\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 94, pp. 1--31.\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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/94\hfil Solitary waves] {Solitary waves for \\ Maxwell-Schr\"odinger equations} \author[G. M. Coclite, V. Georgiev\hfil EJDE-2004/94\hfilneg] {Giuseppe Maria Coclite, Vladimir Georgiev} % in alphabetical order \address{Giuseppe Maria Coclite \hfill\break C.M.A. (Centre of Mathematics for Applications), P.O. Box 1053 Blindern, 0316 Oslo, Norway} \email{giusepc@math.uio.no} \address{Vladimir Georgiev \hfill\break Dipartimento di Matematica, Universit\`a degli Studi di Pisa, Via F. Buonarroti 2, 56100 Pisa, Italy} \email{georgiev@dm.unipi.it} \date{} \thanks{Submitted May 21, 2004. Published July 30, 2004.} \thanks{Supported by Research Training Network (RTN) HYKE and by grant HPRN-CT-2002-00282 \hfill\break\indent from the European Union.} \subjclass[2000]{35Q55, 35Q60, 35Q40} \keywords{Maxwell - Schr\"odinger system; solitary type solutions; \hfill\break\indent variational problems} \begin{abstract} In this paper we study solitary waves for the coupled system of Schr\"odinger- Maxwell equations in the three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed $L^2$ norm. We study the asymptotic behavior and the smoothness of these solutions. We show also that the eigenvalues are negative and the first one is isolated. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} %\allowdisplaybreaks \section{Introduction}\label{intro} The classical correspondence rules in quantum mechanics are \begin{equation}\label{eq.corr1} E \to i \hbar \partial_t , \quad p \to -i \hbar \nabla, \quad \nabla=(\nabla_1,\nabla_2,\nabla_3), \quad \nabla_j = \partial{x_j}, \quad j =1,2,3, \end{equation} where $E$ is the energy and $p=(p_1,p_2,p_3)$ is the momentum (see for example \cite[Section 4, Chapter V]{B69}). Using these rules, we can derive some basic wave equations in quantum mechanics from the corresponding laws of classical mechanics. For example, the classical relation \begin{equation}\label{eq.E=cl} E = \frac{p^2}{2m} + V(x), \quad p^2 = p_1^2 + p_2^2+p_3^2, \end{equation} represents the energy as a sum of kinetic energy $p^2/2m$ and a potential energy term $V(x)$. The well - known Schr\"odinger equation for the wave function $\psi(t,x)$ can be written as \begin{equation}\label{eqS1} i \hbar \partial_t \psi = - \frac{ \hbar^2}{2m} \Delta \psi + V(x) \psi \end{equation} and this equation is a consequence of \eqref{eq.corr1} and the relation \eqref{eq.E=cl}. Here $ \hbar$ is the Plank constant, $m$ is the mass of the field $\psi$ and $V(x)$ is a given external potential. For the case of potential created by the nucleus of the some atoms (see Section 4, Chapter V in \cite{B69} for example) we have a Coulomb potential \begin{equation}\label{eq.Vdef} V(x) = - \frac{e^2 Z}{|x|}, \end{equation} where $e$ is the electron charge, while $Z$ is the number of protons in the nucleus. The interaction between the electromagnetic field and the wave function related to a quantistic non-relativistic charged particle (considered as classical fields) is described by the Maxwell - Schr\"odinger system. More precisely, let $ \psi = \psi (x,t) $ be the wave function and let $\mathcal{A}=(A_0,A_1,A_2,A_3)$ be the electromagnetic potentials of a charged non- relativistic particle. Then the corresponding Maxwell - Schr\"odinger system (in Lorentz gauge) has the form (see the next section for the derivation of this system) \begin{equation} \begin{gathered} \frac{1}{c^2}\partial_{tt} \mathcal{A} - \Delta \mathcal{A} = \mathcal{J}, \\ i\hbar\partial_{t,\mathcal{A}} \psi + \frac{\hbar^2}{2m} \Delta_{ \mathcal{A}} \psi -V(x) \psi= 0, \\ \frac{1}{c}\partial_t A_0 + \sum_{k=1}^3 \partial_{x_k} A_k = 0, \end{gathered} \label{eq.MSdin} \end{equation} where $c$ is the light velocity (in vacuum), \begin{equation} \begin{gathered} \partial_{t, \mathcal{A}} = \partial_t +i\frac{e}{\hbar} A_0, \quad \Delta_{ \mathcal{A}} = \sum_{k=1}^3 \partial_{k, \mathcal{A}}^2, \\ \partial_{k, \mathcal{A}} = \partial_{x_k} +i\frac{e}{\hbar c} A_k, \quad \mathcal{J}= (J_0,J_1,J_2,J_3), \\ J_0 = 4\pi e |\psi|^2, \quad J_k = 4\pi \frac{\hbar e}{mc} \mathop{\rm Im}( \bar{\psi}\partial_{k, \mathcal{A}} \psi). \end{gathered} \label{eq.def132} \end{equation} We choose units in which $$ \hbar = c =1, \quad\alpha = \frac{e^2}{4\pi} \approx \frac{1}{137}. $$ Also for simplicity we take $m=1$. We consider special solitary type solutions to the system \eqref{eq.MSdin} of the form $$ \psi (x,t) = u(x) e^{-i \omega t/\hbar},\quad x\in \mathbb{R}^3 , t\in \mathbb{R},$$ and $$ A_0 = \varphi (x),\quad\ A_j(x)=0,\quad j=1,2,3,\quad x \in \mathbb{R}^3, $$ where $\omega \in \mathbb{R}$ and $u$ is real valued. Then the system \eqref{eq.MSdin} takes the simpler form \begin{equation}\label{eq.M-S} \begin{gathered} - \frac{1}2\Delta u + e \varphi u + V(x)u = \omega u ,\quad x \in \mathbb{R}^3, \\ -\Delta \varphi = 4\pi e u^2, \quad x \in \mathbb{R}^3, \\ \int_{\mathbb{R}^3} u^2 =N, \end{gathered} \end{equation} where the last equation is due to the probabilistic interpretation of the wave function. In this work we shall assume the following relation between $N$ and $Z$ is satisfied \begin{equation}\label{eq.Coul1} N \leq Z. \end{equation} The equations in (\ref{eq.M-S}) have a variational structure, in fact they are the Euler - Lagrange equations related to the functional: \begin{equation} F(u,\varphi)=\frac{1}4 \int_{\mathbb{R}^3}\vert\nabla u\vert^2\,dx + \frac{e}2 \int_{\mathbb{R}^3}\varphi u^2 \,dx + \frac{1}2 \int_{\mathbb{R}^3} V(x) u^2 \,dx - \frac{1}{16\pi}\int_{\mathbb{R}^3}\vert\nabla \varphi \vert^2\,dx . \label{funz.compl}\end{equation} It is easy to see that this functional is well - defined, when $$ u \in H^1(\mathbb{R}^3),\quad \int_{\mathbb{R}^3} \vert \nabla \varphi \vert^2 \,dx< \infty. $$ This functional is strongly indefinite, which means that $F$ is neither bounded from below nor from above and this indefiniteness cannot be removed by a compact perturbation. Moreover, $ F $ is not even. Later on (see \eqref{funz.rid}) we shall introduce a functional $J(u)$ that is bounded from below and such that the critical points of $J$ can be associated with the critical points of $F$. The first natural question is connected with the simplest case $V\equiv 0$ (that is $Z=0$), namely \begin{equation}\label{eq.M-S0} \begin{gathered} - \frac{1}2\Delta u + \varphi e u = \omega u , x \in \mathbb{R}^3, \\ - \Delta \varphi = 4\pi e u^2, \ x \in \mathbb{R}^3. \end{gathered} \end{equation} It is well-known that the similar physical model of Maxwell - Dirac system with zero external field admits solitary solutions (see \cite{EGS}), i.e. nontrivial solutions in the Schwartz class $S( \mathbb{R}^3)$. Our first result is as follows. \begin{theorem}\label{thmain0m} Let $(u,\varphi,\omega)$ be a solution of (\ref{eq.M-S0}) such that $u, \varphi$ are radial and $$u \in H^1 (\mathbb{R}^3),\quad \int_{\mathbb{R}^3} \vert \nabla \varphi \vert^2 \,dx< \infty.$$ Then $u\equiv\varphi\equiv 0$. \end{theorem} The above result shows that the Schr\"odinger - Maxwell equations with zero potential have only trivial solution. This fact justifies the study of the Schr\"odinger - Maxwell equations with nonzero external potential. We shall look for soliton type solutions $u$, i.e. very regular solutions decaying rapidly at infinity. First, we establish the existence of $H^1$ radially symmetric solutions. \begin{theorem}\label{thmain1m} Under the assumptions (\ref{eq.Vdef}) and (\ref{eq.Coul1}), there exists a sequence of real negative numbers $\{\omega_k\}_{k\in\mathbb{N}}$ so that $\omega_k \to 0$ and for any $\omega_k$ there exists a couple $(u_k,\varphi_k) $ of solutions of (\ref{eq.M-S}) such that $$ u_k \in H^1 (\mathbb{R}^3),\quad \int_{\mathbb{R}^3} \vert \nabla \varphi_k \vert^2 \,dx< \infty. $$ Moreover $u_k, \varphi_k$ are radially symmetric functions. \end{theorem} A more precise information about the localization of the eigenvalues $\omega$ is given in the following. \begin{theorem}\label{lemmaspecin} Assume (\ref{eq.Vdef}) and $N < Z$. Let $(u,\varphi,\omega)$ be a nontrivial solution of the equations in (\ref{eq.M-S}) such that $u, \varphi$ radial and $$u \in H^1 (\mathbb{R}^3),\quad \int_{\mathbb{R}^3} \vert \nabla \varphi \vert^2 \,dx< \infty.$$ Then we have \begin{equation}\label{spec2in} \omega<0. \end{equation} \end{theorem} On the other hand, the solutions constructed in Theorem \ref{thmain1m} are only radial ones. Therefore, it remains as an open problem the existence of non-radial solutions. Some qualitative properties of the solutions for the case $N\leq Z$ are described in the following. \begin{theorem}\label{thmain2} Under the assumptions (\ref{eq.Vdef}), if $ (u,\varphi, \omega ) $ is a solution of (\ref{eq.M-S}) with $u$ and $\varphi$ radially symmetric maps and such that $$u \in H^1 (\mathbb{R}^3),\quad \int_{\mathbb{R}^3} \vert \nabla \varphi \vert^2 \,dx< \infty,$$ then \begin{itemize} \item[(a)] $u(r) \in C^\infty([0,1]), \varphi(r) \in C^\infty([0,1])$ \item[(b)] If $N=Z$ then $u \in S( |x| > 1 )$, with $S( |x| > 1)$ being the Schwartz class of rapidly decreasing functions. \end{itemize} \end{theorem} \begin{remark}\label{remthmain2} \rm Property (b) in the above theorem shows that the soliton behavior of the solutions can be established, when the neutrality condition $N=Z$ is satisfied. The physical importance of the neutrality condition is discussed in \cite{Lie} (see (5.2) page 24 in \cite{Lie}). \end{remark} Finally the topological properties of the set of the solutions are stated as follows. \begin{theorem}\label{thmain3} Under the assumptions (\ref{eq.Vdef}) and (\ref{eq.Coul1}), let $ (u,\varphi, \omega ) $ be a solution of (\ref{eq.M-S}) such that $\omega <0$ is the first eigenvalue, $u $ and $\varphi$ are radially symmetric maps and such that $$u \in H^1 (\mathbb{R}^3),\quad \int_{\mathbb{R}^3} \vert \nabla \varphi \vert^2 \,dx< \infty.$$ Then the solution $ (u,\varphi, \omega ) $ is isolated, i.e. there exists a neighborhood $U\subset H^1 (\mathbb{R}^3)$ of $u$, one $W$ of $\varphi$ such that $$ \int_{\mathbb{R}^3} \vert \nabla \phi \vert^2 \,dx< \infty,\quad \phi \in W,$$ and one $\Omega\subset \mathbb{R}$ of $\omega$ such that each $(v,\phi,\lambda) \in U\times W\times \Omega$ with $ (v,\phi,\lambda) \neq (u,\varphi,\omega)$, $v$ and $\phi$ radially symmetric maps satisfying the following $$\int_{\mathbb{R}^3} |v|^2 \,dx =N,$$ is not a solution of (\ref{eq.M-S}). \end{theorem} For the sake of completeness we want to recall that the existence of solitary waves has been studied by Benci and Fortunato (see \cite{BF}) in the case in which the charged particle ``lives'' in a bounded space region $\Omega$. Moreover, the Maxwell equations coupled with nonlinear Klein-Gordon equation, with Dirac equation, with nonlinear Schr\"odinger equation and with the Schr\"odinger equation under the action of some external potential have been studied respectively in \cite{BF1,EGS,C,C1,C2}. Also, we recall the classical papers \cite{AHB,AHB2,CSS}. The plan of the work is the following. In Section 2 we prove some preliminary variational results, that permit to reduce (\ref{eq.M-S}) to a single equation. Moreover we show the variational structure of the problem. In Section 3 we prove some topological properties of the energy functional associated to (\ref{eq.rid}). In Section 4 we prove Theorem \ref{thmain0m} and \ref{lemmaspecin}. In Section 5, 6 and 7 we prove Theorem \ref{thmain1m}, \ref{thmain2} and \ref{thmain3}, respectively. \section{Derivation of the equations} The relations \eqref{eq.corr1} have to be modified as follows (see Section 2, part I in \cite{STZ} \begin{equation}\label{eq.corr1mod} \begin{gathered} E \to i \hbar \partial_{t,\varphi} , \quad p \to -i \hbar \nabla_{\mathbb{A}}, \\ \partial_{t,\varphi} = \partial_t + \frac{i e}{ \hbar}\varphi , \quad \nabla_{ \mathbb{A}} = \nabla - \frac{i e}{\hbar c} \mathbb{A}, \end{gathered}\end{equation} when an external electromagnetic potential $(\varphi,\mathbb{A})$, $\mathbb{A}=(\mathbb{A}_1,\mathbb{A}_2,\mathbb{A}_3)$ is presented. Here $c>0$ is the light speed. Then the relation \eqref{eq.E=cl} leads to the following Schr\"odinger equation with electromagnetic potential and external Coulomb potential \begin{equation}\label{eq.Schemp} i \hbar \partial_{t,\varphi} \psi = - \frac{ \hbar^2}{2m} \nabla_{ \mathbb{A} }^2 \psi + V(x) \psi. \end{equation} The corresponding Lagrangian density (see (6.7), Section 6.2 in \cite{KM}) is \begin{equation}\label{eq.Lag} \mathcal{L}_{\varphi, \mathbb{A}}(\psi) = \frac{i \hbar}{4} \left( \overline{\psi} \ \partial_{t,\varphi} \psi - \psi \ \overline{\partial_{t,\varphi} \psi} \right) - \frac{ \hbar^2}{4m} | \nabla_{ \mathbb{A}} \psi |^2 - \frac{V}2 |\psi|^2. \end{equation} Equation \eqref{eq.Schemp} is then the Euler - Lagrange equation for the functional $$ \int_{\mathbb{R}^{1+3}} \mathcal{L}_{\varphi, \mathbb{A}}(\psi).$$ We have the following charge conservation law for any solution to \eqref{eq.Schemp} \begin{equation}\label{eq.charge} \int_{\mathbb{R}^3} |\psi(t,x)|^2 \,dx = N , \end{equation} where $N$ has the interpretation as number of electrons. Equation \eqref{eq.Schemp} is linear in $\psi$ and the electromagnetic potential is assumed to be a known real - valued function. The description of the interaction between electromagnetic and Schr\"odinger fields involves quantum fields equations for an electrodynamic non - relativistic many body system. A classical approximation of these quantum fields equations gives a simplified nonlinear model for the following Lagrangian density \begin{equation}\label{LM-S} \mathcal{L}_{M-S}(\psi, \varphi, \mathbb{A}) = \mathcal{L}_{\varphi , \ \mathbb{A}}(\psi) + D \ \mathcal{L}_M(A), \end{equation} where $D >0 $ is a suitable constant and \begin{equation}\label{eq.M} \mathcal{L}_M(A) = - \frac{1}4 \sum_{\mu, \nu =0}^3 F_{\mu \nu} F^{\mu \nu} \end{equation} is the Lagrangian density for the free Maxwell equation, i.e. $F_{\mu \nu}$ is the electromagnetic antisymmetric tensor, such that \begin{equation}\label{eq.Fmuni} F_{\mu \nu} = - F_{\nu \mu} = \partial_\mu A_\nu - \partial_\nu A_\mu, \quad \nu , \mu = 0,1,2,3. \end{equation} Here $\partial_0 = c^{-1} \partial_t$, $\partial_j = \partial_{x_j}$, $j=1,2,3$. The four potential $A_\mu$ is defined as follows \begin{equation}\label{eq.FA} A_0 = \varphi , \quad A_j = - \mathbb{A}_j, \quad j=1,2,3. \end{equation} It is easy to compute all components of $F_{\mu \nu}:$ \begin{equation}\label{eq.faj} F_{0 j} = - c^{-1}\partial_t \mathbb{A}_j - \partial_j \varphi, \quad F_{j k} = \partial_k \mathbb{A}_j - \partial_j \mathbb{A}_k,\quad j,k =1,2,3. \end{equation} Since the Minkowski metric with respect to coordinates $$ x^0 = ct,\quad x^j =x_j, \quad j=1,2,3$$ is $g_{\mu \nu} = {\rm diag} (1, -1,-1,-1)$, we find \begin{equation}\label{eq.fauj} F^{0 j} =- F_{0j} , \quad F^{j k} = F_{j k} ,\quad j,k =1,2,3, \end{equation} so \begin{align*} \sum_{\mu, \nu =0}^3 F_{\mu \nu} F^{\mu \nu} &= -2 \sum_{j=1}^3 (F_{0 j})^2 + 2 \sum_{1 \leq j < k \leq 3} (F_{j k})^2 = \\ & = -2 |c^{-1}\partial_t \mathbb{A} + \nabla \varphi |^2 + 2 | \nabla \times \mathbb{A}|^2, \end{align*} where $a \times b$ denotes the vector product in $\mathbb{R}^3$. The Lagrangian density in \eqref{LM-S} for the Maxwell - Schr\"odinger system becomes now \begin{equation}\label{LM-S1} \begin{aligned} \mathcal{L}_{M-S}(\psi, \varphi, \mathbb{A}) &= \frac{i \hbar}{4} \left( \overline{\psi} \ \partial_{t,\varphi} \psi - \psi \ \overline{\partial_{t,\varphi}\ \psi} \right) - \frac{ \hbar^2}{4m} | \nabla_{ \mathbb{A}} \psi |^2 - \frac{V}2 |\psi|^2\\ &\quad + \frac{D}{2 } |c^{-1}\partial_t \mathbb{A} + \nabla \varphi |^2 - \frac{D}{2} | \nabla \times \mathbb{A}|^2 , \end{aligned} \end{equation} where $D>0$ is a dimensionless constant. Taking the variation of the functional $$ \int_{\mathbb{R}^{1+3}} \mathcal{L}_{M-S}(\psi, \varphi, \mathbb{A})$$ with respect to $\bar{\psi}$, we obtain the Scr\"odinger equation \eqref{eq.Schemp} and this is the second equation in \eqref{eq.MSdin}. The variation with respect to $\varphi$ gives the equation \begin{equation}\label{eq.M1} -\frac{e}{2} |\psi|^2 - D \Delta \varphi - \frac{D}{c} \partial_t (\nabla \cdot \mathbb{A}) = 0, \end{equation} while the variation with respect to $ \mathbb{A}$ implies \begin{equation}\label{eq.S2} i \frac{\hbar e }{4m c} ( \nabla \overline{\psi} \psi - \nabla \psi \overline{\psi}) - \frac{ e^2}{2 m c^2} \mathbb{A} |\psi|^2 - \frac{D}{c^2} \partial_t^2 \mathbb{A} + D \Delta \mathbb{A} - D \nabla ( \nabla \cdot \mathbb{A}) - \frac{D}{c} \partial_t \nabla \varphi =0. \end{equation} We shall take (for simplicity) \begin{equation}\label{eq.choiceD} D = \frac{1}{8\pi} \end{equation} and shall assume that the electromagnetic potential satisfies the following Lorentz gauge condition \begin{equation}\label{eq.Lorgauge} \frac{1}{c} \partial_t A^0 + \sum_{k=1}^3 \partial_{x_k} A^k = 0. \end{equation} Then a combination between \eqref{eq.M1} and this Lorentz gauge condition implies \begin{equation}\label{eq.M1nm} - \Delta \varphi + \frac{1}{c^2} \partial_t ^2 \varphi = 4\pi e |\psi|^2, \end{equation} In a similar way from \eqref{eq.S2} we get (using the gauge condition) \begin{eqnarray}\label{eq.S2nm} \frac{ \hbar e}{2 m c} \mathop{\rm Im} \left( \nabla_{k, \mathcal{A}} \psi \ \overline{\psi} \right) - \frac{1}{c^2} \partial_t^2 \mathbb{A}_k + \Delta \mathbb{A}_k =0, \quad \ k=1,2,3. \end{eqnarray} Equations \eqref{eq.M1nm} and \eqref{eq.S2nm} can be rewritten as \begin{align} \frac{1}{c^2}\partial_{tt} \mathcal{A} - \Delta \mathcal{A} = \mathcal{J}, \label{eq.MSdinfirst} \end{align} where \begin{equation} \label{eq.def132adssee} \mathcal{J}= (J_0,J_1,J_2,J_3), \quad J_0 = 4\pi e|\psi|^2,\quad J_k = 4\pi \frac{\hbar e}{mc}\mathop{\rm Im} ( \bar{\psi}\partial_{k, \mathcal{A}} \psi) \end{equation} and this coincides with the first equation in \eqref{eq.MSdin}. \section{The Variational Setting} \label{sec;1} In this section we shall prove a variational principle that permits to reduce (\ref{eq.M-S}) to the study of the critical points of an even functional, which is not strongly indefinite. To this end we need some technical preliminaries. We define the space $ {\bf\mathcal D}^{1,2}(\mathbb{R}^3) $ as the closure of $ C^{\infty }_0(\mathbb{R}^3)$ with respect to the norm $$ \Vert u\Vert_{{\mathcal D}^{1,2}} := \Big( \int_{\mathbb{R}^3}\vert \nabla u\vert^2\,dx \Big)^ {1/2}. $$ The Sobolev - Hardy inequality (see \cite{RS}) implies the following lemma. \begin{lemma} \label{lemma21a} For all $ \rho\in L^{6/5}(\mathbb{R}^3)$ there exists only one $ \varphi \in {\mathcal D}^{1,2}(\mathbb{R}^3)$ {\it such that} $ \Delta\varphi =\rho$. Moreover there results \begin{equation} \Vert\varphi\Vert^2_{{\mathcal D}^{1,2}}\le c\Vert\rho\Vert^2_{L^{6/5}} \label{Poissona}\end{equation} and the map $$ \rho\in L^{6/5}(\mathbb{R}^3)\mapsto\varphi=\Delta^{-1}(\rho)\in {\mathcal D}^{1,2}(\mathbb{R}^3) $$ is continuous. \end{lemma} Moreover, the classical Sobolev embedding and a duality argument guarantee the properties \begin{equation}\label{startx} \begin{gathered} H^1( \mathbb{R}^3 ) \subseteq L^q( \mathbb{R}^3 ) \quad \text{for} 2 \leq q \leq 6 \\ L^{q'} ( \mathbb{R}^3 ) \subseteq \left( H^1( \mathbb{R}^3 )\right)' \quad \text{for} \frac{6}5 \leq q' \leq 2. \end{gathered} \end{equation} Denoting by $H^1_r(\mathbb{R}^3) $ the set of all $H^1$ radial functions. Then the classical Strauss Lemma shows that (see \cite{S} or \cite[Theorem~A.I']{BL}) \begin{equation}\label{eq.strimb} H^1_r(\mathbb{R}^3) \text{ is compactly embedded into} \ \ L^q (\mathbb{R}^3), 2 < q < 6. \end{equation} By Lemma \ref{lemma21a} and by using the Sobolev inequalities, for any given $u \in H^1 (\mathbb{R}^3)$ the second equation of (\ref{eq.M-S}) has the unique solution $$ \varphi=-4\pi e \Delta^{-1} u^2 \big(\in {\mathcal D}^{1,2}(\mathbb{R}^3)\big). $$ For this reason we can reduce the system (\ref{eq.M-S}) to the equations \begin{equation} -\frac{1}2 \Delta u-4\pi e^2 (\Delta^{-1} u^2) u + V(x)u =\omega u,\quad \int_{\mathbb{R}^3} |u|^2 \,dx =N. \label{eq.rid} \end{equation} Observe that (\ref{eq.rid}) is the Euler-Lagrange equation of the functional \begin{equation} J(u)=\frac{1}4 \int_{\mathbb{R}^3}\vert\nabla u\vert^2\,dx+ \pi \ e^2\int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u^2\vert^2\,dx + \frac{1}2 \int_{\mathbb{R}^3} V(x) |u|^2 \,dx, \label{funz.rid} \end{equation} constrained on the manifold $$ B:= \big\{u\in H^1(\mathbb{R}^3) \big\vert \Vert u\Vert_{L^2}^2=N\big\}. $$ Note that the functional $J(u)$ can be defined for complex valued $u$, while its critical points are only real-valued. Given any integer $ k \geq 1$ we set $$ H^k_r(\mathbb{R}^3):=\{u\in H^k(\mathbb{R}^3) : u(x) =u(\vert x\vert), \; x\in\mathbb{R}^3 \}. $$ \begin{lemma} \label{lemma22} There results: \begin{itemize} \item[(i)] $ J $ is even \item[(ii)] $ J $ is $ C^1 $ on $ H^1 (\mathbb{R}^3) $ and its critical points constrained on $B$ are the solutions of (\ref{eq.rid}) \item[(iii)] any critical point of $ J \big|_{H^1_r (\mathbb{R}^3)\cap B} $ is also a critical point of $J\big|_{B}$. \end{itemize} \end{lemma} \begin{proof} The proof of (i) is trivial. Since \begin{equation*}{ \frac{d}{d\lambda} \Big( \int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} |u+ \lambda v|^2 \vert^2\,dx}\Big) \Big|_{\lambda =0}= -4\ \int_{\mathbb{R}^3}(\Delta^{-1} u\vert v)\,dx,\end{equation*} (ii) holds true. Now we prove (iii). Consider the $O(3)$ group action $T_g$ on $H^1(\mathbb{R}^3)$ defined by $$T_g u(x) = u(g(x)),$$ where $g \in O(3)$ and $u \in H^1(\mathbb{R}^3)$. Then the conclusion follows by well known arguments (see for example \cite{S}) because $J$ is invariant under the $T_g$ action, namely $$J (T_g u)=J (u), $$ where $g \in O(3)$ and $u \in H^1(\mathbb{R}^3)$. So, by \cite{P} or \cite[Theorem~1.28]{W}, iii) is proved. \end{proof} \section{Topological Results}\label{sec:2} In this section we shall prove some topological properties of the functional $J$. \begin{lemma}\label{lemma31} The functional $J$ is weakly lower semicontinuous on $H^1_r(\mathbb{R}^3)$. In particular, the operator $$ T: u\in H^1_r(\mathbb{R}^3)\mapsto (\Delta^{-1} u^2) u\in \big(H^1_r(\mathbb{R}^3)\big)' $$ is compact and the functionals \begin{gather*} J_1: u \in H^1_r (\mathbb{R}^3) \mapsto \int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u^2\vert^2\,dx,\\ J_2: u \in H^1 (\mathbb{R}^3) \mapsto\int_{\mathbb{R}^3} V(x)u^2\,dx \end{gather*} are weakly continuous. \end{lemma} \begin{proof} We prove that $T$ is compact. Let $\{u_k\}\subset H^1_r (\mathbb{R}^3)$ be bounded. Passing to a subsequence, there exists $u\in H^1_r (\mathbb{R}^3)$ such that $u_k\rightharpoonup u$ weakly in $H^1_r(\mathbb{R}^3)$. By (\ref{Poissona}) and Sobolev inequalities \eqref{startx} we see that $ \{ \Delta^{-1} u_k^2 \} $ is bounded in $ {\mathcal D}^{1,2}(\mathbb{R}^3)$. Passing to a subsequence, there exists $h \in{\mathcal D}^{1,2}(\mathbb{R}^3) $ such that \begin{equation} \Delta^{-1}u_k^2 \rightharpoonup h \quad{\rm weakly\> in }\>\> {\mathcal D}^{1,2}(\mathbb{R}^3). \label{weak}\end{equation} We have to prove that \begin{equation} (\Delta^{-1} u_k^2)u_k \to hu \quad{\rm in}\>\> (H^1_r(\mathbb{R}^3))'. \label{comp}\end{equation} Denote $$ q=\frac{12}{5},\quad r=\frac{12}{7},\quad \alpha=\frac{q}{r}=\frac{7}{5}, \quad \alpha'=\frac{\alpha}{\alpha-1}=\frac{7}{2}, $$ clearly $2\> L^6 (\mathbb{R}^3). $$ Comparing this result with \eqref{weak}, we conclude that $h =\Delta^{-1} u^2$ and via $$ \Vert (\Delta^{-1} u^2_k) u -hu \Vert_{L^r} \le \Vert \Delta^{-1} u^2_k -h \Vert_{L^{\alpha' r}} \Vert u \Vert_{L^q}, $$ we get \begin{equation} {\Vert (\Delta^{-1} u^2_k) u -hu \Vert_{L^r} \to 0.} \label{proof3} \end{equation} So we have, by (\ref{proof1}), (\ref{proof2}) and (\ref{proof3}), that $$ (\Delta^{-1} u^2_k) u_k \to hu \quad {\rm in }\>\>L^r (\mathbb{R}^3). $$ From the properties \eqref{startx} we arrive at (\ref{comp}). We prove that $J_1$ is weakly continuous. Here it suffices to observe that the operator $$Q: u \in H^1_r (\mathbb{R}^3) \mapsto u^2 \in L^{6/5} (\mathbb{R}^3)$$ is compact, indeed, by the compact embeddings of $ H^1_r (\mathbb{R}^3 ) $, the operator: $$ H^1_r(\mathbb{R}^3)\hookrightarrow \hookrightarrow L^{12/5} (\mathbb{R}^3) {\buildrel Q\over\to} L^{6/5} (\mathbb{R}^3) $$ is compact and, by Lemma \ref{lemma21a}, the following one $\Delta^{-1}: L^{6/5} (\mathbb{R}^3) \to {\mathcal D}^{1,2}(\mathbb{R}^3)$ is continuous. We prove that $J_2$ is weakly continuous. Let $ \{u_k \} \subset H^1 (\mathbb{R}^3) $ and $ u \in H^1 (\mathbb{R}^3)$ such that $$ u_k \rightharpoonup u \quad {\rm weakly\> in}\>\> H^1 (\mathbb{R}^3). $$ Since $u_k \rightharpoonup u$ weakly in $L^2 (\mathbb{R}^3)$, there exists $ C>0 $ such that $$\Vert u_k \Vert_{L^2} \le C, \quad \Vert u \Vert_{L^2} \le C,\quad k\in \mathbb{N}. $$ Fix $ \varepsilon >0 $, there results \begin{equation}\int_{\{ \vert x \vert > z/{ \varepsilon} \}}V(x) u_k^2\,dx \le C\varepsilon , \quad \int_{\{ \vert x \vert >z/{ \varepsilon} \}}V(x) u^2\,dx \le C\varepsilon,\quad k\in \mathbb{N}. \label{st} \end{equation} By the Sobolev inequality, $u_k^2 \rightharpoonup u^2$ weakly in $L^3 (\mathbb{R}^3)$, since $V \in L^{3/2}(\{\vert x \vert \le z/{\varepsilon}\})$, there results $$\int_{\{ \vert x \vert \le z/{ \varepsilon}\}}V (x) u_k^2\,dx \to \int_{\{ \vert x \vert \le z/{ \varepsilon} \}}V(x) u^2\,dx.$$ Then, by the previous one and (\ref{st}), we can conclude $$ \int_{\mathbb{R}^3}V(x) u_k^2\,dx \to \int_{\mathbb{R}^3}V(x) u^2\,dx. $$ Since, by well known arguments, the functional $$ u \in H^1 (\mathbb{R}^3) \mapsto\int_{\mathbb{R}^3}|\nabla u|^2 \,dx $$ is weakly lower semicontinuous. The proof is complete. \end{proof} \begin{lemma}\label{lemma32} The functional $J$ is coercive in $H^1_r (\mathbb{R}^3)$, i. e. for all sequence $\{u_k \} \subset H^1_r (\mathbb{R}^3)$ such that $\Vert u_k\Vert_{H^1} \to +\infty$ there results $\lim_k J(u_k)=+ \infty$. \end{lemma} \begin{proof} Denote $$ B_{H^1_r} =\{ u \in H^1_r (\mathbb{R}^3) \big\vert \Vert u \Vert_{H^1} =1 \}. $$ Let $\{u_k \} \subset H^1_r (\mathbb{R}^3)$ be a sequence, such that $\Vert u_k \Vert_{H^1} \to +\infty$. Take $$ \lambda_k = \Vert u_k \Vert_{H^1}\quad\text{and}\quad {\tilde u}_k = \frac{ u_k}{\lambda_k}. $$ Then obviously, $u_k = \lambda_k {\tilde u}_k$ with $\lambda_k \in \mathbb{R}$ tending to $+\infty$ and ${\tilde u}_k \in B_{H^1_r}$. We have $$ J (u_k ) = a_k\lambda^2_k+b_k\lambda^4_k-c_k\lambda^2_k, $$ with \begin{gather*} a_k = \frac{1}4\int_{\mathbb{R}^3}\vert \nabla \tilde u_k\vert^2\,dx \in \Big[0, \frac{1}4\Big],\\ b_k = \pi e^2\int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} \tilde u_k^2\vert^2\,dx \ge 0,\\ c_k = \frac{1}2 \int_{\mathbb{R}^3}V(x) {\tilde u}_k^2\,dx \ge 0. \end{gather*} Observe that by Sobolev inequality there results \begin{align*} 2 c_k &= \int_{\{ \vert x \vert \le 1 \}} V (x) {\tilde u}_k^2\,dx + \int_{\{ \vert x \vert > 1 \}} V(x) {\tilde u}_k^2\,dx \\ & \le \Vert V \Vert_{L^\frac{3}{2} (\{ \vert x \vert \le 1 \})} \Vert \tilde u_k \Vert_{L^6 }^2 + \sup_{\vert x \vert \ge 1} V(x) \Vert \tilde u_k \Vert_{L^2 }^2 \\ &\le \Big( C \Vert V \Vert_{L^\frac{3}{2} (\{ \vert x \vert \le 1 \})} + \sup_{\vert x \vert \ge 1} V(x) \Big) \Vert \tilde u_k \Vert_{H^1}^2\\ &= \Big( C \Vert V \Vert_{L^\frac{3}{2} (\{ \vert x \vert \le 1 \})} + \sup_{\vert x \vert \ge 1} V(x) \Big), \end{align*} where $ C > 0 $ is the Sobolev embedding constant. Since, by Lemma \ref{lemma31}, $ u \in H^1_r (\mathbb{R}^3) \mapsto \int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u^2\vert^2\,dx $ is weakly continuous and $ B_{H^1_r} $ is bounded in $ H^1_r (\mathbb{R}^3 ) $, there exists $ \alpha >0 $ such that $ b_k \ge \alpha >0. $ Then we can conclude that $$\lim_k J (u_k)=+ \infty ,$$ and so the proof is complete. \end{proof} The two previous lemma guarantee that $J$ is bounded from below. Alternatively, we can give a direct proof of this fact. \begin{lemma}\label{lemma341} The functional $J$ is bounded from below on $ B$. \end{lemma} \begin{proof} For each $u\in B$ there results \begin{equation} J(u) \ge \frac{1}4 \int_{\mathbb{R}^3}\vert\nabla u\vert^2\,dx - \frac{1}2 \int_{\mathbb{R}^3} V(x) |u|^2 \,dx. \label{pb1}\end{equation} By Kato's Inequality (see (7.13) page 35 in \cite{Lie}) and since $u\in B$ $$ \int_{\mathbb{R}^3} V(x) |u|^2 \,dx= Z \int_{\mathbb{R}^3} \frac{|u|^2}{|x|} \,dx\le CZ\Vert u\Vert_{L^2} \Vert\nabla u\Vert_{L^2}=CNZ\Vert\nabla u\Vert_{L^2}, $$ for some constant $C>0$. So, by (\ref{pb1}), \begin{equation}\label{pb10} J(u) \ge \frac{1}4 \Vert\nabla u\Vert^2_{L^2} - \frac{CNZ}2\Vert\nabla u\Vert_{L^2}. \end{equation} Since the map $$x\in{\mathbb{R}}\mapsto \frac{1}4 x^2 - \frac{NCZ}{2}x$$ is bounded from below, by (\ref{pb10}), the claim is done. \end{proof} \section{Spectral Results} \label{sec:4} The main result of this section is as follows. \begin{proposition}\label{lemmaspec} Let $(u,\omega)\in H^1_r(\mathbb{R}^3)\times\mathbb{R}$ be solution of the equation in (\ref{eq.rid}). If \begin{gather}\label{spec1} 0<\int_{\mathbb{R}^3} u^2 \,dx\le N, \\ \label{eq.vwq} V(x) = - \frac{Ze^2}{|x|}, \end{gather} then \begin{gather}\label{spec2} \omega<0 \quad \text{provided $Z>N$}, \\ \label{spec2'} \omega \le 0 \quad\text{ provided $Z=N$}. \end{gather} \end{proposition} This proposition implies that Theorem \ref{lemmaspecin} is valid. However, to prove the above proposition some Lemmas are needed. \begin{lemma}\label{lemmaag0} Let $u\in C^2(\{|x|\ge R\})$ be a solution of \begin{equation}\label{eq.Agmon} \Delta u + p(x) u=0,\quad |x|\ge R,\end{equation} for some $R>0$, if $p\in C(\mathbb{R}^3)$ and there exist $\alpha, R_0>R$ such that \begin{equation}\label{eq.cornic} p(|x|) \geq 0, \quad |x|\ge R_0, \end{equation} \begin{equation}\label{eq.Agmon1} \frac{\partial p}{\partial r} + \frac{2(1-\alpha)}{|x|} p \ge 0,\quad |x|\ge R_0,\end{equation} then \begin{equation}\label{eq.Agmon2} \liminf_{R\to +\infty} \frac{1}{R^{\alpha}}\int_{\{R_0\le |x| \le R\}} p(x) u^2(x)\,dx >0.\end{equation} \end{lemma} The proof of the above lemma is a direct consequence of \cite[Theorem 3]{A}. \begin{lemma}\label{lemmaag} Let $u\in H^1_r(\mathbb{R}^3), v\in L^1(\mathbb{R}^3)\cap L^{6/5}(\mathbb{R}^3)$ radial, $\omega > 0$ or $\omega =0$ and \begin{equation}\label{eq.Agmon3}v\ge 0,\quad \int_{\mathbb{R}^3} v \,dx < Z.\end{equation} If $u,v$ satisfy the equation \begin{equation}\label{eq.Agmon4}-\frac{1}2 \Delta u-4\pi e^2(\Delta^{-1} v) u + V(x)u =\omega u,\end{equation} then $u\equiv 0$. \end{lemma} \begin{proof} Assume, by absurd, that there exist $u\not\equiv 0$ and $\omega \ge 0$ satisfying (\ref{eq.Agmon3}) and (\ref{eq.Agmon4}). Denote $$ p(x) := 8\pi e^2 (\Delta^{-1}v)(x) - 2 V(x) +2\omega, \quad x\in \mathbb{R}^3, $$ clearly $u$ is solution of (\ref{eq.Agmon}). We shall apply Lemma \ref{lemmaag0} for this take $\alpha$, $0<\alpha<\frac{1}{2}$. By \cite{L2} or Lemma \ref{l.Lio} in the Appendix, \begin{equation}\label{eq.Lions} 4\pi (\Delta^{-1}v)(x) = -\int_{\mathbb{R}^3}\frac{v(y)}{\max\{|x|,|y|\}}\,dy , \quad x\in \mathbb{R}^3, \end{equation} so \begin{equation}\label{eq.viz231} p(|x|) = 2 e^2 \int_{|y| \geq |x|} \big( \frac{1}{|x|} - \frac{1}{|y|}\big) v(y)\, dy +2\omega + 2\frac{Z-N}{|x|} e^2 \ge 0. \end{equation} For $r= |x|$, there results \begin{equation} \label{eq.Agmon5} \begin{aligned} \frac{\partial p}{\partial r}(x) + \frac{2(1-\alpha)}{|x|}p(x) &=8\pi e^2 \big(\frac{\partial (\Delta^{-1}v)}{\partial r}(x) + \frac{2(1-\alpha)}{|x|}(\Delta^{-1}v) (x)\big)\\ &\quad- 2\big(\frac{\partial V}{\partial r}(x) + \frac{2(1-\alpha)}{|x|}V (x)\big)+ \frac{4(1-\alpha)\omega}{|x|}. \end{aligned} \end{equation} Moreover, by (\ref{eq.Vdef}), \begin{equation} \label{eq.Agmon6} -\frac{\partial V}{\partial r}(x) - \frac{2(1-\alpha)}{|x|}V(x) = -\frac{Z}{r^2} + \frac{2(1-\alpha)Z}{r^2} = \frac{(1-2\alpha)Z}{r^2}. \end{equation} So using this relation and Lemma \ref{l.Lio1a} from the Appendix, we find \begin{equation}\label{eq.Agmon7} \begin{aligned} &4\pi \big(\frac{\partial \Delta^{-1}v}{\partial r}(x) + \frac{2(1-\alpha)}{|x|}\Delta^{-1}v (x)\big)\\ &= \int_{|y| < r}\frac{v(y)}{|x|^2}\,dy - \frac{2(1-\alpha)}{r}\int_{\mathbb{R}^3}\frac{v(y)}{\max\{|x|,|y|\}}\,dy \\ &=\int_{|y| < r}\frac{v(y)}{|x|^2}\,dy - \frac{2(1-\alpha)}{r}\int_{\{|y|\le r\}}\frac{v(y)}{\max\{|x|,|y|\}}\,dy \\ &\quad -\frac{2(1-\alpha)}{r}\int_{\{|y|\ge r\}}\frac{v(y)}{\max\{|x|,|y|\}}\,dy \\ &=\int_{\{|y|\le r\}}\frac{v(y)}{\max\{|x|^2,|y|^2\}}\,dy - \frac{2(1-\alpha)}{r^2}\int_{\{|y|\le r\}}v(y)\,dy\\ &\quad -\frac{2(1-\alpha)}{r}\int_{\{|y|\ge r\}}\frac{v(y)}{|y|}\,dy\\ &\ge\int_{\{|y|\le r\}}\frac{v(y)}{r^2}\,dy - \frac{2(1-\alpha)}{r^2}\int_{\{|y|\le r\}}v(y)\,dy -\frac{2(1-\alpha)}{r^2}\int_{\{|y|\ge r\}}v(y)\,dy \\ &\ge-\frac{(1-2\alpha)}{r^2}\int_{\mathbb{R}^3}v(y)\,dy- \frac{2(1-\alpha)}{r^2}\int_{\{|y|\ge r\}}v(y)\,dy. \end{aligned} \end{equation} By (\ref{eq.Agmon5}), (\ref{eq.Agmon6}) and (\ref{eq.Agmon7}), \begin{equation} \label{eq.Agmon8} \begin{aligned} &\frac{\partial p}{\partial r}(x) + \frac{2(1-\alpha)}{|x|} p(x)\\ &\ge 2\frac{(1-2\alpha)}{r^2}\Big(Z-\int_{\mathbb{R}^3}v(y)\,dy\Big) +4\frac{(1- \alpha)}{r}\Big(\omega - \frac{1}{r} \int_{\{|y|\ge r\}}v(y)\,dy\Big). \end{aligned} \end{equation} If $\omega >0$, then there exists $R_0> 0$ such that $$ \frac{1}{|x|}\int_{\{|y|\ge |x|\}}v(y)\,dy\le \frac{\omega}2 , \quad |x|\ge R_0. $$ If $\omega = 0$ and $Z>N$, then for any $\varepsilon>0$ one can find $R_0> 0$ such that $$\int_{\{|y|\ge |x|\}}v(y)\,dy\le \varepsilon, \quad |x|\ge R_0. $$ In both cases, by (\ref{eq.Coul1}), (\ref{eq.Agmon3}) and (\ref{eq.Agmon8}), since $0<\alpha<\frac{1}{2}$, we have \begin{equation}\label{eq.Agmon9} \frac{\partial p}{\partial r}(x) + \frac{2(1-\alpha)}{|x|} p(x)\ge 0, \quad |x|\ge R_0. \end{equation} By (\ref{eq.Agmon}) and Lemma \ref{lemmaag0}, the formula (\ref{eq.Agmon2}) holds true. On the other hand, we have \begin{equation}\label{eq.Agmon10} \int_{\mathbb{R}^3} u^2 (\Delta^{-1}v)\,dx\le \Vert u\Vert^2_{L^{12/5}} \Vert \Delta^{-1}v \Vert_{L^6}\end{equation} and, as in Lemma \ref{lemma32}, \begin{equation}\label{eq.Agmon11} \int_{\mathbb{R}^3} u^2 | V | \,dx\le \Vert V \Vert_{L^{3/2}(\{|x|\le 1\})}\Vert u\Vert^2_{L^6} + Z \Vert u\Vert^2_{L^2}, \end{equation} so, by (\ref{eq.Agmon10}) and (\ref{eq.Agmon11}), \begin{equation} \label{eq.Agmon12} \begin{aligned} &\int_{\{R_0\le |x| \le R\}} p u^2 \,dx\\ &\le \int_{\mathbb{R}^3} p u^2 \,dx \\ &= 2\Big(4\pi e^2 \int_{\mathbb{R}^3}(\Delta^{-1}v)u^2\,dx+ \int_{\mathbb{R}^3}V u^2\,dx+\int_{\mathbb{R}^3} \omega u^2\,dx\Big)\\ &\le 8\pi e^2\Vert u\Vert^2_{L^{12/5}} \Vert \Delta^{-1}v \Vert_{L^6}+2\Vert V \Vert_{L^{3/2}(\{|x|\le 1\})}\Vert u\Vert^2_{L^6} + 2Z \Vert u\Vert^2_{L^2}+2\omega \Vert u\Vert^2_{L^2}. \end{aligned} \end{equation} Then \begin{equation}\label{eq.Agmon13} \lim_{R\to +\infty} \frac{1}{R^{\alpha}}\int_{\{R_0\le |x| \le R\}} p(x) u^2(x)\,dx =0,\end{equation} and this is absurd, since (\ref{eq.Agmon13}) contradicts (\ref{eq.Agmon2}), this concludes the proof. \end{proof} \begin{corollary}\label{cor.spec} If $V\equiv 0$ and the assumptions of Lemma \ref{lemmaag} are satisfied, then $u\equiv 0$. \end{corollary} \begin{proof} Suppose, by absurd, that there is $u\not\equiv 0$ solution of (\ref{eq.Agmon4}), multiplying by $u$ and integrating on $\mathbb{R}^3$, we get $\omega>0$. We are going to apply the Agmon's result of Lemma \ref{lemmaag0}. For this we have to verify the condition (\ref{eq.Agmon1}) for $|x|$ large enough, $0<\alpha<\frac{1}{2}$ and $$ p(x) := 2 (\Delta^{-1}v)(x) +2\omega, \quad x\in \mathbb{R}^3. $$ The argument of the previous lemma (with $Z=0$) gives \begin{align*} &\frac{\partial p}{\partial r}(x) + \frac{2(1-\alpha)}{|x|} p(x)\\ &\ge 4\frac{(1-\alpha)\omega}{r}-2\frac{(1-2\alpha)}{r^2} \int_{\mathbb{R}^3}v(y)\,dy-4\frac{(1- \alpha)}{r^2} \int_{\{|y|\ge r\}}v(y)\,dy. \end{align*} So, for $R_0>0$ sufficiently large $$ \frac{\partial p}{\partial r}(x) + \frac{2(1-\alpha)}{|x|} p(x)\ge0,\quad |x|\ge R_0. $$ By (\ref{eq.Agmon12}), with $Z=0$ we have $$ \frac{1}{R^{\alpha}}\int_{\{R_0\le |x| \le R\}} p u^2 \,dx\le \frac{2}{R^{\alpha}}\left(\Vert u\Vert^2_{L^{12/5}} \Vert \Delta^{-1}v \Vert_{L^6}+\omega \Vert u\Vert^2_{L^2}\right). $$ This is absurd, because it contradicts (\ref{eq.Agmon2}), then $u\equiv 0$. \end{proof} \begin{proof}[Proof of Theorem \ref{thmain0m}] Denote $v(x):= u^2(x)$, $x\in \mathbb{R}^3$. By the Sobolev inequalities $$ v\in L^1(\mathbb{R}^3)\cap L^{6/5}(\mathbb{R}^3), $$ it is radial and, by the constraint in (\ref{eq.rid}), it satisfies also (\ref{eq.Agmon3}). Since $\omega\ge 0$, the claim is direct consequence of the previous corollary and of the equivalence between (\ref{eq.M-S}) and (\ref{eq.rid}). \end{proof} \begin{lemma}\label{Lemmaag} If $u\in H^1_r(\mathbb{R}^3)$ is a solution of \eqref{eq.rid}, such that \begin{gather}\label{norm}\int_{\mathbb{R}^3} u^2 \,dx < Z, \\ \label{norm1}\omega\ge 0, \end{gather} or \begin{gather}\label{normqw} \int_{\mathbb{R}^3} u^2 \,dx = Z,\\ \label{norm1qw}\omega > 0, \end{gather} then $u\equiv 0$. \end{lemma} \begin{proof} Denote $v(x):= u^2(x)$, for $x\in \mathbb{R}^3$. By the Sobolev inequalities $$ v\in L^1(\mathbb{R}^3)\cap L^r(\mathbb{R}^3),\quad \frac{6}5 0$, i.e. any sequence $\{ u_k \} \subset H^1_r (\mathbb{R}^3) \cap B $ such that $ \{ J(u_k)\} $ is bounded and \begin{equation} J(u_k)\le -\varepsilon, \quad J \big|_{H^1_r (\mathbb{R}^3)\cap B} ' (u_k) \to 0, \label{assPS}\end{equation} contains a converging subsequence.\end{lemma} \begin{proof} Fix $\varepsilon >0$. Let $\{u_k \} \subset H^1_r (\mathbb{R}^3)\cap B$ be such that $\{ J (u_k) \} $ is bounded and satisfies (\ref{assPS}). First of all observe that, by (iii) of Lemma \ref{lemma22}, there results $$ J \big|_{H^1_r (\mathbb{R}^3)\cap B} ' (u) =0 \Longleftrightarrow J \big|_{B} ' (u)=0,$$ then we can assume $$J \big|_{B}' (u_k) \to 0. $$ Since $J(u_k) \le -\varepsilon$, by Lemma \ref{lemma32}, $\{ u_k\}$ is bounded in $H^1_r ( \mathbb{R}^3 )$, passing to a subsequence, there exists $u \in H^1_r (\mathbb{R}^3 )$ such that \begin{equation} u_k \rightharpoonup u \quad \quad{\rm weakly \>in}\>\> H^1_r(\mathbb{R}^3). \label{4} \end{equation} We shall prove that \begin{equation} u_k \to u \quad \quad{\rm in}\>\> H^1_r(\mathbb{R}^3).\label{6}\end{equation} By definition, there exists $\{\omega_k\}\subset \mathbb{R}$ such that $$ J \big|_{B}' (u_k)= J' (u_k) - \omega_k u_k,\quad k\in\mathbb{N}. $$ Observe that, since $\{u_k\}\subset B$, we have \begin{align*} &N\omega_k \\ &= \langle J \big|_{ B}'(u_k), u_k\rangle - \langle J'(u_k), u_k\rangle \\ &= \langle J \big|_{ B}'(u_k), u_k\rangle-\frac{1}2 \int_{\mathbb{R}^3}\vert\nabla u_k\vert^2\,dx -4\pi e^2\int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u_k^2\vert^2\,dx + \int_{\mathbb{R}^3}V(x) \vert u_k\vert^2\,dx \\ &= \langle J \big|_{ B}'(u_k), u_k\rangle-2 J(u_k) - 2\pi e^2\int_{\mathbb{R}^3}\vert\nabla\Delta^{-1}u_k^2\vert^2\,dx, \end{align*} by Lemma \ref{lemma31} and (\ref{assPS}), $\{\omega_k\}$ is bounded in $\mathbb{R}$ and so passing to a subsequence there results \begin{gather} \omega_k \to \omega\,, \label{W}\\ -\frac{1}2 \Delta u-4\pi e^2(\Delta^{-1} u^2) u - V(x)u =\omega u\,. \label{5} \end{gather} If $\omega<0$, by Lemma \ref{lemma31}, (\ref{4}), (\ref{W}) and (\ref{5}), \begin{equation} \label{5'} \begin{aligned} &\frac{1}2\int_{\mathbb{R}^3}\vert \nabla u_k \vert^2\,dx - \omega \int_{\mathbb{R}^3} u_k^2\,dx\\ &= \big\langle J \big|_{ B}'(u_k) ,\, u_k \big\rangle - 4 \pi e^2 \int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u_k^2\vert^2 \,dx +\int_{\mathbb{R}^3}V(x) u_k^2\,dx + (\omega_k -\omega) \int_{\mathbb{R}^3} u_k^2\,dx\\ &\to - 4 \pi e^2\int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u^2\vert^2\,dx + \int_{\mathbb{R}^3}V(x) u^2\,dx \\ &=\frac{1}2\int_{\mathbb{R}^3}\vert \nabla u \vert^2\,dx - \omega \int_{\mathbb{R}^3} u^2 \,dx, \end{aligned} \end{equation} and then (\ref{6}) follows. Now we consider the case $\omega\ge 0$. If $\Vert u \Vert_{L^2}=0$, by Lemma \ref{lemma31}, we have $$ 0= J(u) \le \liminf_{k} J(u_k)\le -\varepsilon, $$ that is absurd. If $0<\Vert u \Vert_{L^2}^2< N$ then $u$ is solution of the equation in (\ref{eq.rid}), (\ref{norm}) and (\ref{norm1}) hold. So, by Lemma \ref{Lemmaag} , we have $u\equiv 0$ and also this is absurd. Finally, if $\Vert u \Vert_{L^2}^2= N$, we have, from \eqref{4} \begin{equation}\label{5''} u_k\to u \quad \quad{\rm in}\>\> L^2(\mathbb{R}^3), \end{equation} then \eqref{6} is direct consequence of \eqref{W}, \eqref{5'} and \eqref{5''}. This concludes the proof. \end{proof} \begin{remark}\label{rem.ripar} \rm Let $ \rho\in L^1(\mathbb{R}^3)\cap L^r(\mathbb{R}^3)$, with $ { \frac{6}5 }0, \end{gather*} where \begin{gather*} {\mathcal A} := \big\{ A \subset H^1_r (\mathbb{R}^3 )\cap B : A \text{ is closed and symmetric} \big\}, \\ \Omega_k :=\big\{h: S^{k-1}\to H^1_r(\mathbb{R}^3)\cap B : h \text{is continuous and odd}\big\}, \quad k \in \mathbb{N} \backslash \{ 0 \},\\ \Omega_{k,\lambda}:=\big\{h: S^{k-1}\to H^1_r(\mathbb{R}^3)\cap B_{\lambda}: h \text{is continuous and odd}\},\quad k \in \mathbb{N} \backslash \{ 0 \big\}, \lambda>0,\\ B_\lambda := \big\{u\in H^1(\mathbb{R}^3): \Vert u\Vert_{L^2}=\lambda\big\},\quad \lambda>0 \end{gather*} and $\gamma$ is the Krasnoselskii Genus (see e. g. \cite[Definition 1.1]{AR}). \begin{lemma}\label{lemma37} There results \begin{equation} c_k \le \tilde c_k \le\tilde c_{k,\lambda},\label{mono}\end{equation} for each $k\in \mathbb{N} \backslash \{ 0 \}$ and $0<\lambda\le \sqrt{N}$. \end{lemma} \begin{proof} Fix $k\in \mathbb{N} \backslash \{ 0\}$. We prove that \begin{equation} c_k \le \tilde c_k .\label{mono1}\end{equation} Let $h\in \Omega_k$, since $h$ is continuous and odd the set $J(h(S^{k-1}))$ is closed and symmetric. Moreover $h(S^{k-1})\subset B$ and, by the invariance property of the Genus, there results $$\gamma\big(h\big(S^{k-1}\big)\big)\ge \gamma\big(S^{k-1}\big)= k.$$ So we have $c_k \le \sup J\big(h\big(S^{k-1}\big)\big)$ and then (\ref{mono1}) is proved. Now, we prove that \begin{equation} \tilde c_k \le\tilde c_{k,\lambda},\quad 0<\lambda\le \sqrt{N}. \label{mono2}\end{equation} Fix $0<\lambda\le \sqrt{N}$ and define $$ h_\lambda (\xi)(x)= \frac{1}{\lambda^5} h (\xi)\Big(\frac{x}{\lambda^4}\Big), \quad h \in \Omega_k, \xi \in S^{k-1}. $$ Let $h \in\Omega_k$ and $\xi \in S^{k-1}$ such that \begin{equation}\frac{3}{2 N^2} \int_{\mathbb{R}^3}\vert \nabla u \vert^2\,dx -\int_{\mathbb{R}^3}V(x) |u|^2 \,dx \ge 0,\label{pos}\end{equation} where $u:= h(\xi)$. Set $$ \nu := \frac{1}{\lambda^4}, \quad u_\nu (x):= h_\lambda (x)=\nu^{5/4} (x) u(\nu x) $$ and observe that, by (\ref{rip2}), there results \begin{gather*} \int_{\mathbb{R}^3} |u_\nu|^2 \,dx =\frac{1}{\nu^{1/2}} \int_{\mathbb{R}^3} |u|^2 \,dx= \lambda^2N, \\ \int_{\mathbb{R}^3}\vert \nabla u_\nu \vert^2\,dx =\nu^{3/2}\int_{\mathbb{R}^3}\vert \nabla u \vert^2\,dx, \\ \int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u_\nu^2(x)\vert^2\,dx =\int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u^2(x)\vert^2\,dx, \\ \int_{\mathbb{R}^3}V(x) |u_\nu|^2 \,dx =\nu^{1/2}\int_{\mathbb{R}^3}V(x) |u|^2 \,dx. \end{gather*} Consider the map $$ f(\nu):= J(u_\nu)=\frac{\nu^{3/2}}4 \int_{\mathbb{R}^3} \vert\nabla u\vert^2\,dx+ \pi e^2\int_{\mathbb{R}^3} \vert\nabla\Delta^{-1} u^2\vert^2\,dx - \frac{\nu^{1/2}}2 \int_{\mathbb{R}^3} V(x) |u|^2 \,dx, $$ there results $$ \frac{df}{d\nu} (\nu) = \frac{3\nu^{1/2}}8 \int_{\mathbb{R}^3}\vert\nabla u\vert^2\,dx - \frac{1}{4\nu^{1/2}} \int_{\mathbb{R}^3} V(x) |u|^2 \,dx. $$ Clearly $$ \frac{df}{d\nu} (\nu)\ge 0 \Longleftrightarrow\frac{3\nu}{2} \int_{\mathbb{R}^3}\vert\nabla u\vert^2\,dx- \int_{\mathbb{R}^3} V(x) |u|^2 \,dx\ge 0 $$ and then, by (\ref{pos}), $f$ is increasing for $\nu \ge 1/N^2$, namely $$ J(h(\xi)) = J(u) \le J(u_\nu) =J(h_\lambda(\xi)). $$ Since, if there exists $\xi'\in S^{k-1}, \xi\not=\xi'$ such that $h(\xi')$ does not satisfy (\ref{pos}), we have $J(h(\xi'))\le J(h(\xi))$, then $$ \sup J\big(h\big(S^{k-1}\big)\big)\le\sup J\big(h_\lambda\big(S^{k-1}\big)\big). $$ This concludes the proof of (\ref{mono2}). \end{proof} \begin{lemma}\label{lemma371} For all $k \in \mathbb{N} \backslash \{ 0 \}$, there exist a subspace $ V_k \subset H^1_r (\mathbb{R}^3) $ of dimension $ k$ and $\nu > 0 $ such that $$\int_{\mathbb{R}^3} \Big( \frac{1} 2 \vert \nabla u \vert^2 - V(x) u^2 \Big) \,dx \le -\nu, $$ for all $u \in V_k \cap B$. \end{lemma} \begin{proof} Let $u$ be a smooth map with compact support such that $$ \int_{\mathbb{R}^3} \vert u \vert^2 \,dx = N,\quad {\rm supp} (u)\subset B_2 (0) \backslash B_1 (0),$$ where $$ B_{\rho} ( x ) := \big\{ y\in \mathbb{R}^3 \big| \vert x - y \vert < \rho \big\},\quad x \in \mathbb{R}^3, \rho >0. $$ Denote $$ u_{\lambda} (x):=\lambda^{3/2} u(\lambda x), \quad \lambda >0, x \in \mathbb{R}^3, $$ there results $$ \int_{\mathbb{R}^3} \vert u \vert^2 \,dx =\int_{\mathbb{R}^3} \vert u_{\lambda} \vert^2 \,dx =N, \quad {\rm supp} (u_{\lambda})\subset B_{2/\lambda}(0) \backslash B_{1/\lambda} (0). $$ We have \begin{align*} \int_{\mathbb{R}^3} \Big( \frac{1} 2 \vert \nabla u_{\lambda} \vert^2 - V(x) u_{\lambda}^2 \Big)\,dx &=\int_{\mathbb{R}^3} \Big( \lambda^2 \frac{1}2 \vert \nabla u \vert^2 - V \big( {x/\lambda } \big) u^2 \Big)\,dx \\ &\le \lambda^2 \int_{\mathbb{R}^3} \frac{1} 2 \vert \nabla u \vert^2 \,dx -N \inf_{{\rm supp} (u)/ \lambda} V \\ &\le \lambda^2 \int_{\mathbb{R}^3} \frac{1} 2 \vert \nabla u \vert^2 \,dx -\frac{z\lambda}{2}N. \end{align*} There exists $ {\lambda}_0 >0 $ such that $$ \int_{\mathbb{R}^3} \Big(\frac{1} 2 \vert \nabla u_{{\lambda}_0} \vert^2 - V(x) u_{{\lambda}_0}^2 \Big)\,dx <0. $$ Let $ k \in \mathbb{N} \backslash \{ 0 \} $ and $ u_1, u_2, \dots ,u_k $ be smooth maps with compact supports such that $$ \int_{\mathbb{R}^3} \vert u_i \vert^2 \,dx =1,\quad \mathop{\rm supp} (u_i)\subset B_{2i} (0) \backslash B_i (0) , \quad i = 1, 2, \dots , k. $$ Using an analogous argument we are able to find $ \lambda_1, \lambda_2, \dots , \lambda_k >0$ such that $$ \int_{\mathbb{R}^3} \Big( \frac{1} 2 \vert \nabla u_{i_{{\lambda}_i}} \vert^2 - V(x) u_{i_{{\lambda}_i}}^2 \Big)\,dx <0, \quad i = 1, 2, \dots , k\,. $$ Let $0 < \bar \lambda < \min \{\lambda_1, \lambda_2, \dots , \lambda_k \}$ and let $ V_k $ be the subspace spanned by $ u_{1_{\bar \lambda}}$, $u_{2_{\bar \lambda}}$, \dots, $u_{k_{\bar \lambda}}$. Since the supports of this maps are pairwise disjoint, $ V_k $ has dimension $ k. $ Since for all $ i = 1, 2, \dots , k $ and $ \lambda \le \lambda_i $, there results $$ \int_{\mathbb{R}^3} \Big( \frac{1} 2 \vert \nabla u_{i_\lambda} \vert^2 - V(x) u_{i_\lambda}^2 \Big) < 0 $$ and $ V_k \cap B $ is compact, the claim is proved. \end{proof} \begin{lemma}\label{lemma3711} There results \begin{equation} c_k <0,\label{monoo}\end{equation} for each $k\in \mathbb{N} \backslash \{ 0 \}$. \end{lemma} \begin{proof} Let $ k \in \mathbb{N} \backslash \{ 0 \} $, by Lemma \ref{lemma371}, there exist $ V_k \subset H^1_r (\mathbb{R}^3)$ subspace of dimension $ k $ and $ \nu > 0$ such that, for all $ u \in V_k \cap B$, $$ \int_{\mathbb{R}^3} \Big( \frac{1} 2 \vert \nabla u \vert^2 - V (x)u^2 \Big) \,dx \le -\nu. $$ Let $ \lambda >0 $ and define $$h_\lambda : V_k \cap B \to H^1_r (\mathbb{R}^3), \quad h_\lambda (u)= \lambda^{1/2} u. $$ Fixed $ u \in V_k \cap B$ and $ 0 < \lambda < \sqrt{N}$, there results \begin{equation} J ( h_\lambda (u)) \le -{\lambda / 2} \nu + c \lambda^2 \le -{\lambda/2} \nu + c \lambda^2, \label{16} \end{equation} where $c$ is a positive constant. Then there exists $ 0 < \bar \lambda <\sqrt{N} $ such that for all $ u \in V_k \cap B $ there results $ J(h_{\bar\lambda} (u)) <0$. Since $h_{\bar\lambda}\in \Omega_{\bar\lambda}$ and $V_k\cap B \simeq S^{k-1}$, by Lemma \ref{lemma37} and the compactness of $S^{k-1}$, we have $$ c_k \le \tilde c_k \le\tilde c_{k,\bar\lambda}\le \sup J ( h_{\bar\lambda} (V_k \cap B))<0\,. $$ The proof is complete. \end{proof} \begin{corollary}\label{lemma3712} There results \begin{equation} \inf _{u\in H^1_r(\mathbb{R}^3)\cap B} J(u) <0.\label{monooo}\end{equation} \end{corollary} The proof of this corollary is a direct consequence of the previous Lemma. \begin{lemma}\label{lemma381} Let $k \in \mathbb{N}, E \subset H^1(\mathbb{R}^3 )$ be a subspace of dimension $k$ and $A\in {\mathcal A}$, if \begin{equation} \gamma(A)\ge k+1 \label{11}\end{equation} then \begin{equation}A\cap E^{\bot}\not= \emptyset.\label{111}\end{equation} \end{lemma} \begin{proof} Assume, by absurd that (\ref{111}) is false, there results \begin{equation} P(A)\subset E \backslash \{ 0 \},\label{1111} \end{equation} where $P: H^1(\mathbb{R}^3 )\to E$ is the orthogonal projection on $E$. So we have \begin{equation}\gamma (P(A))\le k.\label{11111}\end{equation} On the other side, since $P$ is continuous and odd, by the invariance property of the Genus there results $$k+1 \le \gamma(A)\le \gamma (P(A)).$$ Since this is in contradiction with (\ref{11111}), the proof is complete.\end{proof} \begin{lemma}\label{lemma38} The functional $J$ has a sequence $\{ u_k \}_{k \in \mathbb{N}} \subset H^1_r (\mathbb{R}^3 )\cap B$ of critical points such that $\omega_k<0$ and $\omega_k\to 0$, where $\{ \omega_k \}_{k \in \mathbb{N}} \subset \mathbb{R}$ is the sequence of the Lagrange multipliers associated to the critical points. \end{lemma} \begin{proof} By Lemmas \ref{lemma33} and \ref{lemma3711} (see \cite[Theorem 9.1]{R}) there exists a sequence $\{ u_k \}_{k \in \mathbb{N}} \subset H^1_r ( \mathbb{R}^3 )\cap B$ of critical points of the functional $J$. Call $\{ \omega_k \}_{k \in \mathbb{N}} \subset \mathbb{R}$ the sequence of the Lagrange multipliers associated to this critical points, namely $$J'(u_k)-\omega_ku_k=0, \quad k \in \mathbb{N}\backslash \{ 0 \}.$$ By Lemma \ref{lemmaag}, there results $\omega_k<0$ for $k \in \mathbb{N}\backslash \{ 0 \}$. We have to prove that \begin{equation}\omega_k\to 0.\label{161}\end{equation} Let $\{V_k\}$ be a sequence of subspaces of $ H^1_r (\mathbb{R}^3)$, such that $$ \dim (V_k)=k,\quad \bigcup_{k\in \mathbb{N}\backslash \{ 0 \}}V_k \text{ is dense in } H^1_r (\mathbb{R}^3 ). $$ Moreover, let $\{A_k\}\subset {\mathcal A}$ such that \begin{equation} \gamma (A_k)\ge k,\quad c_k\le \sup J(A_k)\le \frac{c_k}{2},\quad k \in \mathbb{N}\backslash \{ 0 \}.\label{162}\end{equation} Call $$ W_k := V_{k-1}^{\bot},\quad k \in \mathbb{N}\backslash \{ 0 \}, $$ by Lemma \ref{lemma381}, there results $W_k\cap A_k \not= \emptyset$, $k \in \mathbb{N}\backslash \{ 0 \}$. Let $\{v_k \}\subset H^1_r (\mathbb{R}^3 )\cap B$ such that $$ v_k \in W_k\cap A_k,\quad k \in \mathbb{N}\backslash \{ 0 \}, $$ clearly \begin{equation}\label{164} v_k \rightharpoonup 0 \quad \quad\text{weakly in }\>\> H^1_r(\mathbb{R}^3) \end{equation} and, by (\ref{162}), \begin{equation} \sup J(V_k)\le \frac{c_k}{2},\quad k \in \mathbb{N}\backslash \{ 0 \}.\label{163}\end{equation} By (\ref{164}) and Lemma \ref{lemma31} we have \begin{equation} 0\le \liminf_{k} J(v_k) \label{168}\end{equation} and, by (\ref{163}), \begin{equation} \limsup _{k} J(v_k)\le \lim_{k} \frac{c_k}{2}\le 0. \label{169}\end{equation} By (\ref{168}) and (\ref{169}), we deduce $c_k\to 0$. Since $2c_k \le \omega_k <0$, (\ref{161}) is done. \end{proof} \begin{proof}[Proof of Theorem \ref{thmain1m}] Since $F (u, 4 \pi \Delta^{-1} u^2 ) = J (u)$ for all $ u \in H^1 (\mathbb{R}^3)$, by Lemma \ref{lemma22} and the previous one the claim is proved. \end{proof} \section{Proof of Theorem \ref{thmain2}}\label{sec:6} Our next step is to show that the radially symmetric solutions \begin{equation}\label{eq.reg} u \in H^1(\mathbb{R}^3), \quad \nabla \varphi \in L^2( \mathbb{R}^3), \end{equation} to the equation \begin{equation}\label{eq.M-Sep} \begin{gathered} - \frac{1}2\Delta u -e \varphi u - \frac{Z}{|x|} u = \omega u , x \in \mathbb{R}^3, \\ \Delta \varphi = 4\pi e u^2, \ x \in \mathbb{R}^3, \\ \int_{\mathbb{R}^3} u^2 \,dx =N, \end{gathered} \end{equation} constructed in the previous section, are more regular. More precisely, we shall derive the higher regularity \begin{equation}\label{eq.regN} \nabla u \in H^{k-1}(|x|> \varepsilon ) ,\quad \nabla \varphi \in H^{k-1}( |x|> \varepsilon ), \end{equation} where $k$ is arbitrary integer and $\varepsilon >0$. \begin{lemma}\label{eq.N2} If the assumption \eqref{eq.reg} is satisfied, then \begin{equation}\label{eq.regN2} \nabla u \in H^1( \mathbb{R}^3) ,\quad \nabla \varphi \in H^{1}( \mathbb{R}^3), \quad u \in L^\infty(\mathbb{R}^3), \quad \varphi \in L^\infty(\mathbb{R}^3). \end{equation} \end{lemma} \begin{proof} The assumption \eqref{eq.reg} and the Sobolev embedding in $ \mathbb{R}^3 $ guarantee that \begin{equation}\label{eq.sob} \varphi \in L^6, \quad u \in L^p, \quad 2 \leq p \leq 6. \end{equation} This property and the H\"older inequality imply that the nonlinear term $\varphi u$ in the first equation in \eqref{eq.M-Sep} is in $L^2$. The fact that $|x|^{-1} u \in L^2$ follows from the Hardy inequality and the fact that $\nabla u \in L^2$. Therefore this equation shows that $\Delta u \in L^2$, so $ u \in H^2$. Using the second equation and the fact that $u^2 \in L^2$ we conclude that $\nabla \varphi \in H^1$. Finally the property $u \in L^\infty$ follows from the estimate $$ \|u\|_{L^\infty} \leq C \|\nabla u\|_{H^1( \mathbb{R}^3)}. $$ This estimate follows from the Fourier representation $$ u(x) = (2 \pi)^{-3} \int_ { \mathbb{R}^3} {\rm e}^{-i x \xi} \hat{u}(\xi) d \xi,$$ the Cauchy inequality and the fact that $$ |\xi|^{-1}(1+|\xi|)^{-1} \in L^2 (\mathbb{R}^3). $$ The Lemma is established. \end{proof} In the same way, proceeding inductively, we obtain the following result. \begin{lemma}\label{eq.Nany} Under assumption \eqref{eq.reg}, for any integer $k \geq 2$ and for any positive number $\varepsilon > 0$ we have \begin{equation}\label{eq.regNany} u \in H^k(|x| > \varepsilon) ,\quad \nabla \varphi \in H^{k-1}( |x| > \varepsilon). \end{equation} \end{lemma} To study more precisely the behavior of the solution $u(x)= u(|x|)$ we introduce polar coordinates $r=|x|$ and set \begin{equation}\label{eq.set} \mathbb{U}(r) = ru(r) , \quad \mathbb{V}(r) = - r \varphi(r). \end{equation} Using the identities $$ \Delta \big( \frac{\mathbb{U}(r)}{r}\big) = \frac{ \mathbb{U}''(r)}{r} ,\quad \Delta \big( \frac{\mathbb{V}(r)}{r} \big)= \frac{ \mathbb{V}''(r)}{r}, $$ where $\mathbb{U}^{\, \prime}(r) = \partial_r \mathbb{U}(r)$, we can rewrite \eqref{eq.M-Sep} in the form \begin{equation}\label{eq.M-Srad} \begin{gathered} - \frac{\mathbb{U}''}2 + e\frac{ \mathbb{V}}r\mathbb{U} - \frac{Z}r \mathbb{U} = \omega \mathbb{U} ,\quad r > 0, \\ - \mathbb{V}'' = 4 \pi e\frac{\mathbb{U}^2}{r}, \quad r>0. \end{gathered} \end{equation} We shall need the following result. \begin{lemma} \label{Sobvr} Let $k \geq 1$ be an integer and $\varepsilon > 0$ be a real number. We have the following properties: \begin{itemize} \item[(a)] if $ u(x) = u(|x|) \in H^k(\mathbb{R}^3)$, then $\mathbb{U}(r) \in H^k(0,\infty)$ \item[(b)] $ u(x) = u(|x|) \in H^k(|x| > \varepsilon)$, if and only if $\mathbb{U}(r) \in H^k(\varepsilon,\infty)$. \end{itemize} \end{lemma} \begin{proof} The proof of (a) follows from the relation $$ \partial_r^k \mathbb{U}(r) = r \partial_r^k u(r) + k \partial_r^{k-1} u(r)$$ valid for any integer $k \geq 1$. Note that the Hardy inequality implies $$ \int_0^\infty |\partial_r^{k-1} u(r)|^2 dr \leq C \|u\|^2_{H^k( \mathbb{R}^3)}.$$ For property (b), we can use the relation $$ \partial_r^k u(r) = \sum_{j=1}^k \frac{c_{k,j}}{r^j} \partial_r^{k-j} \mathbb{U}(r) $$ and the fact that $r^{-j}$ is bounded for $ r \geq \varepsilon>0$. \end{proof} \begin{lemma}\label{regzero} The functions $ \mathbb{U}(r), \mathbb{V}(r)$ are smooth near $r=0$. \end{lemma} \begin{proof} From $ u \in H^2$ (see Lemma \ref{eq.N2}) it follows $u \in L^\infty$, so $$ |\mathbb{U}(r)| = r |u(r)| \leq Cr $$ near $r=0$. In the same way $\varphi \in L^\infty$ (Lemma \ref{eq.N2}) implies that $$ |\mathbb{V}(r)| = r |\varphi(r)| \leq Cr $$ near $r=0$. The system \eqref{eq.M-Sep} shows that $$ |\mathbb{U}''(r)| + |\mathbb{V}''(r)| \leq C,$$ so $ \mathbb{U}(r) , \mathbb{V}(r) \in C^1([0,1])$. Setting $a_1 = \mathbb{U}'(0)$, $b_1 = \mathbb{V}'(0)$, we can make the representation $$ \mathbb{U}(r) = a_1r + \mathbb{U}_1(r), \quad \mathbb{V}(r) = b_1r + \mathbb{V}_1(r), $$ where $\mathbb{U}_1, \mathbb{V}_1 \in o(r)$ satisfy \begin{equation}\label{eq.M-Sap1} \begin{gathered} - \frac{\mathbb{U}_1''}2 + \frac{ \mathbb{V}_1}r\mathbb{U}_1 - \frac{Z}r \mathbb{U}_1 - \omega \mathbb{U}_1 = c_1 + O(r) , \quad r > 0, \\ - \mathbb{V}_1'' - 4 \pi \frac{\mathbb{U}_1^2}{r} = O(r), \quad r>0, \end{gathered} \end{equation} where $c_1=\omega a_1$. These equations imply $$ \mathbb{U}_1''(r) = c_1+ O(r),\quad \mathbb{V}_1''(r) = O(r), $$ so $$ \mathbb{U}_1(r) = \frac{c_1r^2}2 + O(r^3),\quad \mathbb{V}_1(r) = O(r^3) $$ near $r=0$ and these relations imply $\mathbb{U}_1(r) , \mathbb{V}_1(r) \in C^2([0,1])$. Continuing further we obtain inductively $$ \mathbb{U}(r) = a_1r + a_2 r^2+ \dots +a_{k}r^k + \mathbb{U}_k(r), \quad \mathbb{V}(r) = b_1r + b_2 r^2+ \dots +b_{k}r^k + \mathbb{V}_k(r). $$ Here $\mathbb{U}_k, \mathbb{V}_k \in o(r^k)$ satisfy \begin{equation}\label{eq.M-Sapk} \begin{gathered} - \frac{\mathbb{U}_k''}2 + \frac{ \mathbb{V}_k}r\mathbb{U}_k - \frac{z}r \mathbb{U}_k - \omega \mathbb{U}_k = c_k r^{k-1} + O(r^k) , \quad r > 0, \\ - \mathbb{V}_k'' - 4 \pi \frac{\mathbb{U}_k^2}{r} = \tilde{c}_k r^{k-1}+ O(r^k), \quad r>0. \end{gathered} \end{equation} These relations imply $$ \mathbb{U}_k(r) = \frac{c_k r^{k+1}}{k(k+1)} + O(r^{k+2}),\quad \mathbb{V}_k(r) = \frac{\tilde{c}_k r^{k+1}}{k(k+1)} + O(r^{k+2}) $$ near $r=0$ and these relations imply $\mathbb{U}_k(r) , \mathbb{V}_k(r) \in C^{k+1}([0,1])$. \end{proof} Our next step is to obtain the decay of the solution. We look for soliton type solutions $u$ to \eqref{eq.M-S}, i.e. very regular solutions decaying rapidly at infinity. Our next step is to obtain a very rapid decay of the radial field $u(|x|)$ at infinity. \begin{lemma} \label{solitons} If the assumption \eqref{eq.reg} is satisfied, then \begin{equation}\label{regul} \mathbb{U} \in H^k( (1,+\infty)), \quad\mathbb{V}^{\, \prime} \in H^{k-1}( (1,+\infty)),\end{equation} and \begin{equation} |\mathbb{U}^{\prime}(r)|^2+ | \mathbb{U}(r)|^2 \leq \frac{C}{r^k}, \quad \label{eq.estvnN} 0 \leq \mathbb{V}'(r) \leq \frac{C}{r^k}\end{equation} for each integer $k \geq 2, r \geq 1$. \end{lemma} \begin{proof} The Sobolev embedding and Lemma \ref{eq.Nany} imply that \begin{equation}\label{eq.radsob} \begin{gathered} \int_0^{+\infty} |\mathbb{U}(r)|^2 dr + \int_0^{+\infty} |\mathbb{U}' (r)|^2 dr \leq C\|u\|^2_{H^1( \mathbb{R}^3)}, \\ \int_0^{+\infty} |\mathbb{V}^{\, \prime}(r)|^2 dr \leq C\|\varphi\|^2_{{\bf\mathcal D}^{1,2} ( \mathbb{R}^3)}. \end{gathered} \end{equation} Note that we have used the Hardy inequality \begin{align}\label{eq.Hardy} \int_0^{+\infty} |f(r)|^2 dr \leq C \int_0^{+\infty} |f^{\, \prime}(r)|^2 \ r^2 dr \end{align} in the above estimates (see \cite[Theorem 330]{HLP} or \cite[Remark 1, Section 3.2.6]{Tr}). Hence $$ \mathbb{U} \in H^1( (0,+\infty)), \quad\mathbb{V}^{\, \prime} \in L^2( (0,+\infty)).$$ Proceeding further inductively we find \eqref{regul}. The above properties and the Sobolev embedding imply \begin{equation}\label{eq.asymp} \lim_{r \to +\infty} | \mathbb{U}(r)| = 0 ,\quad \lim_{r \to +\infty} |\mathbb{U}^{\, \prime}(r)| = 0, \end{equation} In a similar way we get \begin{equation}\label{eq.asympV} \lim_{r \to +\infty} |\mathbb{V}^{\, \prime}(r)| = 0. \end{equation} We can improve the last property. Indeed, integrating the second equality in \eqref{eq.M-Srad} we find \begin{equation} \label{eq.estv} \mathbb{V}'(r) = \int_r^\infty \frac{ \mathbb{U}^2(\tau)}{\tau} d\tau .\end{equation} Since \begin{equation}\label{eq.uo} \int_r^\infty \mathbb{U}^2(\tau) d\tau \leq C, \end{equation} we get \begin{equation} \label{eq.estvn} 0 \leq \mathbb{V}'(r) \leq \frac{C}r.\end{equation} Our next step is to obtain weighted Sobolev estimates. From the first equation in \eqref{eq.M-Srad} we have \begin{equation}\label{dua} \begin{gathered} \frac{\mathbb{U}''}2(r) + \omega \mathbb{U}(r) = \mathbb{F}(r), \\ \mathbb{F}(r) = \frac{ \mathbb{V}}r\mathbb{U} - \frac{Z}r \mathbb{U}. \end{gathered} \end{equation} Since the initial data for $ \mathbb{U}$ are \begin{equation}\label{eq.indata} \mathbb{U}(0) = 0 , \quad \mathbb{U}'(0) = a_1, \end{equation} we have the following integral equation satisfied by $\mathbb{U}$ \begin{equation}\label{eq.U} \mathbb{U}(r) = \sinh (\sqrt{ -2\omega} r) a_1 + \int_0^r \sinh(\sqrt{ -2\omega} (r-\rho)) \mathbb{F}(\rho) d \rho. \end{equation} It is easy to see that the function $ \mathbb{F}$ satisfies the estimate \begin{equation}\label{eq.F} \mathbb{F}(r) = O(r^{-1}) , \ \ r \geq 1. \end{equation} Then the condition \eqref{eq.asymp} and simple qualitative study of the integral equation in \eqref{eq.U} guarantees that $$ a_1 + \int_0^\infty {\rm e}^{\sqrt{ -2\omega} \rho} \mathbb{F}(\rho) d \rho = 0. $$ This fact enables one to represent $\mathbb{U}$ as follows \begin{eqnarray}\label{eq.Uneq} \mathbb{U}(r) = {\rm e}^{-\sqrt{ -2\omega} r} a_1 - \\ - \int_r^\infty {\rm e}^{\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho - \int_0^r {\rm e}^{-\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho. \end{eqnarray} The first term in the right side of \eqref{eq.Uneq} is exponentially decaying. The second term we can represent as the following sum $$ \int_r^{2r} {\rm e}^{\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho + \int_{2r}^{\infty} {\rm e}^{\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho . $$ It is clear that $$ \int_{2r}^{\infty} {\rm e}^{\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d\rho $$ is decaying exponentially, while $$ \int_r^{2r} {\rm e}^{\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho \leq \frac{C}r \int_r^{2r} {\rm e}^{\sqrt{ -2\omega} (r-\rho)}d \rho = \frac{C_1}r $$ due to \eqref{eq.F}. In a similar way we can treat the last term $$ \int_0^r {\rm e}^{-\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho $$ in \eqref{eq.Uneq}. This term now is a sum of type $$\int_0^{r/2} {\rm e}^{-\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho + \int_{r/2}^r {\rm e}^{-\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho. $$ The term $$\int_0^{r/2} {\rm e}^{-\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d\rho $$ decays exponentially in $r$ and the property \eqref{eq.Uneq} implies that $$ \int_{r/2}^r {\rm e}^{-\sqrt{ -2\omega} (r-\rho)} \mathbb{F}(\rho) d \rho = O( r^{-1}). $$ The above observation and \eqref{eq.Uneq} implies \begin{gather*} \mathbb{U} = O(r^{-1})\,,\\ \mathbb{F}(r) = \frac{ \mathbb{V}}r\mathbb{U} - \frac{Z}r \mathbb{U} = O(r^{-2}). \end{gather*} This estimate implies a stronger version of \eqref{eq.uo} \begin{equation}\label{eq.uq1} \int_r^\infty \mathbb{U}^2(\tau) d\tau \leq \frac{C}r, \end{equation} and from \eqref{eq.estv} we improve \eqref{eq.estvn} as follows \begin{equation} \label{eq.estvn1} 0 \leq \mathbb{V}'(r) \leq \frac{C}{r^2}. \end{equation} This argument shows that combining \eqref{eq.estvn} and \eqref{dua} we can obtain inductively \begin{gather}\label{eq.uN} \sum_{j=0}^k|\mathbb{U}^{(j)}(r)|^2 \leq \frac{C}{r^n}\,,\\ \label{eq.estvnNa} \sum_{j=1}^k|\mathbb{V}^{(j)}(r)|^2 \leq \frac{C}{r^n} \end{gather} for any integers $k \geq 1$ and $n \geq 2$. \end{proof} The proof of Theorem \ref{thmain2} is an immediate consequence of Lemmas \ref{solitons} and (\ref{eq.Nany}), with the change of variables (\ref{eq.set}). \section{Proof of Theorem \ref{thmain3}} \label{sec:7} Define the functional \begin{equation} I(u,\omega):= \frac{1}4 \int_{\mathbb{R}^3}\vert\nabla u\vert^2\,dx+ \pi e^2\int_{\mathbb{R}^3}\vert\nabla\Delta^{-1} u^2\vert^2\,dx - \frac{1}2 \int_{\mathbb{R}^3} V(x) |u|^2 \,dx-\frac{\omega}2 \int_{\mathbb{R}^3} |u|^2 \,dx, \label{funz.2var} \end{equation} for each $(u,\omega)\in H_r^1(\mathbb{R}^3)\times \mathbb{R}$. There results \begin{gather*} \frac{\partial I}{\partial u}(u,\omega) = -\frac{1}{2} \Delta u-4\pi e^2(\Delta^{-1} u^2) u - V(x)u -\omega u,\\ \frac{\partial I}{\partial \omega}(u,\omega) =-\frac{1}2 \int_{\mathbb{R}^3} |u|^2 \,dx,\\ \frac{\partial^2 I}{\partial u^2}(u,\omega)h = -\frac{1}2 \Delta h-4\pi(\Delta^{-1} u^2) h -8\pi e^2\Delta^{-1}( hu) u - V(x)h -\omega h, h\in H^1_r(\mathbb{R}^3),\\ \frac{\partial^2 I}{\partial u\partial \omega}(u,\omega)=-u,\\ \frac{\partial^2 I}{\partial \omega^2}(u,\omega)=0. \end{gather*} Let $\nabla I:H^1_r(\mathbb{R}^3) \times \mathbb{R}\to \big(H^1_r(\mathbb{R}^3)\big)' \times \mathbb{R}$, $$ \nabla I(u,\omega)=\begin{pmatrix} \displaystyle{\frac{\partial I}{\partial u}(u,\omega)}\\[7pt] \displaystyle{\frac{\partial I}{\partial \omega}(u,\omega)}\end{pmatrix}$$ be the Jacobian matrix of $I$ and $H I(u,\omega):H^1_r(\mathbb{R}^3) \times \mathbb{R}\to \big(H^1_r(\mathbb{R}^3)\big)' \times \mathbb{R}$, $$ H I(u,\omega)=\begin{pmatrix} \displaystyle{\frac{\partial^2 I}{\partial u^2}(u,\omega)} & {\frac{\partial^2 I}{\partial u\partial \omega}(u,\omega)}\\[7pt] \displaystyle{\frac{\partial^2 I}{\partial u\partial \omega}(u,\omega)}& {\frac{\partial^2 I}{\partial \omega^2}(u,\omega)} \end{pmatrix} $$ be the Hessian matrix of $I$ in $(u,\omega)$. More precisely \begin{equation} \label{hess} \begin{aligned} &H I(u,\omega)(h,k)\\ &=\begin{pmatrix} \displaystyle{\frac{\partial^2 I}{\partial u^2}(u,\omega)}h+ {\frac{\partial^2 I}{\partial u\partial \omega}(u,\omega)}k \\[7pt] \displaystyle {\frac{\partial^2 I}{\partial u\partial \omega}(u,\omega)}h+ {\frac{\partial^2 I}{\partial \omega^2}(u,\omega)}k\end{pmatrix}\\ &=\begin{pmatrix}\displaystyle-\frac{1}2 \Delta h-4\pi(\Delta^{-1} u^2) h -8\pi e^2(\Delta^{-1}( hu)) u - V(x)h -\omega h -k u \\[7pt] \displaystyle-\int_{\mathbb{R}^3}uh \,dx\end{pmatrix}, \end{aligned} \end{equation} for each $u, h \in H^1_r(\mathbb{R}^3)$ and $k, \omega \in \mathbb{R}$. Finally denote \begin{equation} B':= B\cap H^1_r(\mathbb{R}^3). \label{B'} \end{equation} \begin{lemma}\label{lemma71} Let $u_0\in B'$ (see (\ref{B'})) be a critical point of $ J\big|_{B'} $ that corresponds to the minimum \begin{equation}\label{minimum} \omega_0 = \inf_{u \in H^1 \setminus \{0\}, \|u\|^2_{L^2}=N} J(u),\end{equation} namely $$ 0= J\big|_{B'}' (u_0)= J\big|_{B}' (u_0)= J' (u_0)-\omega_0 u_0.$$ The operator $$ h\in \Big\{ h \in H^1_r(\mathbb{R}^3); \int h(x) u_0(x)\,dx =0\Big\} \longmapsto \frac{\partial^2 I}{\partial u^2}(u_0,\omega_0)h\in \big(H^1_r(\mathbb{R}^3)\big)' $$ has a trivial kernel and \begin{equation} \big\langle \frac{\partial^2 I}{\partial u^2}(u_0,\omega_0)h\big\vert h \big\rangle =0,\quad \int h(x) u_0(x) \,dx =0 \Longrightarrow h\equiv 0. \label{inv} \end{equation} \end{lemma} \begin{proof} Repeating the qualitative argument in the proof of Lemma \ref{solitons}, we see that any solution of $$ \frac{\partial^2 I}{\partial u^2}(u_0,\omega_0)h = 0 $$ decays rapidly at infinity and it is smooth as a function of $r \geq 0$. Another interpretation of the first eigenvalue $\omega_0 = \omega(N) < 0$ is the following one \begin{equation}\label{eq.varset} \omega_0 = N\inf_{u \in H^1 \setminus \{0\}} \frac{J(u)}{\|u\|^2_{L^2}}. \end{equation} Let $$ \Big\langle \frac{\partial^2 I}{\partial u^2}(u_0,\omega_0)h\big\vert h \Big\rangle =0 $$ for some $h \in H^1$ orthogonal (in $L^2$) to $u_0$. Take $u_0 + i\varepsilon h$ with $\varepsilon > 0$ small enough (will be chosen later on). Then a simple calculation implies $$ \frac{J(u_0+i\varepsilon h)}{\|u_0+i\varepsilon h\|^2_{L^2}} = \frac{J(u_0) + o(\varepsilon^2)}{\|u_0\|^2 - \varepsilon^2 \|h\|^2} = \frac{J(u_0)}{N} + \varepsilon^2 \frac{\|h\|^2}{N} J(u_0) + o(\varepsilon^2). $$ Hence, the assumption $\|h\| \neq 0$ will contradict the fact that $\omega_0$ is defined as the minimum in \eqref{eq.varset}. This completes the proof.\end{proof} \begin{lemma}\label{lemma72} Let $u_0\in B'$ (see (\ref{B'})) be a critical point of $ J\big|_{B'} $ that corresponds to the minimum as in the previous Lemma. The operator $$(h, k)\in H^1_r(\mathbb{R}^3)\times \mathbb{R}\mapsto HI(u_0,\omega_0)(h, k)\in \big(H^1_r(\mathbb{R}^3)\big)'\times \mathbb{R} $$ is invertible. \end{lemma} \begin{proof} Let $u_0 \in B'$ be a critical point of $ J\big|_{B'} $ with multiplier $\omega_0$ as in the previous lemma, call $$ A:= \frac{\partial^2 I}{\partial u^2}(u_0,\omega_0). $$ We begin proving that $HI(u_0,\omega_0)$ is injective. Let $h\in H^1_r(\mathbb{R}^3)$ and $k\in \mathbb{R}$ such that \begin{equation}\label{inj} HI(u_0,\omega_0)(h, k)=0, \end{equation} we have to prove that \begin{equation}\label{inj1} h= k= 0. \end{equation} By (\ref{hess}) and (\ref{inj}), we have \begin{equation}\label{inj2} Ah-k u_0=0,\quad -\int_{\mathbb{R}^3} u_0 h \,dx=0. \end{equation} Multiplying the first of (\ref{inj2}) by $h$ and integrating on $\mathbb{R}^3$, we have $$\int_{\mathbb{R}^3}\big(A h\big)h \,dx=- k\int_{\mathbb{R}^3}u_0 h \,dx =0,$$ and by (\ref{inv}) and the definition of $A$ \begin{equation} \label{inj3} h\equiv 0.\end{equation} On the other hand, multiplying the first of (\ref{inj2}) by $u_0$ and integrating on $\mathbb{R}^3$, since $u_0\in B'$, we have \begin{equation}\label{inj4} kN=k \int_{\mathbb{R}^3}u_0^2\,dx= \int_{\mathbb{R}^3}\big(A h\big)u_0 \,dx=0. \end{equation} Since (\ref{inj1}) is direct consequence of (\ref{inj3}) and (\ref{inj4}), $HI(u_0, \omega_0)$ is injective. We prove that $HI(u_0, \omega_0)$ is surjective. Observe that the operator $A$ is selfadjoint, indeed \begin{align*} (Ah, f)_{L^2}&= \frac{1}2\int_{\mathbb{R}^3} (\nabla h, \nabla f)\,dx -4\pi e^2\int_{\mathbb{R}^3}(\Delta^{-1} u^2) hf\,\,dx \\ &\quad +8\pi\int_{\mathbb{R}^3}\big(\nabla\Delta^{-1}( hu),\nabla\Delta^{-1}(f u)\big)\,dx -\int_{\mathbb{R}^3} V(x)hf\,\,dx -\omega\int_{\mathbb{R}^3} hf\,\,dx, \end{align*} for each $h, f$ in $H^1_r(\mathbb{R}^3)$. Moreover, also the operator $HI(u_0, \omega_0)$ is selfadjont, indeed \begin{align*} \big(HI(u_0, \omega_0)(h,k),(f,\alpha)\big)_{L^2\times\mathbb{R}} &=\Big(\big(Ah-ku_0, -(u_0,h)_{L^2}\big), \big(f,\alpha\big)\Big)_{L^2\times\mathbb{R}} \\ &=\big(Ah-ku_0,f\big)_{L^2}-\alpha(u_0,h)_{L^2}\\ &=\big(Ah,f\big)_{L^2}-k\big(u_0,f\big)_{L^2}-\alpha(u_0,h)_{L^2} \end{align*} and \begin{align*} \big(HI(u_0, \omega_0)(f,\alpha),(h,k)\big)_{L^2\times\mathbb{R}} &= \Big(\big(Af-\alpha u_0, -(u_0,f)_{L^2}\big),\big(h,k\big) \Big)_{L^2\times\mathbb{R}}\\ &=\big(Af-\alpha u_0,h\big)_{L^2}-k(u_0,f)_{L^2}\\ &=\big(Af,h\big)_{L^2}-\alpha(u_0,h)_{L^2}-k\big(u_0,f\big)_{L^2}, \end{align*} since $A$ is selfadjoint $$ \big(HI(u_0, \omega_0)(h,k),(f,\alpha)\big)_{L^2\times\mathbb{R}} =\big(HI(u_0,\omega_0)(f,\alpha),(h,k)\big)_{L^2\times\mathbb{R}} $$ for each $h, f$ in $H^1_r(\mathbb{R}^3)$ and $k, \alpha$ in $\mathbb{R}$. Since $HI(u_0, \omega_0)$ is injective and selfadjoint, there results \begin{equation} \label{surj} \begin{aligned} \mathop{\rm Im} \big(HI(u_0, \omega_0)\big) &=\Big(\ker \big(HI(u_0, \omega_0)^*\big)\Big)^{\perp}\\ &=\Big(\ker \big(HI(u_0,\omega_0)\big)\Big)^{\perp}\\ &=H^1_r(\mathbb{R}^3)\times \mathbb{R}, \end{aligned} \end{equation} then $HI(u_0, \omega_0)$ is surjective. The claim is direct consequence of the Closed Graph Theorem. \end{proof} \begin{lemma}\label{lemma73} The critical points of the functional $ J\big|_{B'} $ that correspond to the minimum are isolated, i.e. for each $u\in B'$ critical point of $ J\big|_{B'} $, with the Lagrange multiplier satisfying (\ref{minimum}), there exists a neighborhood $U\subset H^1(\mathbb{R}^3)$ of $u$ such that any element of $B'\cap U$ is not a critical point of it. \end{lemma} \begin{proof} Let $u_0\in B'$ be a critical point of $ J\big|_{B'}$ corresponding to the minimum as in the previous lemmas, then $$0= J\big|_{B'}' (u_0)= J\big|_{B}' (u_0)= J' (u_0)-\omega_0 u_0= \frac{\partial I}{\partial u}(u_0, \omega_0) $$ and since $u_0\in B'$, $$ \frac{\partial I}{\partial \omega}(u_0, \omega_0) =-\frac{1}2 \int_{\mathbb{R}^3}u_0^2 \,dx= -\frac{N}2, $$ we have $$ \nabla I (u_0, \omega_0)=\begin{pmatrix}0\\ -N/2 \end{pmatrix}. $$ By Lemma \ref{lemma72} and the Implicit Function Theorem there exist $U\subset H^1_r(\mathbb{R}^3) $ neighborhood of $u_0$, $\Omega\subset \mathbb{R} $ neighborhood of $\omega_0$, $W\subset \big(H^1_r(\mathbb{R}^3)\big)'\times\mathbb{R} $ neighborhood of $\big(0, -\frac{N}2\big)$ and $G: W\to U\times \Omega$ such that \begin{equation} \label{730} \begin{gathered} G\big(\nabla I (u, \omega)\big)=(u, \omega), \quad (u,\omega)\in U\times\Omega,\\ \nabla I \big(G(f, \alpha )\big) =(f, \alpha), \quad (f, \alpha)\in W. \end{gathered} \end{equation} Assume, by absurd, that $u_0$ is not isolate, namely there exists a sequence $\{u_k\}\subset B'$ of critical points of $ J\big|_{B'} $, such that \begin{equation}\label{731} u_k\not= u_0,\quad u_k\to u_0\quad {\rm in } H^1(\mathbb{R}^3). \end{equation} Moreover, there exists a sequence $\{\omega_k\}\subset \mathbb{R}$ such that $$ 0= J\big|_{B'}'(u_k) = J'(u_k)-\omega_k u_k =\frac{\partial I}{\partial u}(u_k, \omega_k). $$ Since $u_k\in B'$ and by (\ref{731}), we have \begin{equation}\label{732} \omega_k = \big\langle J'(u_k)\big |u_k\big\rangle \to \big\langle J'(u_0)\big |u_0\big\rangle=\omega_0. \end{equation} By (\ref{731}) and (\ref{732}), there exists $k_0\in \mathbb{N}$ such that $(u_k, \omega_k)\in U\times \Omega$ for $k\ge k_0$. Finally, fixed $k\ge k_0$, since $$ \nabla I (u_k, \omega_k)=\begin{pmatrix}0\\ -N/2 \end{pmatrix}, $$ by (\ref{730}), we have $$(u_k, \omega_k) =G\big(\nabla I (u_k, \omega_k)\big) = G\begin{pmatrix}0\\ -N/2 \end{pmatrix}=G\big(\nabla I (u_0, \omega_0)\big)=(u_0, \omega_0). $$ Since this contradicts (\ref{731}), the claim is done. \end{proof} \begin{lemma}\label{lemma74} The first eigenvalue of the operator $ J\big|_{B'}' $ (see (\ref{minimum})) is isolated, i.e. there exists a neighborhood $\Omega\subset\mathbb{R}$ of $\omega_0$ such that any element of $\Omega$ is not an eigenvalue of the previous operator. \end{lemma} \begin{proof} Assume, by absurd, that the first eigenvalue $\omega_0$ is not isolated, namely there exists a sequence $\{\omega_k\}\subset\mathbb{R}$ of eigenvalues such that \begin{equation}\label{740} \omega_k\to \omega_0.\end{equation} By definition, there exists $\{u_k\}\subset B'$ such that \begin{equation}\label{741} 0=J\big|_{B'}'(u_k)= J'(u_k)-\omega_k u_k,\quad k\in \mathbb{N}.\end{equation} Observe that, by Lemma \ref{Lemmaag}, $\omega_k, \omega_0<0$, then there exists $\varepsilon >0$ such that \begin{equation}\label {7411} \omega_k, \omega_0 \le -\varepsilon ,\quad k\in \mathbb{N} .\end{equation} Moreover, by Lemma \ref{lemma341} and since $\{u_k\}\subset B'$ \begin{equation}\label {742} -\infty < \min_{u\in H^1_r(\mathbb{R}^3)}J(u) \le J(u_k)\le \sup _{k} \frac{\omega_k}2 \le -\frac{\varepsilon}2,\end{equation} then $\{J(u_k)\}$ is bounded and, by (\ref{741}), \begin{equation}\label{743} J\big|_{B'}'(u_k)\to 0. \end{equation} By the Palais-Smale Condition (see Lemma \ref{lemma33}) there exists $u_0 \in B'$ such that, passing to a subsequence, $$u_k\to u_0,\quad {\rm in} H^1(\mathbb{R}^3). $$ By (\ref{740}) and (\ref{741}), $$0=J\big|_{B'}'(u_0)= J'(u_0)-\omega_0 u_0, $$ namely $u_0$ is a not isolated critical point of the functional $J\big|_{B'}$. Since this contradicts Lemma \ref{lemma73}, the proof is done. \end{proof} \begin{proof}[Proof of Theorem \ref{thmain3}] Since $F (u, 4 \pi \Delta^{-1} u^2 ) = J (u)$ for all $ u \in H^1 (\mathbb{R}^3)$, by Lemmas \ref{lemma73} and \ref{lemma74} the claim is complete.\end{proof} \section{ Appendix}\label{sect:8} Here we shall prove for completeness the relation \eqref{eq.Lions}. First, for the partial case of space dimensions $n=3$ we need the following relation (a generalization of this relation for space dimensions $n \geq 3$ can be found in \cite{AKT}). \begin{lemma}[see \cite{AKT}] \label{l.Lio1} If $f(x)=f(|x|)$ is an $L^\infty( \mathbb{R}^3)$ function, then for any $r>0$ and $ x \neq 0$ we have the relation \begin{equation}\label{eq.gap} \int_{ \mathbb{S}^2} f(|x+r\omega|) d \omega = \frac{2\pi}{|x|r} \int_{||x|-r|}^{|x|+r} f(\lambda) \lambda d\lambda. \end{equation} \end{lemma} \begin{proof} It is sufficient to consider only the case $x = (0,0,|x|)$ and to pass to polar coordinates $$ \omega_1= \sin \theta \cos \varphi \,, \quad \omega_2=\sin \theta \sin \varphi\,, \quad \omega_3= \cos \theta\,. $$ Then $d\omega = \sin \theta \, d\theta \, d\varphi$ and $$ \int_{ \mathbb{S}^2} f(|x+r\omega|) d \omega = 2\pi \int_0^\pi f \big(\sqrt{|x|^2+r^2 + 2|x|r \cos\, \theta} \big)\, \sin \theta \, d\theta. $$ Making the change of variable $$ \theta \to \lambda = \sqrt{|x|^2+r^2 + 2|x|r \cos\, \theta} ,$$ the proof is complete. \end{proof} Now we are ready to verify \eqref{eq.Lions}. \begin{lemma}\label{l.Lio} If $v(x)=v(|x|)$ is a radial $C_0^\infty( \mathbb{R}^3)$ function, then the solution of the equation $\Delta u = v$ can be represented as follows \begin{equation}\label{eq.maxgap} 4 \pi u(x) = - \int_{ \mathbb{R}^3} v(|y|) \frac{d y}{\max (|x|,|y|)}, \quad x\in \mathbb{R}^3. \end{equation} \end{lemma} \begin{proof} Starting with the classical representation $$ 4 \pi u(x) = \int_{ \mathbb{R}^3} |x-y|^{-1} v(|y|) \,dy, $$ we introduce polar coordinates $ r = |y|$, $\omega = y/|y|$ apply Lemma \ref{l.Lio1} and find $$ u(x) = - \frac{1}{2|x|} \int_0^\infty \Big(\int_{||x|-r|}^{|x|+r} d \lambda \Big) v(r) r \,dr\,. $$ Note that the right side of \eqref{eq.maxgap} becomes $$ - 4\pi \int_0^\infty v(r) \frac{r^2 d r}{\max(|x|,r)}. $$ Using the fact that $$ \frac{1}{|x|r} \int_{||x|-r|}^{|x|+r} d \lambda = \frac{2}{\max (|x|,r)},$$ we obtain \eqref{eq.maxgap} and this completes the proof. \end{proof} Using the relation $$ 4 \pi u(x) = - \int_{0}^r v(\rho) \frac{\rho^2 d \rho}{r} - \int_{r}^\infty v(\rho) \rho d \rho ,\quad r = |x|$$ and differentiating with respect to $r=|x|$, we arrive at the following lemma. \begin{lemma} \label{l.Lio1a} If $v(x)=v(|x|)$ is a radial $C_0^\infty( \mathbb{R}^3)$ function, then the solution of the equation $\Delta u = v$ satisfies the relation \begin{equation}\label{eq.gapa} 4\pi \frac{\partial \Delta^{-1} v}{\partial r}(x) = \int_{|y| < r}\frac{v(y)}{|x|^2}\,dy, \end{equation} for each $x\in \mathbb{R}^3, x\neq 0$. \end{lemma} \subsection*{Acknowledgments} The authors would like to thank Dr. Simone Secchi and Dr. Nicola Visciglia for the critical remarks and stimulating discussions. \begin{thebibliography}{00} \bibitem{AKT} Agemi R., Kubota K., Takamura H.: \emph{On certain integral equations related to nonlinear wave equations}. 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