\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 $$\label{1.1} i \hbar \frac{\partial \psi}{\partial t} = \Big(\frac{\hbar}{i}\nabla -A(x) \Big)^2 \psi + U(x) \psi - K(x) |\psi|^{p-1} \psi, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N,$$ 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 $$\label{p_h} \Big( \frac{\varepsilon}{i} \nabla - A(x)\Big)^2 u + (U(x)-E) u = K(x) |u|^{p-1} u, \quad\hbox{x\,\in \, \mathbb{R}^N},$$ 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 $$\label{limitprima} \Big( \frac{ \nabla}{i} - A(\varepsilon \xi)\Big)^2 u+u + V(\varepsilon \xi) u = K(\varepsilon \xi) |u|^{p-1} u \quad\hbox{in \mathbb{R}^N},$$ (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 $$\label{singnok} \Big( \frac{\varepsilon}{i} \nabla - A( x)\Big)^2 u + \left(V( x)- \gamma(\varepsilon)W( x) \right) u = |u|^{p-2} u, \quad x \in \mathbb{R}^N,$$ 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 $$\label{gespl} \Big( \frac{\varepsilon}{i} \nabla - A( x)\Big)^2 u+u + \left(V( x)- \gamma(\varepsilon)W( x) \right) u = K(x) |u|^{p-1} u, \quad x \in \mathbb{R}^N,$$ 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 $$\label{tp} \Big( \frac{ \nabla}{i} - A(\varepsilon x)\Big)^2 u+u + \left(V(\varepsilon x)- \gamma(\varepsilon)W(\varepsilon x) \right) u = K(\varepsilon x) |u|^{p-1} u \quad\hbox{in }\mathbb{R}^N.$$ 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 $$\label{xinow} \Big( \frac{ \nabla}{i} - A(\varepsilon \xi)\Big)^2 u+u + V(\varepsilon \xi) u = K(\varepsilon \xi) |u|^{p-1} u \quad\hbox{in }\mathbb{R}^N,$$ (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 \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} 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 $$\label{diam} |\nabla |u|(x)| = \Big|\mathop{\rm Re} \Big( \nabla u \frac{\overline{u}}{|u|} \Big)\Big|= \Big|\mathop{\rm Re} \Big( (\nabla u - i A(\varepsilon x) u) \frac{\overline{u}}{|u|} \Big)\Big| \leq |D^{\varepsilon} u(x)|$$ holds a.e. in $\mathbb{R}^N$ and $|u| \in H^1(\mathbb{R}^N, \mathbb{R})$. Furthermore, $$\label{disug} \int_{\mathbb{R}^N} |\nabla |u\|^2 + |u|^2\, dx \leq \int_{\mathbb{R}^N}\left( |D^\varepsilon u|^2 +(1+ V(\varepsilon x)) |u |^2 \right) \,dx \leq c \|u\|^2$$ So, by the change of variable $y=\varepsilon x$, (W1) and (\ref{disug}), we have that \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} 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 $$\label{xinow2} \Big( \frac{ \nabla}{i} - A(\varepsilon \xi)\Big)^2 u+u + V(\varepsilon \xi) u = K(\varepsilon \xi) |u|^{p-1} u \quad\hbox{in }\mathbb{R}^N$$ (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 $$\label{xiconc} - \Delta U_c+U_c + V(\varepsilon \xi) U_c = K(\varepsilon \xi) |U_c|^{p-1} U_c \quad\hbox{in } \mathbb{R}^N.$$ 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 $$\left\{\begin{gathered} - \Delta U^\xi+U^\xi + V(\varepsilon \xi) U^\xi = K(\varepsilon \xi) |U^\xi|^{p-1} U^\xi, \\ U^\xi(0)=\max_{\mathbb{R}^N} U^\xi, \quad U^\xi >0. \end{gathered} \right.$$ If $U$ denotes the unique solution of $$\left\{\begin{gathered} - \Delta U+U=U^p \quad\hbox{in \mathbb{R}^N}, \\ U(0)=\max_{\mathbb{R}^N} U, \quad U >0, \end{gathered} \right.$$ 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 $$\label{zeta} z^{\varepsilon \xi, \sigma}: x \in \mathbb{R}^N \to e^{i \sigma +i A(\varepsilon \xi) \cdot x} \alpha(\varepsilon \xi) U(\beta(\varepsilon \xi)(x- \xi)).$$ 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 $j=1,\dots ,N$, $$\label{zspan} \frac{\partial}{\partial \xi_j}z^{\varepsilon \xi, \sigma} = -\frac{\partial}{\partial x_j}z^{\varepsilon \xi, \sigma} +i z^{\varepsilon \xi, \sigma} A_j (\varepsilon \xi) +O(\varepsilon)\,.$$ So that any $\zeta \in T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon}$ can be written as $$\zeta= i l_1 z^{\varepsilon \xi, \sigma}+ \sum_{j=2}^{N+1} l_j \frac{\partial}{\partial x_{j-1}}z^{\varepsilon \xi, \sigma}+O(\varepsilon)$$ 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 \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} 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 \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} By the change of variable $x= y /\varepsilon$, the definition of $z$ and (\ref{disug}) we have \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} 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 $$\int_{\mathbb{R}^N} \left| \Big( \frac {\nabla }{i} - A(\xi) \Big) u \right|^2+ |u|^2 \geq C \int_{\mathbb{R}^N} \left(|\nabla u|^2 + |u|^2 \right) \quad\hbox{\forall u \in E}.$$ \end{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 $$\label{ellev} | \langle L_{\varepsilon, \sigma, \xi} v | v \rangle | \geq C \|v\|^2, \quad \forall |\xi| \leq \overline \xi, \;\forall \sigma \in [0, 2 \pi],\; \forall v \in {(T_{z^{\varepsilon \xi, \sigma}} Z^{\varepsilon})}^{\bot}.$$ \end{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 $$\label{effeneg} D^2 F^{\varepsilon \xi, \sigma} (z^{\varepsilon \xi, \sigma}) (z^{\varepsilon \xi, \sigma}, z^{\varepsilon \xi, \sigma}) < -c_0 < 0.$$ 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 $$\label{effephi} D^2 F^{\varepsilon \xi, \sigma} (z^{\varepsilon \xi, \sigma}) (\phi, \phi) \geq c_1 \| \phi\|^2 \quad \forall \phi \ \bot \ \mathcal{N}.$$ 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 $$\label{sommaN} \| \phi\|^2= \| \phi_1\|^2+\| \phi_2\|^2+ 2 I_{\phi} + o_R(1) \| \phi\|^2$$ 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 $$\label{effephi1} D^2 F^{\varepsilon \xi} \left[\phi_1, \phi_1 \right] \geq C \| \phi_1\|^2+ o_R(1)\| \phi\|^2.$$ 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 \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} 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 $$\label{alph2} \alpha_2= \left( L_{\varepsilon, \sigma, \xi} \phi_2 | \phi_2 \right) \geq c_3 \| \phi_2\|^2 + o_R(1)\| \phi\|^2 \\ - G(\varepsilon) \left [ \alpha_\varepsilon \| \phi_2\|^2+ o_R(1)\| \phi\|^2 \right]$$ In a quite similar way one shows that \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} 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: $$\label{normaw} \|w\| \leq C \left( \varepsilon |\nabla V(\varepsilon \xi)| + \varepsilon |\nabla K(\varepsilon \xi)|+\varepsilon |J_A (\varepsilon \xi)| +\varepsilon^2+ C(\varepsilon \xi) G(\varepsilon) \right),$$ where $C > 0$. \end{remark} For future reference, it is also convenient to estimate the gradient $\nabla_{\xi} w$. \begin{lemma} It results $$\label{normagrad} \|\nabla_\xi w\| \leq c \left( \varepsilon |\nabla V(\varepsilon \xi)| + \varepsilon |\nabla K(\varepsilon \xi)|+\varepsilon |J_A (\varepsilon \xi)| +O(\varepsilon^2) \right)^{\gamma},$$ 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 \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} By the definition of $\alpha(\varepsilon \xi)$ and $\beta(\varepsilon \xi)$ we get immediately $$\int_{\mathbb{R}^N} |z^{\varepsilon \xi, \sigma}|^{p+1}= C_0 \Lambda(\varepsilon \xi) \left[ K(\varepsilon \xi) \right]^{-1}$$ where we define the auxiliary function $$\label{lambda} \Lambda(x)= \frac{(1+V(x))^{\theta}}{K(x)^{2/(p-1)}}, \quad \theta=\frac{p+1}{p-1}- \frac N2$$ 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 $$\label{funzridotto} \Phi_\varepsilon (\sigma, \xi)= \Psi_\varepsilon (\xi)=C_1 \Lambda(\varepsilon \xi)+O(\varepsilon).$$ Similarly, $$\label{derivfunz} \nabla \Psi_\varepsilon (\xi)=C_1 \nabla \Lambda(\varepsilon \xi)+ \varepsilon^{1+\gamma}O(1)$$ 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 $$\label{inclusioni} N^{\varepsilon} \subset \Theta^{\varepsilon} \subset N^{\varepsilon}_{\delta},$$ 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 $$\label{concentr} u_{\varepsilon, \eta^{*}, \xi^{*} }(x) \approx z^{\varepsilon \xi^{*}, \eta^{*}} \left( \frac{x - \xi^{*}}{\varepsilon} \right)$$ 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}. 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