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\markboth{\hfil Nonlinear Klein-Gordon equations \hfil EJDE--2002/26}
{EJDE--2002/26\hfil Pietro d'Avenia \&  Lorenzo Pisani  \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 26, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations
 %
\thanks{ {\em Mathematics Subject Classifications:} 58E15, 35J50, 35Q40, 35Q60.
\hfil\break\indent 
{\em Key words:} Nonlinear Klein-Gordon equation, solitary waves, 
electromagnetic field, variational methods.  \hfil\break\indent 
\copyright 2002 Southwest Texas
State University. \hfil\break\indent 
Submitted November 14, 2001. Published March 4, 2002.
\hfil\break\indent 
Sponsored by M.I.U.R. (ex 40\% and ex 60\% funds).\hfil\break\indent
 P. d'Avenia is also sponsored by European Social Fund.} }
\date{}
%
\author{Pietro d'Avenia \&  Lorenzo Pisani}
\maketitle

\begin{abstract}
  In this paper we prove the existence of infinitely many 
  radially symmetric  standing waves in equilibrium with their own
  electro-magnetic field.  The interaction is described by means 
  of the minimal coupling rule; on the other hand the Lagrangian
  density for the electro-magnetic field is the second order 
  approximation of the Born-Infeld Lagrangian density.
\end{abstract}

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\section{Electrically charged fields}

Let us consider the nonlinear Klein-Gordon equation
\begin{equation}
\psi_{tt}-\Delta\psi+m_{0}^{2}\psi-\left|  \psi\right|  ^{p-2}\psi
=0,\label{110}
\end{equation}
where $\psi=\psi\left(  x,t\right)  \in\mathbb{C}$, $x\in\mathbb{R}^{3}$,
$t\in\mathbb{R}$, $m_{0}$ is a real constant and $p>2$.
It is known that (\ref{110}) can be used to develop the theory of electrically
charged fields (see \cite{fel}). Of course we can study the interaction of
$\psi$ with its own electro-magnetic field as it has been done in \cite{bf}.
As usual, the electro-magnetic field is described by the gauge potential
$\left(\phi,\mathbf{A}\right)  $
$$
\phi  :\mathbb{R}^{3}\times\mathbb{R}\to \mathbb{R},\quad
\mathbf{A}   :\mathbb{R}^{3}\times\mathbb{R}\to\mathbb{R}^{3}.
$$
Indeed, from $\left(  \phi,\mathbf{A}\right)  $, we obtain the electric field
\begin{equation}
\mathbf{E}=-\nabla\phi-\mathbf{A}_{t}\label{112}
\end{equation}
and the magnetic induction field
\begin{equation}
\mathbf{B}=\nabla\times\mathbf{A.}\label{113}
\end{equation}

The interaction of $\psi$ with the electro-magnetic field is described by the
minimal coupling rule, that is the formal substitution
\begin{gather}
\frac{\partial}{\partial t}  \longmapsto\frac{\partial}{\partial t}
+ie\phi\label{115}\\
\nabla  \longmapsto\nabla-ie\mathbf{A}\label{116}
\end{gather}
where $e$ is the electric charge.

We recall that (\ref{110}) is the Euler-Lagrange equation with respect to the
Lagrangian density
\begin{equation}
\mathcal{L}_{\mathrm{NLKG}}=\frac{1}{2}\Big[  \big|  \frac{\partial\psi
}{\partial t}\big|  ^{2}-\left|  \nabla\psi\right|  ^{2}-m_{0}^{2}\left|
\psi\right|  ^{2}\Big]  +\frac{1}{p}\left|  \psi\right|  ^{p}.\label{120}
\end{equation}
Then, by (\ref{115}) and (\ref{116}), the Lagrangian density (\ref{120})
becomes
\[
\mathcal{L}_{0}=\frac{1}{2}\Big[  \big|  \frac{\partial\psi}{\partial
t}+ie\phi\psi\big|  ^{2}-\left|  \nabla\psi-ie\mathbf{A}\psi\right|
^{2}-m_{0}^{2}\left|  \psi\right|  ^{2}\Big]  +\frac{1}{p}\left|
\psi\right|  ^{p}.
\]
The total action of the system is the sum
\begin{equation}
\mathcal{S}=\iint\left(  \mathcal{L}_{0}+\mathcal{L}_{\mathrm{e.m.f.}}\right)
dxdt\label{125}
\end{equation}
where $\mathcal{L}_{\mathrm{e.m.f.}}$ is the Lagrangian density of the
electro-magnetic field.
In the classical Maxwell theory, with a suitable choice of constants, we have
\[
\mathcal{L}_{\mathrm{e.m.f.}}=\mathcal{L}_{\text{\textrm{M}}}=\frac{1}{8\pi
}\left(  \left|  \mathbf{E}\right|  ^{2}-\left|  \mathbf{B}\right|
^{2}\right)  .
\]
The existence of infinitely many solutions for the Euler-Lagrange equations
associated to
\[
\mathcal{S}=\iint\left(  \mathcal{L}_{0}+\mathcal{L}_{\text{\textrm{M}}
}\right)  dxdt
\]
has been proved by Benci and Fortunato in \cite{bf}.

We recall that Maxwell equations coupled with Schr\"{o}dinger or Dirac
equations have been studied respectively in \cite{bf98}, \cite{coc1},
\cite{coc2} and in \cite{egs}.
It is well known that the classical theory has two difficulties arising from
the divergence of energy (see the first section of \cite{fop}). An attempt to
avoid this divergence is the Born-Infeld theory, where
\begin{equation}
\mathcal{L}_{\mathrm{BI}}=\frac{b^{2}}{4\pi}\Big(  1-\sqrt{1-\frac{1}{b^{2}
}\big(  \left|  \mathbf{E}\right|  ^{2}-\left|  \mathbf{B}\right|
^{2}\big)  }\Big) \label{130}
\end{equation}
being $b\gg1$ the so-called Born-Infeld parameter (see \cite{bi}).

In \cite{fop}, the authors consider the second order expansion of (\ref{130})
for
\[
\beta=\frac{1}{2b^{2}}\to 0.
\]
They obtain the Lagrangian density
\[
\mathcal{L}_{1}=\frac{1}{4\pi}\left[  \frac{1}{2}\left(  \left|
\mathbf{E}\right|  ^{2}-\left|  \mathbf{B}\right|  ^{2}\right)  +\frac{\beta
}{4}\left(  \left|  \mathbf{E}\right|  ^{2}-\left|  \mathbf{B}\right|
^{2}\right)  ^{2}\right]
\]
and they prove some existence results of finite-energy electrostatic solutions.

In this paper we consider
\[
\mathcal{L}_{\mathrm{e.m.f.}}=\mathcal{L}_{1}.
\]
So the total action we study is
\begin{equation}
\mathcal{S}=\iint\left(  \mathcal{L}_{0}+\mathcal{L}_{1}\right)
dxdt.\label{140}
\end{equation}
As in \cite{bf}, we consider
\[
\psi\left(  x,t\right)  =u\left(  x,t\right)  e^{iS\left(  x,t\right)  }
\]
with $u,S\in\mathbb{R}$; therefore
\[
\mathcal{S=S}\left(  u,S,\phi,\mathbf{A}\right)
\]
and the explicit expression of the Lagrangian densities is
\begin{align*}
\mathcal{L}_{0}  & =\frac{1}{2}\left\{  u_{t}^{2}-\left|  \nabla u\right|
^{2}-\left[  \left|  \nabla S-e\mathbf{A}\right|  ^{2}-\left(  S_{t}
+e\phi\right)  ^{2}+m_{0}^{2}\right]  u^{2}\right\}  +\frac{1}{p}\left|
u\right|  ^{p},\\
\mathcal{L}_{1}  & =\frac{1}{4\pi}\left[  \frac{1}{2}\left(  \left|
\mathbf{A}_{t}+\nabla\phi\right|  ^{2}-\left|  \nabla\times\mathbf{A}\right|
^{2}\right)  +\frac{\beta}{4}\left(  \left|  \mathbf{A}_{t}+\nabla\phi\right|
^{2}-\left|  \nabla\times\mathbf{A}\right|  ^{2}\right)  ^{2}\right]  .
\end{align*}

The Euler-Lagrange equations associated to (\ref{140}) are
\begin{gather}
d\mathcal{S}\left[  \delta u\right]    =0\label{210a}\\
d\mathcal{S}\left[  \delta S\right]    =0\label{210b}\\
d\mathcal{S}\left[  \delta\phi\right]  =0\label{210c}\\
d\mathcal{S}\left[  \delta\mathbf{A}\right]  =0.\label{210d}
\end{gather}
By standard calculations, we get
\begin{gather}
\Box u+\left[  \left|  \nabla S-e\mathbf{A}\right|  ^{2}-\left(  S_{t}
+e\phi\right)  ^{2}+m_{0}^{2}\right]  u-\left|  u\right|  ^{p-2}
u=0\label{220a} \\
\frac{\partial}{\partial t}\left[  \left(  S_{t}+e\phi\right)  u^{2}\right]
-\nabla\left[  \left(  \nabla S-e\mathbf{A}\right)  u^{2}\right]
=0\label{220b} \\
\nabla\cdot\left[  \left(  1+\beta\left|  \mathbf{A}_{t}+\nabla\phi\right|
^{2}-\beta\left|  \nabla\times\mathbf{A}\right|  ^{2}\right)  \left(
\mathbf{A}_{t}+\nabla\phi\right)  \right]  =4\pi e\left(  S_{t}+e\phi\right)
u^{2}\label{220c}
\end{gather}
\begin{equation}
\begin{aligned}
\frac{\partial}{\partial t}\left[  \left(  1+\beta\left|  \mathbf{A}
_{t}+\nabla\phi\right|  ^{2}-\beta\left|  \nabla\times\mathbf{A}\right|
^{2}\right)  \left(  \mathbf{A}_{t}+\nabla\phi\right)  \right] &\\
+\nabla\times\left[  \left(  1+\beta\left|  \mathbf{A}_{t}+\nabla\phi\right|
^{2}-\beta\left|  \nabla\times\mathbf{A}\right|  ^{2}\right)  \left(
\nabla\times\mathbf{A}\right)  \right]  &=4\pi e\left(  \nabla S-e\mathbf{A}
\right)  u^{2}.
\end{aligned}
\label{220d}
\end{equation}

 From the physical point of view, it may be interesting to introduce the
following notation: we set
\begin{gather}
\rho=-e\left(  S_{t}+e\phi\right)  u^{2},\label{222}
\\
\mathbf{J}=e\left(  \nabla S-e\mathbf{A}\right)  u^{2}.\label{223}
\end{gather}

Taking into account (\ref{112}) and (\ref{113}), the equations (\ref{220b}),
(\ref{220c}) and (\ref{220d}) become respectively
\begin{gather}
\frac{\partial\rho}{\partial t}+\nabla\mathbf{J}=0\label{225} \\
\nabla\left[  \left(  1+\beta\left(  \left|  \mathbf{E}\right|  ^{2}-\left|
\mathbf{B}\right|  ^{2}\right)  \right)  \mathbf{E}\right]  =4\pi
\rho\label{226} \\
\nabla\times\left[  \left(  1+\beta\left(  \left|  \mathbf{E}\right|
^{2}-\left|  \mathbf{B}\right|  ^{2}\right)  \right)  \mathbf{B}\right]
-\frac{\partial}{\partial t}\left[  \left(  1+\beta\left(  \left|
\mathbf{E}\right|  ^{2}-\left|  \mathbf{B}\right|  ^{2}\right)  \right)
\mathbf{E}\right]  =4\pi\mathbf{J}.\label{227}
\end{gather}
We notice that (\ref{222}) and (\ref{223}) are good definitions respectively
of charge density and current density, indeed the continuity equation
(\ref{225}) is satisfied.

Equations (\ref{226}) and (\ref{227}) are formally identical to equations (24)
and (25) of \cite{fop}, indeed they replace the second pair of Maxwell
equations when we use the second order approximation of the Born-Infeld
Lagrangian density.

\section{Statement of the main result}

In this paper we look for solutions of (\ref{220a})-(\ref{220d}) such that
\begin{gather}
u   =u\left(  x\right)  ,\nonumber\\
S   =\omega t,\nonumber\\
\phi  =\phi\left(  x\right)  ,\label{228}\\
\mathbf{A}   =\mathbf{0}.\label{229}
\end{gather}
We recall that solutions
\[
\psi\left(  x,t\right)  =u\left(  x\right)  e^{i\omega t}
\]
are called \emph{standing waves}. On the other hand, (\ref{228}) and
(\ref{229}) characterize a purely \emph{electrostatic field}.

With the above \emph{ansatz}, equations (\ref{220b}) and (\ref{220d}) are
identically satisfied; (\ref{220a}) and (\ref{220c}) take the form
\begin{gather}
-\Delta u+\left[  m_{0}^{2}-\left(  \omega+\phi\right)  ^{2}\right]  u-\left|
u\right|  ^{p-2}u=0\label{231a}\\
\Delta\phi+\beta\Delta_{4}\phi=4\pi\left(  \omega+\phi\right)  u^{2}
,\label{231c}
\end{gather}
where we have taken $e=1$.
Now we can state our main result.

\begin{theorem}
\label{main} If $\left|  \omega\right|  <\left|  m_{0}\right|  $ and $4<p<6$,
the system of equations (\ref{231a}) and (\ref{231c}) has infinitely many
solutions $\left(  u,\phi\right)  $ such that
\begin{gather}
\int_{\mathbb{R}^{3}}\left|  u\right|
^{2}dx+\int_{\mathbb{R}^{3}}\left|
\nabla u\right|  ^{2}dx   <+\infty\label{232}\\
\int_{\mathbb{R}^{3}}\left|  \nabla\phi\right|
^{2}dx+\int_{\mathbb{R} ^{3}}\left|  \nabla\phi\right|  ^{4}dx
<+\infty.\label{233}
\end{gather}
Moreover the fields $u$ and $\phi$ are radially symmetric.
\end{theorem}

The fact that $u\in H^{1}\left(  \mathbb{R}^{3}\right)  $ is radially
symmetric implies that $u$ decays to 0 at infinity (see \cite{str}), then
$\psi\left(  x,t\right)  =u\left(  x\right)  e^{i\omega t}$ can be called
\emph{solitary wave} .
Moreover, from (\ref{233}) we deduce that the electrostatic field
$\mathbf{E}=-\nabla\phi$ has finite energy (in the sense of \cite{fop}).

\section{Variational setting}

Let $H^{1}(  \mathbb{R}^{3})$ denote the usual Sobolev space with
norm
\[
\left\|  u\right\|  _{H^{1}}=\Big(  \int_{\mathbb{R}^{3}}\left|  u\right|
^{2}dx+\int_{\mathbb{R}^{3}}\left|  \nabla u\right|  ^{2}dx\Big)  ^{1/2}
\]
and $D$ denote the completion of $C_{0}^{\infty}\left(  \mathbb{R}^{3}\right)
$ with respect to the norm
\[
\left\|  \phi\right\|  _{D}=\left\|  \nabla\phi\right\|  _{L^{2}}+\left\|
\nabla\phi\right\|  _{L^{4}}.
\]

If $D^{1,2}\left(  \mathbb{R}^{3}\right)  $ denotes the completion of
$C_{0}^{\infty}\left(  \mathbb{R}^{3}\right)  $ with respect to the norm
\[
\left\|  \phi\right\|  _{D^{1,2}}=\left\|  \nabla\phi\right\|  _{L^{2}},
\]
it is obvious that $D$ is continuously embedded in $D^{1,2}\left(
\mathbb{R}^{3}\right)  $. On the other hand, by well known Sobolev inequality,
$D^{1,2}\left(  \mathbb{R}^{3}\right)  $ is continuously embedded in
$L^{6}\left(  \mathbb{R}^{3}\right)  $.

Moreover it can be easily proved that $D$ is continuously embedded in
$L^{\infty}\left(  \mathbb{R}^{3}\right)  $ (see Proposition 8 of \cite{fop}).

Consider the functional
\begin{multline*}
F\left(  u,\phi\right)  \\
=\int\left[  \frac{1}{2}\left|  \nabla u\right|
^{2}-\frac{1}{8\pi}\left|  \nabla\phi\right|  ^{2}+\frac{1}{2}\left(
m_{0}^{2}-\left(  \omega+\phi\right)  ^{2}\right)  u^{2}-\frac{\beta}{16\pi
}\left|  \nabla\phi\right|  ^{4}-\frac{1}{p}\left|  u\right|  ^{p}\right]  dx.
\end{multline*}
 For the rest of this article, the integration domain is $\mathbb{R}^{3}$.
By the above remarks, for every $\left(  u,\phi\right)  \in H^{1}\left(
\mathbb{R}^{3}\right)  \times D$
\[
F\left(  u,\phi\right)  \in\mathbb{R}.
\]

\begin{proposition} \label{cpf}
The functional $F$ is $C^{1}$ on $H^{1}\left(  \mathbb{R}
^{3}\right)  \times D$ and its critical points are solutions of (\ref{231a})
and (\ref{231c}) and satisfy (\ref{232}) and (\ref{233}).
\end{proposition}

\paragraph{Proof.}
Let $F_{u}'\left(  u,\phi\right)  $ and $F_{\phi}'\left(
u,\phi\right)  $ denote the partial derivatives of $F$ at $\left(
u,\phi\right)  \in H^{1}\left(  \mathbb{R}^{3}\right)  \times D$.
For every $v\in H^{1}\left(  \mathbb{R}^{3}\right)  $ and $w\in D$,
\begin{gather}
F_{u}'\left(  u,\phi\right)  \left[  v\right]  =\int\left\{  \left(
\nabla u\mid\nabla v\right)  +\left[  m_{0}^{2}-\left(  \omega+\phi\right)
^{2}\right]  uv-\left|  u\right|  ^{p-2}uv\right\}  dx\label{240a}\\
F_{\phi}'\left(  u,\phi\right)  \left[  w\right]  =-\int\left\{
\frac{1}{4\pi}\left(  \left(  1+\beta\left|  \nabla\phi\right|  ^{2}\right)
\nabla\phi\mid\nabla w\right)  +\left(  \omega+\phi\right)  u^{2}w\right\}
dx.\label{240b}
\end{gather}
Using standard computations we show that $F_{u}'$ and $F_{\phi
}'$ are continuous; therefore $F$ is $C^{1}$ on $H^{1}\left(
\mathbb{R}^{3}\right)  \times D$.
Moreover, from (\ref{240a}) and (\ref{240b}) we get obviously that the
critical points of $F$ are solutions of (\ref{231a}) and (\ref{231c}).
\hfill$\Box$ \smallskip

The functional $F$ is strongly indefinite, i.e. it is unbounded from above and
from below, even modulo compact perturbations.
Following \cite{bf} we are going to study a functional of the only variable
$u$ whose critical points give rise to critical points of $F$.

\begin{lemma}
\label{unic} For every $u\in H^{1}\left(  \mathbb{R}^{3}\right)  $ there
exists a unique $\phi=\Phi\left[  u\right]  \in D$ solution of (\ref{231c}).
\end{lemma}

\paragraph{Proof.}
For every fixed $u\in H^{1}\left(  \mathbb{R}^{3}\right)  $, the solutions of
(\ref{231c}) are critical points of the functional
\begin{equation}
I\left(  \phi\right)  =\int\left\{  \frac{1}{8\pi}\left|  \nabla\phi\right|
^{2}+\frac{\beta}{16\pi}\left|  \nabla\phi\right|  ^{4}+\omega u^{2}\phi
+\frac{1}{2}\phi^{2}u^{2}\right\}  dx\label{defi}
\end{equation}
defined on $D$.
This functional is coercive; indeed, by the continuous embedding of $D$ in
$L^{\infty}\left(  \mathbb{R}^{3}\right)  $,
\[
I\left(  \phi\right)  \geq\frac{1}{8\pi}\left\|  \nabla\phi\right\|  _{L^{2}
}^{2}+\frac{\beta}{16\pi}\left\|  \nabla\phi\right\|  _{L^{4}}^{4}-c\left\|
u^{2}\right\|  _{L^{1}}\left(  \left\|  \nabla\phi\right\|  _{L^{2}}+\left\|
\nabla\phi\right\|  _{L^{4}}\right)  .
\]
Furthermore $I$ is weakly lower semicontinuous since each term in (\ref{defi})
is continuous and convex.
Therefore $I$ admits a global minimum.
The solution of (\ref{231c}) is unique because the operator
\[
\mathcal{A}=-\Delta-\beta\Delta_{4}+4\pi u^{2}
\]
is strictly monotone. \hfill$\Box$ \smallskip


By Lemma \ref{unic} we can consider the map
\[
\Phi:H^{1}\left(  \mathbb{R}^{3}\right)  \to  D
\]
which is implicitly defined by $F_{\phi}'\left(  u,\phi\right)  =0$.

 From standard arguments we deduce that the map $\Phi$ is $C^{1}$.
Fixed $u\in H^{1}\left(  \mathbb{R}^{3}\right)  $, since $\Phi\left[
u\right]  $ is the solution of (\ref{231c}), we have
\[
\left\langle \frac{1}{4\pi}\Delta\Phi\left[  u\right]  +\frac{\beta}{4\pi
}\Delta_{4}\Phi\left[  u\right]  -\Phi\left[  u\right]  u^{2},\Phi\left[
u\right]  \right\rangle =\left\langle \omega u^{2},\Phi\left[  u\right]
\right\rangle ,
\]
i.e.
\begin{equation}
-\frac{1}{4\pi}\int\left|  \nabla\Phi\left[  u\right]  \right|  ^{2}
dx-\frac{\beta}{4\pi}\int\left|  \nabla\Phi\left[  u\right]  \right|
^{4}dx-\int\left(  \Phi\left[  u\right]  \right)  ^{2}u^{2}dx=\omega\int
u^{2}\Phi\left[  u\right]  dx.\label{250}
\end{equation}
Now we consider the functional
\[
J:H^{1}\left(  \mathbb{R}^{3}\right)  \to \mathbb{R},
\quad
J\left(  u\right)  =F\left(  u,\Phi\left[  u\right]  \right)  .
\]
Using (\ref{250}) we obtain
\begin{align*}
J\left(  u\right)   =& \frac{1}{2}\int\left|  \nabla u\right|  ^{2}dx+\frac
{1}{2}\left(  m_{0}^{2}-\omega^{2}\right)  \int u^{2}dx+\frac{1}{2}\int\left(
\Phi\left[  u\right]  \right)  ^{2}u^{2}dx+\\
& +\frac{1}{8\pi}\int\left|  \nabla\Phi\left[  u\right]  \right|  ^{2}
dx+\frac{3}{16\pi}\beta\int\left|  \nabla\Phi\left[  u\right]  \right|
^{4}dx-\frac{1}{p}\int\left|  u\right|  ^{p}dx.
\end{align*}

\begin{remark} \rm
\label{ccj}Since $F$ and $\Phi$ are $C^{1}$, also $J$ is $C^{1}$. Moreover, if
$u\in H^{1}\left(  \mathbb{R}^{3}\right)  $ is a critical point of $J$, then
$\left(  u,\Phi\left[  u\right]  \right)  $ is a critical point of $F$ (see
Proposition 7 of \cite{bf}).
\end{remark}

Thus, to get solutions of our problem, we look for critical points of $J$.
For every $a\in\mathbb{R}^{3}$ and $v:\mathbb{R}^{3}\to \mathbb{R}$, we
set
\begin{equation}
v_{a}\left(  x\right)  =v\left(  x+a\right)  .\label{t}
\end{equation}
Lemma \ref{unic} implies that for every $u\in H^{1}\left(  \mathbb{R}
^{3}\right)  $,
\[\Phi\left[  u_{a}\right]  =\left(  \Phi\left[  u\right]  \right)  _{a};\]
therefore
$J\left(  u_{a}\right)  =J\left(  u\right)$.


Since $J$ is invariant under translations (\ref{t}), there is still lack of
compactness. For this reason we restrict $J$ to the subspace
\[
H_{r}^{1}\left(  \mathbb{R}^{3}\right)  =\left\{  u\in H^{1}\left(
\mathbb{R}^{3}\right)  \vline u=u\left(  \left|  x\right|  \right)  \right\}
\]
which is a natural constraint for $J$ in the sense of following lemma.

\begin{lemma}
\label{pcv} If $u\in H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $ is a critical
point of $J\vline_{H_{r}^{1}\left(  \mathbb{R}^{3}\right)  }$, then $u$ is a
critical point of $J$.
\end{lemma}

\paragraph{Proof.}
For every field $v$ defined almost everywhere in $\mathbb{R}^{3}$ and for
every $g\in O\left(  3\right)  $ we set
\begin{equation}
\left(  T_{g}v\right)  \left(  x\right)  =v\left(  gx\right)  .\label{r}
\end{equation}
Of course (\ref{r}) defines an action of $O\left(  3\right)  $ on
$H^{1}\left(  \mathbb{R}^{3}\right)  $.

Since $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $ is the set of fixed points
for this action, by the well known Principle of Symmetric Criticality (see
\cite{pal}), it is enough to prove that $J$ is $T_{g}$-invariant, i.e. for
every $u\in H^{1}\left(  \mathbb{R}^{3}\right)  $ and
$g\in O\left(  3\right)$
\begin{equation}
J\left(  T_{g}u\right)  =J\left(  u\right)  .\label{255}
\end{equation}
The main point is to prove that, for every $u\in H^{1}\left(  \mathbb{R}
^{3}\right)  $
\begin{equation}
\Phi\left[  T_{g}u\right]  =T_{g}\Phi\left[  u\right]  .\label{260}
\end{equation}
It is well known that
\[
\Delta T_{g}\Phi\left[  u\right]  =T_{g}\left(  \Delta\Phi\left[  u\right]
\right)  .
\]
Analogously one can show that
\[
\Delta_{4}T_{g}\Phi\left[  u\right]  =T_{g}\left(  \Delta_{4}\Phi\left[
u\right]  \right)  .
\]
Then it is simple to verify that $\Phi\left[  T_{g}u\right]  $ and $T_{g}
\Phi\left[  u\right]  $ solve the same equation
\[
\Delta\phi+\beta\Delta_{4}\phi=4\pi\left(  \omega+\phi\right)  \left(
T_{g}u\right)  ^{2}.
\]
Therefore we obtain (\ref{260}).
So, by the $T_{g}$-invariance of the norms in $H^{1}\left(  \mathbb{R}
^{3}\right)  $, $D$ and $L^{p}\left(  \mathbb{R}^{3}\right)  $ we get
(\ref{255}).
\hfill$\Box$

\section{Proof of Theorem \ref{main}}

By Proposition \ref{cpf}, Remark \ref{ccj} and Lemma \ref{pcv}, the thesis of
Theorem \ref{main} will follow if we show that the functional $J$ has
infinitely many critical points on $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $.

First of all we prove that on $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $ there
is no lack of compactness. Indeed the following lemma holds true.

\begin{lemma}
If $\left|  \omega\right|  <\left|  m_{0}\right|  $ and $4<p<6$, the
functional $J\vline_{H_{r}^{1}\left(  \mathbb{R}^{3}\right)  }$ satisfies the
Palais-Smale condition.
\end{lemma}

\paragraph{Proof.}
Let $\left\{  u_{n}\right\}  \subset H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $
be a Palais-Smale sequence, i.e.
$J\left(  u_{n}\right)  =M_{n}$ is bounded and
\[
J'\vline_{H_{r}^{1}\left(  \mathbb{R}^{3}\right)  }\left(
u_{n}\right)  =\varepsilon_{n}\to 0
\]
in $\left(  H_{r}^{1}\left(  \mathbb{R}^{3}\right)  \right)'$ (the
dual space of $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $).

We want to prove that $\left\{  u_{n}\right\}  $ contains a convergent subsequence.
 From the definition of $J$ we have that
\[
J'\left(  u_{n}\right)  =F_{u}'\left(  u_{n},\Phi\left[
u_{n}\right]  \right)  +F_{\phi}'\left(  u_{n},\Phi\left[
u_{n}\right]  \right)  \Phi'\left[  u_{n}\right]  =F_{u}^{\prime
}\left(  u_{n},\Phi\left[  u_{n}\right]  \right)  =\varepsilon_{n}
\]
since
$F_{\phi}'\left(  u_{n},\Phi\left[  u_{n}\right]  \right)  =0$.
Then
\begin{align*}
J( u_{n})&  -\frac{1}{p}\left\langle J'\left(
u_{n}\right)  ,u_{n}\right\rangle =J\left(  u_{n}\right)  -\frac{1}
{p}\left\langle F_{u}'\left(  u_{n},\Phi\left[  u_{n}\right]  \right)
,u_{n}\right\rangle \\
=&\left(  \frac{1}{2}-\frac{1}{p}\right)  \int\left|  \nabla u_{n}\right|
^{2}dx+\left(  \frac{1}{2}-\frac{1}{p}\right)  \left(  m_{0}^{2}-\omega
^{2}\right)  \int u_{n}^{2}dx+\\
&+\left(  \frac{1}{2}+\frac{1}{p}\right)  \int\left(  \Phi\left[  u_{n}\right]
\right)  ^{2}u_{n}^{2}dx+\frac{2}{p}\omega\int\Phi\left[  u_{n}\right]
u_{n}^{2}dx+\\
&+\frac{1}{8\pi}\int\left|  \nabla\Phi\left[  u_{n}\right]  \right|
^{2}dx+\frac{3}{16\pi}\beta\int\left|  \nabla\Phi\left[  u_{n}\right]
\right|  ^{4}dx.
\end{align*}
On the other hand
\[
J\left(  u_{n}\right)  -\frac{1}{p}\left\langle J'\left(
u_{n}\right)  ,u_{n}\right\rangle =M_{n}-\frac{1}{p}\left\langle
\varepsilon_{n},u_{n}\right\rangle .
\]
Hence we can write
\begin{equation}
c_{1}\int\left|  \nabla u_{n}\right|  ^{2}dx+c_{2}\int u_{n}^{2}dx+A_{n}
=M_{n}-\frac{1}{p}\left\langle \varepsilon_{n},u_{n}\right\rangle \label{320}
\end{equation}
where
\begin{align}
A_{n}  =&\frac{1}{8\pi}\int\left|  \nabla\Phi\left[  u_{n}\right]  \right|
^{2}dx+\frac{3}{16\pi}\beta\int\left|  \nabla\Phi\left[  u_{n}\right]
\right|  ^{4}dx+\label{321}\\
& +\left(  \frac{1}{2}+\frac{1}{p}\right)  \int\left(  \Phi\left[
u_{n}\right]  \right)  ^{2}u_{n}^{2}dx+\frac{2}{p}\omega\int\Phi\left[
u_{n}\right]  u_{n}^{2}dx\nonumber
\end{align}
and $c_{1}$, $c_{2}$ are positive constants.

Equation (\ref{250}) holds also for the pair $\left(  u_{n},\Phi\left[
u_{n}\right]  \right)  $ and so
\begin{align}
A_{n}  =&\frac{1}{4\pi}\left(  \frac{1}{2}-\frac{2}{p}\right)  \int\left|
\nabla\Phi\left[  u_{n}\right]  \right|  ^{2}dx+\frac{\beta}{4\pi}\left(
\frac{3}{4}-\frac{2}{p}\right)  \int\left|  \nabla\Phi\left[  u_{n}\right]
\right|  ^{4}dx+\nonumber\\
& +\left(  \frac{1}{2}-\frac{1}{p}\right)  \int\left(  \Phi\left[
u_{n}\right]  \right)  ^{2}u_{n}^{2}dx.\label{330}
\end{align}
Then, since $4<p<6$, there exists $c_{3}>0$ such that
\begin{equation}
A_{n}\geq c_{3}\left(  \int\left|  \nabla\Phi\left[  u_{n}\right]  \right|
^{2}dx+\int\left|  \nabla\Phi\left[  u_{n}\right]  \right|  ^{4}dx+\int\left(
\Phi\left[  u_{n}\right]  \right)  ^{2}u_{n}^{2}dx\right)  .\label{331}
\end{equation}
Thus, being $A_{n}\geq0$, from (\ref{320}) we obtain that $\left\{
u_{n}\right\}  $ is bounded in $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $.
Moreover, the equation (\ref{320}) implies that $\left\{  A_{n}\right\}  $ is
bounded and so, from (\ref{331}), also $\left\{  \Phi\left[  u_{n}\right]
\right\}  $ is bounded in $D$.
Hence, up to subsequence,
\begin{gather*}
u_{n}   \rightharpoonup u\text{ in }H_{r}^{1}\left(  \mathbb{R}^{3}\right) \\
\Phi\left[  u_{n}\right]    \rightharpoonup\bar{\Phi}\text{ in }D.
\end{gather*}
Now we prove that
$u_{n}\to  u$ in $H_{r}^{1}(\mathbb{R}^{3})$.
We know that
\[
-\Delta u_{n}+\left[  m_{0}^{2}-\left(  \omega+\Phi\left[  u_{n}\right]
\right)  ^{2}\right]  u_{n}-\left|  u_{n}\right|  ^{p-2}u_{n}=\varepsilon_{n},
\]
i.e.
\[
-\Delta u_{n}+\left(  m_{0}^{2}-\omega^{2}\right)  u_{n}=2\omega\Phi\left[
u_{n}\right]  u_{n}+\left(  \Phi\left[  u_{n}\right]  \right)  ^{2}
u_{n}+\left|  u_{n}\right|  ^{p-2}u_{n}+\varepsilon_{n}.
\]
Let
$L:H_{r}^{1}\left(  \mathbb{R}^{3}\right)  \to \left(  H_{r}^{1}\left(
\mathbb{R}^{3}\right)  \right)'$
be the isomorphism
\[
Lu=-\Delta u+\left(  m_{0}^{2}-\omega^{2}\right)  u.
\]
Then
\begin{equation}
u_{n}=2\omega L^{-1}\left(  \Phi\left[  u_{n}\right]  u_{n}\right)
+L^{-1}\left(  \left(  \Phi\left[  u_{n}\right]  \right)  ^{2}u_{n}\right)
+L^{-1}\left(  \left|  u_{n}\right|  ^{p-2}u_{n}\right)  +L^{-1}\left(
\varepsilon_{n}\right)  .\label{333}
\end{equation}

To prove the strong convergence of $\left\{  u_{n}\right\}  ,$ it is
sufficient to prove the strong convergence of each term in the right hand side
of (\ref{333}).
Obviously the sequence $\left\{  L^{-1}\left(  \varepsilon_{n}\right)
\right\}  $ converges strongly.
Since $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $ is compactly embedded into
$L^{q}\left(  \mathbb{R}^{3}\right)  $ for $2<q<6$ (see \cite{str},
\cite{bl}), we have that
\begin{equation}
L^{p'}\left(  \mathbb{R}^{3}\right)  \text{ is compactly embedded into
}\left(  H_{r}^{1}\left(  \mathbb{R}^{3}\right)  \right)  ^{\prime
},\label{sobd}
\end{equation}
where $p'=\frac{p}{p-1}$ is the conjugate exponent of $p$.
Thus, if we set
\[
v_{n}=\left|  u_{n}\right|  ^{p-2}u_{n}
\]
an easy calculation shows that
\begin{equation}
\left\|  v_{n}\right\|  _{L^{p'}}=\left\|  u_{n}\right\|  _{L^{p}
}^{p-1}.\label{335}
\end{equation}
Since $\left\{  u_{n}\right\}  $ is bounded in $H_{r}^{1}\left(
\mathbb{R}^{3}\right)  $, from Sobolev embedding theorem, it is bounded in
$L^{p}\left(  \mathbb{R}^{3}\right)  $ and so, by (\ref{335}), we have that
$\left\{  v_{n}\right\}  $ is bounded in $L^{p'}\left(  \mathbb{R}
^{3}\right)  $.

Then, up to subsequence, $\left\{  v_{n}\right\}  $ converges weakly in
$L^{p'}\left(  \mathbb{R}^{3}\right)  $ and, by (\ref{sobd}),
converges strongly in $\left(  H_{r}^{1}\left(  \mathbb{R}^{3}\right)
\right)'$.
So we conclude that $\big\{  L^{-1}(|u_{n}|^{p-2}u_{n})\big\}$
 converges strongly in $H_{r}^{1}\left(\mathbb{R}^{3}\right)  $.

Finally, it remains to show the strong convergence of $\left\{  L^{-1}\left(
\Phi\left[  u_{n}\right]  u_{n}\right)  \right\}  $ and $\left\{
L^{-1}\left(  \left(  \Phi\left[  u_{n}\right]  \right)  ^{2}u_{n}\right)
\right\}  $.
Since $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $ is compactly embedded into
$L^{3}\left(  \mathbb{R}^{3}\right)  $, we have that $L^{3/2}\left(
\mathbb{R}^{3}\right)  $ is compactly embedded into $\left(  H_{r}^{1}\left(
\mathbb{R}^{3}\right)  \right)'$and then it is sufficient to prove
that $\left\{  \Phi\left[  u_{n}\right]  u_{n}\right\}  $ and $\left\{
\left(  \Phi\left[  u_{n}\right]  \right)  ^{2}u_{n}\right\}  $ are bounded in
$L^{3/2}\left(  \mathbb{R}^{3}\right)  $.
By H\"{o}lder inequality
\begin{gather*}
\left\|  \left(  \Phi\left[  u_{n}\right]  \right)  ^{2}u_{n}\right\|
_{L^{3/2}}   \leq\left\|  \Phi\left[  u_{n}\right]  \right\|  _{L^{6}}
^{3}\left\|  u_{n}\right\|  _{L^{3}}^{3/2}\\
\left\|  \Phi\left[  u_{n}\right]  u_{n}\right\|  _{L^{3/2}}  \leq\left\|
\Phi\left[  u_{n}\right]  \right\|  _{L^{6}}\left\|  u_{n}\right\|  _{L^{2}}.
\end{gather*}
Hence, being $\left\{  u_{n}\right\}  $ and $\left\{  \Phi\left[
u_{n}\right]  \right\}  $ bounded respectively in $H_{r}^{1}\left(
\mathbb{R}^{3}\right)  $ and $D$ and from well known Sobolev inequalities, we
conclude that $\left\{  \Phi\left[  u_{n}\right]  u_{n}\right\}  $ and \\
$\{(\Phi[u_{n}])^{2}u_{n}\}  $ are
bounded in $L^{3/2}(\mathbb{R}^{3})$.
\hfill$\Box$ \smallskip

Since $J$ is \emph{even} and $J\vline_{H_{r}^{1}\left(  \mathbb{R}^{3}\right)
}$ satisfies the Palais-Smale condition, the existence of infinitely many
critical points follows from the equivariant version of the mountain pass
theorem (Theorem 9.12 of \cite{rab}).
Let $J\vline_{H_{r}^{1}\left(  \mathbb{R}^{3}\right)  }$ satisfy the
geometrical assumptions:
\begin{enumerate}
\item[$(\text{G}_{1})$] There exists $\alpha$, $\rho>0$ such
that for every $u\in H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $ with $\left\|
u\right\|  _{H_{r}^{1}}=\rho$
\[
J\left(  u\right)  \geq\alpha
\]

\item[$(\text{G}_{2})$] For every finite dimensional subspace
$V $ of $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $ there exists $R>0$ such
that for every $u\in H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $ with $\left\|
u\right\|  _{H_{r}^{1}}\geq R$
\[
J\left(  u\right)  \leq0.
\]
\end{enumerate}
Then we deduce that $J\vline_{H_{r}^{1}\left(  \mathbb{R}^{3}\right)  }$
has infinitely many critical points.
Therefore we are left to prove the geometrical assumptions $\left(
\text{G}_{1}\right)  $ and $\left(  \text{G}_{2}\right)$.

In the sequel $c_{i}$ denotes a positive constant.
 From the definition of $J$, for every $u\in H_{r}^{1}\left(  \mathbb{R}
^{3}\right)  $
\[
J\left(  u\right)  \geq c_{1}\left\|  u\right\|  _{H^{1}}^{2}-c_{2}\left\|
u\right\|  _{L^{p}}^{p}.
\]
Since $4<p<6$, by well known Sobolev inequalities, we obtain $\left(
\text{G}_{1}\right)  $.

As to $(\text{G}_{2})$, let $V$ be a finite dimensional
subspace of $H_{r}^{1}\left(  \mathbb{R}^{3}\right)  $. For every $u\in V$
\begin{equation}\begin{aligned}
J\left(  u\right)  \leq &c_{3}\left\|  u\right\|  _{H^{1}}^{2}+\frac{1}
{8\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{2}}^{2}+\frac
{3}{16\pi}\beta\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{4}}
^{4}+\\
& +\frac{1}{2}\left\|  \phi\left[  u\right]  u\right\|  _{L^{2}}^{2}-\frac
{1}{p}\left\|  u\right\|  _{L^{p}}^{p}.
\end{aligned}\label{425}
\end{equation}
We know that
\[
\frac{1}{4\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{2}}
^{2}+\frac{\beta}{4\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{4}
}^{4}+\left\|  \phi\left[  u\right]  u\right\|  _{L^{2}}^{2}=-\omega\int
u^{2}\phi\left[  u\right]  dx.
\]
Moreover, by H\"{o}lder and Sobolev inequalities
\[
\left|  \omega\int u^{2}\phi\left[  u\right]  dx\right|  \leq c_{4}\left\|
u\right\|  _{L^{12/5}}^{2}\left\|  \phi\left[  u\right]  \right\|  _{L^{6}
}\leq c_{5}\left\|  u\right\|  _{L^{12/5}}^{2}\left\|  \nabla\phi\left[
u\right]  \right\|  _{L^{2}}
\]
and so
\begin{gather}
\frac{1}{4\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{2}}
^{2}+\frac{\beta}{4\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{4}
}^{4}+\left\|  \phi\left[  u\right]  u\right\|  _{L^{2}}^{2}\leq\label{430}\\
\leq c_{5}\left\|  u\right\|  _{L^{12/5}}^{2}\left\|  \nabla\phi\left[
u\right]  \right\|  _{L^{2}}.\nonumber
\end{gather}
Then
\begin{equation}
\frac{1}{4\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{2}}\leq
c_{5}\left\|  u\right\|  _{L^{12/5}}^{2}\label{431}
\end{equation}
being
\begin{equation}
\frac{1}{4\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{2}}^{2}\leq
c_{5}\left\|  u\right\|  _{L^{12/5}}^{2}\left\|  \nabla\phi\left[  u\right]
\right\|  _{L^{2}}.\label{432}
\end{equation}
Thus, from (\ref{430}), (\ref{431}) and (\ref{432})
\begin{align*}
\frac{1}{4\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{2}}^{2}  &
\leq c_{5}^{2}\left\|  u\right\|  _{L^{12/5}}^{4}\\
\frac{\beta}{4\pi}\left\|  \nabla\phi\left[  u\right]  \right\|  _{L^{4}}^{4}
& \leq c_{5}^{2}\left\|  u\right\|  _{L^{12/5}}^{4}\\
\left\|  \phi\left[  u\right]  u\right\|  _{L^{2}}^{2}  & \leq c_{5}
^{2}\left\|  u\right\|  _{L^{12/5}}^{4}.
\end{align*}
Hence, from (\ref{425}),
\begin{align*}
J\left(  u\right)  \leq &c_{3}\left\|  u\right\|  _{H^{1}}^{2}+\frac{c_{5}^{2}
}{2}\left\|  u\right\|  _{L^{12/5}}^{4}+\frac{c_{5}^{2}}{2}\left\|  u\right\|
_{L^{12/5}}^{4}+\frac{3}{4}c_{5}^{2}\left\|  u\right\|  _{L^{12/5}}^{4}+\\
&-\frac{1}{p}\left\|  u\right\|  _{L^{p}}^{p}=c_{3}\left\|  u\right\|  _{H^{1}
}^{2}+\frac{7}{4}c_{5}^{2}\left\|  u\right\|  _{L^{12/5}}^{4}-\frac{1}
{p}\left\|  u\right\|  _{L^{p}}^{p}
\end{align*}
and, since $V$ is finite dimensional and $p>4$, we obtain $\left(
\text{G}_{2}\right)  $.


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\noindent\textsc{Pietro d'Avenia} (e-mail: pdavenia@dm.uniba.it) \\
\textsc{Lorenzo Pisani} (e-mail: pisani@dm.uniba.it) \\[2pt]
Dipartimento Interuniversitario di Matematica \\
Universit\`{a} degli Studi di Bari \\
Via Orabona, 4 \\
70125 Bari Italy

\end{document}