}>1$ such that
$\overline{c}|W''(0)|0$:
indeed it is sufficient to take $u\in\Upsilon_{F}^{\ast}$ and set
$\widetilde{u}(x)=u(x-\frac{r_0}{h}\frac{x}{|x|})$ if $|x|\geq \frac{r_0}{h}$
and $\widetilde{u}(x)=0$ everywhere else. By lemma \ref{novi} we get
$ch(\widetilde{u})=ch(u)\neq 0$, by which
$\widetilde{u}\in\widetilde{\Lambda}_{F}^{\ast}(h)$ for every $h>0$.
The first object is to minimize the energy functional $J_h$ in the set
$\widetilde{\Lambda}_{F}^{\ast}(h)$. To this aim we need the estimate
provided by the following lemma.
\begin{lemma}\label{limsup}$\limsup_{h\to 0^{+}}h^{N-1}
\widetilde{c}_h^{\ast}\leq \inf_{r\geq r_0}r^{N-1}
\widetilde{{\mathcal{E}}}_{r}^{\ast}$.\end{lemma}
\paragraph{Proof.} Consider $r\geq r_0$ arbitrarily and
$w\in \Upsilon_{F}^{\ast}$. For every $h>0$, define
$\widetilde{w}_h:{\mathbb R}^{N}\to {\mathbb R}^{N+1}$ by setting
$$\widetilde{w}_h(x)=\cases{
0& if $|x|\leq r/h$, \cr
w\big(x-\frac{r}{h\,|x|}x\big) & if $|x|> r/h$ . \cr}
$$
It is obvious that $\widetilde{w}_h\in \Lambda_{F}(h)$ for every $h>0$
and furthermore $\widetilde{w}_h(x)=0$ if $|x|\leq r_0/h$.
By lemma \ref{novi} there follows $ch\left(\widetilde{w}_h,
{\mathbb R}^{N}\setminus \overline{B}\left(0,r_0/h\right)\right)
=ch(\widetilde{w}_h)=ch(w)\neq 0$ so that
$ \widetilde{w}_h\in\widetilde{\Lambda}_{F}^{\ast}(h)$.
Note that by construction it is $J_h(\widetilde{w}_h)
=E_h(\widetilde{w}_h)$ for every $h>0$. From part 2 of lemma
\ref{impo} we have:
$$\limsup_{h\to 0^{+}}
h^{N-1}\widetilde{c}_h^{\ast}\leq \limsup_{h\to 0^{+}}
h^{N-1}E_h(\widetilde{w}_h)\leq r^{N-1}{\mathcal{E}}_{r}(w).
$$
By taking the infimum on the right side of last inequality first
for $w\in \Upsilon_{F}^{\ast}$ and then for $r\geq r_0$ we
achieve the desired conclusion.\hfill $\diamondsuit$
Now we are able to give the main result of this section.
\begin{theorem} \label{imp}
Assume that assumptions a)-g) hold and, in addition, (\ref{assures}).
Then there exists $h_0>0$ such that for every $h\in(0,h_0)$
there is a solution $u_h\in \widetilde{\Lambda}_{F}^{\ast}(h)$
to equation (\ref{equ2}). Furthermore if for every $h\in(0,h_0)$
we set
\begin{equation} \label{related1}v_h(x)=u_h\left(\frac{x}{h}\right),
\end{equation}
then $v_h$ is a solution of (\ref{equ1}) and the family $\{v_h\}$
exhibits the following concentration behaviour as $h\to 0^{+}$:
for each sequence $h_n\to 0^{+}$ there exists a subsequence,
still denoted by $h_n$, such that $|v_{h_n}|$ has a circle of
local maximum points $\{x\in{\mathbb R}^{N}:|x|=r_{h_n}\}$
with $|v_{h_n}(x)|>1$ for $|x|=r_{h_n}$ and $$ r_{h_n}\to
\overline{r},\quad \overline{r}^{N-1}
\widetilde{{\mathcal{E}}}^{\ast}_{\overline{r}}=\inf_{r\geq
r_0}r^{N-1} \widetilde{{\mathcal{E}}}^{\ast}_{r};
$$
also, for every $\delta>0$, it holds
$v_{h_n}\to 0$ as $n\to +\infty$ uniformly in the set
$\{x:\big||x|-\overline{r}\big|\geq\delta\}$.
\end{theorem}
\paragraph{Proof.} Fix $h>0$ arbitrarily and consider $u_{k}^{h}$
a minimizing sequence in $\widetilde{\Lambda}_{F}^{\ast}(h)$ for
$J_h$; it has obviously bounded energy because of
(\ref{coercive}). Then up to a subsequence we have
$u_{k}^{h}\rightharpoonup u_h$ weakly in $H_h$ as $ k\to
+\infty$, for some $u\in F_h$ and, from lemma \ref{compactinj},
\begin{equation} \label{compi}u_{k}^{h}\to u_h\quad
\mbox{uniformly in }\;\;{\mathbb R}^{N}\quad \mbox{as}\quad k\to
+\infty\,.
\end{equation} Proposition 4.2 implies $u_h\in\Lambda_{F}(h)$.
By (\ref{compi}) we deduce
$$|u_h(x)|\leq \frac{\overline{\varepsilon}}{2}\quad \forall x\in
\overline{C}\left(\frac{r_0}{2h},\frac{r_0}{h}\right).
$$
Using lemma \ref{cont} we infer that the sequence
$\left\{ch\left(u_{k}^{h},{\mathbb R}^{N}\setminus\overline{B}
\left(0,\frac{r_0}{h}\right)\right)\right\}$ is definitively constant
and moreover, for $k$ large enough,
\begin{equation} \label{mag12}
ch\left(u_h,{\mathbb R}^{N}\setminus\overline{B}\left(0,
\frac{r_0}{h}\right)\right)=ch\left(u_{k}^{h},{\mathbb R}^{N}
\setminus\overline{B}\left(0,\frac{r_0}{h}\right)\right)\neq 0\,.
\end{equation}
>From now on we focus our attention on a generic sequence $h_n\to
0^{+}$.
For every $n\in{\mathbb N}$ let $\varphi_{h_n}: [0,+\infty)\to
{\mathbb R}$ such that
$$
\varphi_{h_n}(|x|)=|u_{h_n}(x)|\quad \forall x\in{\mathbb R}^{N}.
$$
The weakly lower semi-continuity of $J_h$ leads to the inequality
\begin{equation} \label{wls}
J_{h_n}(u_{h_n})\leq \liminf_{k\to +\infty}J_{h_n}( u_{k}^{h_n})
=\widetilde{c}_{h_n}^{\ast}.
\end{equation}
Taking into account of lemma \ref{limsup}, the sequence
$\{h_n^{N-1}\widetilde{c}_{h_n}^{\ast}\}$, and consequently
$\{h_n^{N-1}J_{h_n}(u_{h_n})\}$, is bounded. Taken
$\overline{\varepsilon}\in(0,1)$ given by hypothesis g), by (\ref{prop})
and (\ref{mag12}) we immediately get
$\sup_{|x|\geq\frac{r_0}{h_ {n}}}|u_{h_n}(x)|>1>\overline{\varepsilon};$
then the sequence $\{u_{h_n}\}$ satisfies all the hypotheses of
lemma \ref{preliminary} with $t\equiv r_0$ and
$\overline{\varepsilon}\equiv \varepsilon$. Then we obtain the existence
of $\ell\in{\mathbb N}$, $R_1,\ldots,R_{\ell}>0$ and $\ell$
sequences of positive numbers $\{r_{h_n}^{1}\},\ldots,\{r_{h_n}
^{\ell}\}$, with $r_{h_n}^{i}>\frac{r_0}{h_n}$, such that, up to
a subsequence
\begin{eqnarray} \label{dim12}
&\{h_nr_{h_n}^{i}\}\quad \mbox{is bounded}\quad \forall
i=1,\ldots,\ell;&\\
&\varphi_{h_n}(r_{h_n}^{i})>\frac{\overline{\varepsilon}}{2}\quad
\forall\,n\in{\mathbb N},\quad \forall i=1,\ldots,\ell ;&
\nonumber\\
&r_{h_n}^{i} \mbox{is the maximum point for }
\varphi_{h_n}\mbox{in }&\nonumber\\
&\big[\frac{r_0}{2h_n},+\infty\big)\setminus
\cup_{j\frac{r_0}{h_n}$,
we have $r_{h_n}^{i}-\frac{r_0}{2h_n}\to +\infty$.
Hence, taking into account of (\ref{allont12}), it makes sense to
assume $C\left(r_{h_n}^{i}-R_{i},r_{h_n}^{i}+R_{i}\right)\cap
C\left(r_{h_n}^{j}-R_{j},r_{h_n}^{j}+R_{j}\right)=\emptyset$
for $ i\neq j$ and $C\left(r_{h_n}^{i}-R_{i},r_{h_n}^{i}+R_{i}\right)
\subset {\mathbb R}^{N}\setminus \overline{B}\left(0,\frac{r_0}{2h_n}
\right)$ for every $n\in{\mathbb N}$.
Then from definition 3.1 and (\ref{summ}) there follows
$$ch\left(u_{h_n},{\mathbb R}^{N}\setminus
\overline{B}\left(0,\frac{r_0}{h_n}\right)\right)=\sum_{i=1}^{\ell}ch(u_{h_n},
C(r_{h_n}^{i}-R_{i},r_{h_n}^{i}+R_{i})).$$
This equality implies the existence of $
\widehat{i}\in\{1,\ldots,\ell\}$ such that
\begin{equation} \label{conta1}
ch(u_{h_n},C(r_{h_n}^{\widehat{i}}-R_{\widehat{i}},
r_{h_n}^{\widehat{i}}+R_{\widehat{i}}))\neq 0.
\end{equation}
In particular, according to (\ref{prop}) and ({\ref{max12}),
this implies that
$\varphi_{h_n}(r_{h_n}^{\widehat{i}})>1$ and, because of
(\ref{magg12}), $|u_{h_n}(x)|\leq\frac{\overline{\varepsilon}}{2}<1$
for every
$x\in \partial C(r_{h_n}^{\widehat{i}}-R_{\widehat{i}},r_{h_n}
^{\widehat{i}}+R_{\widehat{i}})$.
Hence we have \begin{equation}
\label{max123}|u_{h_n}(x)|>1\quad \forall x\in{\mathbb
R}^{N}\quad\mbox{with}\quad
|x|=r_{h_n}^{\widehat{i}}\;\;\forall\,n\in{\mathbb
N}\end{equation}
and \begin{equation} \label{max124}
\left\{x\in{\mathbb R}^{N}:|x|=r_{h_n}^{\widehat{i}}\right\}
\end{equation}
is a circle of local maximum points for $|u_{h_n}|$ for all
$n$ in ${\mathbb N}$. By the
inequality $r_{h_n}^{\widehat{i}}>\frac{r_0}{h_n}$ and by
(\ref{dim12}) we get the existence of $\overline{r}\geq0$
verifying, up to subsequence,
\begin{equation}
\label{conve}h_nr_{h_n}^{\widehat{i}}\to \overline{r}\quad
\mbox{as}\quad n\to +\infty,\quad \overline{r}\geq
r_0.\end{equation}
Now for every $R>2R_{\widehat{i}}$ and $n\in{\mathbb N}$ consider
$\eta_{h_n}^{R}\in C_0^{\infty}({\mathbb R}^{N},{\mathbb R})$ a
radial function so that
$$\displaylines{
\eta_{h_n}^{R}\equiv0\;\;\mbox{on}\;\;{\mathbb R}^{N}\setminus
C(r_{h_n}^{\widehat{i}}-R,r_{h_n}^{\widehat{i}}+R),\cr
\eta_{h_n}^{R}\equiv
1\;\;\mbox{on}\;\; C\left(r_{h_n}^{\widehat{i}}-\frac{R}{2},r_{h_n}
^{\widehat{i}}+\frac{R}{2}\right), \quad
0\leq\eta_{h_n}^{R}\leq1,\quad
|\nabla \eta_{h_n}^{R}|\leq \frac{c}{R}
}$$
where $c$ is a constant independent of $R$ and $n$.
Hereafter, for the sake of simplicity, for every $u\in\Lambda_{F}(h)$ and
$A\subset {\mathbb R}^{N}$ measurable we set
$$\left(J_h\right)_{\big|A}(u)=\frac{1}{2}\int_{A}\left(|\nabla u|^{2}
+V_h(x)|u|^{2}\right)\,dx+\frac{1}{p}\int_{A}|\nabla u|^{p}\,dx
+\int_{A}G(hx,u)\,dx.
$$
The key steps in the proof of the theorem are the following 6 claims.
\paragraph{Claim 1.} For every couple of positive numbers
$\delta, M>0$ there exists $R=R(\delta,M)>2R_{\widehat{i}}$ and
$\overline{n}\in{\mathbb N}$ such that for every $n\geq\overline{n}$
and for every $w_n\in\Lambda_{F}(h_n)$ satisfying
\begin{equation} \label{vera}|w_n(x)|\leq\overline{\varepsilon}\;\;for
\;\;x\in C_n^{R},\quad h_n^{N-1}J_{h_n}(w_n)\leq M,
\end{equation}
there results
$$h_{n }^{N-1}J_{h_n}(\eta_{h_n}^{R}w_n)\leq h_{n }^{N-1}
(J_{h_n})_{\big|C_n^{R}} (w_n)+\delta,
$$
where
$$C_n^{R}\equiv C\big(r_{h_n}^{\widehat{i}}-R,r_{h_n}^
{\widehat{i}}-\frac{R}{2}\big)\cup C\big(r_{h_n}^{\widehat{i}}
+\frac{R}{2},r_{h_n}^{\widehat{i}}+R\big).
$$
{\bf Proof of Claim 1.}
Fix $\delta, M>0$ arbitrarily. The basic tool is the following assertion:
for every $R>2R_{\widehat{i}}$ there exists $n_{R}>0$ such that for every
$n\geq n_{R}$ and for every $w_n\in\Lambda_{F}(h_n)$ satisfying
(\ref{vera}), it holds:
\begin{equation} \label{grad1}
h_n^{N-1}\int_{C_n^{R}}|\nabla\eta_{h_n}^{R}w_n |^{2}\,dx\leq
h_n^{N-1}\int_{C_n^{R}}|\nabla w_n|^{2}\,dx+C_1
\big(\frac{1}{R}+\frac{1}{R^{2}}\big)\end{equation}
and \begin{equation} \label{grad2}
h_n^{N-1}\int_{C_n^{R}}|\nabla\eta_{h_n}^{R}w_n|^{p}\,dx
\leq h^{N-1}\int_{C_n^{R}}|\nabla w_n|^{p}\,dx+C_{2}
\big(\frac{1}{R^{(p-1)/p}}+\frac{1}{R^{p-1}}\big)
\end{equation}
where $C_1$ and $C_{2}$ are constants depending only on $\overline{r}$,
$M$, $\overline{\varepsilon}$, $p$ and $N$.
Before proving the above assertion, observe how, assuming
(\ref{grad1}) and (\ref{grad2}), the thesis of claim 1 easily follows.
Indeed hypothesis g) and the first of (\ref{vera}) give
$W(\eta_{h_n}^{R}w_n)\leq W(w_n)$ in $ C_n^{R}$ and this implies
\begin{eqnarray} \label{co2}
\lefteqn{\frac{1}{2}\int_{{\mathbb R}^{N}}V_{h_n}(x)|
\eta_{h_n}^{R}w_n|^{2}\,dx+\int_{{\mathbb R}^{N}}W(\eta_{h_n}^{R}
w_n)\,dx}\\
&\leq& \frac{1}{2}\int_{{\mathbb R}^{N}}V_{h_n}(x)|w_n|^{2}
\,dx+\int_{{\mathbb R}^{N}}W(w_n)\,dx.\nonumber
\end{eqnarray}
Hence, taking $R=R(\delta,M)>2R_{\widehat{i}}$ sufficiently large and
combining (\ref{grad1}), (\ref{grad2}) and (\ref{co2}) we obtain the
thesis.
The rest of this proof will be devoted (\ref{grad1})
and (\ref{grad2}). Let us begin with (\ref{grad1}). Observe
\begin{eqnarray*}
\lefteqn{h_n^{N-1}\int_{C_n^{R}}|\nabla\eta_{h_n}^{R}w_n |^{2}
\,dx }\\
&\leq& h_n^{N-1}\Big(\int_{C_n^{R}}|\nabla w_n|^{2}\,dx
+\frac{c^{2}}{R^{2}}\int_{C_n^{R}}| w_n|^{2}\,dx+2\frac{c}{R}
\int_{C_n^{R}}|\nabla w_n||w_n|\,dx\Big).
\end{eqnarray*}
The H\"older inequality yields
\begin{eqnarray*}
h_n^{N-1}\int_{C_n^{R}}|\nabla \eta_{h_n}^{R}w_n|^{2}\,dx
&\leq& h_n^{N-1} \big(\int_{C_n^{R}}|\nabla w_n|^{2}\,dx+
\frac{c^{2}}{R^{2}}\int_{C_n^{R}}| w_n|^{2}\,dx\big)\\
&&+h_n^{N-1}2\frac{c}{R}\big(\int_{C_n^{R}}|\nabla w_n|^{2}\,dx
\big)^{1/2}\big(\int_{C_n^{R}}|w_n|^{2}\,dx\big)^{1/2}
\end{eqnarray*}
by which, since $h_n^{N-1}\int_{C_n^{R}}|\nabla w_n|^{2}\,dx,\leq2M$
and $h_n^{N-1}\int_{C_n^{R}}|w_n|^{2}\,dx\leq \frac{2M}{V_0}$,
(\ref{grad1}) immediately follows.
As regards (\ref{grad2}) the proof is more delicate; we will use the
following numerical inequality
$$(a+b)^{p}\leq a^{p}+(p+1)a^{p-1}b+C_{p}b^{p}
$$
which is valid if
$p\geq 3$, $C_{p}\geq \frac{1}{p}(p+1)^{p-1}(p-2)^{p-2}$ and $a,b\geq 0$.
Then we get
\begin{eqnarray*}
\lefteqn{h_n^{N-1}\int_{C_n^{R}}|\nabla\eta_{h_n}^{R}w_n|^{p}
\,dx
\;\leq\; h_n^{N-1}\int_{C_n^{R}}|\nabla w_n|^{p}\,dx }\\
&&+ h_n^{N-1}\Big((p+1)\frac{c}{R}\int_{C_n^{R}}|
\nabla w_n|^{p-1}|w_n|\,dx+C_{p}\frac{c^{p}}{R^{p}}
\int_{C_n^{R}}|w_n|^{p}\,dx\Big).
\end{eqnarray*}
Again from H\"older inequality, we obtain
\begin{eqnarray}
\lefteqn{h_n^{N-1}\int_{C_n^{R}}|\nabla\eta_{h_n}^{R}w_n|^{p}\,dx
\;\leq\; h_n^{N-1}\int_{C_n^{R}}|\nabla w_n|^{p}\,dx
+h_n^{N-1}\Big((p+1)\frac{c}{R} }\nonumber\\
&&\times \Big(\int_{C_n^{R}}|\nabla w_n|^{p}\,dx\Big)^{(p-1)/p}
\overline{\varepsilon}\,(\mathop{\rm meas}C_n^{R})^{1/p}+C_{p}
\frac{c^{p}}{R^{p}}
\overline{\varepsilon}^{p}\mathop{\rm meas}(C_n^{R})\Big).\quad \label{fol}
\end{eqnarray}
Taking into account of (\ref{conve}), an elementary calculus shows
$$\lim_{n\to +\infty}h_n^{N-1}\mathop{\rm meas}(C_n^{R})
=\omega_{N}\overline{r}^{N-1}R.$$ Then, by inserting this in (\ref{fol})
and proceeding as for the other inequality we conclude. Claim 1 is
completely proved.
Now for every $n\in{\mathbb N}$ put $\widetilde{u}_{h_n}^{R}\equiv
\eta_{h_n}^{R}u_{h_n}$.
\paragraph{Claim 2.} For every $R>2R_{\widehat{i}}$ there exists
$\widehat{n}\in{\mathbb N}$ such that for $n\geq\widehat{n}$,
$$|u_{h_n}(x)|\leq\frac{\overline{\varepsilon}}{2}\quad\forall
x\in C_n^{R},\quad\widetilde{u}^{R}_{h_n}\in\Lambda_{F}^{\ast}(h_n).
$$
{\bf Proof of Claim 2.}
According to (\ref{allont12}) for $n$ sufficiently large,
$$C(r_{h_n}^{\widehat{i}}-R,r_{h_n}^{\widehat{i}}+R)\cap
C(r_{h_n}^{i}-R_{i},r_{h_n}^{i}+R_{i})=\emptyset \quad
\forall i\in\{1,\ldots,\ell\},\;\;i\neq \widehat{i}
$$
and, since $r_{h_n}^{\widehat{i}}>\frac{r_0}{h_n}$,
$(r_{h_n}^{\widehat{i}}-R,r_{h_n}^{\widehat{i}}+R)\subset
\big[\frac{r_0}{2h_n},+\infty\big)$.
Then, by (\ref{magg12}),
\begin{equation} \label{liu}
|u_{h_n}(x)|\leq \frac{\overline{\varepsilon}}{2}\quad \forall
x\in C(r_{h_n}^{\widehat{i}}-R,r_{h_n}^{\widehat{i}}+R)\setminus
C(r_{h_n}^{\widehat{i}}-R_{\widehat{i}},r_{h_n}^{\widehat{i}}
+R_{\widehat{i}}).
\end{equation}
It is easy to show that $C_n^{R}\subset C(r_{h_n}^{\widehat{i}}
-R,r_{h_n}^{\widehat{i}}+R)\setminus C(r_{h_n}^{\widehat{i}}
-R_{\widehat{i}},r_{h_n}^{\widehat{i}}+R_{\widehat{i}})$,
then the first part of the thesis follows.
By construction there results $\widetilde{u}_{h_n}^{R}\in F_{h_n}$.
On the other hand $\widetilde{u}_{h_n}^{R}\equiv 0$ in
${\mathbb R}^{N}\setminus C(r_{h_n}^{\widehat{i}}-R,
r_{h_n}^{\widehat{i}}+R)$ and, since $R>2 R_{\widehat{i}}$,
it is $\widetilde{u}_{h_n}^{R}\equiv u_{h_n}$ in
$C(r_{h_n}^{\widehat{i}}-R_{\widehat{i}},r_{h_n}^{\widehat{i}}
+R_{\widehat{i}})$, hence we easily deduce
$\widetilde{u}_{h_n}^{R}\in\Lambda_{F}(h_n)$; furthermore from
(\ref{conta1}) and (\ref{liu}) we get
$$ch(\widetilde{u}_{h_n}^{R})=ch( \widetilde{u}_{h_n}^{R},
C(r_{h_n}^{\widehat{i}}-R_{\widehat{i}},r_{h_n}^{\widehat{i}}
+R_{\widehat{i}}))=ch( u_{h_n},C(r_{h_n}^{\widehat{i}}
-R_{\widehat{i}},r_{h_n}^{\widehat{i}}+R_{\widehat{i}}))\neq 0,
$$
by which, provided $n$ is sufficiently large,
$\widetilde{u}_{h_n}^{R}\in\Lambda_{F}^{\ast}(h_n)$.
\paragraph{Claim 3.} $\overline{r}^{N-1}
\widetilde{{\mathcal{E}}}_{\overline{r}}^{\ast}=\inf_{r>r_0}r^{N-1}
\widetilde{{\mathcal{E}}}_{r}^{\ast}$
\noindent{\bf Proof of claim 3.} Assume on the contrary that
$\overline{r}^{N-1}\widetilde{{\mathcal{E}}}_{\overline{r}}^{\ast}\neq
\inf_{r>r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}$. In
particular, because of (\ref{conve}), it would be
$\overline{r}^{N-1}\widetilde{{\mathcal{E}}}_{\overline{r}}^{\ast}>
\inf_{r>r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}$; then
take $\delta>0$ such that
\begin{equation} \label{ca01}
\overline{r}^{N-1}\widetilde{{\mathcal{E}}}_{\overline{r}}^{\ast}
>\inf_{r>r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}+\delta.
\end{equation}
Using (\ref{wls}) and lemma \ref{limsup}, choose $M>0$ such that
$h_n^{N-1}J_{h_n}(u_{h_n})< M$, for $n\in{\mathbb N}$. Then
let $R=R\left(\frac{\delta}{4}, M\right)$ given by claim 1.
According to claims 1 and 2 there exists $\overline{n}\in{\mathbb N}$
such that for every $n\geq\overline{n}$:
\begin{eqnarray} \label{ca1}
&\widetilde{u}_{h_n}^{R}\in \Lambda_{F}^{\ast}(h_n),\quad
|u_{h_n}(x)|\leq\frac{\overline{\varepsilon}}{2}\;\;\forall x\in C_n^{R}, &\\
& h_n^{N-1}J_{h_n}(\widetilde{u}_{h_n}^{R})
\leq h_n^{N-1}J_{h_n}(u_{h_n})+\frac{\delta}{4}\,.&\nonumber
\end{eqnarray}
For every $n\in{\mathbb N}$ let $w_{h_n}:{\mathbb R}^{N}\to
{\mathbb R}^{N+1}$ defined as $$ w_{h_n}(x)=\cases{0& if
$|x|\leq R$, \cr
\widetilde{u}_{h_n}^{R}\big(x+(r_{h_n}^{\widehat{i}}-2R)
\frac{x}{|x|}\big) & if $|x|>R$. \cr}
$$
It is obvious that $w_{h_n}\in \Upsilon_{F}$ for every
$n\geq\overline{n}$.
Since $\widetilde{u}_{h_n}^{R}(x)=w_{h_n}
\left(x-(r_{h_n}^{\widehat{i}}-2R)\frac{x}{|x|}\right)$
if $|x|\geq r_{h_n}^{\widehat{i}}-R$ and
$\widetilde{u}_{h_n}^{R}\equiv 0$ in $B(0,r_{h_n}^{\widehat{i}}-R)$,
from lemma \ref{novi} we obtain
$ch(w_{h_n})=ch(\widetilde{u}_{h_n}^{R})\neq 0$, which implies
$w_{h_n}\in \Upsilon_{F}^{\ast}$.
Note that, since $r_{h_n}^{\widehat{i}}>\frac{r_0}{h_n}$,
provided that $\overline{n}$ is large enough we have
$C(r_{h_n}^{\widehat{i}}-R,r_{h_n}^{\widehat{i}}+R)
\subset {\mathbb R}^{N}\setminus B\left(0,\frac{r_0}{2h_n}\right)$
for every $n\geq \overline{n}$ so that, by construction,
$E_{h_n}(\widetilde{u}_{h_n}^{R})=J_{h_n}(\widetilde{u}_{h_n}^{R})$
for all $n\geq \overline{n}$. From part 1 of lemma \ref{impo} we obtain
$$\liminf_{n\to+\infty}h_n^{N-1}J_{h_n}(\widetilde{u}_{h_n}^{R})
\geq \overline{r}^{N-1}\liminf_{n\to +\infty}
\widetilde{{\mathcal{E}}}_{\overline{r}}(w_{h_n}).
$$
Then it makes sense to assume $\overline{n}$ sufficiently large so that
for every $n\geq\overline{n}$
\begin{equation} \label{ca3}h_n^{N-1}J_{h_n}
(\widetilde{u}_{h_n}^{R})\geq\overline{r}^{N-1}\liminf_{n\to+\infty}
{\mathcal{E}}_{\overline{r}}(w_{h_n})-\frac{\delta}{2},
\end{equation}
and, by lemma \ref{limsup},
\begin{equation} \label{ca5}
h_n^{N-1}\widetilde{c}_{h_n}^{\ast}\leq \inf_{r\geq r_0}r^{N-1}
\widetilde{{\mathcal{E}}}_{r}^{\ast}+\frac{\delta}{4}.
\end{equation}
Now (\ref{wls}), (\ref{ca1}), (\ref{ca3}) and (\ref{ca5}) yield, for
$n\geq\overline{n}$,
\begin{eqnarray*}
\inf_{r\geq r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}
&\geq& h_n^{N-1} \widetilde{c}_{h_n}^{\ast}-\frac{\delta}{4}
\geq h_n^{N-1} J_{h_n}(u_{h_n})-\frac{\delta}{4}\\
&\geq& h_n^{N-1}J_{h_n}(\widetilde{u}_{h_n}^{R})-\frac{\delta}{2}
\geq \overline{r}^{N-1}\liminf_{n\to +\infty}
\widetilde{{\mathcal{E}}}_{\overline{r}}(w_{h_n})-\delta \\
&\geq& \overline{r}^{N-1}\widetilde{ {\mathcal{E}}}_{\overline{r}}^{\ast}
-\delta\,,
\end{eqnarray*}
where we have used the fact that each $ w_{h_n}$ belongs to
$\Upsilon_{F}^{\ast}$, the above inequalities and the definition of
$\widetilde{{\mathcal{E}}}_{r}^{\ast}$. The contradiction follows and the
proof of claim 3 is now complete.
Then Claim 3, (\ref{assures}), and (\ref{conve}) imply $\overline{r}>r_0$;
therefore, by (\ref{conve}) we get
\begin{equation} \label{conve3}r_{h_n}^{\widehat{i}}-\frac{r_0}{h_n}
\to +\infty\quad\mbox{as}\quad n\to +\infty.
\end{equation}
\paragraph{Claim 4.} For every $\delta>0$ there exists
$R=R(\delta)>2R_{\widehat{i}}$ such that, for large $n$,
$$s\in\big[\frac{r_0}{2h_n},+\infty\big)\setminus
[r_{h_n}^{\widehat{i}}-R,r_{h_n}^{\widehat{i}}+R]\Rightarrow
\varphi_{h_n}(s)\leq \delta.
$$
{\bf Proof of claim 4.}
Fix $\delta>0$ arbitrarily. According to lemma \ref{limsup} choose
$M>0$ such that $$\sup_{n\in{\mathbb N}}h_n^{N-1}\widetilde{c}_{h_n}
^{\ast}0$ such that
\begin{equation} \label{equs}
\forall n\in{\mathbb N},\quad\forall s,s'\in\big[\frac{r_0}{2h_n},
+\infty\big):\quad |s-s'|\leq\sigma\Rightarrow |\varphi_{h_n}(s)
-\varphi_{h_n}(s')|\leq\frac{\delta}{2}.
\end{equation}
As Claim 1 assures the existence of $R>2R_{\widehat{i}}$ and
$\overline{n}\in{\mathbb N}$ such that for $n\geq \overline{n}$ and for
all $w_n\in\Lambda_{F}(h_n)$ satisfying
$$ h_n^{N-1}J_{h_n}(w_n)\leq M,\quad |w_n(x)|
\leq\overline{\varepsilon}\;\;\mbox{in} \;\;C_n^{R},
$$
there results
\begin{equation} \label{app2}
h_{n }^{N-1}J_{h_n}(\eta_{h_n}^{R}w_n)\leq h_n^{N-1}
(J_{h_n})_{\big| C_n^{R}}(w_n)+V_0\frac{\delta^{2}}{32}
\omega_{N}\left(\frac{r_0}{2}\right)^{N-1}\sigma.
\end{equation}
Furthermore, by Claim 2, we may assume
\begin{equation} \label{mam1}
\widetilde{u}_{h_n}^{R}\in\Lambda_{F}^{\ast}(h_n),\quad
|u_{h_n}(x)|\leq\frac{\overline{\varepsilon}}{2}\quad
\forall x\in C_n^{R}\quad \forall n\geq\overline{n}.
\end{equation}
Observe that $\lim_{n\to +\infty}\frac{1}{h_n}\left((\frac{r_0}{2}
+h_n\sigma)^{N}-(\frac{r_0}{2})^{N}\right)=
N\sigma (\frac{r_0}{2})^{N-1}$. Then, taking also into account of
(\ref{conve3}), without loss of generality we may assume
$\overline{n}$ chosen sufficiently large so that for every
$n\geq \overline{n}$
\begin{equation} \label{outl}
\frac{(\frac{r_0}{2}+h_n\sigma)^{N}-(\frac{r_0}{2})^{N}}{h_n}
>\frac{N}{2}\sigma\left(\frac{r_0}{2}\right)^{N-1},\quad
\mbox{and}\quad r_{h_n}^{\widehat{i}}-\frac{r_0}{h_n}>R+\sigma.
\end{equation}
Arguing by contradiction, assume that there exists $\nu \geq \overline{n}$
and $s_{\nu}\in [\frac{r_0}{2h_{\nu}},+\infty)\setminus [r_{h_{\nu}}^
{\widehat{i}}-R,r_{h_{\nu}}^{\widehat{i}}+R]$ such that
$\varphi_{h_{\nu}}(s_{\nu})> \delta$. (\ref{equs}) allows us to infer
$$\forall s\in [s_{\nu}-\sigma,s_{\nu}+\sigma]:\quad \varphi_{h_{\nu}}
(s_{\nu})> \frac{\delta}{2}.
$$
We recall that $u_{h_{\nu}}$ has been obtained as weak limit of a
minimizing sequence $\{u^{h_{\nu}}_{k}\}_{k\in{\mathbb N}}$ in
$\widetilde{\Lambda}_{F}^{\ast}(h_{\nu})$. Lemma \ref{compactinj} yields
$u_{k}^{h_{\nu}}\to u_{h_{\nu}}$ uniformly in ${\mathbb R}^{N}$, then
for $k$ large enough there results
$$
|u_{k}^{h_{\nu}}(x)|> \frac{ \delta}{2}\;\;\forall x\in C(s_{\nu}
-\sigma,s_{\nu}+\sigma),\quad |u_{k}^{h_{\nu}}(x)|
\leq\overline{\varepsilon}\;\;\forall x\in C_{\nu}^{R}
$$
and, since $J_{h_{\nu}}(u_{k}^{h_{\nu}})\to \widetilde{c}_{h_{\nu}}^{\ast}$,
it follows that $h_{\nu}^{N-1}J_{h_{\nu}}(u_{k}^{h_{\nu}})r_{h_{\nu}}^{\widehat{i}}+R$ and
then $C(s_{\nu},s_{\nu}+\sigma)\subset{\mathbb R}^{N}\setminus
B(0,r_{h_{\nu}}^{\widehat{i}}+R)$, or $s_{\nu}\in[\frac{r_0}{2h_{\nu}},
r_{h_{\nu}}^{\widehat{i}}-R)$; in the latter case, by the second relation
of (\ref{outl}), we get that the length of the interval
$(s_{\nu}-\sigma,s_{\nu}+\sigma)\cap [\frac{r_0}{2h_{\nu}},r_{h_{\nu}}
^{\widehat{i}}-R\big)$ is bigger then $\sigma$. Hence in both cases it
happens that for large $k$ $|u_{k}^{h_{\nu}}|$ is bigger than
$\frac{\delta}{2}$ in an ring contained in
${\mathbb R}^{N}\setminus \left(B(0,\frac{r_0}{2h_{\nu}})\cup C_{\nu}
^{R}\right)$ whose Lebesgue measure is bigger then
$\frac{\omega_{N}}{N}((\frac{r_0}{2h_{\nu}}+\sigma)^{N}
-(\frac{r_0}{2h_{\nu}})^{N})$; from the definition of $J_{h_{\nu}}$
we obtain $$J_{h_{\nu}}(u^{h_{\nu}}_{k})\geq (J_{h_{\nu}})_{\big|
C_{\nu}^ {R}}(u^{h_{\nu}}_{k})+V_0\frac{\delta^{2}}{8}
\frac{\omega_{N}}{N}\Big(\big(\frac{r_0}{2h_{\nu}}+\sigma\big)^{N}
-\big(\frac{r_0}{2h_{\nu}}\big)^{N}\Big).
$$
Inserting the last inequality in (\ref{app3}), one obtains
\begin{eqnarray*}
\lefteqn{h_{\nu }^{N-1}J_{h_{\nu}}(\eta_{h_{\nu}}^{R}u^{h_{\nu}}_{k}) }\\
&\leq& h_{\nu}^{N-1} J_{h_{\nu}}(u^{h_{\nu}}_{k})+V_0
\frac{\delta^{2}}{32}\omega_{N}\left(\frac{r_0}{2}\right)^{N-1}
\sigma-V_0\frac{\delta^{2}}{8}\frac{\omega_{N}}{N}\frac{(\frac{r_0}{2}
+h_{\nu}\sigma)^{N}-(\frac{r_0}{2})^{N}}{h_{\nu}}.
\end{eqnarray*}
By the first of (\ref{outl}) we conclude with the following inequality
which holds for $k$ sufficiently large
\begin{equation} \label{we}
J_{h_{\nu}}(\eta_{h_{\nu}}^{R}u^{h_{\nu}}_{k})0$, it is obvious that $\gamma$ is
independent on $k$. Then, by passing to the limit as $k\to +\infty$ in
(\ref{we}), we achieve the contradiction $\widetilde{c}^{\ast}_{h_{\nu}}
\leq \widetilde{c}^{\ast}_{h_{\nu}}-\gamma$.
Thus it remains to prove (\ref{what}).
By construction we obtain $\eta_{h_{\nu}}^{R}u^{h_{\nu}}_{k}\in
F_{h_{\nu}}$. Note that, since $u^{h_{\nu}}_{k}\to u_{h_{\nu}}$
uniformly in ${\mathbb R}^{N}$, we also have
\begin{equation} \label{unic}
\eta_{h_{\nu}}^{R}u^{h_{\nu}}_{k}\to\eta_{h_{\nu}}^{R}u_{h_{\nu}}
\equiv\widetilde{u}_{h_{\nu}}^{R}\quad \mbox{uniformly as}\quad k\to
+\infty.
\end{equation}
Since $\inf_{x\in{\mathbb R}^{N}}| \widetilde{u}_{h_{\nu}}^{R}(x)
-\overline{\xi}|>0$, by (\ref{unic}) for $k$ large enough
$\eta_{h_{\nu}}^{R}u^{h_{\nu}}_{k}\neq \overline{\xi}$, by which
$\eta_{h_{\nu}}^{R}u^{h_{\nu}}_{k}\in\Lambda_{F}(h_{\nu})$.
Combining (\ref{unic}) with lemma \ref{cont} we deduce
$$ch\left(\eta_{h_{\nu}}^{R}u^{h_{\nu}}_{k}, {\mathbb R}^{N}\setminus
\overline{B}\left(0,\frac{r_0}{h_{\nu}}\right)\right)
=ch\left(\widetilde{u}_{h_{\nu}}^{R}, {\mathbb R}^{N}\setminus
\overline{B}\left(0,\frac{r_0}{h_{\nu}}\right)\right)
$$
at least for $k$ sufficiently large. The second inequality of (\ref{outl})
leads to the inclusion $C(r_{h_{\nu}}^{\widehat{i}}-R,r_{h_{\nu}}
^{\widehat{i}}+R)\subset{\mathbb R}^{N}\setminus \overline{B}
\left(0,\frac{r_0}{h_{\nu}}\right)$, while
$\widetilde{u}_{h_{\nu}}^{R} \equiv 0 $ in
$ {\mathbb R}^{N}\setminus C(r_{h_{\nu}}^{\widehat{i}}-R,r_{h_{\nu}}
^{\widehat{i}}+R)$; hence, taking into account of (\ref{mam1}),
we obtain $ch\left(\widetilde{u}_{h_{\nu}}^{R}, {\mathbb R}^{N}\setminus
\overline{B}\left(0,\frac{r_0}{h_{\nu}}\right)\right)
=ch(\widetilde{u}_{h_{\nu}}^{R})\neq 0$.
On the other hand it is obvious that $$|\eta_{h_{\nu}}^{R}u^{h_{\nu}}_{k}
|\leq |u^{h_{\nu}}_{k}|<\frac{\overline{\varepsilon}}{2}\quad
\forall x\in \overline{C}\left(\frac{r_0}{2h_{\nu}},\frac{r_0}
{h_{\nu}}\right),\quad \forall k\in{\mathbb N}
$$ and this concludes the proof.
\paragraph{Claim 5.} For large $n$ there results
$u_{h_n}\in\widetilde{\Lambda}_{F}^{\ast}(h_n)$, and
$J_{h_n}(u_{h_n})=\widetilde{c}_{h_n}^{\ast}$.
Furthermore
$$x\in B\left(0,\frac{r_0}{2h_n}\right)\Rightarrow |u_{h_n}(x)|
\leq \frac{\alpha}{\sqrt{2}},$$ where
$\alpha\in(0,\overline{\varepsilon})$ has been defined at the beginning
of section 6. As a corollary, for $n$ sufficiently large $u_{h_n}$ is
a solution to equation (\ref{equ2}).
\noindent{\bf Proof of Claim 5.}
By applying last step we have the existence of $R=R(\alpha)>0$ such that,
for large $n$
$$
x\in{\mathbb R}^{N}\setminus \left(B\left(0,\frac{r_0}{2h_n}\right)
\cup C\left(r_{h_n}^{\widehat{i}}-R, r_{h_n}^{\widehat{i}}+R\right)
\right)\Rightarrow |u_{h_n}(x)|\leq \frac{\alpha}{(2(N+1))^{1/2}}.
$$
In particular by (\ref{conve3}) we may assume
$ C\left(\frac{r_0}{2h_n},\frac{r_0}{h_n}\right)\cap C
\left(r_{h_n}^{\widehat{i}}-R, r_{h_n}^{\widehat{i}}+R\right)
=\emptyset$ so that for $n$ sufficiently large there results
\begin{equation} \label{formula1}
|u_{h_n}(x)|\leq \frac{\alpha}{(2(N+1))^{1/2}}
<\frac{\overline{\varepsilon}}{2}\quad \forall x\in
C\left(\frac{r_0}{2h_n},\frac{r_0}{h_n}\right).
\end{equation}
On the other hand in the first part of the proof we have already showed
that for large $n$ $u_{h_n}\in \Lambda_{F}(h_n)$ and
$ch\left(u_{h_n},{\mathbb R}^{N}\setminus \overline{B}
\left(0,\frac{r_0}{h_n}\right)\right)\neq 0$.
The definition of $\widetilde{\Lambda}_{F}^{\ast}(h_n)$ leads to
$u_{h_n}\in\widetilde{\Lambda}_{F}^{\ast}(h_n)$ for $n$
sufficiently large. But $u_{h_n}$ is the weak limit of a minimizing
sequence $u_{k}^{h_n}$ in $\widetilde{\Lambda}_{F}^{\ast}(h_n)$
for the functional $J_{h_n}$. Then the weakly lower semi-continuity of
$J_{h_n}$ implies
$$
\widetilde{c}_{h_n}^{\ast}\leq J_{h_n}(u_{h_n})
\leq \liminf_{k\to +\infty}J_{h_n}(u_{k}^{h_n})
=\widetilde{c}_{h_n}^{\ast}
$$
i.e. $u_{h_n}$ is the desired minimizing function. Then $u_{h_n}$
is a local minimum for the functional $J_{h_n}$ in
$\Lambda_{F}(h_n);$ as we have pointed out at the beginning of the
section, $u_{h_n}$ turns out to be a critical point of $J_{h_n}$.
Then the function $u_{h_n}$ solves the equation
\begin{equation} \label{equ6}-\Delta u_{h_n}+V_{h_n}(x)u_{h_n}
-\Delta_{p} u_{h_n}+G'(h x,u_{h_n})=0\quad \mbox{in}\quad
{\mathbb R}^{N}.
\end{equation}
To simplify notation, for every $a>0$ we consider the
function $T_{a}:{\mathbb R}^{N+1}\to {\mathbb R}^{N+1}$ defined by
$$
T_{a}(z)=\pmatrix{ \mathop{\rm sign} z_1\,\max
\{\left|z_1\right|-\frac{a}{(2(N+1))^{1/2}} ,\,0\}\cr
\vdots\cr \mathop{\rm sign} z_{N+1}\max
\{\left|z_{N+1}\right|-\frac{a}{(2(N+1))^{1/2}} , 0\}\cr},
\quad z=\pmatrix{ z_1\cr \vdots \cr z_{N+1}\cr}
\in{\mathbb R}^{N+1}.
$$
Since each $u_{h_n}$ solves (\ref{equ6}), we can choose as a test
function
$$\psi_{h_n}(x)=\cases{
T_{\alpha}(u_{h_n}(x))& if $x\in B\left(0,\frac{r_0}{2h_n}\right)$,\cr
0 & if $ x\in{\mathbb R}^{N}\setminus B\left(0,\frac{r_0}{2h_n}\right)$.
\cr}$$
By (\ref{formula1}) we easily infer that $\psi_{h_n}$ is continuous on
the circle $\{x\in{\mathbb R}^{N}:|x|=r_0/2h_n\}$, hence it
is continuous everywhere. It is standard to check that
$\psi_{h_n}\in H_{h_n}$. By construction there results
$\nabla u_{h_n}\equiv \nabla \psi_{h_n}$ a.e. in ${\mathbb R}^{N}$.
Hence, after integration by parts, one gets
\begin{eqnarray} \label{boc}
\int_{B\left(0,\frac{r_0}{2h_n}\right)}\left(|\nabla \psi_{h_n}|
^{2}+|\nabla \psi_{h_n}|^{p}\right)\,dx
+\int_{B(0,\frac{r_0}{2h_n})}
\Big(V_{h_n}(x)u_{h_n}&&\\
+W'(u_{h_n})K_{\alpha}\left(|u_{h_n}|^{2}\right)
+2W(u_{h_n})K_{\alpha}'\left(|u_{h_n}|^{2}\right)u_{h_n}
\Big)\psi_{h_n}\,dx&=&0\,. \nonumber
\end{eqnarray}
We want to write the function in the second integral in a more suitable
way. First observe that, denoting by $\psi_{h_n}^{i}$ the component
functions of $\psi_{h_n}$, it holds
$u_{h_n}\psi_{h_n}= |\psi_{h_n}|^{2}+\frac{\alpha}{(2(N+1))^{1/2}}
\sum_{i=1}^{N+1}|\psi_{h_n}^{i}|$; hence for every
$x\in{\mathbb R}^{N}$, we have
\begin{eqnarray*}
\lefteqn{\Big(V_{h_n}(x)u_{h_n}(x)+W'(u_{h_n}(x))K_{\alpha}
\left(|u_{h_n}(x)|^{2}\right) }\\
\lefteqn{+2W(u_{h_n})K_{\alpha}'
\left(|u_{h_n}(x)|^{2}\right)u_{h_n}(x)\Big)\psi_{h_n}(x) }\\
&\geq&
V_{h_n}(x)|\psi_{h_n}(x)|^{2}+\frac{\alpha}{(2(N+1))^{1/2}}
V_{h_n}(x)\\
&&\times \sum_{i=1}^{N+1}|\psi_{h_n }^{i}(x)|-|W'(u_{h_n}(x))|
K_{\alpha}(|u_{h_n}(x)|^{2})\,|\psi_{h_n }(x)|\\
&&+2W(u_{h_n}(x))K_{\alpha}'\left(|u_{h_n}(x)|^{2}\right)
|\psi_{h_n}(x)|^{2} \\
&&+2W(u_{h_n}(x))K_{\alpha}'\left(|u_{h_n}(x)|^{2}\right)
\frac{\alpha}{(2(N+1))^{1/2}} \sum_{i=1}^{N+1}|\psi_{h_n}^{i}(x)| \\
&\geq& \big( V_0-2\sup_{|\xi|\leq\alpha}|W(\xi)|\frac{2\overline{c}}
{\alpha^{2}}\big)| \psi_{h_n}(x)|^{2}
+\Big(V_0\frac{\alpha}{(2(N+1))^{1/2}}
-\sup_{|\xi|\leq\alpha}|W'(\xi)| \\
&&-2\sup_{|\xi|\leq\alpha}|W(\xi)|
\frac{2c_{\alpha}}{\alpha^{2}}\frac{\alpha}{(2(N+1))^{1/2}}\Big)
\sum_{i=1}^{N+1}|\psi_{h_n}^{i}(x)|\\
&=&\left( V_0-\sup_{|\xi|\leq\alpha}|W(\xi)|
\frac{4\overline{c}}{\alpha^{2}}\right)|\psi_{h_n}(x)|^{2} \\
&&+\left( V_0\frac{\alpha}{(2(N+1))^{1/2}+2}-\sup_{|\xi|
\leq\alpha}|W'(\xi)|\right)\sum_{i=1}^{N}|\psi_{h_n}^{i}(x)|\\
&&+\frac{2\alpha}{(2(N+1))^{1/2}}\left(\frac{1}{(2(N+1))^{1/2}+2} V_0
-\sup_{|\xi|\leq\overline{\varepsilon}}|W(\xi)|\frac{2\overline{c}}
{\alpha^{2}}\right)\sum_{i=1}^{N}|\psi_{h_n}^{i}(x)|.
\end{eqnarray*}
To obtain the last equality we have used the obvious identity
$$\frac{V_0}{(2(N+1))^{1/2}}=\frac{V_0}{(2(N+1))^{1/2}+2}
+\frac{2V_0}{(2(N+1))^{1/2}((2(N+1))^{1/2}+2)}\,.$$
Then (\ref{alpha12})
implies that the terms in the last inequality, which multiply
$|\psi_{h_n}(x)|^{2}$ and $\sum_{i=1}^{N+1}|\psi_{h_n}^{i}(x)|$
are nonnegative. Then both integrals in (\ref{boc}) are nonnegative,
hence they must be all zero. We conclude in particular
$\psi_{h_n}\equiv 0$ by which $|u_{h_n}(x)|\leq\frac{\alpha}{\sqrt{2}}$
for every $x\in B\left(0,\frac{r_0}{2h_n}\right)$.
Hence $u_{h_n}$ solves equation (\ref{equ2}).
\paragraph{Claim 6.} End of the proof.
To conclude we have just only to combine the results obtained in
the last steps. In particular by (\ref{max123}), (\ref{max124}),
(\ref{conve}), claims 3, 4 and 5 we have that for small $h>0$
there is a solution $u_h$ of equation (\ref{equ2}); furthermore,
setting $v_h(x)=u_h\left(\frac{x}{h}\right)$ and using a
re-scaling argument we infer that for $h$ small enough,
\begin{eqnarray}
& Equation (\ref{equ1}) \mbox{ has a solution } v_h,&\nonumber\\
&|v_h| \mbox{ has a circle of local maximum points }\{x\in{\mathbb
R}^{N}:|x|=r_h\},&\\
&|v_h(x)|>1\;\; \forall x\;\;\mbox{with}\;\;|x|=r_h;& \label{orr2} \\
& r_h>r_0,\;\;r_h^{N-1}\widetilde{{\mathcal{E}}}_{r_h}
^{\ast}\to \inf_{r\geq
r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast},&\nonumber\\
&\limsup_{h\to 0^{+}}\sup_{x\not\in
C(r_h-\delta,r_h+\delta)}|v_h(x)|\leq\alpha\quad\forall
\delta>0\,.
\end{eqnarray}
Note that in order to deduce (\ref{orr2}) we have
used the continuity of the function $r>r_0\mapsto
r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}$. The construction of
the family of solutions $\{v_h\}$ depends on the particular
$\alpha\in\left(0,\overline{\varepsilon}\right)$ chosen at the
beginning of Section 8. To emphasize this fact we denote this
family as $\{v_h^{\alpha}\}$. Let $\alpha_{j}$ be any sequence
of positive numbers such that $\alpha_{j}\to 0$. We have already
observed that each $\alpha'\leq\alpha$ still verifies
(\ref{alpha1}) and (\ref{alpha2}). Then we can repeat the same
arguments we have used for $\alpha$ and obtain that there is a
decreasing sequence of positive numbers $h_{j}\to 0$ such that for
all $01$ for $|x|=r_h^{j}$; furthermore
$r_h^{j}>r_0$ and $$\inf_{r\geq
r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}
-\frac{1}{j}<(r_h^{j})^{N-1}\widetilde{{\mathcal{E}}}_{r_h^{j}}
^{\ast}< \inf_{r\geq
r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}+\frac{1}{j}\quad
\forall h\in(0,h_{j})$$
and
$$\sup_{x\not\in C(r_h^{j}-\alpha_{j},r_h^{j}+\alpha_{j})}|v_h
^{j}(x)|\leq
\alpha_{j}\quad\forall h\in(0,h_{j}).
$$
Then for $h_{j}\geq h>h_{j-1}$, we just define
$v_h=v_h^{\alpha_{j}}$ and $r_h=r_h^{j}$.
Then by the definition above we clearly have
$$r_h^{N-1}\widetilde{{\mathcal{E}}}_{r_h}^{\ast}\to
\inf_{r\geq r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}$$
and
\begin{equation} \label{stu} \limsup_{h\to 0^{+}}\sup_{x\not\in
C(r_h-\delta,r_h+\delta)}|v_h(x)|=0\quad \forall
\delta>0.
\end{equation}
Observe that, by (\ref{cor}), the function
$r>r_0\mapsto r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}$ is coercive;
hence the family $\{r_h\}$ is bounded in ${\mathbb R}^{N}$ for small
$h>0$; hence, considered a generic sequence $h_n\to 0^{+}$, up to a
subsequence, it converges to a point $\overline{r}$.
The continuity of the function $ r>r_0\mapsto r^{N-1}
\widetilde{{\mathcal{E}}}_{r}^{\ast}$ implies
$$ \overline {r}^{N-1}\widetilde{{\mathcal{E}}}_{\overline{r}}^{\ast}
=\inf_{r\geq r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}.
$$
Finally taking into account of (\ref{stu}), it is easy to show that
$\{v_{h_n}\}$ decays uniformly outside every fixed neighbourhood of
the circle $\{|x|=\overline{r}\}$.\hfill $\diamondsuit$
\paragraph{Remark} By i) of lemma \ref{remind} we immediately obtain that
the function $r\geq r_0\mapsto r^{N-1}\widetilde{{\mathcal{E}}}_{r}
^{\ast}$ is continuous. Moreover, by (\ref{cor}), we get
$r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}\geq r^{N-1}\nu\to +\infty$
as $r\to+\infty$ for some $\nu>0$. This fact, combined with
(\ref{assures}), implies that the infimum
$\inf_{r\geq r_0}r^{N-1}{\mathcal{E}}_{r}^{\ast}$ is attained by some
$\overline{r}>r_0$. If such minimum point $\overline{r}$ is unique,
then all the bound states $\{v_h\}$ we have found in last theorem
concentrate at the circle $\{x\in{\mathbb R}^{N}:|x|=\overline{r}\}$
as $h\to 0^{+}$ in the sense specified in theorem \ref{imp}.
In general $r_h^{N-1}\widetilde{{\mathcal{E}}}_{r_h}^{\ast}\to
\inf_{r\geq r_0}r^{N-1}\widetilde{{\mathcal{E}}}_{r}^{\ast}$ as
$h\to 0^{+}$, where the circle $\{x\in{\mathbb R}^{N}:|x|=r_h\}$
is a set of local maxima of $|v_h|$ for small $h$.
\paragraph{Remark} We conclude by observing that if we require the function
${\mathcal{V}}$ to be sufficiently big in $r_0$, then (\ref{assures})
holds true. Indeed, fixed $\overline{r}>r_0$ and
${\mathcal{V}}(\overline{r})$ arbitrarily, then iii) of lemma
\ref{remind} assures that provided that ${\mathcal{V}}(r_0)$ is large
enough the inequality $\widetilde{{\mathcal{E}}}_{r_0}^{\ast}
>(\frac{\overline{r}}{r_0})^{N-1}
\widetilde{{\mathcal{E}}}_{\overline{r}}^{\ast}$ is satisfied.
\paragraph{Acknowledgments}
I wish to thank Professors Marino Badiale and Vieri Benci for their
helpful comments and suggestions.
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\noindent{\sc Teresa D'Aprile} \\
Scuola Normale Superiore\\
Piazza dei Cavalieri 7, 56126 Pisa, Italy\\
e-mail: aprilet@cibs.sns.it
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