\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 05, pp. 1--20.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/05\hfil Multiplicity of symmetric solutions] {Multiplicity of symmetric solutions for a nonlinear eigenvalue problem in $\mathbb{R}^n$} \author[Daniela Visetti\hfil EJDE-2005/05\hfilneg] {Daniela Visetti} \address{Dipartimento di Matematica Applicata U.~Dini'', Universit\a degli studi di Pisa, via Bonanno Pisano 25/B, 56126 Pisa, Italy} \email{visetti@mail.dm.unipi.it} \date{} \thanks{Submitted October 22, 2004. Published January 2, 2005.} \thanks{Supported by M.U.R.S.T., project Metodi variazionali e topologici nello studio di \hfill\break\indent fenomeni non lineari''.} \subjclass[2000]{35Q55, 45C05} \keywords{Nonlinear Schr\"odinger equations; nonlinear eigenvalue problems} \begin{abstract} In this paper, we study the nonlinear eigenvalue field equation $$-\Delta u+V(|x|)u+\varepsilon(-\Delta_p u+W'(u))=\mu u$$ where $u$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$ with $n\geq 3$, $\varepsilon$ is a positive parameter and $p>n$. We find a multiplicity of solutions, symmetric with respect to an action of the orthogonal group $O(n)$: For any $q\in\mathbb{Z}$ we prove the existence of finitely many pairs $(u,\mu)$ solutions for $\varepsilon$ sufficiently small, where $u$ is symmetric and has topological charge $q$. The multiplicity of our solutions can be as large as desired, provided that the singular point of $W$ and $\varepsilon$ are chosen accordingly. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this paper, we find infinitely many solutions of the nonlinear eigenvalue field equation $$\label{Pe} -\Delta u+V(|x|)u+\varepsilon(-\Delta_p u+W'(u))=\mu u\, ,$$ where $u$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$ with $n\geq 3$, $\varepsilon$ is a positive parameter and $p\in\mathbb{N}$ with $p>n$. The choice of the nonlinear operator $-\Delta_p+W'$ is very important. The presence of the $p$-Laplacian comes from a conjecture by Derrick (see \cite{D}). He was looking for a model for elementary particles, which extended the features of the sine-Gordon equation in higher dimension; he showed that equation $$- \Delta u + W'(u) = 0$$ has no nontrivial stable localized solutions for any $W\in C^1$ on $\mathbb{R}^n$ with $n\geq 2$. He proposed then to consider a higher power of the derivatives in the Lagrangian function and this has been done for the first time in \cite{BFP}. So the $p$-Laplacian is responsible for the existence of nontrivial solutions. As concerns $W'$, it denotes the gradient of a function $W$, which is singular in a point: this fact constitutes a sort of topological constraint and permits to characterize the solutions of \eqref{Pe} by a topological invariant, called topological charge (see \cite{BFP}). The free problem $$- \Delta u - \varepsilon\Delta_6 u + W'(u) = 0$$ has been studied in \cite{BFP}, while the concentration of the solutions has been considered in \cite{BBD}. In \cite{BMV} and \cite{BMV2} the authors have studied problem \eqref{Pe} respectively in a bounded domain and in $\mathbb{R}^n$. In \cite{BDFP} the authors have proved the existence of infinitely many solutions of the free problem, which are symmetric with respect to the action of the orthogonal group $O(n)$. In this paper, we find a multiplicity of solutions, symmetric with respect to the action considered in \cite{BDFP}, of problem \eqref{Pe} in $\mathbb{R}^n$: For any $q\in\mathbb{Z}$ we prove the existence of finitely many pairs $(u,\mu)$ solutions of problem \eqref{Pe} for $\varepsilon$ sufficiently small, where $u$ is symmetric and has topological charge $q$. The multiplicity of the solutions can be as large as one wants, provided that the singular point $\xi_\star =(\xi_0,0)$ ($\xi_0\in\mathbb{R}$, $0\in\mathbb{R}^n$) of $W$ and $\varepsilon$ are chosen accordingly. The basic idea is to consider problem \eqref{Pe} as a perturbation of the linear problem when $\epsilon=0$. In terms of the associated energy functionals, one passes from the non-symmetric functional $J_\epsilon$ (defined in (\ref{funzionale})) to the symmetric functional $J_0$. Non-symmetric perturbations of a symmetric problem, in order to preserve critical values, have been studied by several authors. We recall only \cite{BB}, which seems to be the first work on the subject, and the papers \cite{B} and \cite{BG}. In fact, the existence result is a result of preservation for the functional $J_\epsilon$ of some critical values of the functional $J_0$, constrained on the unitary sphere of $L^2(\mathbb{R}^n,\mathbb{R}^{n+1})$. Since the topological charge divides the domain $\Lambda$ of the energy functional $J_\epsilon$ into connected components $\Lambda_q$ with $q\in\mathbb{Z}$, the solutions are found in each connected component and in two different ways: as minima and as min-max critical points of the energy functional constrained on the unitary sphere of $L^2(\mathbb{R}^n,\mathbb{R}^{n+1})$. More precisely we can state: \medskip \textit{Given $q\in\mathbb{Z}$, for any $\xi_\star =(\xi_0,0)$ (with $\xi_0>0$ and $0\in\mathbb{R}^n$) and for any $\varepsilon>0$, there exist $\mu_1(\varepsilon)$ and $u_1(\varepsilon)$ respectively eigenvalue and eigenfunction of the problem \eqref{Pe}, such that the topological charge of $u_1(\varepsilon)$ is $q$.} \textit{Moreover, given $q\in\mathbb{Z}\setminus\{ 0\}$ and $k\in\mathbb{N}$, we consider $\xi_\star =(\xi_0,0)$ with $\xi_0$ large enough and $0\in\mathbb{R}^n$. Let $\lambda_j$ be the eigenvalues of the linear problem \eqref{Pe} with $\epsilon=0$. Then for $\varepsilon$ sufficiently small and for any $j\leq k$ with $\lambda_{j-1}<\lambda_j$, there exist $\mu_j(\varepsilon)$ and $u_j(\varepsilon)$ respectively eigenvalue and eigenfunction of the problem \eqref{Pe}, such that the topological charge of $u_j(\varepsilon)$ is $q$.} \section{Functional setting} \subsection*{Statement of the problem} We consider from now on the field equation $$\label{Pe1} -\Delta u+V(|x|)u+\varepsilon^r(-\Delta_p u+W'(u))=\mu u\, ,$$ where $u$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$ with $n\geq 3$, $\epsilon$ is a positive parameter and $p,\,r\in\mathbb{N}$ with $p>n$ and $r>p-n$ (for technical reasons we prefer to re-scale the parameter $\epsilon$). The function $V$ is real and we denote with $W'$ the gradient of a function $W:\mathbb{R}^{n+1}\setminus\{\xi_\star \}\to\mathbb{R}$, where $\xi_\star$ is a point of $\mathbb{R}^{n+1}$, different from the origin, which for simplicity we choose on the first component: $$\xi_\star =(\xi_0,0)\, , \label{xistar}$$ % with $\xi_0\in\mathbb{R}$, $\xi_0>0$ and $0\in\mathbb{R}^n$. Throughout the paper, we assume the following hypotheses on the function $V: [0,+\infty)\to\mathbb{R}$: \begin{itemize} \item[(V1)] $\displaystyle\lim_{r\to +\infty}V(r)=+\infty$ \item[(V2)] $V(|x|)e^{-|x|}\in L^p(\mathbb{R}^n,\mathbb{R})$ \item[(V3)] $\mathop{\rm ess\,inf}_{r\in[0,+\infty)} V(r)>0$ \end{itemize} The assumptions on the function $W:\mathbb{R}^{n+1}\setminus\{\xi_\star\}\to\mathbb{R}$ are as follows: \begin{itemize} \item[(W1)] $W\in C^1(\mathbb{R}^{n+1}\setminus\{ \xi_\star \},\mathbb{R})$ \item[(W2)] $W(\xi)\geq 0$ for all $\xi\in\mathbb{R}^{n+1}\setminus \{\xi_\star\}$ and $W(0)=0$ \item[(W3)] There exist two constants $c_1,\, c_2>0$ such that $$\xi\in\mathbb{R}^{n+1},\; 0<|\xi|0 such that$$ \xi\in\mathbb{R}^{n+1},\; 0\leq |\xi|0$such that for every$u\in E$% \begin{gather*} \| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})} \leq C_0 \| u\|_E\, , \\ |u(x)-u(y)| \leq C_1 |x-y|^{1-\frac{n}{p}} \| u\|_{W^{1,p}(\mathbb{R}^n, \mathbb{R}^{n+1})}\, ; \end{gather*} % \item If$u\in E$then$\lim_{|x|\to\infty} u(x) = 0$. \end{enumerate} \end{proposition} \subsection*{The energy functional$J_\epsilon$} In the space$E$, by Proposition \ref{prp-E}, it is possible to consider the open subset % $$\Lambda = \{ u\in E : \xi_\star\not\in u(\mathbb{R}^n) \}\, . \label{Lambda}$$ On$\Lambda$we consider the functional $$J_\epsilon(u) = \int_{\mathbb{R}^n} \big[ \frac{1}{2}|\nabla u|^2 + \frac{1}{2}V(|x|)|u|^2 + \frac{\epsilon^r}{p}|\nabla u|^p + \epsilon^r W(u) \big] \, dx\, , \label{funzionale}$$ which is the energy functional associated to the problem \eqref{Pe1}. It is easy to verify the following lemma (see Lemma 2.3 of \cite{BMV2}). \begin{lemma} \label{lmm-C^1} The functional$J_\epsilon$is of class$C^1$on the open set$\Lambda$of$E$. \end{lemma} \subsection*{The topological charge} On the open set$\Lambda$a topological invariant can be defined. Let$\Sigma$be the sphere of center$\xi_\star$and radius$\xi_0$in$\mathbb{R}^{n+1}$. Let$P$be the projection of$\mathbb{R}^{n+1}\setminus\{\xi_\star\}$onto$\Sigma$: $$P(\xi) = \xi_\star+\frac{\xi-\xi_\star}{|\xi-\xi_\star|}\, . \label{P}$$ \begin{definition} \label{def-ch} \rm For any$u\in\Lambda$,$u=(u^1,\dots,u^{n+1})$the open and bounded set $$K_u = \{ x\in\mathbb{R}^n : u^1(x)>\xi_0 \}$$ is called support of$u$. Then the topological charge of$u$is the number $$\mathop{\rm ch}(u) = \deg(P\circ u,K_u,2\xi_\star)\, .$$ \end{definition} To use some properties of the topological charge, we need to recall the following result, whose proof can be found in \cite{BFP}. \begin{proposition} If a sequence$\{u_m\}\subset\Lambda$converges to$u\in\Lambda$uniformly on$A\subset\mathbb{R}^n$, then also$P\circ u_m$converges to$P\circ u$uniformly on$A$. \end{proposition} This proposition permits to prove the continuity of the charge with respect to the uniform convergence: \begin{theorem} \label{trm-chcontinua} For every$u\in\Lambda$there exists$r=r(u)>0$such that, for every$v\in\Lambda$$$\| v-u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}\leq r \Longrightarrow \mathop{\rm ch}(v)=\mathop{\rm ch}(u)\, .$$ \end{theorem} \subsection*{The connected components of$\Lambda$} The topological charge divides the open set$\Lambda$into the following sets, each of them associated to an integer number$q\in\mathbb{Z}$: % $$\Lambda_q = \{ u\in\Lambda : \mathop{\rm ch}(u)=q\}\, . \label{Lambdaq}$$ % By Theorem \ref{trm-chcontinua}, we can conclude that the sets$\Lambda_q$are open in$E$. Moreover it is easy to see that $\Lambda = \bigcup_{q\in\mathbb{Z}} \Lambda_q\, , \quad \Lambda_p\cap\Lambda_q = \emptyset \mbox{ if } p\neq q$ % and each$\Lambda_q$is a connected component of$\Lambda$. \section{Symmetry and compactness properties} \subsection*{Action of$O(n)$} We consider the following action of the orthogonal group$O(n)$on the space of the continuous functions$C(\mathbb{R}^n,\mathbb{R}^{n+1})$: % $$\begin{array}{rccc} T: & O(n) \times C(\mathbb{R}^n,\mathbb{R}^{n+1}) & \longrightarrow & C(\mathbb{R}^n,\mathbb{R}^{n+1}) \\ & (g,u) & \longmapsto & T_gu \end{array} \label{azione}$$ % where % $$T_gu(x) = (u^1(gx),g^{-1}\tilde u(gx))\, , \label{azione2}$$ % with % $$u(x) = (u^1(x),\tilde u(x)) = (u^1(x),u^2(x),\dots,u^{n+1}(x))\, . \label{u1tildeu}$$ % In particular$O(n)$acts on the space$E$and so one can prove the following result. \begin{lemma} The open subset$\Lambda\subset E$and the energy functional$J_\epsilon$are invariant with respect to the action (\ref{azione}-\ref{u1tildeu}). \end{lemma} \begin{remark} \label{rmk1} \rm More precisely every connected component$\Lambda_q$of$\Lambda$is invariant with respect to the action (\ref{azione}-\ref{u1tildeu}) of the orthogonal group$O(n)$. Moreover for any$u\in E$and for any$g\in O(n)$$$\| T_gu\|_E = \| u\|_E\, .$$ \end{remark} Let$F$denote the subspace of the fixed points with respect to the action (\ref{azione}-\ref{u1tildeu}) of$O(n)$on$E$: % $$F = \{ u\in E : \forall g\in O(n)\; T_gu=u \}\, . \label{F}$$ % \begin{remark} \label{rmk-Fchiuso} \rm The set$F$is a closed subspace. \end{remark} The set $$\Lambda^F = \Lambda\cap F$$ is a natural constraint for the energy functional$J_\epsilon$. In fact, if$u\in\Lambda^F$is a critical point for$J_\epsilon \big|_{\Lambda^F}$, it is a global critical point (see \cite{BDFP}): \begin{lemma} \label{lem3.2} For every$u\in\Lambda^F$and$v\in E$, we have $$J_\epsilon '(u)(v) = J_\epsilon '(u)(Pv)\, ,$$ being$P$the projection of$E$onto$F$. \end{lemma} We denote by$\Lambda^F_q$the subset of the invariant functions of topological charge$q$: $$\Lambda^F_q = \Lambda_q\cap F\, .$$ \subsection*{Results of compactness} Next proposition provides a compact embedding for the subspace of the invariant functions of$E$into$L^s(\mathbb{R}^n,\mathbb{R}^{n+1})$: \begin{proposition} \label{prp-imm.comp.} The space$F$equipped with the norm$\|\cdot\|_E$is compactly embedded into$L^s(\mathbb{R}^n,\mathbb{R}^{n+1})$for every$s\in[2,\frac{2n}{n-2})$. \end{proposition} The proof is a consequence of \cite[Proposition 4]{BDFP} and of Theorem \ref{trm-L2}. We set $$S = \{ u\in E: \| u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}=1\}\, . \label{S}$$ % To get some critical points of the functional$J_\epsilon$on the$C^2$manifold$\Lambda\cap S$we use the following version of Palais-Smale condition. For$J_\epsilon\in C^1(\Lambda,\mathbb{R})$, the norm of the derivative at$u\in S$of the restriction$\hat J_\epsilon = \left. J_\epsilon \right|_{\Lambda\cap S}$is defined by $$\| \hat{J}_\epsilon '(u)\|_\star = \min_{t\in\mathbb{R}} \| J_\epsilon '(u)-tg'(u)\|_{E^*}\, ,$$ where$g:E\to\mathbb{R}$is the function defined by$g(u)=\int_{\mathbb{R}^n} |u(x)|^2 dx$. \begin{definition} \rm The functional$J_\epsilon$is said to satisfy the Palais-Smale condition in$c\in\mathbb{R}$on$\Lambda\cap S$(on$\Lambda_q\cap S$, for$q\in\mathbb{Z}$) if, for any sequence$\{ u_i\}_{i\in\mathbb{N}}\subset\Lambda\cap S$($\{ u_i\}_ {i\in\mathbb{N}}\subset\Lambda_q\cap S$) such that$J_\epsilon(u_i)\to c$and$\|\hat{J}_\epsilon '(u_i)\|_\star\to 0$, there exists a subsequence which converges to$u\in\Lambda\cap S$($u\in\Lambda_q\cap S$). \end{definition} To obtain the Palais-Smale condition, we need a few technical lemmas (see \cite{BMV2} and \cite{BFP}). \begin{lemma} \label{lmm1} Let$\{ u_i\}_{i\in\mathbb{N}}$be a sequence in$\Lambda_q$(with$q\in\mathbb{Z}$) such that the sequence$\{ J_\epsilon(u_i)\}_{i\in\mathbb{N}}$is bounded. We consider the open bounded sets % $$Z_i = \{ x\in\mathbb{R}^n : |u_i(x)|>c_3\}\, . \label{Zi}$$ % Then the set$\cup_{i\in\mathbb{N}} Z_i\subset\mathbb{R}^n$is bounded. \end{lemma} \begin{lemma} \label{lmm2} Let$\{ u_i\}_{i\in\mathbb{N}}\subset\Lambda$be a sequence weakly converging to$u$and such that$\{ J_\epsilon(u_i)\}_{i\in\mathbb{N}}\subset\mathbb{R}$is bounded, then$u\in\Lambda$. \end{lemma} \begin{lemma} \label{lmm3} For any$a>0$, there exists$d>0$such that for every$u\in\Lambda$$$J_\epsilon(u) \leq a \quad\Rightarrow\quad \inf_{x\in\mathbb{R}^n} |u(x)-\xi_\star| \geq d\, .$$ \end{lemma} Now it is possible to prove (see \cite{BMV2}) that the functional$J_\epsilon$satisfies the Palais-Smale condition on$\Lambda\cap S$for any$c\in\mathbb{R}$and$0<\epsilon\leq 1$. As a consequence the following proposition holds: \begin{proposition} \label{prp-PSF} The functional$J_\epsilon$satisfies the Palais-Smale condition on$\Lambda^F\cap S$(on$\Lambda^F_q\cap S$for$q\in\mathbb{Z}$) for any$c\in\mathbb{R}$and$0<\epsilon\leq 1$. \begin{proof} Given a Palais-Smale sequence$\{ u_m\}_{m\in\mathbb{N}}$for$J_\epsilon$on$\Lambda^F\cap S\subset\Lambda\cap S$, it strongly converges to a function$u\in\Lambda\cap S$by Proposition 2.1 of \cite{BMV2}. As the subspace$F$is closed (see Remark \ref{rmk-Fchiuso}),$u\in \Lambda^F$. \end{proof} \end{proposition} \section{Eigenvalues of the Schr\"odinger operator} \subsection*{Existence of the eigenvalues} We define the following subspace of invariant functions with respect to the action of$O(n)$(see (\ref{azione}-\ref{u1tildeu})): % $$\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1}) = \{ u\in\Gamma(\mathbb{R}^n,\mathbb{R}^{n+1}) : \forall g\in O(n)\; T_gu=u \}\, . \label{GammaF}$$ % By Proposition \ref{prp-imm.comp.} the identical embedding of$\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$into$L^2(\mathbb{R}^n,\allowbreak\mathbb{R}^{n+1})$is continuous and compact. Then there exists a monotone increasing sequence$\{\tilde\lambda_m\}_{m \in\mathbb{N}}$of eigenvalues $$0<\tilde\lambda_1\leq\tilde\lambda_2\leq\dots\leq\tilde\lambda_m \stackrel{m\to\infty}{\longrightarrow} +\infty$$ with $$\tilde\lambda_m = \inf_{E_m\in\mathcal{E}_m} \max_{v\in E_m,\, v\neq 0} \frac{\| v\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| v\|^2_{L^2(\mathbb{R}^n, \mathbb{R}^{n+1})}}\,,$$ where$\mathcal{E}_m$is the family of all$m$-dimensional subspaces$E_m$of$\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$. Also there exists a sequence$\{\varphi_m\}_{m\in\mathbb{N}}\subset\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$of eigenfunctions, orthonormal in$L^2(\mathbb{R}^n,\mathbb{R}^{n+1})$, such that $$(\varphi_m,v)_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})} = \tilde\lambda_m (\varphi_m,v)_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}\, ,\quad \forall v\in \Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})\, ,\quad \forall m\in\mathbb{N}\, .$$ \subsection*{Regularity of the eigenfunctions} The eigenfunctions$\varphi_m$have been found in the space$\Gamma_F(\mathbb{R}^n, \mathbb{R}^{n+1})$. Nevertheless they possess some more regularity properties, as it can be shown using the following theorem: \begin{theorem} \label{trm-decadimento} If$V(x)\to +\infty$as$|x|\to\infty$, then for any$z\in H^1(\mathbb{R}^n,\mathbb{R})$such that $$-\Delta z + V(x)z = \lambda z$$ the following estimate holds: % $$|z(x)| \leq C_a e^{-a|x|}\, , \label{exp}$$ % where$a>0$is arbitrary and$C_a>0$depends on$a$. \end{theorem} For the proof of this theorem, see \cite[p.~169]{BS}. \begin{proposition} \label{prp-regolarita'} The eigenfunctions$\varphi_m\in\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})$of the Schr\"odinger operator$-\Delta +V(|x|)$belong to the Banach space$E$. \end{proposition} \begin{proof} We prove the result for the real-valued eigenfunctions$e_m$so that the statement of the proposition follows immediately. By the regularity result of Agmon-Douglis-Nirenberg, if$z\in\Gamma_F(\mathbb{R}^n,\mathbb{R})$is such that$-\Delta z-\lambda z=-Vz$and if$Vz\in L^2(\mathbb{R}^n,\mathbb{R})\cap L^p(\mathbb{R}^n,\mathbb{R})$, then$z\in W^{2,p}(\mathbb{R}^n,\mathbb{R})$. So we only have to verify that$Vz\in L^2(\mathbb{R}^n,\mathbb{R})\cap L^p(\mathbb{R}^n, \mathbb{R})$. By Theorem \ref{trm-decadimento} and$\big(V_2\big)$we get $$\int_{\mathbb{R}^n} |V(|x|)z(x)|^p dx \leq C \left\| V(|x|)e^{-|x|} \right\|^p_{L^p(\mathbb{R}^n,\mathbb{R})} < +\infty\, .$$ Moreover, if$R>0$is such that for$x\in\mathbb{R}^n\setminus B_{\mathbb{R}^n}(0,R)V(|x|)>1, we have % \begin{align*} &\int_{\mathbb{R}^n} |V(|x|)z(x)|^2 dx\\ & < C \Big( \int_{B_{\mathbb{R}^n}(0,R)} |V(|x|)|^2 e^{-p|x|} dx + \int_{\mathbb{R}^n\setminus B_{\mathbb{R}^n}(0,R)} |V(|x|)|^p e^{-p|x|} dx \Big) < + \infty\, . \end{align*} \end{proof} \subsection*{Useful properties} We give here another variational characterization of the eigenvalues (see for example \cite{Courant-Hilbert} and \cite{Micheletti}) and we introduce the subspaces spanned by the eigenfunctions. \begin{definition} \rm Form\in\mathbb{N}$we consider the following subspaces of$\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1}): % \begin{align} F_m & = \mathop{\rm span}[\varphi_1,\ldots,\varphi_m]\, ,\label{Fm3}\\ F_m^\perp & = \{ u\in\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1}): (u,\varphi_i)_ {L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}=0 \mbox{ for } 1\leq i\leq m\}\, . \label{Fmperp3} \end{align} \end{definition} \begin{lemma} The following properties hold: % $$\tilde\lambda_m = \min_{{u\in\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1}),\; u\neq 0 \atop (u,\varphi_i)_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}=0} \atop \forall i=1,\dots,m-1} \frac{\| u\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| u\|^2_{L^2 (\mathbb{R}^n,\mathbb{R}^{n+1})}} \label{tildelambdam}$$ % and % $$(\varphi_i,\varphi_j)_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})} = \tilde\lambda_i\delta_{ij} \quad \forall i,j\in\mathbb{N}\, . \label{(fii,fij)GammaF}$$ % Moreover, % \begin{eqnarray} u\in F_m\, , u \neq 0 & \Longrightarrow & \tilde\lambda_1 \leq \frac{\| u\|^2_{\Gamma_F (\mathbb{R}^n,\mathbb{R}^{n+1})}}{\| u\|^2_{L^2(\mathbb{R}^n, \mathbb{R}^{n+1})}} \leq \tilde\lambda_m\, , \label{disvar1F} \\ u\in F_m^\perp\, , u \neq 0 & \Longrightarrow & \frac{\| u\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}} {\| u\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \geq \tilde\lambda_{m+1}\, . \label{disvar2F} \end{eqnarray} % \end{lemma} \section{Min-max values} \subsection*{The functions\Phi^q_\epsilon$} We introduce here a particular class of functions in$E$, which are invariant with respect to the action of the orthogonal group$O(n)$. Let us consider the functions$\varphi:\mathbb{R}^n\to\mathbb{R}^{n+1}$defined in the following way (see \cite{CF}): % $$\varphi(x) = \begin{cases} \begin{pmatrix} \varphi_1(|x|) \\ \varphi_2(|x|)\frac{x}{|x|} \end{pmatrix} &\mbox{for }x\neq 0 \\ \begin{pmatrix} \varphi_1(0) \\ 0 \end{pmatrix} & \mbox{for }x=0 \end{cases} \label{varphi}$$ % where$\varphi_i:[0,+\infty)\to\mathbb{R}$for$i=1,2$. In fact for any$g\in O(n)$and$x\in\mathbb{R}^n$$$T_g\varphi(x) = \varphi(x)\, .$$ By Proposition \ref{prp-regolarita'}, the set$F_m$defined in (\ref{Fm3}) is a subset of$E$. Then, for any$m\in\mathbb{N}$, let$S(m)$denote the$m$-dimensional sphere: % $$S(m)=F_m\cap S\, , \label{S(m)3}$$ % where$S$has been defined in (\ref{S}). Fixed an integer$k\in\mathbb{N}$, we introduce the number % $$M_k=\sup_{u\in S(k)} \| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}\, . \label{Mk3}$$ % Then we choose the first coordinate$\xi_0$of the point$\xi_\star= (\xi_0,0)$in such a way that % $$\xi_0>2M_k\, . \label{xi0>2Mk3}$$ % We can now introduce for any$q\in\mathbb{Z}\setminus\{ 0\}$the functions$\Phi^q_a:\mathbb{R}^n\to\mathbb{R}^{n+1}$of type (\ref{varphi}): % $$\Phi^q_a(x) = \begin{cases} \begin{pmatrix} \Phi^q_{a,1}(|x|) \\ \Phi^q_{a,2}(|x|)\frac{x}{|x|} \end{pmatrix} & \mbox{for }x\neq 0 \\ \begin{pmatrix} \Phi^q_{a,1}(0) \\ 0 \end{pmatrix} & \mbox{for }x=0 \end{cases} \label{Phiqa}$$ where $$\begin{gathered} \Phi^q_{a,1}(|x|) = \begin{cases} 2\xi_0[\cos(\pi|x|)+1] & \mbox{for }R_1\leq |x|\leq R_2 \\ 0 & \mbox{for } 0\leq |x|\leq R_1 \mbox{ or } |x|\geq R_2 \end{cases} \\ \Phi^q_{a,2}(|x|) = a|x|e^{-|x|}\sin(\pm\pi|x|) \end{gathered} \label{Phiqa12}$$ % with \begin{itemize} \item[(i)]$a>0$\item[(ii)] The sign in the argument of the sine in$\Phi^q_{a,2}$is equal to the sign of$q$, \item[(iii)]$R_1$is a constant depending on the parity of$q$: $$R_1 = R_1(q) = \begin{cases} 0 & \mbox{if q is odd,} \\ 1 & \mbox{if q is even,} \end{cases}$$ \item[(iv)]$R_2$is a positive constant depending on$q$: $$R_2 = R_2(q) = \begin{cases} |q| & \mbox{if q is odd,} \\ |q|+1 & \mbox{if q is even.} \end{cases}$$ \end{itemize} Next lemma computes the topological charge of the functions just defined (see \cite{BDFP}). \begin{lemma} \label{lmm-PhiqainLambdaq} For any$q\in\mathbb{Z}\setminus\{ 0\}$, the functions$\Phi^q_a$defined in (\ref{Phiqa}), (\ref{Phiqa12}), with the hypotheses$(i)$-$(iv)$, belong to$E$and have topological charge $$\mathop{\rm ch}\left(\Phi^q_a\right) = q\, .$$ \end{lemma} \begin{proof} The functions$\Phi^q_a$belong to the space$E$. If we consider the components \begin{gather*} f_1(x^1,x^2,\dots,x^n) = \Phi^q_{a,1}(|x|)\, , \\ f_i(x^1,x^2,\dots,x^n) = \Phi^q_{a,2}(|x|)\frac{x^i}{|x|}\, , \end{gather*} % where$2\leq i\leq n+1, we have % \begin{gather*} |\nabla_x f_1|^2 = |{\Phi^q_{a,1}}'(|x|)|^2 \\ |\nabla_x f_i|^2 \leq C \left( |{\Phi^q_{a,2}}'(|x|)|^2 + \frac{|\Phi^q_{a,2}(|x|)|^2}{|x|^2} \right) \end{gather*} % and consequently % \begin{align*} \sum_{i=1}^{n+1} \|\nabla f_i\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^n)} & \leq C\int_0^\infty \left( |{\Phi^q_{a,1}}'(r)|^2 + |{\Phi^q_{a,2}}'(r)|^2 + \frac{|\Phi^q_{a,2}(r)|^2}{r^2} \right)r^{n-1}\, dr \\ & < +\infty \\ \sum_{i=1}^{n+1} \|\nabla f_i\|^p_{L^p(\mathbb{R}^n,\mathbb{R}^n)} & \leq C\int_0^\infty \left( |{\Phi^q_{a,1}}'(r)|^p + |{\Phi^q_{a,2}}'(r)|^p + \frac{|\Phi^q_{a,2}(r)|^p}{r^p} \right) r^{n-1}\, dr \\ & < +\infty \end{align*} % Moreover the following inequalities hold: \begin{align*} &\int_{\mathbb{R}^n} V(|x|)\sum_{i=0}^n |f_i(x)|^2 dx\\ & \leq C \int_{R_1}^{R_2} V(r)(\Phi^q_{a,1}(r))^2 r^{n-1} dr + \int_{\mathbb{R}^n} V(|x|) (\Phi^q_{a,2}(|x|))^2 dx \\ & \leq C' + \| V(|x|)e^{-|x|}\|_{L^p(\mathbb{R}^n,\mathbb{R})} \| a^2|x|^2e^{-|x|}\|_{L^q(\mathbb{R}^n,\mathbb{R})} < + \infty\, , \end{align*} % whereq=\frac{p}{p-1}$. The functions$\Phi^q_a$belong to the space$\Lambda$. In fact, if$\Phi^q_{a,2}(|x|)=0$, then$|x|\in\mathbb{N}\cup\{ 0\}$and hence$\Phi^q_{a,1}(|x|)\in\{ 0,4\xi_0\}$, so that$\Phi^q_a(\mathbb{R}^n)\not\ni \xi_\star$. The functions$\Phi^q_a$have topological charge$q$. Let$P$be the projection introduced in (\ref{P}) of$\mathbb{R}^{n+1}$onto the sphere$\Sigma$of center$\xi_\star$and radius$\xi_0$in$\mathbb{R}^{n+1}; then $$P\circ\Phi^q_a(x) = \begin{pmatrix} \frac{\Phi^q_{a,1}(|x|)-\xi_0}{\sqrt{(\Phi^q_{a,1}(|x|)-\xi_0)^2+ (\Phi^q_{a,2}(|x|))^2}} + \xi_0 \$3pt] \frac{\Phi^q_{a,2}(|x|)}{\sqrt{(\Phi^q_{a,1}(|x|)-\xi_0)^2+ (\Phi^q_{a,2}(|x|))^2}} \frac{x}{|x|} \end{pmatrix} If K_{\Phi^q_a} is the support of \Phi^q_a, we can consider on it the local coordinates obtained by the stereographic projection of the sphere \Sigma from the origin onto the plane \Pi=\{\xi^1=2\xi_0\}: \begin{array}{ccccc} p & : & \Sigma & \longrightarrow & \Pi \\ & & (\xi^1,\xi^2,\dots,\xi^{n+1}) & \longmapsto & 2\xi_0 \left( \frac{\xi^2}{\xi^1},\frac{\xi^3}{\xi^1},\dots, \frac{\xi^{n+1}}{\xi^1} \right). \end{array} Then the function \Phi^q_a in the new coordinates becomes \overline\Phi^q_a(x) = p\circ P\circ\Phi^q_a(x) = f^q_a(|x|)\frac{x}{|x|}\, , where % $$f^q_a(|x|) = \frac{\Phi^q_{a,2}(|x|)}{\Phi^q_{a,1}(|x|)-\xi_0+ \xi_0\sqrt{(\Phi^q_{a,1}(|x|)-\xi_0)^2+(\Phi^q_{a,2}(|x|))^2}}\, . \label{fqa}$$ % The topological charge is therefore \mathop{\rm ch}(\Phi^q_a) = \deg\big(\overline\Phi^q_a,K_{\Phi^q_a}, 0\big)\, . Let \delta be a positive parameter, \delta<\frac{3}{4} and let i_1,\, i_2\in\mathbb{N}\cup\{ 0\}, with % \[ i_1 = R_1\, , \quad i_2 = \max\{ i\in\mathbb{N}\cup\{0\} : 2i+1\leq R_2\}\, .$ % Then the sets$$ K_i = \{ x\in\mathbb{R}^n : 2i-\delta<|x|<2i+\delta\}\, , $$for i\in\mathbb{N}\cup\{ 0\}, i_1\leq i\leq i_2, are disjoint and their union satisfies the inclusion:$$ \bigcup_{i=i_1}^{i_2} K_i \subset K_{\Phi^q_a}\, . $$Moreover this subset of K_{\Phi^q_a} contains all the zeros of the function \overline\Phi^q_a, that is:$$ \left\{ x\in K_{\Phi^q_a} : \overline\Phi^q_a=0 \right\} \subset \bigcup_{i=i_1}^{i_2} K_i\, . $$By the excision and the additive properties of the topological degree we can write$$ \deg (\overline\Phi^q_a,K_{\Phi^q_a},0) = \sum_{i=i_1}^{i_2} \deg(\overline\Phi^q_a,K_i,0)\, . $$To conclude we want to prove that$$ \deg(\overline\Phi^q_a,K_i,0) = \begin{cases} \mathop{\rm sign}(q) & \mbox{for } i=0\, , \\ 2\mathop{\rm sign}(q) & \mbox{for } i\in\mathbb{N}\, . \\ \end{cases} $$In fact consider the function$$ v_0(x) = \frac{f^q_a(\delta)}{\delta}x\, , $$where f^q_a(|x|) is defined in (\ref{fqa}). Since v_0 coincides with \overline\Phi^q_a on the boundary of K_0, i.e. for any x\in \partial K_0$$ \overline\Phi^q_a(x) = f^q_a(|x|)\frac{x}{|x|} = v_0(x)\, , $$the degrees of the two functions coincide too, so$$ \deg(\overline\Phi^q_a,K_0,0) = \deg(v_0,K_0,0) = \mathop{\rm sign}(q)\, . $$Finally, for 1\leq i\leq i_2, set % $K_i^+ = \{ x\in\mathbb{R}^n : |x|<2i+\delta \}\, , \quad K_i^- = \{ x\in\mathbb{R}^n : |x|<2i-\delta \}\, ;$ % then the degrees satisfy$$ \deg\big(\overline\Phi^q_a,K_i,0\big) = \deg\big(\overline\Phi^q_a,K_i^+,0\big) - \deg\big(\overline\Phi^q_a,K_i^-,0\big)\, . $$Analogously to the previous argument, we introduce the functions: % $v_i^+(x) = \frac{f^q_a(2i+\delta)}{2i+\delta} x\, , \quad v_i^-(x) = \frac{f^q_a(2i-\delta)}{2i-\delta} x\, .$ % As v_i^\pm coincides with \overline\Phi^q_a on the boundary of K_i^\pm, we conclude that % \begin{gather*} \deg\big(\overline\Phi^q_a,K_i^+,0\big) = \deg(v_i^+,K_i^+,0) = \mathop{\rm sign}(q)\, , \\ \deg\big(\overline\Phi^q_a,K_i^-,0\big) = \deg(v_i^-,K_i^-,0) = -\mathop{\rm sign}(q)\,. \end{gather*} % This completes the proof. \end{proof} The following corollary is now immediate. \begin{corollary} For all q\in\mathbb{Z} the connected component \Lambda^F_q is not empty. \end{corollary} \begin{lemma} \label{lmm-Phiq} Fixed q\in\mathbb{Z}\setminus\{0\}, there exists \hat a_q>0 such that for every a\geq\hat a_q the functions \Phi^q_a have the following properties: \begin{itemize} \item[(i)] The distance of \Phi^q_a from the point \xi_\star is \xi_0, i.e.$$ d(\Phi^q_a,\xi_\star) = \inf_{x\in\mathbb{R}^n} |\Phi^q_a(x)-\xi_\star| = \xi_0\, . $$\item[(ii)] If we expand \Phi^q_a of a factor t\geq 1, t\Phi^q_a \in\Lambda^F and$$ d(t\Phi^q_a,\xi_\star) = \inf_{x\in\mathbb{R}^n} |t\Phi^q_a(x)-\xi_\star| = \xi_0\, . $$\end{itemize} \end{lemma} \begin{proof} (i) We prove that there exists a sufficiently large such that$$ |\Phi^q_a(x)-\xi_\star| \geq \xi_0 $$for all x\in\mathbb{R}^n. For x\in\mathbb{R}^n with 0\leq |x|\leq R_1 or |x|\geq R_2, it is immediate that$$ |\Phi^q_a(x)-\xi_\star|^2 = a^2|x|^2e^{-2|x|}\sin^2(\pi |x|) + \xi_0^2 \geq \xi_0^2\, . As for x\in\mathbb{R}^n with R_1\leq |x|\leq R_2 there holds: % \begin{align*} |\Phi^q_a(x)-\xi_\star|^2 & = \xi_0^2[2\cos(\pi|x|)+1]^2 + a^2|x|^2e^{-2|x|}\sin^2(\pi|x|) \\ & = \big( 4\xi_0^2-a^2|x|^2e^{-2|x|} \big) \cos^2(\pi|x|) + 4\xi_0^2\cos(\pi|x|) + \xi_0^2 + a^2|x|^2e^{-2|x|}\, . \end{align*} % Let f_a:[0,+\infty)\to\mathbb{R} be the function f_a(r) = \left( 4\xi_0^2-a^2r^2e^{-2r} \right) \cos^2(\pi r) + 4\xi_0^2\cos(\pi r) + a^2r^2e^{-2r}\, . $$We consider the polynomial$$ P(y) = P_\alpha(y) = \left( 4\xi_0^2-\alpha^2 \right) y^2 + 4\xi_0^2 y + \alpha^2\, , $$where \alpha = \alpha_a(r) = are^{-r}, on the interval [-1,+1]. Now, if \alpha^2=4\xi_0^2, the only zero of P(y) is y=-1 and therefore on [-1,1] P(y) is nonnegative. On the contrary, if \alpha^2\neq 4\xi_0^2, the zeros of P(y) are:$$ y_{1,2} = \frac{-2\xi_0^2\pm\left(\alpha^2-2\xi_0^2\right)} {4\xi_0^2-\alpha^2} = \begin{cases} -1 \\ \frac{\alpha^2}{\alpha^2-4\xi_0^2} \end{cases} $$For \alpha^2>4\xi_0^2 we have y_1=-1 and y_2>1, so P(y)\geq 0 on [-1,1]. For 2\xi_0^2\leq\alpha^2<4\xi_0^2, we have y_2\leq -1, so P(y)\geq 0 and consequently$$ a^2r^2e^{-2r}\geq 2\xi_0^2 \Longrightarrow f_a(r)\geq 0\, . $$If we consider % $$a \geq \frac{\sqrt{2}\xi_0}{R_2e^{-R_2}} \label{cond.a}$$ % and R_1=1 (i.e. q even), there always holds \alpha^2\geq 2\xi_0^2. If on the contrary q is odd and so R_1=0, for a as in (\ref{cond.a}) (\alpha_a(r))^2<2\xi_0^2 for 0\leq r4\xi_0^2 we have y_1=-1 and y_2>1, then \widetilde P(y)\geq 0 in [-1,1]. For 2\xi_0^2\leq\alpha^2<4\xi_0^2, there holds y_2\leq -1, so \widetilde P(y)\geq 0 and consequently$$ a^2r^2e^{-2r}\geq 2\xi_0^2 \Longrightarrow \widetilde f_a(r)\geq 0\, . $$Now, with the choice of a done in (i) and R_1=1 (q even), \alpha^2\geq 2\xi_0^2. Finally, if R_1=0 and a is as in (i), \alpha^2<2\xi_0^2 for 0\leq r2M_k, where$$ M_k=\sup_{u\in S(k)} \| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}\, , $$and 0\in\mathbb{R}^n. There exist \hat\rho_q>0 and \overline\epsilon_q, with 0< \overline\epsilon_q\leq 1, such that for all 0<\epsilon\leq \overline\epsilon_q we have \begin{itemize} \item[(i)] \|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})} \leq 1 for all u\in S(k), \item[(ii)] \displaystyle\inf_{\epsilon\in (0,\overline\epsilon_q] \atop u\in S(k)} \|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}>0, \item[(iii)] \displaystyle\inf_{{x\in\mathbb{R}^n \atop \epsilon\in (0, \overline\epsilon_q]} \atop u\in S(k)} \left| \frac{\Phi^q_\epsilon(x) + \hat\rho_q u(x)}{\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n, \mathbb{R}^{n+1})}} -\xi_\star \right| >\frac{\xi_0}{2}, \item[(iv)] \displaystyle\frac{\Phi^q_\epsilon +\hat\rho_q u} {\|\Phi^q_\epsilon + \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}\in\Lambda_q \cap S for all u\in S(k). \end{itemize} \end{lemma} \begin{proof} \noindent (i) For any \rho>0 and 0<\epsilon\leq 1 we have$$ \|\Phi^q_\epsilon +\rho u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}\leq \epsilon^{\frac{n}{2}}\|\Phi^q\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}+\rho\, . $$Let \overline\epsilon_q be such that % $$\begin{gathered} \overline\epsilon_q < \left( \frac{1}{\|\Phi^q\|_{L^2(\mathbb{R}^n, \mathbb{R}^{n+1})}} \right)^\frac{2}{n}\, , \\ \overline\epsilon_q \leq 1\, . \end{gathered} \label{overlineepsilonq}$$ % Then there exists \hat\rho_q>0 such that \|\Phi^q_{\overline\epsilon_q} \|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}+\hat\rho_q\leq 1. \noindent (ii) As \|\Phi^q_\epsilon+\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})} \geq\hat\rho_q-\|\Phi^q_\epsilon\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}, reducing if necessary \overline\epsilon_q, we get \|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}>0. \noindent (iii) By (ii) of Lemma \ref{lmm-Phiq} we deduce that for all u\in S(k)$$ \inf_{x\in\mathbb{R}^n \atop \epsilon\in (0,\overline\epsilon_q]} \left| \frac{\Phi^q_\epsilon(x)}{\|\Phi^q_\epsilon +\hat\rho_q u\|_ {L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} -\xi_\star \right| = \xi_0\, . $$To get (iii) it is sufficient to prove that, reducing if necessary \overline\epsilon_q, for all \epsilon\leq\overline\epsilon_q$$ \sup_{u\in S(k)} \frac{\hat\rho_q\| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}} {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} < \frac{\xi_0}{2}\, . We observe that % \begin{align*} \sup_{u\in S(k)} \frac{\hat\rho_q\| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}} {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} & \leq \frac{\hat\rho_q M_k}{\displaystyle\inf_{u\in S(k)} \|\Phi^q_\epsilon + \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \\ & \leq \frac{M_k}{1-\frac{\epsilon^\frac{n}{2}}{\hat\rho_q} \|\Phi^q\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}\, . \end{align*} % Since M_k<\frac{\xi_0}{2}, for \overline\epsilon_q sufficiently small we have (iii). \noindent (iv) follows immediately from (iii). \end{proof} \subsection*{The values c^q_{\epsilon,j}} Using the properties of the functions \Phi^q_\epsilon seen in Lemma \ref{lmm-Phiqepsilon}, it is possible to introduce the following subsets of \Lambda^F\cap S: \begin{definition} \label{def-M} \rm Fixed k\in\mathbb{N}, q\in\mathbb{Z}\setminus\{ 0\} and 0<\epsilon\leq \overline\epsilon_q, where \overline\epsilon_q is defined in Lemma \ref{lmm-Phiqepsilon}, we set % $$\mathcal{M}^q_{\epsilon,j} = \big\{ \frac{\Phi^q_\epsilon +\hat\rho_q u} {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} : u\in S(j) \big\} \label{M3}$$ % with j\leq k and \hat\rho_q defined in Lemma \ref{lmm-Phiqepsilon}. We pose by convention \mathcal{M}^q_{\epsilon,0}=\emptyset. \end{definition} \begin{remark} \label{rmk-M3} \rm We outline the following properties of the sets \mathcal{M}^q_{\epsilon,j}: \begin{itemize} \item[(i)] \mathcal{M}^q_{\epsilon,j-1}\subset\mathcal{M}^q_{\epsilon,j}; \item[(ii)] \mathcal{M}^q_{\epsilon,j}\subset \Lambda^F_q\cap S; \item[(iii)] \mathcal{M}^q_{\epsilon,j} is a compact set; \item[(iv)] \mathcal{M}^q_{\epsilon,j} is a sub-manifold of \Lambda^F_q for 0<\epsilon\leq\overline\epsilon_q (see Lemma \ref{lmm-Phiqepsilon}). \end{itemize} \end{remark} Next definition introduces the min-max values c^q_{\epsilon,j}. \begin{definition} \rm Fixed k\in\mathbb{N}, for all q\in\mathbb{Z}\setminus\{ 0\}, j\leq k and 0<\epsilon\leq\overline{\epsilon}_q (\overline\epsilon_q is defined in Lemma \ref{lmm-Phiqepsilon}), we define the following values: % $$c^q_{\epsilon,j} = \inf_{h\in\mathcal{H}^q_{\epsilon,j}} \sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_\epsilon \big( h(v) \big)\, , \label{cqepsilonj3}$$ % where \mathcal{H}^q_{\epsilon,j} are the following sets of continuous transformations: \mathcal{H}^q_{\epsilon,j} = \left\{ h:\Lambda^F_q\cap S\to\Lambda^F_q\cap S : h\,\mbox{continuous},\, h\big|_{\mathcal{M}^q_{\epsilon,j-1}}= \mathrm{id}_{\mathcal{M}^q_{\epsilon,j-1}} \right\}\, . $$\end{definition} We observe that \mathcal{H}^q_{\epsilon,j+1} \subset \mathcal{H}^q_{\epsilon,j}. \begin{lemma} Fixed k\in\mathbb{N}, for all q\in\mathbb{Z}\setminus\{ 0\}, j0 such that the sublevel J^{c+k} is complete. \end{itemize} Then there exist \delta>0 and a deformation \eta:[0,1]\times E \longrightarrow E such that: \begin{itemize} \item[(a)] \eta (0,u)=u for all u\in E, \item[(b)] \eta (t,u)=u for all t\in [0,1] and u such that |J(u)-c|\geq 2\delta, \item[(c)] J(\eta (t,u)) is non-increasing in t for any u\in E, \item[(d)] \eta (1,J^{c+\delta})\subset J^{c-\delta}. \end{itemize} \end{lemma} To apply Lemma \ref{lmm-def} on each connected component \Lambda^F_q, with q\in\mathbb{Z}\setminus\{ 0\}, intersected with the unitary sphere S we need the completeness of the sub-levels of the functional J_\epsilon. It is simple to verify next: \begin{lemma} \label{lmm-compl} For any q\in\mathbb{Z}, \epsilon\in (0,1] and c\in\mathbb{R}, the subset \Lambda^F_q\cap S\cap J_\epsilon ^c of the Banach space E is complete. \end{lemma} Now we get easily the minimum values of the functional J_\epsilon on each set \Lambda^F_q\cap S: \begin{theorem} \label{trm-minimo} Given q\in\mathbb{Z}, for any \xi_\star =(\xi_0,0) with \xi_0>0 and 0\in\mathbb{R}^n and for any \epsilon>0, there exists a minimum for the functional J_\epsilon on the subset \Lambda^F_q\cap S of \Lambda \cap S. \end{theorem} \begin{proof} For any t\geq 1 we have that t\Phi^q\in\Lambda^F_q (see (iii) of Remark \ref{rmk-Phiqepsilon}) and in particular the function \frac{\Phi^q}{\|\Phi^q\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} is in \Lambda^F_q \cap S. This means that \Lambda^F_q\cap S is not empty for all q\in\mathbb{Z}, since it is obvious that \Lambda^F_0\cap S\neq\emptyset. The claim follows by the fact that \Lambda^F_q\cap S is not empty, the functional J_\epsilon is bounded from below and satisfies the Palais-Smale condition on \Lambda^F_q\cap S (see Proposition \ref{prp-PSF}). \end{proof} \begin{remark} \rm We point out that to have this result there is no need to require that the first coordinate \xi_0 of the point \xi_\star is sufficiently large (see (\ref{xi0>2Mk3})). In fact this assumption is necessary to have properties (iii) and (iv) of Lemma \ref{lmm-Phiqepsilon}, while here we only have to show that \Lambda^F_q\cap S is not empty for all q\in\mathbb{Z}. \end{remark} \subsection*{Critical values} The next theorem is an existence and multiplicity result of solutions for the problem \big(P_\epsilon\big). \begin{theorem} \label{trm-main} Given q\in\mathbb{Z}\setminus\{ 0\} and k\in\mathbb{N}, we consider \xi_\star = (\xi_0,0) with \xi_0>2M_k, where$$ M_k=\sup_{u\in S(k)} \| u\|_{L^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})}\, , $$and 0\in\mathbb{R}^n. Then there exists \hat\epsilon_q\in(0,1] such that for any \epsilon\in (0,\hat\epsilon_q] and for any 2\leq j\leq k with \tilde\lambda_{j-1}< \tilde\lambda_j, we get that c^q_{\epsilon,j} is a critical value for the functional J_\epsilon restricted to the manifold \Lambda^F_q\cap S. Moreover c^q_{\epsilon,j-1}p-n), we have that the third term of the last inequality of (\ref{dis4}) tends to zero when \epsilon tends to zero. Regarding the last term, we verify that$$ \int_{\mathbb{R}^n} W \left( \frac{\Phi^q_\epsilon +\hat\rho_q u} {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right) dx is bounded uniformly with respect to \epsilon\in (0, \overline{\epsilon}] and u\in S(k). In fact by definition of \Phi^q_\epsilon and by the exponential decay of the eigenfunctions (see Theorem \ref{trm-decadimento}) there exists a ball B_{\mathbb{R}^n}(0,R) such that, if we write u=\sum_{m=1}^j a_m\varphi_m with \sum_{m=1}^j a_m^2=1, for all x\in\mathbb{R}^n\setminus B_{\mathbb{R}^n}(0,R) the following inequalities hold % \begin{align*} \left| \frac{\Phi^q_\epsilon(x)+\hat\rho_q u(x)}{\|\Phi^q_\epsilon + \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right| & = \frac{\hat\rho_q |u(x)|}{\|\Phi^q_\epsilon + \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \\ & \leq \frac{C\,\hat\rho_q \Big( \sum_{m=1}^j |a_m| \Big) e^{-|x|}}{\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n, \mathbb{R}^{n+1})}} \\ & \leq M e^{-|x|} < c_3 \end{align*} % where the constant M does not depend on u\in S(j) nor on \epsilon for \epsilon small enough (see the point (ii) of Lemma \ref{lmm-Phiqepsilon}). By \big(W_4\big) we get \left| W \left( \frac{\Phi^q_\epsilon(x) +\hat\rho_q u(x)} {\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right) \right| \leq c_4 \frac{|\Phi^q_\epsilon(x) +\hat\rho_q u(x)|^2} {\|\Phi^q_\epsilon +\hat\rho_q u\|^2_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}}$for any$x\in\mathbb{R}^n\setminus B_{\mathbb{R}^n}(0,R)$. Concluding we have % \begin{eqnarray*} \lefteqn{\left| \int_{\mathbb{R}^n} W \left( \frac{\Phi^q_\epsilon +\hat\rho_q u}{\|\Phi^q_\epsilon +\hat\rho_q u\|_{L^2 (\mathbb{R}^n,\mathbb{R}^{n+1})}} \right) dx \right|} \\ & \leq & c_4 + \int_{B_{\mathbb{R}^n}(0,R)} \left| W \left( \frac{\Phi^q_\epsilon + \hat\rho_q u}{\|\Phi^q_\epsilon + \hat\rho_q u\|_{L^2(\mathbb{R}^n,\mathbb{R}^{n+1})}} \right) \right| dx \end{eqnarray*} % where the integral on the right hand side is bounded by$(iii)$of Lemma \ref{lmm-Phiqepsilon}. So we have the claim. \noindent\textbf{Step 3} \emph{We prove that$c^q_{\epsilon,j}\geq\tilde\lambda_j$.} By Step 1 and by the positivity of$Wwe get % \begin{align*} c^q_{\epsilon,j} & \geq \inf_{h\in\mathcal{H}^q_{\epsilon,j}} \sup_{v\in\mathcal{M}^q_{\epsilon,j}} \| h(v)\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})} \\ & \geq \inf_{h\in\mathcal{H}^q_{\epsilon,j}} \sup_{{v\in\mathcal{M}^q_{\epsilon,j}} \atop {P_{F_{j-1}}h(v)=0}} \| h(v)\|^2_{\Gamma_F(\mathbb{R}^n,\mathbb{R}^{n+1})}\\ & \geq \tilde\lambda_j \end{align*} % In fact by Step 1 for allh\in\mathcal{H}^q_{\epsilon,j}$we have that the set$h(\mathcal{M}^q_{\epsilon,j})$intersects the set$\{ u\in F: (u,\varphi_i)=0\,\forall i=1,\dots,j-1\}$and so from (\ref{disvar2F}) we get the claim. \noindent\textbf{Step 4} \emph{If$\tilde\lambda_{j-1}<\tilde\lambda_j$, then for$\epsilon$small enough we have: % \begin{eqnarray} c^q_{\epsilon,j-1} & < & c^q_{\epsilon,j}\, , \label{dis5} \\ \sup_{v\in\mathcal{M}^q_{\epsilon,j-1}} J_\epsilon(v) & < & c^q_{\epsilon,j}\, . \label{dis6} \end{eqnarray} % } By Step 2 and 3 we obtain for$\epsilon$small enough % \begin{gather*} c^q_{\epsilon,j-1} \leq \tilde\lambda_{j-1} + \sigma(\epsilon) < \tilde\lambda_j \leq c^q_{\epsilon,j}\, , \\ \sup_{v\in\mathcal{M}^q_{\epsilon,j-1}} J_\epsilon(v) \leq \tilde\lambda_{j-1} + \sigma(\epsilon) < \tilde\lambda_j \leq c^q_{\epsilon,j}\, . \end{gather*} % \noindent\textbf{Step 5} \emph{If$\tilde\lambda_{j-1}<\tilde\lambda_j$, then$c^q_{\epsilon,j}$is a critical value for the functional$J_\epsilon$on the manifold$\Lambda_q^F\cap S$.} By contradiction we suppose that$c^q_{\epsilon,j}$is a regular value for$J_\epsilon$on$\Lambda_q^F\cap S$. By Proposition \ref{prp-PSF} and Lemmas \ref{lmm-def} and \ref{lmm-compl}, there exist$\delta>0$and a deformation$\eta:[0,1]\times\Lambda_q^F\cap S\rightarrow \Lambda_q^F\cap S$such that \begin{gather*} \eta (0,u)=u \quad \forall u\in \Lambda_q^F\cap S\, , \\ \eta (t,u)=u \quad \forall t\in [0,1],\, \forall u\in J_\epsilon^{c^q_{\epsilon,j}-2\delta}\, , \\ \eta (1,J_\epsilon^{c^q_{\epsilon,j}+\delta})\subset J_\epsilon^{c^q_{\epsilon,j}-\delta}\, . \end{gather*} By (\ref{dis6}) we can suppose % $$\sup_{v\in\mathcal{M}^q_{\epsilon,j-1}} J_\epsilon(v) < c^q_{\epsilon,j} -2\delta\, . \label{dis7}$$ % Moreover by definition of$c^q_{\epsilon,j}$there exists a transformation$\hat h\in\mathcal{H}^q_{\epsilon,j}$such that$\sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_\epsilon \big( \hat h(v) \big)< c^q_{\epsilon,j}+\delta$. Now by the properties of the deformation$\eta$and by (\ref{dis7}) we get$\eta \big( 1,\hat h( .) \big) \in\mathcal{H}^q_{\epsilon,j}$and$\sup_{v\in\mathcal{M}^q_{\epsilon,j}} J_\epsilon \big( \eta(1,\hat h(v)) \big) K$, such that among the first$k$eigenvalues$\tilde\lambda$there are$K$jumps''$\tilde\lambda_j<\tilde\lambda_{j+1}$, so that Theorem \ref{trm-main} gives$K$critical values. \item For all$q\in\mathbb{Z}\setminus\{ 0\}$,$\epsilon\in (0,1]$the critical values$c^q_{\epsilon,j}$tend to the eigenvalues$\tilde\lambda_j$when$\epsilon$tends to zero. \end{enumerate} \end{remark} \subsection*{Acknowledgements} This paper arises from the Ph.D. thesis (see \cite{tesi}) of the author, who heartily thanks her supervisors V.~Benci and A.~M.~Micheletti. \begin{thebibliography}{00} \bibitem{BBD} \textsc{M.~Badiale, V.~Benci and T.~D'Aprile,} \textit{Existence, multiplicity and concentration of bound states for a quasilinear elliptic field equation}, Calculus of Variations and Partial Differential Equations \textbf{12} 3 (2001), 223--258. \bibitem{BB} \textsc{A.~Bahri and H.~Berestycki,} \textit{A perturbation method in critical point theory and applications}, Transactions of the American Mathematical Society {\bf 267} 1 (1981), 1--32. \bibitem{BDFP} \textsc{V.~Benci, P.~D'Avenia, D.~Fortunato and L.~Pisani,} \textit{Solitons in several space dimensions: Derrick's problem and infinitely many solutions}, Archive for Rational Mechanics and Analysis \textbf{154} 4 (2000), 297--324. \bibitem{BF} \textsc{V.~Benci and D.~Fortunato,} \textit{Discreteness conditions of the spectrum of Schr\"odinger operators}, Journal of Mathematical Analysis and Applications \textbf{64} 3 (1978), 695--700. \bibitem{BFMP} \textsc{V.~Benci, D.~Fortunato, A.~Masiello and L.~Pisani,} \textit{Solitons and the electromagnetic field}, Mathematische Zeitschrift \textbf{232} 1 (1999), 73--102. \bibitem{BFP} \textsc{V.~Benci, D.~Fortunato and L.~Pisani,} \textit{Soliton-like solution of a Lorentz invariant equation in dimension 3}, Reviews in Mathematical Physics \textbf{10} 3 (1998), 315--344. \bibitem{BMV} \textsc{V.~Benci, A.M.~Micheletti and D.~Visetti,} \textit{An eigenvalue problem for a quasilinear elliptic field equation}, Journal of Differential Equations {\bf 184} 2 (2002), 299--320. \bibitem{BMV2} \textsc{V.~Benci, A.M.~Micheletti and D.~Visetti,} \textit{An eigenvalue problem for a quasilinear elliptic field equation on$\mathbb{R}^n\$}, Topological Methods in Nonlinear Analysis \textbf{17} 2 (2001), 191--211. \bibitem{BS} \textsc{F.A.~Berezin and M.A.~Shubin,} \textit{The Schr\"odinger Equation}, Kluwer Academic Publishers 1991. \bibitem{B} \textsc{P.~Bolle,} \textit{On the Bolza problem}, Journal of Differential Equations {\bf 152} 2 (1999), 274--288. \bibitem{BG} \textsc{P.~Bolle, N.~Ghoussoub and H.~Tehrani,} \textit{The multiplicity of solutions in non-homogeneous boundary value problems}, Manuscripta Mathematica \textbf{101} 3 (2000), 325--350. \bibitem{CF} \textsc{C.~Cid and P.L.~Felmer,} \textit{A note on static solutions of a Lorentz invariant equation in dimension 3}, Letters in Mathematical Physics \textbf{53} 1 (2000), 1--10. \bibitem{Courant-Hilbert} \textsc{R.~Courant and D.~Hilbert,} \textit{Methods of Mathematical Physics}, vol. 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