\documentclass[reqno]{amsart}

\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 37, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/37\hfil Multiple semiclassical states]
{Multiple semiclassical states for singular magnetic
 nonlinear  Schr\"{o}dinger equations}

\author[S. Barile\hfil EJDE-2008/37\hfilneg]
{Sara Barile}

\address{Sara Barile \newline
Dipartimento di Matematica \\
Politecnico di Bari \\
Via Orabona 4, I-70125 Bari, Italy}
\email{s.barile@dm.uniba.it}

\thanks{Submitted November 26, 2007. Published March 14, 2008.}
\subjclass[2000]{35J10, 35J60, 35J20, 35Q55, 58E05}
\keywords{Nonlinear Schr\"{o}dinger equations;
external magnetic field; \hfill\break\indent
singular potentials; semiclassical limit}

\begin{abstract}
 By means of a finite-dimensional reduction, we show
 a multiplicity result of semiclassical solutions
 $u: \mathbb{R}^N \to\mathbb{C}$
 to the singular nonlinear Schr\"o\-dinger equation
  \begin{equation*}
 \Big( \frac{\varepsilon}{i} \nabla -  A(x)\Big)^2 u +
 u+(V(x)-\gamma(\varepsilon)W(x)) u = K(x) | u|^{p-1} u, \quad x \in
 \mathbb{R}^N,
 \end{equation*}
 where $N \geq 2$, $1 < p < 2^{*}-1$, $A(x), V(x)$ and $K(x)$
 are bounded potentials. Such solutions concentrate  near
 (non-degenerate) \textit{local} extrema or  a
 (non-degenerate) \textit{manifold} of stationary points of an
 auxiliary function $\Lambda$ related to the unperturbed electric
 field $V(x)$ and the coefficient $K(x)$ of the nonlinear term.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction and main results}\label{intro}

In recent years, much attention has been devoted to the search of
standing waves solutions of the type
$\psi(x, t) =\exp (-i \frac{E}{\hbar} t) u(x)$, $E\in \mathbb{R}$,
$u : \mathbb{R}^N \to\mathbb{C}$ to the time-dependent NLS equations
(Nonlinear Schr\"{o}dinger equations) with potentials
\begin{equation} \label{1.1}
i \hbar \frac{\partial \psi}{\partial t}
= \Big(\frac{\hbar}{i}\nabla -A(x) \Big)^2 \psi +  U(x) \psi
-  K(x) |\psi|^{p-1} \psi, \quad (t, x) \in  \mathbb{R} \times \mathbb{R}^N,
\end{equation}
where $i$ is the imaginary unit and $\hbar$ is the Planck
constant.  The function  $A: \mathbb{R}^N \to\mathbb{R}^N$ denotes
a magnetic potential , $U: \mathbb{R}^N \to\mathbb{R}$ represents an
electric potential and the nonlinear term grows subcritically,
namely for $p > 1$ if $N=2$ and $ 1 < p < (N+2) / (N-2)$ if
$N \geq 3$.

 This leads to solve the complex semilinear elliptic equation
\begin{equation} \label{p_h}
\Big( \frac{\varepsilon}{i} \nabla -  A(x)\Big)^2 u + (U(x)-E) u = K(x)
|u|^{p-1} u,  \quad\hbox{$x\,\in \, \mathbb{R}^N$},
\end{equation}
where $\varepsilon=\hbar$ and $V(x)+1=U(x) -E$  is strictly positive on
the whole $\mathbb{R}^N$, whose solutions are usually referred as
semi-classical ones since their existence is proved by letting
$\varepsilon \to0$ thus performing the transition from Quantum to
Classical Mechanics. It has been also investigated the problem of
finding a family $\{u_\varepsilon \}$ of such solutions which exhibits a
\textit{concentration behavior} around a special point, namely,
solutions with a spike shape, a  ma\-xi\-mum point converging to a
point located around a prescribed region, while vanishing as $\varepsilon
\to0$ everywhere else in the domain. Such special point has been
proved to be a critical point of the potential $V(x)$ and the
study of single  and multiple spike solutions to (\ref{p_h}) and
related problems has attracted  considerable attention in recent
years.  In the case $A=0$, different approaches have been carried
out in order to study  one-bump or multi-bump semi-classical bound
states (solutions with finite energy) and different cases have
been covered (see \cite{ ABC, byjj, bjpeak, caonouss, cl, cingnol,
clPot, df1, df2, DFpeak, fw, gui, jjtanaka, li, oh, ra, wa, wz}).
In the case $A \neq 0$, the first existence result is due to
 Esteban and Lions \cite{EL} for $\varepsilon >0$ fixed by means of concentration-compactness arguments.
Later,  Kurata \cite{ku} has showed, in the semiclassical limit, the existence and the concentration of a least energy solution near global minima of $V$ under suitable assumptions linking the magnetic and the electric potentials in the case $K(x)=1$. Furthermore, he has proved that the magnetic potential only contributes to the phase factor of the complex solution but it doesn't influence the concentration of its modulus.  A first multiplicity result for solutions of (\ref{p_h})  has been proved by Cingolani in \cite{ci}, by means of topological arguments that allow to relate   the number of the solutions  to the  richness of the set $M$ of global minima of an auxiliary function $\Lambda$ defined as
\begin{equation*}
\Lambda(x)= \frac{(1+V(x))^{\theta}}{K(x)^{2/(p-1)}}, \quad
\theta=\frac{p+1}{p-1}- \frac N2,
\end{equation*}
 (see (\ref{lambda}) in Section \ref{finitered} for
details) on the whole $\mathbb{R}^N$ since  $K(x) > 0$ for all $x
\in \mathbb{R}^N$,  which coincide with global minima of $V(x)$ if
$K(x)=1$. In \cite{cs}, Cingolani and Secchi  have treated the
more general case in which $\Lambda$ has a non-degenerate manifold
of stationary points. For bounded electric and magnetic
potentials, they have proved a multiplicity result following the
new perturbation approach introduced in the paper \cite{AMMASE}
due to Ambrosetti, Malchiodi and Secchi  in the case $A=0$ (see
also \cite{am05}).  Precisely, by means of a  finite-dimensional
reduction, the complex valued solutions to (\ref{p_h}) (after the
change of variable $x \to\varepsilon x$)  are found \textit{near} least
energy solutions of the complex limiting equation
\begin{equation} \label{limitprima}
\Big( \frac{ \nabla}{i}  -  A(\varepsilon \xi)\Big)^2 u+u + V(\varepsilon \xi)
u = K(\varepsilon \xi) |u|^{p-1} u \quad\hbox{in $\mathbb{R}^N$},
\end{equation}
(see Remark \ref{near}) where $\varepsilon \xi $ is in a neighborhood of
$M$. In such sense, here and in what follows, as $\varepsilon \to0$,
solutions of (\ref{p_h}) concentrate around stationary points of
$\Lambda$ (see  Proof of Theorem \ref{cinque1}). Furthermore, the
boundedness of the electromagnetic potentials assures that the
variational setting $H^1(\mathbb{R}^N, \mathbb{C})$ of (\ref{limitprima})
becomes equivalent to the variational framework in which
(\ref{p_h}) is set up. Then, such result has been improved in
\cite{cs1} to degenerate and topologically non-trivial critical
points of $\Lambda$ dropping the boundedness of the magnetic
potential. Necessary conditions for a sequence of  solutions to
(\ref{p_h}) to concentrate, in different senses, around a given
point have been established by Secchi and Squassina in \cite{ss}.
For multi-peaks, we refer to \cite{badapeng, caotang, csjj} and
for the critical case to \cite{ariolisz, Barcs, chsz}. The
asymptotic evolution has been recently studied in  \cite{cauchy}.
Dealing with singular magnetic NLS equations, we cite  a  recent
paper by Barile \cite{Bar}  where the author  has  obtained
a multiplicity result of complex-valued solutions to
\begin{equation} \label{singnok}
\Big( \frac{\varepsilon}{i} \nabla -  A( x)\Big)^2 u + \left(V( x)-
\gamma(\varepsilon)W( x) \right) u =  |u|^{p-2} u,  \quad x \in
\mathbb{R}^N,
\end{equation}
where $2 < p < 2^*$,  $\gamma : [0, +\infty) \to[0,
+\infty)$ and $W: \mathbb{R}^N \to[0, +\infty)$ is a  measurable
potential satisfying (W1) like $\frac{1}{|x|}, \frac{1}{|x|^2}$
(see \cite{lazzo} in the case $A=0$).  The introduction of
singular potentials  has important physical interest since they
appear in many fields such as  Quantum Mechanics and Astrophysics
\cite{frankal, landlif},  Chemistry \cite{cattoal, lionschim},
Cosmology  \cite{beresteb}  and  Differential Geometry
\cite{aubin} thus being the object of a wide recent mathematical
research (e.g. \cite{chabr, chavgarcia, EL, felterr, fergazz,
garcper, ruizwill, smets}). Furthermore, in such a case, it  has a
certain relevance from the mathematical point of view since it
allows to perturbe the potential $V(x)$ which is supposed to be
bounded below so that the resulting potential $V_\varepsilon(x)=V( x)-
\gamma(\varepsilon)W( x)$ may be unbounded below and eventually above.
Following the  variational approach used in \cite{ci}, it is
proved that
 the number of the solutions to (\ref{singnok}) can still be related to the topology of the global minima set of the unperturbed potential $V(x)$, provided the perturbation $\gamma(\varepsilon)$ is small with respect to the coefficient $\varepsilon^2$ of the differential term, in the sense that for any $\delta > 0$ there exists $\eta^{**}(\delta) >0$ such that
$$
\limsup_{\varepsilon \to0} \frac{\gamma(\varepsilon)}{\varepsilon^2} <
\eta^{**}(\delta).
$$
 Thus such result can be seen as  a quite natural but important
generalization of  the one in \cite{ci} to the case of unbounded
electric potentials and  $K(x)=1$.
Our purpose, in this work, is to extend such multiplicity result to
\begin{equation} \label{gespl}
\Big( \frac{\varepsilon}{i} \nabla -  A( x)\Big)^2 u+u + \left(V( x)-
\gamma(\varepsilon)W( x) \right) u = K(x) |u|^{p-1} u,  \quad x \in
\mathbb{R}^N,
\end{equation}
in the more general case in which the auxiliary function $\Lambda$
has a manifold $M$ of stationary points, not necessarily global
minima and, for bounded magnetic and electric potentials $A(x)$
and $V(x)$, following the  perturbation approach used in \cite{cs}.
Really, we are able to prove that the result in the spirit of \cite{cs}
holds after  the introduction of the new term $- \gamma(\varepsilon)W( x)$
which may be unbounded below, thus generalizing it to the case of
electric potentials eventually unbounded.

Without loss of generality we can assume that $V(0)=0$ and
$K(0)=1$. Performing the change of variable  $x \mapsto   \varepsilon x$,
the problem becomes that of finding some functions
$u: \mathbb{R}^N \to\mathbb{C}$ such that
\begin{equation} \label{tp}
\Big( \frac{ \nabla}{i} -  A(\varepsilon x)\Big)^2 u+u + \left(V(\varepsilon
x)- \gamma(\varepsilon)W(\varepsilon x) \right) u = K(\varepsilon x) |u|^{p-1} u
\quad\hbox{in }\mathbb{R}^N.
\end{equation}
 Of course, if $u$ is a solution of (\ref{tp}), then
$u( \cdot / \varepsilon)$ is a solution of (\ref{gespl}).
 Since  (\ref{tp}) is invariant under the multiplicative action
of $S^1$, solutions of (\ref{tp}) naturally appear as \textit{orbits}
 so that we simply speak about solutions.
The complex-valued solutions to (\ref{tp}) are found \textit{near}
least energy solutions of the equation
\begin{equation} \label{xinow}
\Big( \frac{ \nabla}{i}  -  A(\varepsilon \xi)\Big)^2 u+u + V(\varepsilon \xi)
u = K(\varepsilon \xi) |u|^{p-1} u \quad\hbox{in }\mathbb{R}^N,
\end{equation}
(see Remark \ref{near}) where $\varepsilon \xi $ is in a neighborhood of $M$.
The least energy of (\ref{xinow}) have the form
$$
z^{\varepsilon \xi, \sigma}: x \in \mathbb{R}^N \to e^{i \sigma
+i A(\varepsilon \xi) \cdot x} \bigg( \frac{1+V(\varepsilon \xi)}{K(\varepsilon \xi)}
 \bigg)^{1/(p-1)}  U((1+V(\varepsilon \xi))^{1/2}(x-\xi)),
$$
(see Section \ref{frame}) where $\varepsilon \xi $ belongs to $M$ and
$\sigma \in [0,2 \pi ]$. As in \cite{cs} (see also \cite{AMMASE}),
 the proof relies on a suitable finite-dimensional reduction and
critical points of the Euler functional $f_\varepsilon$ associated to
problem (\ref{tp}) are found \textit{near} critical points of
a finite-dimensional functional $\Phi_\varepsilon$  which is defined on
a suitable neighborhood of $M$ (see (\ref{funzridotto}) and
(\ref{derivfunz})). This allows to use Ljusternik-Schnirelman
category in the case $M$ is a set of local maxima or minima of
$\Lambda$. We remark again that the case of maxima cannot be
handled by using direct variational arguments as in \cite{Bar, ci}.
We present a special case of our results.

We will use the following assumptions:
\begin{itemize}
 \item[(K1)] $K \in L^{\infty}(\mathbb{R}^N) \cap C^2(\mathbb{R}^N)$
  is strictly positive and $K''$ is bounded;
 \item[(V1)] $V \in L^{\infty}(\mathbb{R}^N) \cap C^2(\mathbb{R}^N)$
  satisfies $\inf_{x \in \mathbb{R}^N} (1+V(x)) > 0$, and $V''$ is bounded;
 \item[(W1)] $W: \mathbb{R}^N \to[0, +\infty)$ is a measurable function
  such that, for some $\alpha_1 > 0$ and $\alpha_2 \geq 0$,
$$
\int_{\mathbb{R}^N} W(x) |v|^2 \leq \alpha_1 \| \nabla |v| \|_2^2
 + \alpha_2 \| v \|_2^2
$$
 for any $v$ such that $|v| \in H^1(\mathbb{R}^N,\mathbb{R})$;
 \item[(A1)]  $A \in L^{\infty}(\mathbb{R}^N, \mathbb{R}^N)
 \cap C^1(\mathbb{R}^N, \mathbb{R}^N)$, and the Jacobian $J_A$
 of $A$ is globally bounded in $\mathbb{R}^N$;
 \item[(G1)]  $\gamma: [0, +\infty) \to[0, +\infty)$ is a function
which depends on $\varepsilon$ such that
 $ G(\varepsilon):=\frac{\gamma(\varepsilon)}{\varepsilon^2}=O(\varepsilon)$.
 \end{itemize}

\begin{theorem}\label{main}
Assume {\rm (K1), (V1), (W1), (A1), (G1)}.
 If the auxiliary function $\Lambda$ has a non-degenerate critical
point $x_0 \in \mathbb{R}^N$, then for $\varepsilon > 0$ small, the
problem \eqref{tp} has at least a (orbit of) solution concentrating
near $x_0$.
\end{theorem}

Furthermore, if $M$ is a set of critical points non-degenerate
in the sense of Bott (see \cite{bott}) we can prove the existence
of (at least) cup long  of $M$, denoted by $ l(M)$, solutions
concentrating near points of $M$. For the definition of the cup long,
refer to Section \ref{statem}.

\begin{theorem}\label{gener}
As in Theorem \ref{main}, assume {\rm (K1), (V1), (W1), (A1), (G1)}.
If the auxiliary function $\Lambda$ has a smooth, compact,
non-degenerate manifold of critical points $M$, then  for $\varepsilon > 0$
small, the problem \eqref{tp} has at least $l(M)$ (orbits of)
 solutions concentrating near points of $M$.
\end{theorem}


We remark that the presence of an external magnetic field
 produces a phase in the complex wave which depends on the value
of $A$ near $M$, but does not seem to influence the location of the
peaks of the modulus of the complex wave.


\subsection*{Notation} 1. The complex conjugate of any
number $z\in\mathbb{C}$ will be denoted by $\bar z$. 2. The real part of a
number $z\in\mathbb{C}$ will be denoted by $\mathop{\rm Re} z$. 3. The ordinary inner
product between two vectors $a,b\in {\mathbb{R}^N}$ will be
denoted by $a \cdot b$. 4. We omit the symbol $dx$ in integrals
over $\mathbb{R}^N$ when no confusion can arise. 5. $C$ denotes a
generic positive constant, which may vary inside a chain of
inequalities. 6. We use the Landau symbols. For example $O(\varepsilon)$
is a generic function such that $\limsup_{\varepsilon\to 0}
O(\varepsilon)/\varepsilon < \infty$, and $o(\varepsilon)$ is a function such that
$\lim_{\varepsilon\to 0} o(\varepsilon)/\varepsilon=0$.

\section{The variational framework}\label{frame}

We work in the real Hilbert space $E$ obtained as the completion
of $C_{0}^{\infty}(\mathbb{R}^N, \mathbb{C})$ with respect to the norm
$\| \cdot \|$ associated to the inner product
\begin{equation*}
(u|v) \equiv \mathop{\rm Re} \int_{\mathbb{R}^N} \nabla u \cdot
\overline{\nabla v} + u \overline v.
\end{equation*}
Solutions to (\ref{tp}) are, under some conditions we are going to
point out, critical points of the functional formally defined on $E$ as
 \begin{equation} \label{funct}
\begin{aligned}
f_\varepsilon (u)&= \frac 12 \int_{\mathbb{R}^N} \bigg(
\Big| \Big( \frac 1i \nabla - A(\varepsilon x) \Big) u  \Big|^2 +|u|^2
+(V(\varepsilon x)-\gamma(\varepsilon)W(\varepsilon x)) |u|^2  \bigg)\, dx   \\
&\quad - \frac{1}{p+1} \int_{\mathbb{R}^N} K(\varepsilon x) |u|^{p+1} \, dx.
\end{aligned}
\end{equation}
In the following, we shall assume that the functions $V$, $W$, $K$ and
$A$ satisfy assumptions (V1), (W1), (K1) and (A1).
In particular, by the boundedness of the magnetic and electric potentials,
  the norm $\| \cdot \|^2$  is equivalent to the usual norm
\begin{equation*}
 \|u \|_\varepsilon^2 \equiv  \int_{\mathbb{R}^N}
\left( |D^\varepsilon u|^2  +(1+ V(\varepsilon x)) |u |^2 \right) \,dx < \infty
\end{equation*}
on the real Hilbert space $E_\varepsilon$, defined by the closure of
$C_0^\infty (\mathbb{R}^N, \mathbb{C})$ under the scalar product
\begin{equation*}
(u|v)_\varepsilon \equiv \mathop{\rm Re} \int_{\mathbb{R}^N} \left(D^\varepsilon u
\overline{D^\varepsilon v} +(1+ V(\varepsilon x)) u \overline v\right) \,dx,
\end{equation*}
 where $D^\varepsilon u = (D_1^\varepsilon u,\dots , D_N^\varepsilon u)$ and
$D_j^\varepsilon = i^{-1}  \partial_j -A_j(\varepsilon x)$. Indeed,
\[
\int_{\mathbb{R}^N} \bigg( \Big| \Big( \frac 1i \nabla - A(\varepsilon
x) \Big) u  \Big|^2   \bigg)\, dx
= \int_{\mathbb{R}^N}
\bigg(  | \nabla u |^2 + | A(\varepsilon x) u |^2 - 2 \mathop{\rm Re} \Big( \frac
{\nabla u}{i} \cdot A(\varepsilon x) \overline u \Big) \bigg) \, dx,
\]
and the last integral is finite thanks to the Cauchy-Schwartz
inequality and the boundedness of $A$. The functional spaces
$E$ and $E_\varepsilon$ are isomorphic so, roughly speaking, we can
say that the above variational frameworks  become equivalent.
This allows us to prove that the integral involving $W$ is
finite by assumption (W1) as we need  that for all
$u \in E$ it results $|u| \in H^1(\mathbb{R}^N, \mathbb{R})$.
 Since $A$ is real valued, it is easy to deduce that
(see, for example, \cite{jt,rs}) for any $u\in E_{\varepsilon}$,
the  diamagnetic inequality
\begin{equation} \label{diam}
 |\nabla |u|(x)| =  \Big|\mathop{\rm Re} \Big( \nabla u \frac{\overline{u}}{|u|}
 \Big)\Big|= \Big|\mathop{\rm Re} \Big( (\nabla u - i A(\varepsilon x) u)
\frac{\overline{u}}{|u|}     \Big)\Big| \leq |D^{\varepsilon} u(x)|
\end{equation}
 holds a.e. in $\mathbb{R}^N$ and
$|u| \in H^1(\mathbb{R}^N, \mathbb{R})$.  Furthermore,
\begin{equation} \label{disug}
\int_{\mathbb{R}^N} |\nabla |u\|^2 + |u|^2\, dx \leq
\int_{\mathbb{R}^N}\left( |D^\varepsilon u|^2  +(1+ V(\varepsilon x)) |u |^2
\right) \,dx \leq c \|u\|^2
\end{equation}
So, by the change of variable $y=\varepsilon x$, (W1) and (\ref{disug}),
we have that
\begin{equation} \label{maggW1}
\begin{aligned}
\gamma(\varepsilon) \int_{\mathbb{R}^N} W(\varepsilon x) |u|^2
&\leq  \frac{\gamma(\varepsilon)}{\varepsilon^N}
 \Big[ \alpha_1 \int_{\mathbb{R}^N}  \left| \nabla \left|
u \Big( \frac y \varepsilon \Big)\right|\right|^2
+ \alpha_2 \int_{\mathbb{R}^N} \left| u \Big(\frac y \varepsilon \Big)\right|^2
 \Big]  \\
& \leq  \frac{\gamma(\varepsilon)}{\varepsilon^2} \Big[ \alpha_1 \int_{\mathbb{R}^N}
| \nabla |u(x)\|^2  + \alpha_2 \varepsilon^2 \int_{\mathbb{R}^N}
\left| u(x) \right|^2   \Big] \quad
(\text{with } x=\frac y \varepsilon) \\
& \leq  \frac{\gamma(\varepsilon)}{\varepsilon^2} \alpha_{\varepsilon}
\Big[  \int_{\mathbb{R}^N} | \nabla |u(x)\|^2  +  \int_{\mathbb{R}^N}
\left| u(x) \right|^2   \Big]  \\
& \leq  G(\varepsilon) \alpha_{\varepsilon} c \|u\|^2
\end{aligned}
\end{equation}
is finite for $\varepsilon$ small, where
$ \alpha_\varepsilon:=\max  \{\alpha_1, \alpha_2 \varepsilon^2 \} \to\alpha_1$
 as $\varepsilon \to 0$. It follows  that $f_\varepsilon$ is actually well defined
on $E$ for $\varepsilon$ small enough. In order to find possibly multiple
critical points of (\ref{funct}), we follow the approach of
\cite{AMMASE, cs}. Since we need to find complex-valued solutions,
some further remarks are due.

Let $\xi \in \mathbb{R}^N$ which will be fixed suitable later on:
we look for solutions to  (\ref{tp}) ``close'' to a particular
solution of the equation
\begin{equation} \label{xinow2}
\Big( \frac{ \nabla}{i}  -  A(\varepsilon \xi)\Big)^2 u+u + V(\varepsilon \xi)
u = K(\varepsilon \xi) |u|^{p-1} u \quad\hbox{in }\mathbb{R}^N
\end{equation}
(see Remark \ref{near}). More precisely, we denote by $U_c :
\mathbb{R}^N \to\mathbb{C}$ a least-energy solution to the scalar problem
\begin{equation} \label{xiconc}
- \Delta U_c+U_c + V(\varepsilon \xi) U_c = K(\varepsilon \xi) |U_c|^{p-1} U_c
\quad\hbox{in } \mathbb{R}^N.
\end{equation}
By energy comparison (see \cite{ku}), one has that
$$
U_c(x)= e^{i \sigma} U^{\xi}(x-y_0)
$$
for some choice of $\sigma \in [0, 2 \pi ]$ and $y_0 \in
\mathbb{R}^N$, where $U^\xi : \mathbb{R}^N \to\mathbb{R}$ is the unique
solution of
\begin{equation}
\left\{\begin{gathered}
- \Delta U^\xi+U^\xi
+ V(\varepsilon \xi) U^\xi = K(\varepsilon \xi) |U^\xi|^{p-1} U^\xi, \\
U^\xi(0)=\max_{\mathbb{R}^N} U^\xi, \quad
U^\xi >0.
\end{gathered} \right.
\end{equation}
If $U$ denotes the unique solution of
\begin{equation}
\left\{\begin{gathered}
- \Delta U+U=U^p \quad\hbox{in $\mathbb{R}^N$}, \\
U(0)=\max_{\mathbb{R}^N} U, \quad
U >0,
\end{gathered} \right.
\end{equation}
then some elementary and direct computations prove that
$U^\xi(x)=\alpha(\varepsilon \xi) U(\beta(\varepsilon \xi)x)$, where
\[
 \alpha(\varepsilon \xi)= \bigg( \frac{1+V(\varepsilon \xi)}{K(\varepsilon \xi)} \bigg)^{1/(p-1)},
\quad
\beta(\varepsilon \xi)= \big( 1+V(\varepsilon \xi)   \big)^{1/2},
\]
and the function $u(x)= e^{i A(\varepsilon \xi) \cdot x} U_c(x)$ actually
solves (\ref{xinow2}).

For $\xi \in \mathbb{R}^N$ and $\sigma \in [0, 2 \pi ]$, we set
\begin{equation} \label{zeta}
z^{\varepsilon \xi, \sigma}: x \in \mathbb{R}^N \to e^{i \sigma +i A(\varepsilon
\xi) \cdot x} \alpha(\varepsilon \xi) U(\beta(\varepsilon \xi)(x- \xi)).
\end{equation}
Sometimes, for convenience, we shall identify $[0, 2 \pi ]$
and $S^1 \subset \mathbb{C}$, through $\eta= e^{i \sigma}$.
Introduce the functional $F^{\varepsilon \xi, \sigma}: E \to\mathbb{R}$ defined by
 \begin{align*}
F^{\varepsilon \xi, \sigma} (u)
&= \frac 12 \int_{\mathbb{R}^N} \bigg( \Big|
 \Big( \frac{ \nabla u}{i} - A(\varepsilon \xi) u \Big)   \Big|^2 +|u|^2
+V(\varepsilon \xi) |u|^2  \bigg)\, dx   \\
&\quad- \frac{1}{p+1} \int_{\mathbb{R}^N} K(\varepsilon \xi) |u|^{p+1} \, dx,
\end{align*}
whose critical points correspond to solutions of (\ref{xinow2}).
The set
$$
Z^\varepsilon = \{ z^{\varepsilon \xi, \sigma}| \xi \in \mathbb{R}^N \wedge
\sigma \in [0, 2 \pi ] \} \cong S^1 \times \mathbb{R}^N
$$
is a regular manifold of critical points for the functional
$F^{\varepsilon \xi, \sigma}$. From elementary differential geometry
it follows that
 \[
T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon}
= \operatorname{span}_{\mathbb{R}}\Big\{\frac{\partial}{\partial \sigma}
z^{\varepsilon \xi, \sigma}= i z^{\varepsilon \xi, \sigma}, \frac{\partial}{\partial
\xi_1}z^{\varepsilon \xi, \sigma},\dots  ,
\frac{\partial}{\partial \xi_N}z^{\varepsilon \xi, \sigma}  \Big\}
  \]
 where we mean by the symbol  $\operatorname{span}_{\mathbb{R}}$ that
all the linear combinations must have real coefficients.
We remark that, for $j=1,\dots ,N$,
 \begin{equation} \label{zspan}
  \frac{\partial}{\partial \xi_j}z^{\varepsilon \xi, \sigma}
= -\frac{\partial}{\partial x_j}z^{\varepsilon \xi, \sigma}
+i z^{\varepsilon \xi, \sigma} A_j (\varepsilon \xi) +O(\varepsilon)\,.
  \end{equation}
So that any $\zeta \in  T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon}$ can be written as
 \begin{equation}
  \zeta= i l_1 z^{\varepsilon \xi, \sigma}+ \sum_{j=2}^{N+1} l_j \frac{\partial}{\partial x_{j-1}}z^{\varepsilon \xi, \sigma}+O(\varepsilon)
  \end{equation}
for some real coefficients $l_1, l_2,\dots ,l_{N+1}$.

The next lemma shows that $\nabla f_\varepsilon(z^{\varepsilon \xi, \sigma})$ gets
small when $\varepsilon \to0$, namely  $z^{\varepsilon \xi, \sigma}$ is an
``almost solution'' of (\ref{tp}).

\begin{lemma}\label{grad}
For all $\xi \in \mathbb{R}^N$, all $\eta \in S^1$ and all
$\varepsilon > 0$ small, one has that
 \begin{align*}
 \|\nabla f_\varepsilon(z^{\varepsilon \xi, \sigma})\|
 &\leq C \Big( \varepsilon |\nabla V(\varepsilon \xi)|+ \varepsilon |\nabla K(\varepsilon \xi)|
  +\varepsilon |J_A (\varepsilon \xi)|  \\
 &\quad +  \varepsilon |\operatorname{div} A(\varepsilon \xi)|+ \varepsilon^2
+ C(\varepsilon \xi) G(\varepsilon) \Big),
 \end{align*}
 for some constant $C > 0$.
\end{lemma}

\begin{proof}
 From
\begin{equation} \label{somma}
\begin{aligned}
f_\varepsilon(u)&= F^{\varepsilon \xi, \eta} (u)+\frac 12 \int_{\mathbb{R}^N}
\bigg( \Big|  \frac {\nabla u}{i} - A(\varepsilon x) u  \Big|^2
- \Big| \frac {\nabla u}{i} - A(\varepsilon \xi) u  \Big|^2 \bigg)  \\
&\quad +\frac 12 \int_{\mathbb{R}^N} \left[V(\varepsilon x)-V(\varepsilon \xi) \right] |u|^2  - \frac {\gamma(\varepsilon)}{2} \int_{\mathbb{R}^N} W(\varepsilon x)) |u|^2  \\
&\quad - \frac{1}{p+1} \int_{\mathbb{R}^N} \left[K(\varepsilon x)-K(\varepsilon
\xi)\right] |u|^{p+1}
\end{aligned}
\end{equation}
and since $z^{\varepsilon \xi, \eta} $ is a critical point of $F^{\varepsilon \xi, \eta}$,
one has (with $ z= z^{\varepsilon \xi, \eta}  $)
\begin{align*}
&\langle \nabla f_\varepsilon(z) | v \rangle \\
&= \varepsilon \mathop{\rm Re} \int_{\mathbb{R}^N} \frac 1i
(\operatorname{div} A(\varepsilon x)) z \overline{v} + 2 \mathop{\rm Re}
 \int_{\mathbb{R}^N} \left(  A(\varepsilon \xi)- A(\varepsilon x)  \right)z \cdot
\overline{\Big(\frac {\nabla }{i} - A(\varepsilon \xi)\Big) v }   \\
&\quad + \mathop{\rm Re} \int_{\mathbb{R}^N} \left[V(\varepsilon x)-V(\varepsilon \xi) \right] z \overline v  - \gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N} W(\varepsilon x) z \overline v   \\
&\quad - \mathop{\rm Re} \int_{\mathbb{R}^N} \left[K(\varepsilon x)-K(\varepsilon \xi)\right]
|z|^{p-2} z \overline v.
\end{align*}
 From the assumption that $|D^2V(x)|\leq {\rm const.}$
and direct calculations one infers
$$
\int_{\mathbb{R}^N} \left |V(\varepsilon x)-V(\varepsilon \xi) \right|^2
|z^{\varepsilon \xi, \sigma}|^2 \leq c_1 \varepsilon^2 |\nabla V(\varepsilon \xi)|^2 +c_2 \varepsilon^4,
$$
and similar estimates hold for the terms involving $K$ (see \cite{cs}).
In particular, after the change of variable $y=\varepsilon x$, by
 H\"{o}lder inequality and (W1) we have
\begin{equation} \label{tau12}
\begin{aligned}
\gamma(\varepsilon)\int_{\mathbb{R}^N} W(\varepsilon x) | z^{\varepsilon \xi, \sigma} | |
\overline v|
& \leq   \frac{\gamma(\varepsilon)}{\varepsilon^N}
\bigg[\int_{\mathbb{R}^N} W(y)  \Big|z^{\varepsilon \xi, \sigma} \Big(\frac y
\varepsilon \Big) \Big|^2 \bigg]^{1/2}
\bigg[\int_{\mathbb{R}^N} W(y) \Big|v \Big( \frac{y}{\varepsilon} \Big) \Big|^2  \bigg]^{1/2}  \\
& \leq  \frac{\gamma(\varepsilon)}{\varepsilon^N}
\bigg[ \underbrace{\alpha_1 \int_{\mathbb{R}^N} \Big|\nabla \Big|z^{\varepsilon \xi, \sigma}
\Big(\frac y \varepsilon \Big) \Big| \Big|^2   +   \alpha_2   \int_{\mathbb{R}^N}
\left|z^{\varepsilon \xi, \sigma} (\frac y \varepsilon )  \right|^2 }_{\tau_1} \bigg]^{1/2}
 \\
&\quad \times  \bigg[ \underbrace{ \alpha_1 \int_{\mathbb{R}^N}
\left|\nabla \left|v \left( \frac{y}{\varepsilon} \right) \right| \right|^2
+ \alpha_2 \int_{\mathbb{R}^N} \left|v \left( \frac{y}{\varepsilon} \right) \right|^2 }_{\tau_2}
    \bigg]^{1/2}
\end{aligned}
\end{equation}
By the change of variable $x= y /\varepsilon $, the definition of $z$
and (\ref{disug}) we have
\begin{equation} \label{tau1}
\begin{aligned}
\tau_1
 &=   \frac{\varepsilon^N}{\varepsilon^2} \Big[ \alpha_1 \alpha(\varepsilon \xi)^2
\beta(\varepsilon \xi)^{2-N} \int_{\mathbb{R}^N} |\nabla U|^2
+ \alpha_2 \alpha(\varepsilon \xi)^2 \beta(\varepsilon \xi)^{-N} \varepsilon^2
\int_{\mathbb{R}^N}  U ^2 \Big]  \\
& \leq   \frac{\varepsilon^N}{\varepsilon^2} \alpha_\varepsilon
\underbrace{ \underbrace{\alpha(\varepsilon \xi)^2
\beta(\varepsilon \xi)^{-N}}_{C^1(\varepsilon \xi)} \underbrace{\max
\{1, \beta(\varepsilon \xi)^{2}\}}_{C^2(\varepsilon \xi)}}_{C^{1,2}(\varepsilon \xi)}
\|U\|^2
\end{aligned}
\end{equation}
where $C^1(\varepsilon \xi),C^2(\varepsilon \xi) \to1$ as $\varepsilon \to0$ and
\begin{align*}
\tau_2  =  \frac{\varepsilon^N}{\varepsilon^2} \Big[  \alpha_1
\int_{\mathbb{R}^N} |\nabla |\overline v\|^2+ \alpha_2 \varepsilon^2
\int_{\mathbb{R}^N}  |\overline v| ^2   \Big]
 \leq   \frac{\varepsilon^N}{\varepsilon^2} \alpha_\varepsilon c \|v\|^2
\end{align*}
so that
\begin{align*}
\gamma(\varepsilon)\int_{\mathbb{R}^N} W(\varepsilon x) | z^{\varepsilon \xi, \sigma} |\
| \overline v|  \leq  \frac{\gamma(\varepsilon)}{\varepsilon^2}\alpha_\varepsilon C'(\varepsilon
\xi) c' \|U\|\  \|v\| \leq G(\varepsilon)\alpha_\varepsilon C'(\varepsilon \xi) c''  \|v\|
\end{align*}
where $ C'(\varepsilon \xi)=  \left(C^{1,2}(\varepsilon \xi) \right)^{1/2} $.
It then follows that
  \begin{align*}
  \|\nabla f_\varepsilon(z^{\varepsilon \xi, \sigma})\|
&\leq C  \Big( \varepsilon |\nabla V(\varepsilon \xi)|+ \varepsilon |\nabla K(\varepsilon \xi)|+\varepsilon |J_A (\varepsilon \xi)| \\
 &\quad+  \varepsilon |\operatorname{div} A(\varepsilon \xi)|+ \varepsilon^2 + C(\varepsilon \xi)G(\varepsilon)
\Big),
 \end{align*}
where $ C(\varepsilon \xi)= \alpha_\varepsilon C'(\varepsilon \xi)$. The lemma is proved.
\end{proof}

\section{The invertibility of $D^2 f_\varepsilon$ on $(T Z^\varepsilon )^{\bot}$}\label{sez3}

To apply the perturbation method, we need to exploit some
non-degeneracy pro\-per\-ties of the solution $z^{\varepsilon \xi, \sigma}$ as
 a critical point of  $F^{\varepsilon \xi, \sigma}$.
Let $L_{\varepsilon, \sigma, \xi}:{(T_{z^{\varepsilon \xi, \sigma}}
Z^{\varepsilon})}^{\bot} \to{(T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon})}^{\bot} $
be the operator defined by
$$
\langle L_{\varepsilon, \sigma, \xi} v | w \rangle
= D^2 f_\varepsilon( z^{\varepsilon \xi, \sigma})(v,w)
$$
for all $v, w \in {(T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon})}^{\bot}$.
Recall the following elementary result which will play a fundamental
role in the present section.

\begin{lemma} \label{lem3.1}
Let $M \subset \mathbb{R}^N$ be a bounded set. Then there exists a
constant $C > 0$ such that for all $\xi \in M$ one has
\begin{equation}
 \int_{\mathbb{R}^N} \left| \Big( \frac {\nabla }{i} - A(\xi) \Big)
u  \right|^2+ |u|^2 \geq  C \int_{\mathbb{R}^N}
\left(|\nabla u|^2 + |u|^2 \right) \quad\hbox{$\forall u \in E$}.
\end{equation}
\end{lemma}

For the proof, we refer to \cite{cs}. At this point we shall
prove the following result.

\begin{lemma} \label{lemma3.2}
Given $\overline{\xi} > 0$, there exists $ C> 0$ such that for $\varepsilon > 0$
small enough one has
\begin{equation} \label{ellev}
| \langle L_{\varepsilon, \sigma, \xi} v | v \rangle | \geq C \|v\|^2, \quad
\forall |\xi| \leq \overline \xi, \;\forall \sigma \in [0, 2 \pi],\;
\forall v \in {(T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon})}^{\bot}.
\end{equation}
\end{lemma}

\begin{proof}
We follow the arguments in \cite{cs} with some modifications due to
the presence of the terms involving $W$. Recall that
 \[
T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon} =
 \operatorname{span}_{\mathbb{R}}\bigg\{ \frac{\partial}{\partial
\xi_1}z^{\varepsilon \xi, \sigma},\dots  ,\frac{\partial}{\partial \xi_N}
z^{\varepsilon \xi, \sigma}, i z^{\varepsilon \xi, \sigma}  \bigg\},
 \]
 define
  \[
\mathcal{N}  = \operatorname{span}_{\mathbb{R}}
\bigg\{ \frac{\partial}{\partial x_1}z^{\varepsilon \xi, \sigma},\dots ,
\frac{\partial}{\partial x_N}z^{\varepsilon \xi, \sigma}, z^{\varepsilon \xi, \sigma} ,
 i z^{\varepsilon \xi, \sigma} \bigg\}.
 \]
As in \cite{AMMASE,cs},  it suffices to prove (\ref{ellev}) for all
$v \in  \operatorname{span}_{\mathbb{R}}\{z^{\varepsilon \xi, \sigma}, \phi \}$,
where $\phi \  \bot \   \mathcal{N}$. More precisely, we shall prove
that for some constants $C_1 >0$, $C_2 >0$, for all $\varepsilon$ small
 enough and all $|\xi| \leq  \overline \xi$ we have
\begin{gather} \label{elleneg}
 \langle L_{\varepsilon, \sigma, \xi} z^{\varepsilon \xi, \sigma}
| z^{\varepsilon \xi, \sigma} \rangle  \leq -C_1 < 0, \\
 \label{ellepos}
 \langle L_{\varepsilon, \sigma, \xi} \phi | \phi \rangle  \geq C_2 \|\phi\|^2
\quad \forall \  \phi \  \bot \   \mathcal{N}.
\end{gather}
 From the expression for the second derivative of $F^{\varepsilon \xi, \sigma}$
and the fact that $ z^{\varepsilon \xi, \sigma}$, as a solution of (\ref{xinow2}),
is a mountain pass critical point of  $F^{\varepsilon \xi, \sigma}$, we can find
some $c_0 >0$ such that for all $\varepsilon > 0$ small, all
$|\xi| \leq \overline \xi$ and all $\sigma \in [0, 2 \pi]$ it results
\begin{equation} \label{effeneg}
D^2 F^{\varepsilon \xi, \sigma} (z^{\varepsilon \xi, \sigma}) (z^{\varepsilon \xi, \sigma},
z^{\varepsilon \xi, \sigma}) < -c_0 < 0.
\end{equation}
Recalling (\ref{somma}), we find
\begin{align*}
 \langle L_{\varepsilon, \sigma, \xi} z^{\varepsilon \xi, \sigma} | z^{\varepsilon \xi, \sigma}
\rangle
&= D^2 F^{\varepsilon \xi, \sigma} (z^{\varepsilon \xi, \sigma}) (z^{\varepsilon \xi, \sigma},
z^{\varepsilon \xi, \sigma}) \\
&\quad + \int_{\mathbb{R}^N} \bigg( \Big|
  \Big( \frac {\nabla }{i} - A(\varepsilon x) \Big) z^{\varepsilon \xi, \sigma} \Big|^2
- \Big| \Big(\frac {\nabla }{i} - A(\varepsilon \xi) \Big) z^{\varepsilon \xi, \sigma}
 \Big|^2 \bigg)  \\
&\quad + \int_{\mathbb{R}^N} \left[V(\varepsilon x)-V(\varepsilon \xi) \right]
 |z^{\varepsilon \xi, \sigma}|^2  - \gamma(\varepsilon) \int_{\mathbb{R}^N} W(\varepsilon x))
 |z^{\varepsilon \xi, \sigma}|^2  \\
&\quad -  \int_{\mathbb{R}^N} \left[K(\varepsilon x)-K(\varepsilon \xi)\right]
 |z^{\varepsilon \xi, \sigma}|^{p+1}.
\end{align*}
Since, following the computations in (\ref{tau12}) and  (\ref{tau1}),
\begin{align*}
\gamma(\varepsilon) \int_{\mathbb{R}^N} W(\varepsilon x)) |z^{\varepsilon \xi, \sigma}|^2
& \leq  \frac{\gamma(\varepsilon)}{\varepsilon^N}
\bigg[ \underbrace{\alpha_1 \int_{\mathbb{R}^N}
 \Big|\nabla \Big|z^{\varepsilon \xi, \sigma}
 \Big(\frac y \varepsilon \Big) \Big| \Big|^2
+   \alpha_2   \int_{\mathbb{R}^N} \Big|z^{\varepsilon \xi, \sigma}
\Big(\frac y \varepsilon \Big) \Big|^2 }_{\tau_1}   \bigg]  \\
& \leq   \frac{\gamma(\varepsilon)}{\varepsilon^2} \alpha_\varepsilon C^{1,2}(\varepsilon \xi)
\|U\|^2 \leq G(\varepsilon) C' \alpha_\varepsilon C^{1,2}(\varepsilon \xi)
\end{align*}
for $\varepsilon$ small enough, we infer that
\begin{align*}
 \langle L_{\varepsilon, \sigma, \xi} z^{\varepsilon \xi, \sigma} | z^{\varepsilon \xi, \sigma}
\rangle
& \leq  D^2 F^{\varepsilon \xi, \sigma} (z^{\varepsilon \xi, \sigma}) (z^{\varepsilon \xi, \sigma},
z^{\varepsilon \xi, \sigma})+ c_1  \varepsilon |\nabla V(\varepsilon \xi)| \\
&\quad +  c_2 \varepsilon |\nabla K(\varepsilon \xi)|+ c_3 \varepsilon |J_A (\varepsilon \xi)|
+ c_4 \varepsilon^2 +c_5 C(\varepsilon \xi)G(\varepsilon)
\end{align*}
where $C(\varepsilon \xi)= \alpha_\varepsilon C^{1,2}(\varepsilon \xi)  $.
Hence (\ref{elleneg}) follows. The proof of (\ref{ellepos}) is more
 involved.  As before, since $z^{\varepsilon \xi, \sigma}$ is a critical
point for   $F^{\varepsilon \xi, \sigma}$ of mountain-pass type, by standard
 results (see \cite{chang}) there results
 \begin{equation} \label{effephi}
D^2 F^{\varepsilon \xi, \sigma} (z^{\varepsilon \xi, \sigma}) (\phi, \phi) \geq c_1 \| \phi\|^2
 \quad \forall \phi \ \bot \  \mathcal{N}.
\end{equation}
Let $R >> 1$  and consider a radial smooth function $\chi_1:
\mathbb{R}^N \to\mathbb{R}$ such
\begin{gather*}
\chi_1(x)=1, \quad\hbox{for $|x| \leq R$;} \quad  \chi_1(x)=0,
\quad\hbox{for }|x| \geq 2R; \\
| \nabla \chi_1(x)| \leq \frac 2R, \quad\hbox{for } R \leq|x| \leq 2R.
\end{gather*}
 We also set $\chi_2(x)=1-\chi_1(x)$. Given $\phi$ let us consider
the functions
$$
\phi_i(x)=\chi_i(x-\xi) \phi(x), \quad i=1,2.
$$
Due to the definition of $\chi$,  straightforward computations yield
\begin{equation} \label{sommaN}
\| \phi\|^2= \| \phi_1\|^2+\| \phi_2\|^2+ 2 I_{\phi} + o_R(1) \| \phi\|^2
\end{equation}
where $I_\phi= \int_{\mathbb{R}^N} \chi_1 \chi_2 (\phi^2+ |\nabla \phi|^2)$
and $o_R(1)$ is a function which tends to $0$, as $R \to+\infty$.

At this point, let us evaluate the three terms in the equation below:
$$
\left( L_{\varepsilon, \sigma, \xi} \phi | \phi \right)
= \underbrace{( L_{\varepsilon, \sigma, \xi} \phi_1 | \phi_1 )}_{\alpha_1}
+\underbrace{ ( L_{\varepsilon, \sigma, \xi} \phi_2 | \phi_2 )}_{\alpha_2}
+ 2 \underbrace{( L_{\varepsilon, \sigma, \xi} \phi_1 | \phi_2 )}_{\alpha_3}.
$$
 One has
\begin{align*}
\alpha_1&= \langle L_{\varepsilon, \sigma, \xi} \phi_1 | \phi_1 \rangle = D^2 F^{\varepsilon \xi, \sigma} (z^{\varepsilon \xi, \sigma})(\phi_1, \phi_1)  \\
&\quad + \int_{\mathbb{R}^N} \left[V(\varepsilon x)-V(\varepsilon \xi) \right] |\phi_1|^2  - \gamma(\varepsilon) \int_{\mathbb{R}^N} W(\varepsilon x)) |\phi_1|^2  \\
&\quad -  \int_{\mathbb{R}^N} \left[K(\varepsilon x)-K(\varepsilon \xi)\right] |\phi_1|^{p+1} \\
&\quad + \int_{\mathbb{R}^N} \bigg( \Big| \Big( \frac {\nabla }{i} -
A(\varepsilon x) \Big) \phi_1 \Big|^2- \Big| \Big(\frac {\nabla }{i}
- A(\varepsilon \xi) \Big) \phi_1 \Big|^2 \bigg).
 \end{align*}
 Using (\ref{effephi}) (for details, see \cite{cs}), we infer
  \begin{equation} \label{effephi1}
D^2 F^{\varepsilon  \xi} \left[\phi_1, \phi_1 \right]
\geq C \| \phi_1\|^2+ o_R(1)\| \phi\|^2.
\end{equation}
Using arguments already carried out before, one has
$$
\int_{\mathbb{R}^N} \left[V(\varepsilon x)-V(\varepsilon \xi) \right] |\phi_1|^2
\leq \varepsilon c R \| \phi\|^2
$$
and similarly for the terms containing $K$. In particular, by the
change of variable $y=\varepsilon x$, assumption (W1) and the definition
of $\phi_1$ we have
\begin{align*}
&  \gamma(\varepsilon) \int_{\mathbb{R}^N} W(\varepsilon x) |\phi_1|^2  \leq   \frac{\gamma(\varepsilon)}{\varepsilon^2} \left[  \alpha_1  \int_{\mathbb{R}^N} |\nabla |\chi_1(x-\xi) \phi\|^2+ \alpha_2 \varepsilon^2 \int_{\mathbb{R}^N}  |\chi_1(x-\xi) \phi| ^2   \right] \\
&  =    \frac{\gamma(\varepsilon)}{\varepsilon^2} \bigg[  \alpha_1  \Big( \int_{R \leq |y|
\leq 2R} |\nabla \chi_1(y)|^2  | \phi(y+\xi)|^2
 + \int_{|y| \leq 2R} | \chi_1(y)|^2  | \nabla \phi(y+\xi)|^2  \\
&\quad  +  2 \int_{R \leq |y| \leq 2R}  \chi_1(y) \phi(y+\xi)
 \nabla \chi_1(y) \cdot \nabla \phi(y+\xi) \Big)\\
&  \quad + \alpha_2 \varepsilon^2 \int_{| y| \leq 2R}  |\chi_1(y) \phi(y+\xi)| ^2   \bigg] \\
&  \leq  G(\varepsilon)  \left [ \alpha_{\varepsilon} \| \phi_1\|^2+ o_R(1)\| \phi\|^2 \right]
 \end{align*}
 It follows that
  \begin{equation} \label{alph1}
\begin{aligned}
 \alpha_1= \left( L_{\varepsilon, \sigma, \xi} \phi_1 | \phi_1 \right)
& \geq  c_1 \| \phi_1\|^2- c_2 \varepsilon R \| \phi\|^2+ o_R(1)\| \phi\|^2  \\
& \quad -   G(\varepsilon)  \left[ \alpha_\varepsilon \| \phi_1\|^2+ o_R(1)\| \phi\|^2
\right]\,.
\end{aligned}
\end{equation}
Let us now estimate $\alpha_2$.  In particular,
\begin{align*}
&\gamma(\varepsilon) \int_{\mathbb{R}^N} W(\varepsilon x) |\phi_2|^2\\
& \leq   \frac{\gamma(\varepsilon)}{\varepsilon^2}
\Big[  \alpha_1  \int_{\mathbb{R}^N} |\nabla |\chi_2(x-\xi) \phi\|^2
+ \alpha_2 \varepsilon^2 \int_{\mathbb{R}^N}  |\chi_2(x-\xi) \phi| ^2   \Big] \\
&=    \frac{\gamma(\varepsilon)}{\varepsilon^2} \Big[  \alpha_1  \Big( \int_{R \leq |y|
\leq 2R} |\nabla \chi_2(y)|^2  | \phi(y+\xi)|^2 + \int_{|y| \geq R} |
\chi_2(y)|^2  | \nabla \phi(y+\xi)|^2   \\
&\quad +   2 \int_{R \leq |y| \leq 2R} \! \chi_2(y) \phi(y+\xi)
\nabla \chi_2(y) \cdot \nabla \phi(y+\xi) \Big)
+ \alpha_2 \varepsilon^2 \int_{| y| \geq R}  |\chi_2(y) \phi(y+\xi)| ^2   \Big] \\
& \leq  G(\varepsilon)  \left [ \alpha_\varepsilon \| \phi_2\|^2+ o_R(1)\| \phi\|^2 \right].
 \end{align*}
One finds
 \begin{equation} \label{alph2}
 \alpha_2= \left( L_{\varepsilon, \sigma, \xi} \phi_2 | \phi_2 \right)
\geq c_3 \| \phi_2\|^2 + o_R(1)\| \phi\|^2  \\
  - G(\varepsilon)  \left [ \alpha_\varepsilon \| \phi_2\|^2+ o_R(1)\| \phi\|^2 \right]
\end{equation}
In a quite similar way one shows that
 \begin{equation} \label{alph3}
\begin{aligned}
 \alpha_3 &= \left( L_{\varepsilon, \sigma, \xi} \phi_1 | \phi_2 \right) \geq c_4 I_{\phi} + o_R(1)\| \phi\|^2  \\
 &\quad - G(\varepsilon) \left[   \left( \alpha_\varepsilon \| \phi_1\|^2+ o_R(1)\|
\phi\|^2  \right)^{1/2}   \left ( \alpha_\varepsilon \| \phi_2\|^2
+ o_R(1)\| \phi\|^2 \right)^{1/2} \right]
\end{aligned}
\end{equation}
Indeed, by the change of variable $y=\varepsilon x$, assumption (W1)
and H\"{o}lder inequality
\begin{align*}
& \gamma(\varepsilon) \int_{\mathbb{R}^N} W(\varepsilon x) |\phi_1(x)|
|\overline{\phi_2(x)} | \\
&  \leq   \frac{\gamma(\varepsilon)}{\varepsilon^N} \Big[ \Big(\int_{\mathbb{R}^N}
W(y) \left|\phi_1 \left(\frac y \varepsilon \right) \right|^2 \Big)^{1/2}
  \Big(\int_{\mathbb{R}^N} W(y) \left| \overline{\phi_2
\left( \frac y \varepsilon \right)} \right|^2  \Big)^{1/2}  \Big] \\
&  \leq    \frac{\gamma(\varepsilon)}{\varepsilon^2} \Big[  \Big( \alpha_1
\int_{\mathbb{R}^N} |\nabla |\chi_1(x-\xi) \phi\|^2+ \alpha_2 \varepsilon^2
\int_{\mathbb{R}^N}  |\chi_1(x-\xi) \phi| ^2 \Big)^{1/2}   \\
& \quad  \times    \Big( \alpha_1  \int_{\mathbb{R}^N} |\nabla |
\chi_2(x-\xi) \phi\|^2+ \alpha_2 \varepsilon^2 \int_{\mathbb{R}^N}  |
\chi_2(x-\xi) \phi| ^2 \Big)^{1/2} \Big] \\
&  \leq  G(\varepsilon) \left[   \left( \alpha_\varepsilon \| \phi_1\|^2+ o_R(1)\|
 \phi\|^2  \right)^{1/2}   \left ( \alpha_\varepsilon \| \phi_2\|^2
+ o_R(1)\| \phi\|^2 \right)^{1/2} \right]
\end{align*}
where in the last inequality we have used previous calculations.
Finally, (\ref{alph1}), (\ref{alph2}), (\ref{alph3}) and the fact
 that $I_{\phi} \geq 0$, yield
\begin{align*}
\left( L_{\varepsilon, \sigma, \xi} \phi | \phi \right)
&=  \alpha_1+\alpha_2+2 \alpha_3 \\
& \geq  c_5 \left[  \| \phi_1\|^2+ \| \phi_2\|^2+2I_{\phi}  \right]-c_6 R \varepsilon \| \phi\|^2+ o_R(1)\| \phi\|^2 \\
&\quad -  G(\varepsilon) \alpha_\varepsilon \Big[  \| \phi_1\|^2+ \| \phi_2\|^2 +2
\left(  \| \phi_1\|^2+ o_R(1)\| \phi\|^2  \right)^{1/2}\\
&\quad\times \big(  \| \phi_2\|^2+ o_R(1)\| \phi\|^2 \big)^{1/2}
 +  o_R(1)\| \phi\|^2  \Big]
\end{align*}
Recalling (\ref{sommaN}), we infer that
 \begin{align*}
\left( L_{\varepsilon, \sigma, \xi} \phi | \phi \right)
&\geq  c_7 \| \phi\|^2-c_8 R \varepsilon \| \phi\|^2+ o_R(1)\| \phi\|^2 \\
&\quad -  G(\varepsilon) \alpha_\varepsilon \Big[  \| \phi_1\|^2+ \| \phi_2\|^2
 +2  \left(  \| \phi_1\|^2+ o_R(1)\| \phi\|^2  \right)^{1/2}\\
&\quad\times   \left (  \| \phi_2\|^2+ o_R(1)\| \phi\|^2 \right)^{1/2}
 +       o_R(1)\| \phi\|^2  \Big]
\end{align*}
Taking $R=\varepsilon^{-1/2}$, and choosing $\varepsilon$ small, equation (\ref{ellepos})
follows. This completes the proof.
\end{proof}

\section{The finite-dimensional reduction}\label{finitered}

In this section we will show that the existence of critical points
of $f_\varepsilon$ can be reduced to the search of critical points of an
auxiliary finite-dimensional functional. The proof will be carried
out in two subsections dealing, respectively, with a Liapunov-Schmidt
reduction, and with the behavior of the auxiliary finite dimensional
functional.

\subsection{A Liapunov-Schmidt type reduction}
The main result of this section is the following lemma.

\begin{lemma}\label{implicitw}
For $\varepsilon > 0$ small, $|\xi| \leq \overline \xi$ and
$ \sigma \in [0, 2 \pi]$, there exists a unique
$w=w(\varepsilon, \sigma, \xi) \in {(T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon})}^{\bot}$
such that $\nabla f_\varepsilon( z^{\varepsilon \xi, \sigma}+w) \in T_{z^{\varepsilon \xi,
\sigma}} Z^{\varepsilon}$. Such a $w(\varepsilon, \sigma, \xi)$ is of  class $C^2$,
respectively $C^{1,p-1}$, with respect to $\xi$, provided that $p \geq 2$,
respectively $1 < p < 2$. Moreover, the functional
$\Phi_\varepsilon (\sigma, \xi)=f_\varepsilon( z^{\varepsilon \xi, \sigma}+w(\varepsilon, \sigma, \xi))$
has the same regularity as $w$ and satisfies:
$$
\nabla \Phi_\varepsilon (\sigma_0, \xi_0)=0 \Longleftrightarrow
\nabla f_\varepsilon( z_{ \xi_0}+w(\varepsilon, \sigma_0, \xi_0))=0.
$$
\end{lemma}

For the proof of the above lemma, we refer to \cite[Lemma 4.1]{cs}.


\begin{remark}\label{indipdasigma} \rm
Since $f_\varepsilon (z^{\varepsilon \xi, \sigma})$ is independent of $\sigma$,
the implicit function $w$ is constant with respect to that
variable. Consequently, there exists a functional $\Psi_\varepsilon:
\mathbb{R}^N \to\mathbb{R}$ such that
$$
\Phi_\varepsilon (\sigma, \xi)= \Psi_\varepsilon (\xi), \quad\forall \sigma
\in [0, 2 \pi], \; \forall \xi \in \mathbb{R}^N.
$$
For this reason, in the sequel we will omit the dependence of $w$
on $\sigma$, even it is defined over $S^1 \times \mathbb{R}^N$.
\end{remark}

\begin{remark} \rm
 From the proof of  Lemma \ref{implicitw} (see \cite{cs})
and Lemma \ref{grad}, it follows that:
\begin{equation} \label{normaw}
\|w\| \leq C \left( \varepsilon |\nabla V(\varepsilon \xi)|
+ \varepsilon |\nabla K(\varepsilon \xi)|+\varepsilon |J_A (\varepsilon \xi)|
+\varepsilon^2+ C(\varepsilon \xi) G(\varepsilon) \right),
\end{equation}
where $C > 0$.
\end{remark}

 For future reference, it is also convenient to estimate the gradient
$  \nabla_{\xi} w$.


\begin{lemma}
It results
\begin{equation} \label{normagrad}
\|\nabla_\xi w\| \leq c \left( \varepsilon |\nabla V(\varepsilon \xi)|
+ \varepsilon |\nabla K(\varepsilon \xi)|+\varepsilon |J_A (\varepsilon \xi)| +O(\varepsilon^2)
\right)^{\gamma},
\end{equation}
where $\gamma=\min\{ 1,p-1\}$ and $c > 0$ is some constant.
\end{lemma}

\begin{proof}
 For the details, we refer to  \cite[Lemma 4]{AMMASE} and
  \cite[Lemma 4.2]{cs}. We will denote by  $\dot{w}_{i}$
the components of $  \nabla_{\xi} w$ and
$\dot{z}_{i}= \partial_{\xi_{i}} z$. Since $w$ satisfies the equation
$\langle  P \nabla f_\varepsilon( z^{\varepsilon \xi, \sigma}+w), v \rangle =0$
for all $v \in {(T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon})}^{\bot}$
(with $P=$ the projection onto ${(T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon})}^{\bot}$),
 we find that  $\dot{w}_{i}$ verifies
$$
\partial_{\xi_{i}} \left(\langle L_{\varepsilon, \sigma, \xi} w, v \rangle
+ \langle P \nabla f_\varepsilon( z), v \rangle + \langle
 R(z, w),v \rangle \right)=0
$$
with $ R(z, w) =\|o(w)\|$. Taking into account   \cite[Lemma 4.2]{cs},
 we limit to estimate the $ \partial_{\xi_{i}}$ of $\nabla W_{\varepsilon}(z)[v]$,
namely
$$
\partial_{\xi_{i}} \Big( -\gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N} W(\varepsilon x) z
 \overline v\Big) =-\gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N} W(\varepsilon x)
\dot{z}_{i}   \overline v.
$$
As in (\ref{tau12}), by  (W1) and the expression of $\dot{z}_{i}$
in (\ref{zspan}) we get
$$
\gamma(\varepsilon) \bigg|  \int_{\mathbb{R}^N} W(\varepsilon x)  \dot{z}_{i}
 \overline v \bigg| \leq  \gamma(\varepsilon)   \int_{\mathbb{R}^N} W(\varepsilon x)
\left| \dot{z}_{i} \right|    \left| \overline v \right|
  \leq  \widetilde{C}(\varepsilon \xi) \varepsilon  G(\varepsilon) \| v \|
$$
where $\widetilde{C}(\varepsilon \xi)$  depends on
 $\alpha$ and $\beta$. From  \cite[Lemma 4]{AMMASE},
Inequality (\ref{normagrad}) follows without effort.
\end{proof}

\subsection{The finite-dimensional functional}
The purpose of this subsection is to give an explicit form to the finite
dimensional functional
$\Phi_\varepsilon (\sigma, \xi)= \Psi_\varepsilon (\xi)=f_\varepsilon (z^{\varepsilon \xi, \sigma}
+ w(\varepsilon, \xi))$. For brevity, we set in the sequel
$z=z^{\varepsilon \xi, \sigma}$ and $w=w(\varepsilon, \xi)$.
Since $z$ satisfies (\ref{xinow2}) and $K''$ is bounded we get
\begin{equation} \label{sviluppo}
\begin{aligned}
\Phi_\varepsilon (\sigma, \xi)
&=  f_\varepsilon (z^{\varepsilon \xi, \sigma}+ w(\varepsilon, \sigma, \xi))  \\
&=  K(\varepsilon \xi) \left( \frac 12-\frac {1}{p+1} \right) \int_{\mathbb{R}^N} |z|^{p+1}+ \frac12 \int_{\mathbb{R}^N} \left|  A(\varepsilon \xi)- A(\varepsilon x)  \right|^2 z^2  \\
&\quad + \mathop{\rm Re} \int_{\mathbb{R}^N} \left(  A(\varepsilon \xi)- A(\varepsilon x)  \right) z \cdot \left(  A(\varepsilon \xi)- A(\varepsilon x)  \right) \overline w  \\
&\quad +  \varepsilon \mathop{\rm Re} \int_{\mathbb{R}^N} \frac 1i z \overline{w}
\operatorname{div} A(\varepsilon x)+ \frac12 \int_{\mathbb{R}^N}
\left|\left( \frac {\nabla }{i} - A(\varepsilon x)   \right) w \right|^2  \\
&\quad + \mathop{\rm Re} \int_{\mathbb{R}^N} \left[V(\varepsilon x)-V(\varepsilon \xi) \right] z \overline w + \frac12 \int_{\mathbb{R}^N} \left[V(\varepsilon x)-V(\varepsilon \xi) \right] |w|^2  \\
&\quad + \frac12 \int_{\mathbb{R}^N} \left[V(\varepsilon x)-V(\varepsilon \xi) \right] |z|^2 + \frac12 V(\varepsilon \xi) \int_{\mathbb{R}^N} |w|^2  \\
&\quad -   \frac{\gamma(\varepsilon)}{2}  \int_{\mathbb{R}^N} W(\varepsilon x) |z|^2 - \gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N} W(\varepsilon x) z \overline w  \\
&\quad - \frac{\gamma(\varepsilon)}{2}  \int_{\mathbb{R}^N} W(\varepsilon x) |w|^2
  \\
&\quad - \frac{1}{p+1} \mathop{\rm Re}  \int_{\mathbb{R}^N} K(\varepsilon x) \left( |z+w|^{p+1}-|z|^{p+1} -(p+1)|z|^{p-1} z  \overline w  \right)  \\
&\quad + \mathop{\rm Re} K(\varepsilon \xi) \int_{\mathbb{R}^N} |z|^{p-1} z \overline
w+O(\varepsilon^2).
\end{aligned}
\end{equation}
By the definition of $\alpha(\varepsilon \xi)$ and $\beta(\varepsilon \xi)$ we get
immediately
\begin{equation}
\int_{\mathbb{R}^N} |z^{\varepsilon \xi, \sigma}|^{p+1}= C_0 \Lambda(\varepsilon
\xi) \left[ K(\varepsilon \xi) \right]^{-1}
\end{equation}
where we define the auxiliary function
\begin{equation} \label{lambda}
\Lambda(x)= \frac{(1+V(x))^{\theta}}{K(x)^{2/(p-1)}}, \quad
\theta=\frac{p+1}{p-1}- \frac N2
\end{equation}
for all $x \in \mathbb{R}^N$ since, by (K1), $K$ is strictly
positive on $\mathbb{R}^N$ and $C_0= \int_{\mathbb{R}^N}
|U|^{p+1}$. Now we can estimate the various terms in
(\ref{sviluppo}) by means of (\ref{normaw}) and (\ref{normagrad})
as in \cite{cs}. In particular,
\begin{gather*}
\gamma(\varepsilon)  \int_{\mathbb{R}^N} W(\varepsilon x) |z|^2 \leq
\frac{\gamma(\varepsilon)}{\varepsilon^2}\alpha_\varepsilon C^{1,2}(\varepsilon \xi) \| U \|^2
\leq G(\varepsilon) C(\varepsilon \xi) C_1,
\\
\gamma(\varepsilon) \int_{\mathbb{R}^N} W(\varepsilon x) | z | |\overline w | \leq
\frac{\gamma(\varepsilon)}{\varepsilon^2} \alpha_\varepsilon C'(\varepsilon \xi) C'_2  \| U \| \|
w \| \leq G(\varepsilon)C''(\varepsilon \xi) C_2  \| w \|,
\\
\gamma(\varepsilon)  \int_{\mathbb{R}^N} W(\varepsilon x) |w|^2 \leq
\frac{\gamma(\varepsilon)}{\varepsilon^2} \alpha_\varepsilon C_3 \| w \|^2 \leq G(\varepsilon)
\alpha_\varepsilon C_3 \| w \|^2
\end{gather*}
where
$$
C(\varepsilon \xi)= \alpha_{\varepsilon} C^{1,2}(\varepsilon \xi)
= \alpha_{\varepsilon} \alpha^2(\varepsilon \xi) \beta(\varepsilon \xi)^{-N}
\max \{1, \beta^2(\varepsilon \xi)\}
$$
and
$C''(\varepsilon \xi)= \alpha_{\varepsilon} C'(\varepsilon \xi)$ with
$C'(\varepsilon \xi)= (C^{1,2}(\varepsilon \xi))^{1/2}$. So it results
\begin{equation} \label{funzridotto}
\Phi_\varepsilon (\sigma, \xi)= \Psi_\varepsilon (\xi)=C_1 \Lambda(\varepsilon \xi)+O(\varepsilon).
\end{equation}
Similarly,
\begin{equation} \label{derivfunz}
\nabla \Psi_\varepsilon (\xi)=C_1 \nabla \Lambda(\varepsilon \xi)+ \varepsilon^{1+\gamma}O(1)
\end{equation}
where $C_1= ( \frac 12-\frac{1}{p+1}) C_0$. Indeed, taking account
of the result in \cite{cs},
we limit  to consider
\begin{align*}
 \nabla_{\xi} W_{\varepsilon}(z+w)
&=  \nabla_{\xi} \left( -\frac{\gamma(\varepsilon)}{2} \int_{\mathbb{R}^N} W(\varepsilon x) |z+w|^2 \right)= \langle  W'_{\varepsilon}(z+w), ( \nabla_{\xi} z + \nabla_{\xi} w )   \rangle   \\
&= -   \gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N} W(\varepsilon x)  (z+w )\  \overline{\nabla_{\xi} z + \nabla_{\xi} w }  \\
&=   - \gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N} W(\varepsilon x) z  \overline{\nabla_{\xi} z} - \gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N} W(\varepsilon x)  z   \overline{ \nabla_{\xi} w}  \\
&\quad-  \gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N} W(\varepsilon x) w
\overline{\nabla_{\xi} z }-  \gamma(\varepsilon) \mathop{\rm Re} \int_{\mathbb{R}^N}
W(\varepsilon x) w  \overline{ \nabla_{\xi} w}.
\end{align*}
whose last four terms can be estimated as in (\ref{tau12})
by means of (\ref{normaw}) and (\ref{normagrad}) again so that
 (\ref{derivfunz}) holds.

\section{Statement and proof of the main results}\label{statem}

In this section we obtain existence and multiplicity of solutions
to (\ref{gespl}) by means of the finite-dimensional reduction
performed in the previous section. Recalling  Lemma \ref{implicitw},
we have to look for critical points of $\Phi_\varepsilon$ as a function of
the variables $(\sigma, \xi ) \in [0, 2 \pi  ] \times \mathbb{R}^N$
(or, equivalently,  $(\eta, \xi ) \in S^1 \times \mathbb{R}^N$).

We use the following notation: given a set
$M \subset \mathbb{R}^N$ and a number $ \delta> 0$,
$$
M_{\delta}:= \{ x \in \mathbb{R}^N: \operatorname{dist}(x, \Omega)
 < \delta  \}.
$$
If $M \subset N$, $\operatorname{cat}(M,N)$ denotes
the Ljusternik-Schnirelman category of $M$ with respect to $N$,
namely the least integer $k$ such that $M$ can be covered by $k$
closed subsets of $N$, contractible to a point in $N$.
We set $\operatorname{cat}(M)=\operatorname{cat}(M,M)$.
We start with the following result, which deals with local extrema.

\begin{theorem}\label{cinque1}
Suppose we are in the hypotheses of Theorem \ref{main}. Assume
moreover that there is a compact set $M \subset \mathbb{R}^N$ over
which $\Lambda$ achieves an isolated strict local minimum, resp.
maximum, with value $a$, resp. $b$, in the sense that for some
$\delta > 0$,
$$
b:= \inf_{x \in \partial M_\delta} \Lambda(x) > a, \quad\text{resp. }
a:= \sup_{x \in \partial M_\delta} \Lambda(x) < b.
$$
Then there exists $ \varepsilon_\delta > 0$ such that \eqref{tp} has at
least $\operatorname{cat}(M, M_\delta)$ (orbits of) solutions
concentrating near $M_\delta$, for all $0 < \varepsilon < \varepsilon_\delta$.
\end{theorem}

 For the sake of completeness, we rewrite the proof as in
\cite{ AMMASE, cs}.

\begin{proof}
Recall that $\Phi_\varepsilon(\eta, \xi)= \Psi_\varepsilon (\xi)$ and choose
$\overline{\xi} > 0$ such that $M_{\delta} \subset \{ x \in
\mathbb{R}^N |\  |x| < \overline{\xi}\}$. Set $N^{\varepsilon}
=\{ \xi \in \mathbb{R}^N |\ \varepsilon \xi \in M \} $,
$N^{\varepsilon}_{\delta}= \{ \xi \in \mathbb{R}^N |\ \varepsilon \xi \in
M_{\delta} \}  $ and $ \Theta^{\varepsilon}= \{ \xi \in
\mathbb{R}^N |\ \Psi_{\varepsilon}(\xi) \leq C_1 \frac{a+b}{2}\} $.
 From (\ref{funzridotto}) we get some $\varepsilon_{\delta} > 0$ such that
\begin{equation} \label{inclusioni}
  N^{\varepsilon} \subset \Theta^{\varepsilon} \subset N^{\varepsilon}_{\delta},
  \end{equation}
  for all $ 0 < \varepsilon < \varepsilon_{\delta}$.   To apply standard category theory,
it suffices to prove that $\Theta_{\varepsilon}$ cannot touch
$\partial N^{\varepsilon}_{\delta}$ so that $\Theta_{\varepsilon}$ is compact.
But if $\varepsilon \xi \in \partial N^{\varepsilon}_{\delta}$, one has
$\Lambda(\varepsilon \xi) \geq b$ by the definition of $\delta$, and so
$$
 \Psi_{\varepsilon}(\xi) \geq C_1 \Lambda(\varepsilon \xi) +o_{\varepsilon}(1)
\geq C_1 b + o_{\varepsilon}(1).
$$
On the other hand, for all $\xi \in \Theta^{\varepsilon}$ one has also
$ \Psi_{\varepsilon}(\xi) \leq C_1 \frac{a+b}{2} $.
 From (\ref{inclusioni}) and elementary properties of
the Ljusternik-Schnirelman category we can conclude that
$\Psi_{\varepsilon}$ has at least
$$
\operatorname{cat}( \Theta^{\varepsilon},\Theta^{\varepsilon} )
\geq   \operatorname{cat}( N^{\varepsilon}, N^{\varepsilon}_{\delta})
  =  \operatorname{cat}( N ,N_{\delta} )
$$
critical points in $\Theta^{\varepsilon}$, which correspond to at least
$ \operatorname{cat}( M, M_{\delta} )$ orbits of solutions
to \eqref{tp}. Now, let $(\eta^{*}, \xi^{*}) \in S^1 \times M_{\delta}$
 a critical point of $\Phi_{\varepsilon}$. By Lemma \ref{implicitw},
this point localizes a solution
$ u_{\varepsilon, \eta^{*}, \xi^{*} }(x)= z^{\varepsilon  \xi^{*}, \eta^{*}}(x)
+ w (\varepsilon, \eta^{*}, \xi^{*})  $ of  \eqref{tp}.
By the change of variable which allowed us to pass from (\ref{gespl})
   to \eqref{tp} we find that
\begin{equation} \label{concentr}
  u_{\varepsilon, \eta^{*}, \xi^{*} }(x)  \approx     z^{\varepsilon  \xi^{*}, \eta^{*}}
 \left( \frac{x - \xi^{*}}{\varepsilon} \right)
\end{equation}
satisfies   (\ref{gespl}) where $\approx $ stands for the concept
of ``near'' or  ``close'' whose sense is  explained in the following
Remark \ref{near}. The concentration statement follows as in
\cite{AMMASE} from standard arguments.  The proof of the second
part follows with analogous arguments.
\end{proof}


\begin{remark}\label{near} \rm
By means of a Liapunov-Schmidt type reduction, we have found that
a solution of \eqref{tp} has the form   $u_{\varepsilon,
\eta^{*}, \xi^{*} }(x)= z^{\varepsilon  \xi^{*}, \eta^{*}}(x) + w (\varepsilon,
\eta^{*}, \xi^{*})  $. From this and the properties of the
function  $w (\varepsilon, \eta^{*}, \xi^{*})$, it follows that   $
\|u_{\varepsilon, \eta^{*}, \xi^{*} }-  z^{\varepsilon  \xi^{*}, \eta^{*}}\| \to0$
as $\varepsilon \to0$. In this sense, we say that the complex solutions
$u_{\varepsilon, \eta^{*}, \xi^{*} }(x)$ of \eqref{tp} are found
``near''  or  ``close'' to the least energy solutions $z^{\varepsilon
\xi^{*}, \eta^{*}}$ of (\ref{xinow}) and this corresponds, after
the change of variable, to (\ref{concentr}).
\end{remark}


Observe  that Theorem \ref{main} in the Introduction is an immediate
corollary of the previous one when $x_0$ is either a nondegenerate
local maximum or minimum for $\Lambda$. When $\Lambda$ has a maximum,
the direct variational approach and the arguments in  \cite{Bar}
cannot be applied.

 To treat the general case, we refer to some topological concepts as
the \textit{cup long} of a set $M \subset \mathbb{R}^N$  which is by
definition
 $$
l(M)=  1+ \sup \{ k \in \mathbb{N} \mid ( \exists \alpha_1,\dots,
\alpha_N \in \check{H}^*(M) \setminus \{ 1 \})
(\alpha_1 \cup \dots  \cup \alpha_k \neq 0)\},
$$
where $\check{H}^*(M)$ is the Alexander cohomology of $M$ with real
coefficients and $\cup$ denotes the cup product. In some cases as
 $M=S^{N-1}$, $T^N$, we have $l(M)= \operatorname{cat}(M)$,
but in general $l(M) \leq  \operatorname{cat}(M)$.
Furthermore we recall the following definition which dates
back to  Bott \cite{bott}:

\begin{definition} \label{def5.3} \rm
We say that $M$ is non-degenerate for a $C^2$ function $I:
\mathbb{R}^N \to\mathbb{R}$ if $M$ consists of Morse theoretically
non-degenerate critical points for the restriction
$I_{|M^{\bot}}$.
\end{definition}

To prove our existence result,  we recall the following result
which is an adaptation of
\cite[Theorem 6.4, Chapter II]{chang}
and fits into the frame of the Conley theory \cite{conley}.

\begin{theorem}\label{funznali}
 Let $ I \in C^1(V)$ and $J \in C^2(V)$ be two functionals defined
on the Riemannian manifold $V$, and let $\Sigma \subset V$ be a smooth,
compact, non-degenerate manifold of critical points of $J$.
Denote by $ \mathcal{U } $ a neighborhood of $\Sigma$.
 If $\| I-J\|_{C^1( \mathcal{U }  )}$ is small enough,
then the functional $I$ has at least $l(\Sigma)$ critical
points contained in $ \mathcal{U}$.
\end{theorem}

At this point, we can prove an existence and multiplicity result
for (\ref{gespl}).

\begin{theorem} \label{thm5.5}
Let {\rm (K1), (V1), (W1), (A1), (G1)} hold.
If the auxiliary function $\Lambda$ has a smooth, compact,
non-degenerate manifold of critical points $M$, then for $\varepsilon > 0$
small, the problem \eqref{tp} has at least $l(M)$ (orbits)
of solutions concentrating near points of $M$.
\end{theorem}

\begin{proof}
By Remark \ref{indipdasigma}, we have to find critical points of
$\Psi_{\varepsilon}=\Psi_{\varepsilon}(\xi)$. Since $M$ is compact, we can choose
and fix $\overline{\xi} >0$ so that $|x| < \overline{\xi} $ for
all $x \in M$. $\{ \eta^* \} \times M$ is obviously a
non-degenerate critical manifold. We set $V= \mathbb{R}^N$,
$J=\Lambda$, $\Sigma=M$ and $I(\xi)=\Psi_{\varepsilon}(\eta, \xi / \varepsilon)$.
Select $\delta > 0$ so that $M_{\delta} \subset  \{ x \in
\mathbb{R}^N |\  |x| < \overline{\xi} \}$, and no critical
points of $\Lambda$ are in $M_{\delta}$, except for those of $M$.
Set $\mathcal{U}=M_{\delta}$. By the definition of
(\ref{funzridotto}) and (\ref{derivfunz}) and hypotheses (K1)
and (V1), it follows that $ J \in C^2( \overline{\mathcal{U }}
)$. As concerns as the regularity of the functional $I$,
 we  have  to prove that the functional
\begin{align*}
\widetilde{W}(\xi)
&=\widetilde{W}_{\varepsilon} ( \eta,  \xi / \varepsilon ) \\
&= -\frac{\gamma(\varepsilon)}{2}  \int_{\mathbb{R}^N} W(\varepsilon x)
 |z^{\xi, \eta}|^2 - \gamma(\varepsilon) \mathop{\rm Re}
 \int_{\mathbb{R}^N} W(\varepsilon x)  z^{\xi, \eta}
 \overline {w(\varepsilon, \eta, \xi / \varepsilon)}\\
&\quad - \frac{\gamma(\varepsilon)}{2}  \int_{\mathbb{R}^N} W(\varepsilon x)
 |w(\varepsilon, \eta, \xi / \varepsilon)|^2
\end{align*}
is of class  $C^1( V )$.  Indeed, by its definition,
$z^{\xi,\eta}$ depends on the functions $\alpha(\xi)$ and $\beta(\xi)$ so
on the potentials $V(\xi)$ and $K(\xi)$ which are both in
$C^1(\mathbb{R}^N)$ (with respect to $\xi$) by hypotheses (K1)
and (V1). Furthermore, by Lemma \ref{implicitw}, $w$ is of
class $C^2$ (if $p \geq 2$) or $C^{1, p-1}$ (if $ 1 < p < 2$) and
the result follows without effort. Again by  (\ref{funzridotto})
and (\ref{derivfunz}), it results that $I$ is close to $J$ in
$C^1( \overline{\mathcal{U }}  )$ when $\varepsilon$ is very small. We can
apply Theorem \ref{funznali} to find at least $l(M)$ critical
points $\{ \xi_1,\dots ,\xi_{l(M)} \}$ for
$\Psi_{\varepsilon}$, provided $\varepsilon$ is small enough. Hence the orbits
$S^1 \times \{ \xi_1 \},\dots ,S^1 \times
\{\xi_{l(M)} \}$ consist of critical points for
$\Phi_{\varepsilon}$ which produce solutions of  \eqref{tp}. The
concentration statement follows as in \cite{AMMASE}.
\end{proof}

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\end{thebibliography}


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