\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 83, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/83\hfil Remarks on least energy solutions]
{Remarks on least energy solutions for
quasilinear elliptic problems in $\mathbb{R}^N$}

\author[J. M. do \'O \&  E. S. Medeiros\hfil EJDE--2003/83\hfilneg]
{Jo\~ao Marcos do \'O \&  Everaldo S. Medeiros}  % in alphabetical order

\address{Departamento de Matem\'atica, Univ. Fed. Para\'\i ba \\
58059-900 Jo\~ao Pessoa, PB, Brazil}
\email[Jo\~ao Marcos do \'O]{jmbo@mat.ufpb.br}
\email[Everaldo S. Medeiros]{everaldo@mat.ufpb.br}


\date{}
\thanks{Submitted June 3, 2003. Published August 11, 2003.}
\subjclass[2000]{35J20, 35J60}
\keywords{Variational methods, minimax methods,
 superlinear elliptic problems, \hfill\break\indent
 p-Laplacian,
 ground-states, moutain-pass solutions}


\begin{abstract}
  In this work we establish some properties of the solutions
  to the quasilinear second-order problem
  $$
   -\Delta_p w=g(w)\quad \mbox{in }  \mathbb{R}^N
  $$
  where $\Delta_p u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$
  is the $p$-Laplacian  operator and $ 1<p\leq N $. We study a
  mountain pass characterization of least energy solutions of this
  problem. Without assuming the monotonicity of the function
  $t^{1-p}g(t)$, we show that the Mountain-Pass value gives the
  least energy level. We also prove the exponential decay of the
  derivatives of the solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{coro}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In this paper, we consider the quasilinear elliptic problem
\begin{equation}
-\Delta_p w=g(w)\quad  \mbox{in } \mathbb{R}^N, \label{eq:1.1}
\end{equation}
where $\Delta_p u=\mathop{\rm div} (|\nabla u|^{p-2}\nabla u)$
is the $p$-Laplacian operator and $1<p\leq N$. Using variational
methods, more precisely by a constrained minimization argument, we
show the existence of ground states solutions (or least energy
solutions) for the problem \eqref{eq:1.1} in both cases, $ 1 < p <
N $ and $ p =N $. As it is well known, in the case $ 1 < p < N $
the nonlinearities are required to have polynomial growth at
infinity in order to define the associated functionals in Sobolev
spaces. Coming to the case $ p=N $, much faster growth is allowed
for the nonlinearity and the Trudinger-Moser inequality in $ p=N$
replaces the Sobolev imbedding theorem used for $ 1<p<N$.

In our study, we prove also that the Mountain-Pass value gives the
least energy level and  we obtain the exponential decay of the
derivatives of the solutions of problem \eqref{eq:1.1}.

The knowledge of ground states plays a role in several
applications in elliptic problems. For example in the study of
various  types of spike solutions, ground state serves as scaled
limit profile of the solution near the spike \cite{NT}.

There has been recently a good amount of work on this class of
problem \eqref{eq:1.1} in the semilinear case which corresponds to
the case $ p=2 $, see for example \cite{BL,BGK,Jean}. In these
papers was investigated the existence of ground state solutions
using the minimization argument. The characterization of the least
energy level was investigated by Ding Ni \cite{DN} and Rabinowitz
\cite{Rabi}, under the monotonicity condition of the function
$g(t)/t$. Recently, Jeanjean and Tanaka \cite{Jean} have obtained
this kind of result without this monotonicity assumption.

The study of the exponential decay of the solutions, in the
semilinear case, was considered by Strauss \cite{St}, Berestycki
and Lions \cite{BL}, among others. Gongbao and Shusen \cite{Su}
have showed the exponential decay for weak solution of a class of
p-Laplacian equations. Under severe restrictions about the
structure of the operator and the nature of the solutions, some
exponential decay results have been obtained recently by Rabier
and Stuart \cite{RS}. However, on these works the decay of
derivatives for the degenerate case was not shown. In the present
paper we prove the exponential decay of first derivatives for all
radial solution of problem \eqref{eq:1.1} by using an
appropriated test function.

The operator $-\Delta_p$ with $p\neq2$ arises from a variety of
physical phenomena. It is used in non-Newtonian fluids, in some
reaction-diffusion problems, as well as in flow through porous
media. It also appears in nonlinear elasticity, glaciology and
petroleum extraction \cite{Diaz}.

Several papers have appeared recently about the p-Laplacian
problems involving unbounded domains, among others Serrin-Tang
\cite{ST}, Serrin-Zhou \cite {SZ}, Do \'O \cite{Do},
Hebey-Demengel \cite{HB} and Jianfu and Xi Ping \cite{JX}. We
referred to their references for other related results.\\

For easy reference we state now the assumptions that will be
assumed through  this paper.
\begin{itemize}
\item[(G1)]  $g\in C(\mathbb{I}, \mathbb{R})$ and is odd;

\item[(G2)]  when $1<p<N$ we assume that
$$
\lim_{u\to+\infty}\frac{g(u)}{u^{p^*-1}}=0\quad
\mbox{where}\quad p^*=\frac{Np}{N-p};
$$
when $p=N$ we assume that
$$
|g(u)|\leq C[\exp(\alpha_0|u|^{\frac{N}{N-1}})-S_{N-2}(\alpha_0,u)],
$$
for some constants $\alpha_0$, $C>0$, where
$$
S_{N-2}(\alpha_0,u)=\sum_{k=0}^{N-2}\frac{\alpha_0^k}{k!}|u|^{\frac{Nk}{N-1}};
$$

\item[(G3)] when $1<p<N$ we suppose that
$$
-\infty<\liminf_{u\to0^+}\frac{g(u)}{u^{p-1}}\leq\limsup_{u\to0^+}
\frac{g(u)}{u^{p-1}}=-\nu<0,
$$
and for $p=N$
$$
\lim_{u\to0}\frac{g(u)}{|u|^{N-1}}=-\nu<0.
$$
\item[(G4)]  There exists $\zeta>0$ such that
$G(\zeta)>0$, where $G(u)=\int_0^{u}g(t)dt$.
\end{itemize} \smallskip

\noindent\textbf{Example.} Let $1<p<N$ and consider the function
\[
g(u)=\lambda|u|^{q-2}u-\mu|u|^{p-2}u,
\]
where $\lambda,\mu$ are positive constants and $1<p<q<p^*-1$. It
is  not difficult to see that $ g $ satisfies the assumptions
(G1)--(G4). \smallskip


\noindent\textbf{Example}
Assume that $p=N$ and. consider the function
$$
g(u)=-\mu|u|^{N-2}u+|u|^{N-1}ue^{\beta |u|^{\frac{N}{N-1}}},
$$
where $\beta>0$ and $\mu>0$. We can see that $ g $ satisfies the
assumptions (G1)--(G4). \smallskip


\noindent\textbf{Notation.}
In this paper we make use of the following
notation.
\begin{itemize}
\item For $1 \leq p \leq \infty$, $L^p(U)$,  denotes  Lebesgue
spaces with the norm  $\| u \|_{L^p(U)}$ \item
$W^{1,p}(\mathbb{R}^N)$  denote Sobolev spaces with the norm
$\|u\|_{W^{1,p}(\mathbb{R}^N)}$ \item $W^{1,p}_r(\mathbb{R}^N)$
denotes the subspace of $W^{1,p}(\mathbb{R}^N)$ formed by the
radial functions \item $C^{k,\alpha}(U)$, with $k$ a non-negative
integer and $ 0 \leq \alpha <1$, denotes H\"{o}lder spaces \item
$C$, $C_0$,  $C_{1}$, $C_{2}$, \dots denote (possibly different)
positive constants \item $|A|$ denotes Lebesgue measure of the set
$A\subset \mathbb{R}^N$ \item $\omega_{N-1}  $ is the
$(N-1)$-dimensional measure of the $N-1$ unit sphere in $
\mathbb{R}^N $.
\end{itemize}

\subsection*{Variational Formulation}

We begin by recalling the following Trundiger-Moser
type inequality which is crucial for our variational
argument. the Trudinger-Moser inequality for
$ p=N$ replaces the Sobolev imbedding theorem used for $ 1<p<N$.

\begin{lemma}\label{Trudinger-Moser}
If $N\geq 2$, $\alpha >0$ and $u\in W^{1,N}(\mathbb{R}^N)$, then
\begin{equation}
\int_{\mathbb{R}^N}\Big[ \exp \big( \alpha | u | ^{\frac N{N-1}}\big)
-S_{N-2}\big( \alpha ,u\big) \Big]\,dx <\infty .
\label{TM1}
\end{equation}
Moreover, if $\| \nabla u\| _{L^N(\mathbb{R}^N)}^N\leq 1$,
 $\| u\| _{L^N(\mathbb{R}^N)}\leq M<\infty $, and $\alpha <\alpha
_N=N\omega_{N-1}^{\frac 1{N-1}}$, then there exists a constant
$C$, which depends only on $N,M$ and $\alpha $,
such that
\begin{equation}
\int_{\mathbb{R}^N}\big[ \exp \big( \alpha | u| ^{\frac N{N-1}}\big)
-S_{N-2}( \alpha ,u) \big] \,dx \leq C( N,M,\alpha ) .  \label{TM2}
\end{equation}
\end{lemma}

The proof of this lemma can be found in  \cite[Lemma 1]{Do}.

\begin{lemma}
Suppose that $g$ satisfies (G1)--(G3). Then the associated
energy functional of problem \eqref{eq:1.1},
$I:W^{1,p}(\mathbb{R}^N)\to \mathbb{R}$, given by
\[
I(u)=\frac 1p\int_{\mathbb{R}^N}|\nabla u|^p
dx-\int_{\mathbb{R}^N}G(u) \,dx
\]
is well defined and of class $C^1$ with
\[
I'(u)v=\int_{\mathbb{R}^N}|\nabla u|^{p-2}\nabla u\nabla
v\, dx-\int_{\mathbb{R}^N}g(u)v \,dx,\quad \forall v\in W^{1,p}(\mathbb{R}^N).
\]
Consequently, critical points of the functional $I$ are precisely
the weak solutions of problem \eqref{eq:1.1}.
\end{lemma}

\begin{proof}
{\bf Case: $ 1<p<N$.} As a consequence of assumptions (G1)--(G3), with the
 aid of the Holder and Sobolev inequalities, we see that $I$ and $I'(u)$ are
well defined on $W^{1,p}(\mathbb{R}^N)$.

\noindent{\bf Case: $p=N$.} From (G2) it follows that
\begin{equation}
| G(u) | \leq C\big[ \exp \big( \alpha _1| u| ^{\frac
N{N-1}}\big) -S_{N-2}( \alpha _1,u) \big] ,\label{V1}
\end{equation}
for some constants $\alpha _1$, $C>0$. Thus, by Lemma \ref{Trudinger-Moser}%
, we have $G(u) \in L^1(\mathbb{R}^N)$ for all $u\in
W^{1,N}(\mathbb{R}^N)$.

 Furthermore, using standard arguments \cite{BL,Djairo} as well as
the fact that for any
given strong convergent sequence $(u_n)$ in $W^{1,N}(\mathbb{R}^N)
$ there is a subsequence $(u_{n_k})$ and there exists $h\in
W^{1,N}(\mathbb{R}^N) $ such
 that $|u_{n_k}(x)| \leq h(x)$ almost everywhere in
$\mathbb{R}^N$, we see that $I$ is a $C^1$ functional on
$W^{1,N}(\mathbb{R}^N)$.
\end{proof}

\begin{remark} \rm
Recall that if $g:\mathbb{R}\to \mathbb{R}$ is a
continuous function such that $g(0)=0$, and $w$ is a solution of
\eqref{eq:1.1} with $w\in L^\infty_{loc}(\mathbb{R}^N),|\nabla
w|\in L^p(\mathbb{R}^N)$ and $G(w)\in L^1(\mathbb{R}^N)$. Then $w$
satisfies the Pohozaev-Pucci-Serrin identity \cite{SPucci},
\begin{equation}
(N-p)\int_{\mathbb{R}^N}|\nabla w|^p\,dx=Np\int_{\mathbb{R}^N}G(w)\,dx.
\end{equation}
\end{remark}

Let
\begin{equation}\label{m}
m:=\inf \big\{I(u) :u\in W^{1,p}(\mathbb{R}^N)\backslash\{0\}
\mbox{ and $u$ is a solution of \eqref{eq:1.1}}
\big\}.
\end{equation}
By a {\it least energy solution  (or ground state)} of
\eqref{eq:1.1} we mean a minimizer of $m$. Therefore, if $w$ is
a minimizer of $(\ref{m})$ and $\bar w$ is any solution of
\eqref{eq:1.1} then $I(w)\leq I(\bar w)$.


In the case $1<p<N$, we consider the  constrained
minimization problem
\begin{equation}\label{M}
M:=\inf \Big\{\int_{\mathbb{R}^N}|\nabla u|^p\,dx  :u \in
W^{1,p}(\mathbb{R}^N) \mbox{ and }  \int_{\mathbb{R}^N}G(u)\,dx=1
\Big\},
\end{equation}
introduced by Coleman-Glazer and Martin \cite{CGM}.

Next, we establish the existence of a {\it least
energy solution} for \eqref{eq:1.1}.

\begin{theorem}\label{least}
Let $1<p<N$. Under the hypotheses {\rm (G1)--(G4)},
the minimization problem (\ref{M}) has a solution
$u\in W^{1,p}(\mathbb{R}^N)$ which is positive.
\end{theorem}

The proof of this theorem follows the same pattern as the proof of Theorem 2 in
Berestycki an Lions \cite{BL}.


\begin{remark}\rm
Let $u$ be given by Theorem \ref{least}. By Lagrange
Multipliers Theorem there exists a multiplier $\mu$ such that
(in the weak sense)
$$
-\Delta_p u=\mu g(u)\quad \mbox{in } \mathbb{R}^N.
$$
Then after some appropriated scaling $w(x)=u(\mu^{1/(1-p)} x)$ is
a weak solution of \eqref{eq:1.1}.
\end{remark}

In the case $p=N$, we consider the minimization problem
\begin{equation}\label{N}
N:=\inf \Big\{\int_{\mathbb{R}^N}|\nabla u|^p\,dx :u\in
W^{1,p}(\mathbb{R}^N) \mbox{ and } \int_{\mathbb{R}^N}G(u)\,dx=0
\Big\},
\end{equation}
which is motivated by the fact that if $p=N$, from the
Pohozaev-Pucci-Serrin identity,
$$
\int_{\mathbb{R}^N}G(u)\,dx =0.
$$


Now we state a result about the existence of {\it
least energy solution} for \eqref{eq:1.1}. Its
The proof  follows the same method as in Theorem 1
by Berestycki-Gallouet-Kavian \cite{BGK}.

\begin{theorem}\label{p=N}
Let $p=N$. Under the hypotheses {\rm (G1)--(G4)}
the minimization problem $(\ref{N})$ has a solution $u\in
W^{1,N}(\mathbb{R}^N)$ which is positive.
\end{theorem}

 In Section 2, we show that under the assumptions (G1)--(G3), the functional $I$
has the {\it Mountain Pass Geometry} (see Lemma \ref{mountain}
below). In particular, we can conclude that the set
$$
\Gamma= \big\{ \gamma \in C([0,1],W^{1,p}(\mathbb{R}^N))  :
\gamma(0)=0 \mbox{ and } I(\gamma(1))<0 \big\},
$$
is not empty and the {\it mountain pass level}

\begin{equation} \label{eq:c}
c:=\inf_{g\in\Gamma}\max_{0\leq t\leq1} J(\gamma(t)),
\end{equation}
is positive.

\begin{remark} \rm
Under the hypotheses that the function
\begin{equation}\label{crescimento}
s\mapsto g(s)/s
\end{equation}
is increasing for $s>0$, Ding and Ni \cite{DN} obtained the
 characterization
\begin{equation}
c=m=\inf_{v\in W^{1,p}(\mathbb{R}^N)\setminus\{0\}}\max_{t>0}
I(tv) \label{eq:c}.
\end{equation}
\end{remark}

 Without the monotonicity assumption (\ref{crescimento}), we prove that the
level of the mountain pass is a critical value and the corresponding critical
points are least energy solutions.

\begin{theorem}\label{jean}
Let $1<p\leq N$ and assume that {\rm (G1)--(G3)}. Then
$c=m$.
Furthermore, for each least energy solution $w$ of \eqref{eq:1.1},
there exists a path $\gamma\in\Gamma$ such that
$w\in\gamma([0,1])$ and
$$
\max_{t\in[0,1]} I(\gamma(t))=I(w).
$$
\end{theorem}

It has been established in \cite{LD,SZ} that for  $ 1  < p <2$,
positive solutions of problems like \eqref{eq:1.1} are radially
symmetric around some point. In the next result, we obtain the
exponential decay of positive radial solutions of \eqref{eq:1.1}
and their derivatives.

\begin{theorem}\label{theo:1}
Problem \eqref{eq:1.1} has a positive radial solution $w\in
C^{1,\alpha}(\mathbb{R}^N)\cap W_r^{1,p}(\mathbb{R}^N)$ such that
\begin{itemize}
\item[(i)] There exists $r_o>0$ such that $w'(r)\leq0$ for $r\geq
r_o$ and $w\in C^2(r_o,\infty)$

\item[(ii)] $w$ and its first derivatives decay exponentially, i.e.,
there exist $C>0$, $\delta>0$ such that
     \begin{equation}
      |D^\alpha w(x)|\leq Ce^{-\delta|x|}, \quad \mbox{if }  |\alpha|\leq1
      \label{eq:D.1}
      \end{equation}

\item[(iii)] Moreover, $w$
       is a solution with \textit{minimal energy}, i.e., $ 0<I(w)\leq
       I(v)$ for any positive solution $v$ of \eqref{eq:1.1}.
    \end{itemize}
\end{theorem}

In the classical case, when $p=2$, Problem \eqref{eq:1.1} reduces to
$$
-\Delta u=g(u)\quad \mbox{in}\quad \mathbb{R}^N
$$
which has been treated by several authors \cite{BL,BGK,CGM,St}. Our result
can be considered as an extension of the classical case.

\section{Characterization of Mountain Pass Level}

 The main goal of this section is to present the proof of
Theorem \ref{jean}. For this end we use arguments similar in
spirit to those addressed in \cite{Jean}. We divide the prove in
two steps.

First, we prove the {\it Mountain Pass Geometry} for the energy
functional $I$. More precisely, we have the following lemma.

\begin{lemma}[Geometrical Mountain-Pass structure] \label{mountain}
The functional $I$ satisfies the following three conditions:
\begin{itemize}
\item[(i)] $I(0)=0$.
\item[(ii)] There exist $\rho $, $\alpha >0$,
such that $I(u)\geq \alpha $ if
$\|u\|_{W^{1,p}(\mathbb{R}^N)}=\rho$.
\item[(iii)] There is $u_o\in W^{1,p}(\mathbb{R}^N)$ such that
$\|u_0\|_{W^{1,p}(\mathbb{R}^N)}>\rho$ and $I(u_0)<0$.
\end{itemize}
\end{lemma}

\begin{proof}
Statement (i) is trivial. To show {\bf (ii)}, we consider two cases:

\noindent {\bf Case : $1<p<N$.} By our assumptions (G1)--(G3), for any
$\epsilon>0$ there exists $C_\epsilon>0$ such that
\begin{equation}\label{morena}
g(s)\leq(\epsilon-\nu)s^{p-1}+C_\epsilon s^{p^*-1}, \quad
\mbox{for } s\geq0.
\end{equation}
Since $g$ is an odd function, we have
$$
G(s)\leq\frac{1}{p}(\epsilon-\nu)|s|^p+C'_\epsilon |s|^{p^*}, \quad
\mbox{for all } s\in \mathbb{R}.
$$
In view of embedding $W^{1,p}(\mathbb{R}^N)\hookrightarrow
L^{p^*}(\mathbb{R}^N)$ we have
\begin{align*}
I(u)&\geq\frac{1}{p}\int_{\mathbb{R}^N}|\nabla u|^p\,dx+
 \frac{\nu-\epsilon}{p}\int_{\mathbb{R}^N}|u|^p\,dx-\frac{1}{p^*}C'_\epsilon\int_{\mathbb{R}^N}|u|^{p^*}\,dx\\
&\geq\frac{1}{p}\min\{1,\nu-\epsilon\}\|u\|^p_{W^{1,p}(\mathbb{R}^N)}-
 \frac{1}{p^*}C'_\epsilon\|u\|_{L^{p^*}(\mathbb{R}^N)}^{p^*}\\
&\geq\frac{1}{p}\min\{1,\nu-\epsilon\}\|u\|^p_{W^{1,p}(\mathbb{R}^N)}-
 C''_\epsilon\|u\|_{W^{1,p}(\mathbb{R}^N)}^{p^*},
\end{align*}
for all $u\in W^{1,p}(\mathbb{R}^N)$. This implies (ii).

\noindent{\bf Case: $p=N$.} From (G3), given $\epsilon>0$ there is
$\delta>0$ such that
\[
G(u)\leq \frac{\epsilon-\nu}N|u|^N, \quad \mbox{if } | u|\leq\delta.
\]
On the other hand, for $q>N$, by (G2), there is a constant
$C=C(q,\delta )$ such that
\[
G(u)\leq C| u| ^q\big[ \exp ( \beta | u| ^{\frac
N{N-1}}) -S_{N-2}( \beta ,u) \big],\quad
\mbox{if } | u| \geq\delta.
\]
These two estimates yield
\[
G(u)\leq \frac{\epsilon-\nu}N| u| ^N+C| u| ^q\big[ \exp \big(
\beta | u| ^{\frac N{N-1}}\big) -S_{N-2}( \beta ,u)
\big].
\]
 In what follows we make use of the inequality (to be proved later)
\begin{equation}
\int_{\mathbb{R}^N}| u| ^q\big[ \exp ( \beta | u|
^{\frac N{N-1}}) -S_{N-2}( \beta ,u) \big]\,dx
\leq C(\beta ,N)\| u\| _{W^{1,N}(\mathbb{R}^N)}^q, \label{M1}
\end{equation}
provided that $\| u\|_{W^{1,N}(\mathbb{R}^N)}\leq M$, where $M$ is
sufficiently small. Under this assumption, we have
\begin{align*}
I( u) & \geq \dfrac 1N\int_{\mathbb{R}^N}|\nabla u|^N \,dx-\dfrac{(\epsilon-\nu}N)\| u\|
_{L^N(\mathbb{R}^N)}^N-C\| u\|
_{W^{1,N}(\mathbb{R}^N)}^q\\
&\geq C_1 \| u\|
_{W^{1,N}(\mathbb{R}^N)}^N-C\| u\|
_{W^{1,N}(\mathbb{R}^N)}^q.
\end{align*}
Thus, since $\varepsilon >0$ and $q>N$, we may choose $\alpha $, $\rho >0$ such that $I( u) \geq \alpha$ if $\|
u\| _{W^{1,N}(\mathbb{R}^N)}=\rho $. Hence (ii) holds.


Now, we prove inequality (\ref{M1}). We may assume $u\geq 0$,
since we can replace $u$ by $| u| $ without causing any increase
in the integral of the gradient. Here, we make use of Schwarz
symmetrization method. We begin by recalling some basic
properties: let $1\leq p\leq \infty $ and $u\in L^p(\mathbb{R}^N)$
such that $u\geq 0$. Thus, there is a unique nonnegative function
$u^{*}\in L^p(\mathbb{R}^N)$, called the
Schwarz symmetrization of $u$, such that it depends only on $| x| $,
$u^{*}$ is a decreasing function of $| x| ;$ for all $\lambda>0$
\[
| \left\{ x:u^{*}( x) \geq \lambda \right\} | =|
\left\{ x:u( x) \geq \lambda \right\} |
\]
and there exists $R_\lambda >0$ such that $\left\{ x:u^{*}\geq
\lambda \right\} $ is the ball $B[0,R_\lambda ]$ of radius
$R_\lambda $ centered at origin. Moreover, if $G:[0,+\infty
)\to [0,+\infty )$ is a continuous and increasing function
such that $G( 0) =0.$ Then, we have
\[
\int_{\mathbb{R}^N}G(u^{*}(x))
dx=\int_{\mathbb{R}^N}G( u( x))\,dx.
\]
Moreover, if $u\in W^{1,p}(\mathbb{R}^N) $ then $u^{*}\in
W^{1,p}(\mathbb{R}^N) $ and
\[
\int_{\mathbb{R}^N}| \nabla u^{*}| ^p(x) \,dx\leq
\int_{\mathbb{R}^N}| \nabla u| ^p(x) \,dx.
\]
Thus, we can write
\[
\int_{\mathbb{R}^N}\big[ \exp ( \alpha | u| ^{\frac
N{N-1}}) -S_{N-2}( \alpha ,u) \big] \,dx
=\int_{\mathbb{R}^N}\big[ \exp ( \alpha | u^{*}|
^{\frac N{N-1}}) -S_{N-2}( \alpha ,u^{*}) \big]\,dx ,
\]
 Letting $R(\beta,u)=\exp ( \beta | u| ^{\frac N{N-1}}) -S_{N-2}(\beta ,u)$,
 we have
\[
\int_{\mathbb{R}^N}R(\beta ,u)| u| ^q \,dx=\int_{\mathbb{R}^N}R(\beta ,u^{*})| u^{*}| ^q \,dx
\]
and
\begin{equation}
\int_{\mathbb{R}^N}R(\beta ,u^{*})| u^{*}| ^q \,dx=\int_{| x| \leq \sigma }R(\beta ,u^{*})| u^{*}| ^q \,dx+\int_{| x| \geq \sigma }R(\beta ,u^{*})| u^{*}| ^q \,dx, \label{eu}
\end{equation}
where $\sigma $ is a number to be determined later.


Let us recall two elementary inequalities. Using the fact that the function $%
h:(0,+\infty )\to {\mathbb{R}}$ given by $h( t)
=[( t+1) ^{\frac N{N-1}}-t^{\frac N{N-1}}-1]/t^{\frac
1{N-1}}$ is bounded, we have a positive constant $A=A(N)$ such
that
\begin{equation}
( u+v) ^{\frac N{N-1}}\leq u^{\frac N{N-1}}+Au^{\frac
1{N-1}}v+v^{\frac N{N-1}},\quad \forall u,v\geq 0.  \label{TM3}
\end{equation}
If $\gamma $ and $\gamma '$ are positive real numbers such that $%
\gamma +\gamma '=1$, then for all $\varepsilon >0$, we
have
\begin{equation}
u^\gamma v^{\gamma \prime }\leq \varepsilon u+\varepsilon ^{-\frac
\gamma {\gamma '}}v,\quad \forall u,v\geq 0,  \label{TM4}
\end{equation}
because $g:[0,+\infty )\to {\mathbb{R}}$, given by $g(
t) =t^\gamma -\varepsilon t$, is bounded.

Let $v( x) =u^{*}( x) -u^{*}(rx_0) $ where $x_0$ is some fixed unit vector in
$\mathbb{R}^N$. Notice that $v\in W_0^{1,N}( B(
0,r) )$. Here, $B(0,r)$ denotes the ball of radius
$r$ centered at the origin of $\mathbb{R}^N$. Now, from (\ref{TM3}) and
(\ref{TM4}), we have, respectively,
\[
| u^{*}| ^{\frac N{N-1}}=| v+u^{*}( rx_0) | ^{\frac
N{N-1}}\leq v^{\frac N{N-1}}+Av^{\frac 1{N-1}}u^{*}(
rx_0) +u^{*}( rx_0) ^{\frac N{N-1}},
\]
\[
v^{\frac 1{N-1}}u^{*}( rx_0) =( v^{\frac N{N-1}})
^{1/N}( u^{*}( rx_0) ^{\frac N{N-1}}) ^{\frac{N-1}N}
\leq \frac \varepsilon Av^{\frac N{N-1}}
+( \frac \varepsilon A)^{\frac 1{1-N}}u^{*}( rx_0)^{\frac N{N-1}},
\]
and hence,
\[
| u^{*}| ^{\frac N{N-1}}\leq ( 1+\varepsilon ) v^{\frac
N{N-1}}+K(\varepsilon ,N)u^{*}( rx_0) ^{\frac N{N-1}},
\]
where $K(\varepsilon ,N)=A^{\frac N{N-1}}\varepsilon ^{\frac 1{1-N}}+1$. Therefore,
\[
\int_{| x| \leq r}\exp ( \alpha | u^{*}| ^{\frac
N{N-1}}) \leq \exp \big( K(\varepsilon ,N)u^{*}(
rx_0) ^{\frac N{N-1}}\big) \int_{| x| \leq r}\exp
\big( \alpha | ( 1+\varepsilon ) v| ^{\frac N{N-1}}\big) ,
\]
which, in view of Trudinger-Moser inequality, implies,
\begin{equation}
\int_{| x| \leq r}\exp \big( \alpha | u^{*}| ^{\frac
N{N-1}}\big) <\infty ,\quad \forall u\in W^{1,N}(\mathbb{R}^N)
,\quad \forall \alpha >0.  \label{TM5}
\end{equation}
Furthermore, taking $\epsilon >0$ such that
$( 1+\varepsilon) \alpha <\alpha _N$, we obtain
\begin{equation}\label{TM6}
\begin{aligned}
\int_{|x|\leq r}\exp ( \alpha|u^{*}|^{\frac N{N-1}}) &\leq C(N)
\frac{\omega_{N-1}}Nr^N\exp( K(\epsilon ,N)u^{*}( rx_0) ^{\frac
N{N-1}}) \\
&\leq C( N) \frac{\omega_{N-1}}Nr^N
\exp(( \frac{NM^N}{\omega_{N-1}}) ^{\frac 1{N-1}}\frac{K(\epsilon ,N)}{r^{\frac
N{N-1}}}),
\end{aligned}
\end{equation}
for all $u\in W^{1,N}(\mathbb{R}^N) $ such that $\| \nabla
u\| _{L^N(\mathbb{R}^N)}^N\leq 1$ and $\| u\|_{L^N(\mathbb{R}^N)}\leq M$,
where in the last inequality we have
used Radial Lemma A.IV in \cite{BL}:
\[
 | u^{*}( x)| \leq | x| ^{-1}\big( \frac N{\omega_{N-1}}\big) ^{1/N}
 \| u^{*}\| _{L^N(\mathbb{R}^N) },\quad \forall x\neq 0.
\]
 Now, we estimate (\ref{eu}). Using the H\"{o}lder
inequality we obtain
\begin{align*}
\int_{|x|\leq \sigma }R(\beta ,u^{*})|u^{*}|^q \,dx
& \leq
\int_{|x|\leq \sigma }[\exp (\beta|u^{*}|^{\frac N{N-1}})]|u^{*}|^q \,dx \\
& \leq \Big( \int_{| x| \leq \sigma }\exp (\beta r| u^{*}|
^{\frac N{N-1}}) \,dx \Big) ^{1/r}
\Big( \int_{| x|\leq \sigma }| u^{*}| ^{qs} \,dx\Big) ^{1/s},
\end{align*}
where $1/r+1/s=1$. In view, of (\ref{TM6}) we get
$$
\int_{| x| \leq \sigma }\exp (\beta r| u^{*}| ^{\frac
N{N-1}}) \,dx \leq C(\beta ,N)
$$
if $\|u\|_{W^{1,N}(\mathbb{R}^N)}\leq $ $M$, where $M$ is such
that $\beta rM^{\frac N{N-1}}<\alpha _N$. Thus, using the continuous imbedding
$W^{1,N}(\mathbb{R}^N)\hookrightarrow L^{qs}(\mathbb{R}^N)$, we
have
\begin{equation}
\int_{| x| \leq \sigma }R(\beta ,u^{*})| u^{*}| ^q \,dx
\leq C(\beta ,N)\|u\|_{W^{1,N}(\mathbb{R}^N)}^q. \label{M2}
\end{equation}
On the other hand, the Radial Lemma leads to
\begin{align*}
& \int_{|x|\geq \sigma }|u^{*}|^{\frac N{N-1}k}| u^{*}| ^q \,dx\\
& \leq \Big( \big( \frac N{\omega_{N-1}}\big) ^{1/N}
\| u^{*}\| _{L^N(\mathbb{R}^N)}\Big) ^{\frac N{N-1}k}
\int_{| x| \geq \sigma }\frac{| u^{*}| ^q}{|
x| ^{\frac N{N-1}k}} \,dx \\
& \leq \Big( \big( \frac N{\omega_{N-1}}\big) ^{1/N}
\| u^{*}\| _{L^N(\mathbb{R}^N)}\Big) ^{\frac N{N-1}k}
\Big( \int_{| x | \geq \sigma } \frac{dx}{| x|
^{\frac N{N-1}kr}}\Big) ^{\frac 1r}
\Big( \int_{| x| \geq \sigma }| u^{*}| ^{qs} \,dx \Big) ^{1/s} \\
& \leq \omega_{N-1}\sigma ^N\Big( \dfrac{( \frac
N{w_{N-1}}) ^{1/N}\| u^{*}\|
_{L^N(\mathbb{R}^N)}}{\sigma ^r}\Big) ^{\frac N{N-1}k}
\| u\| _{L^{sq}(\mathbb{R}^N)}^q \\
& \leq C(N,M)\| u\| _{W^{1,N}(\mathbb{R}^N)}^q,
\end{align*}
for all $k\geq N$, where  $\sigma ^r=M_0( \frac N{\omega_{N-1}}) ^{1/N}$ and
$\| u\|_{L^N(\mathbb{R}^N)}\leq M_0=\lambda _1(N)^{1/N}M$. We also have
that if $\| u^{*}\| _{W^{1,N}(\mathbb{R}^N)}^q\leq M$,
\begin{align*}
\int_{| x| \geq \sigma }| u^{*}| ^N| u^{*}| ^q \,dx
&\leq \Big( \int_{| x| \geq \sigma }| u^{*}| ^{Nr} \,dx\Big) ^{1/r}
\Big(  \int_{| x| \geq \sigma }| u^{*}| ^{qs} \,dx\Big) ^{1/s} \\
& \leq \| u^{*}\| _{L^{Nr}(\mathbb{R}^N)\|
u^{*}\| _{L^{qs}(\mathbb{R}^N)}^q }\\
& \leq C( N,M) \| u^{*}\|_{W^{1,N}(\mathbb{R}^N)}^q,
\end{align*}
which is shown via the continuous imbedding
$ W^{1,N}(\mathbb{R}^N)\hookrightarrow L^{Nr}(\mathbb{R}^N)$.
Therefore,
\begin{equation}
\int_{| x| \geq\sigma}R_N(\beta ,u^{*})| u^{*}| ^q \,dx\leq C(N,M)\exp (\beta )\| u\|
_{W^{1,N}(\mathbb{R}^N)}^q. \label{M3}
\end{equation}
Finally, the combination of estimates (\ref{M2})-(\ref{M3}) and
(\ref{eu}) implies that (\ref{M1}) is holds.\\

Now we prove (iii). Since $I(0)=0$, by (ii) we have
$I(u)>0$ for all $0<\|u\|_{W^{1,p}(\mathbb{R}^N)}\leq\rho_0$.
Thus, ir suffices to show that $\Gamma\neq\emptyset$. This will
be done in the next Lemma.
\end{proof}


\begin{lemma}\label{fim}
There exists $\gamma$ in the set
$$
\Gamma= \big\{ \gamma \in C([0,1],W^{1,p}(\mathbb{R}^N) :
\gamma(0)=0 \mbox{ and } I(\gamma(1))<0 \big\},
$$
such that
\begin{equation}
w\in\gamma([0,1])\quad \mbox{and}\quad \max_{t\in[0,1]}I(\gamma(t))=m,
\end{equation}
where $w$ is a given least energy.
\end{lemma}

\begin{proof}
Let $w$ be a given {\it least energy solution} of
(\ref{eq:1.1}.
In the case $1<p<N$, we consider the curve $\gamma:[0, \infty
)\to W^{1,p}(\mathbb{R}^N)$ defined by
\[\gamma(t)(x) = \begin{cases}
   w(x/t)& \mbox{if }  t>0,\\
  0 & \mbox{if }   t=0.
\end{cases}
\]
It is not difficult to see that
\begin{itemize}
\item[(i)]  $\|\gamma(t)\|^p_{W^{1,p}(\mathbb{R}^N)}=t^{N-p}\|\nabla
w\|_{L^p(\mathbb{R}^N)}^p+t^N\|w\|_{L^p(\mathbb{R}^N)}^p$

\item[(ii)]  $ I(\gamma(t))=\frac{t^{N-p}}{p}\|\nabla
w\|_{L^p(\mathbb{R}^N)}^p-t^N\int_{\mathbb{R}^N}G(w)dx =
\frac{t^N}{p} \|\nabla w\|^p_{L^p(\mathbb{R}^N)} \Big(
\frac{1}{t^p} - \frac{N-p}{N} \Big)$, where in the above term we
have used the Pohozaev-Pucci-Serrin identity.
\end{itemize}
Using (i), we have
\[
\lim_{t \to 0} \| \gamma (t) \|_{W^{1,p}(\mathbb{R}^N)} =
0,
\]
which implies that $ \gamma $ is continuous. From (ii)
and $ 1 < p < N $, we obtain a value $ L > 0 $ such that $I(\gamma(L)) < 0 $.
These facts together with a suitable scale
change in $ t $, imply that there exists the desired path
$\gamma\in \Gamma$.

In the case $p=N$, we choose real numbers $ 0 < t_0 < 1 <  t_1 <\theta_1 $
 so that a curve $\gamma$, constituted of three
pieces defined below, gives a desired path:
\[
 \gamma(\theta) = \begin{cases}
  \theta \omega_{t_0} & \mbox{if }  \theta \in [0, t_0],\\
  \theta \omega_{\theta} & \mbox{ if }   \theta \in  [t_0, t_1],\\
  \theta \omega_{t_1} & \mbox{ if }  \theta \in [t_1, \theta_1],
\end{cases}
\]
where $w_t(x)=w(x/t)$. Since $w$ is a weak solution we have
$$
\int_{\mathbb{R}^N}g(w)w\,dx=\|\nabla w\|^N_{L^N(\mathbb{R}^N)} >0.
$$
Thus we can find $\theta_1>1$ such that
$$
\int_{\mathbb{R}^N}g(\theta w)w\,dx>0\quad \mbox{for all }
\theta\in [1, \theta_1].
$$
Next we set $\varphi(s)=g(s)/s^{N-1}$. By assumption (G3) we
have $\varphi\in C(\mathbb{R}, \mathbb{R})$. Therefore,
\begin{equation}\label{Mari}
\int_{\mathbb{R}^N}\varphi(\theta w)w^N\,dx>0\quad \mbox{for all }
 \theta\in [1, \theta_1].
\end{equation}
Now note that
\begin{align*}
\frac{d}{d \theta}I(\theta w_t)
& =  I'(\theta w_t)w_t \\
& =  \theta^{N-1} \Big(\|\nabla w_t\|_{L^N(\mathbb{R}^N)}^N
-\int_{\mathbb{R}^N}\varphi(\theta w_t)w_t^{N}\,dx\Big)\\
& =  \theta^{N-1} \Big(\|\nabla w \|_{L^N(\mathbb{R}^N)}^N
-t^N \int_{\mathbb{R}^N}\varphi(\theta w)w^{N}\,dx\Big).
\end{align*}
Choosing $t_0\in(0,1)$ sufficiently small, we have
\begin{equation}\label{Maria2}
\|\nabla
w_t\|_{L^N(\mathbb{R}^N)}^N-t_0^N\int_{\mathbb{R}^N}\varphi(\theta
w)w^N\,dx>0\quad \mbox{for all } \theta\in [1, \theta_1].
\end{equation}
By $(\ref{Mari})$, we can also choose $t_1>1$ such that
for all $\theta\in [1,\theta_1]$,
\begin{equation}\label{Maria3}
\|\nabla w\|_{L^N(\mathbb{R}^N)}^N-t_1^N\int_{\mathbb{R}^N}\varphi(\theta
w)w^2\,dx\leq -\frac{1}{\theta_1-1}\|\nabla w\|^N_{L^N(\mathbb{R}^N)}\,.
\end{equation}
Thus we can see by (\ref{Maria2}) that the function
$I(\gamma(\theta))$ is increasing on the interval $ [0,t_0]$ and
takes its maximal at $\theta=1$. By Pohozaev-Pucci-Serrin identity
we have $\int_{\mathbb{R}^N}G(w)=0$. Consequently
$$
I(w_{t_1})=I(w)=\frac{1}{N}\|\nabla w\|_{L^N(\mathbb{R}^N)}^N.
$$
Now note that
\begin{align*}
I(\theta_1w_{t_1})&=
I(w_{t_1})+\int_1^{\theta_1}\frac{d}{dt}I(\theta w_{t_1})d\theta\\
&\leq\frac{1}{N}\|\nabla w\|_{L^N(\mathbb{R}^N)}^N-\frac{1}{\theta_1-1}
\int_1^{\theta_1}\|\nabla w\|^N_{L^N(\mathbb{R}^N)}d\theta\\
&<(\frac{1}{N}-1)\|\nabla w\|_{L^N(\mathbb{R}^N)}^N<0.
\end{align*}
Thus, we have obtained the desired curve.
\end{proof}

As consequence of Lemma \ref{fim} we have the following important
step of the proof of Theorem \ref{jean}.

\begin{coro}\label{jean1}
With $c$ and $m$ as defined in \eqref{eq:c} and \eqref{m}, we have
$ c\leq m $.
\end{coro}

In view of the Pohozaev-Pucci-Serrin identity we have

\begin{lemma}\label{lima}
For $1<p\leq N$, we obtain
\begin{equation}\label{P}
m=\inf_{u\in\mathcal{P}}I(u),
\end{equation}
where
\[
\mathcal{P}=\Big\{u\in W^{1,p}(\mathbb{R}^N)\setminus\{0\} :
(N-p)\int_{\mathbb{R}^N}|\nabla u|^p\,dx = Np
\int_{\mathbb{R}^N}G(u)\,dx\Big\}.
\]
\end{lemma}

\begin{proof}
For the case $1<p<N$, we introduce the set
$$
\mathcal{S}= \Big\{ u\in W^{1,p}(\mathbb{R}^N) :\int_{\mathbb{R}^N}G(u)\,dx=1
\Big\},
$$
which is in one-to-one correspondence with the set $ \mathcal{P}$ via
the map $\Phi:\mathcal{S}\to\mathcal{P}$:
$\Phi(u)(x)=u(x/t_u)$ with
$t_u=\big(\frac{N-p}{Np}\big)^{1/p}\|\nabla u\|_{L^p(\mathbb{R}^N)}$.
Thus,
\[
\inf_{u\in\mathcal{P}}I(u)=\inf_{u\in\mathcal{S}} I(\Phi(u)).
\]
Next we prove that
$\inf_{u\in\mathcal{S}} I(\Phi(u)) = m$.
 From Theorem \ref{least}, there exists $ u_0 \in W^{1,p}(\mathbb{R}^N)$
such that
\[
M = \inf_{u \in \mathcal{S} } \int_{\mathbb{R}^N} | \nabla u |^p \,dx
= \int_{\mathbb{R}^N} | \nabla u_0 |^p \,dx .
\]
After a suitable scale change, $ \Phi(u_0) $ becomes a least
energy solution; that is, $ I(\Phi(u_0)) = m$. By the
Pohozaev-Pucci-Serrin identity,
\begin{align*}
I(\Phi(u_0))&=\frac{1}{p}t_{u_0}^{N-p}\|\nabla
u_0\|_{L^p(\mathbb{R}^N)}^p-t_{u_0}^N\int_{\mathbb{R}^N} G(u_0)\,dx\\
&=\frac{1}{N}\Big(\frac{N-p}{Np}\Big)^{(N-p)/{p}}\|\nabla
u_0\|_{L^p(\mathbb{R}^N)}^N \\
& =  \inf_{u\in
S}\frac{1}{N}\Big(\frac{N-p}{Np}\Big)^{\frac{N-p}{p}}\|\nabla
u\|_{L^p(\mathbb{R}^N)}^N.
\end{align*}
Thus we have (\ref{P}) in the case $1<p<N$.

For the case  $p=N$, we have
$$
\mathcal{P}=\Big\{u\in W^{1,p}(\mathbb{R}^N)\setminus\{0\} :
\int_{\mathbb{R}^N}G(u)\,dx =0\Big\}.
$$
Thus
\begin{align*}
 \inf_{u \in \mathcal{P}} I(u) & =  \frac{1}{N} \inf \Big\{
\int_{\mathbb{R}^N} |\nabla u|^N\,dx :u\in
W^{1,p}(\mathbb{R}^N) \mbox{ and }
\int_{\mathbb{R}^N}G(u)\,dx =0 \Big\} \\
& =  \frac{1}{N} \int_{\mathbb{R}^N}|\nabla u_0|^N\,dx \,,
\end{align*}
where in the last equality we have used Theorem \ref{p=N}. On the
other hand
\[
\int_{\mathbb{R}^N}|\nabla u_0|^N\,dx  = N I(u_0) = Nm.
\]
Thus we have (\ref{P}) in the case $ p=N $. Therefore the proof of
Lemma~\ref{P} is complete.
\end{proof}

To complete the proof of  Theorem \ref{jean}, in view of
Corollary~\ref{jean1} and Lemma \ref{lima}, it only remains to
prove that $ m \leq c $, which is a consequence of the following
result.

\begin{lemma}\label{I}
For all $\gamma\in\Gamma$,
$\gamma([0,1])\cap\mathcal{P}\neq\emptyset$.
\end{lemma}

\begin{proof} \textbf{Case: $1<p<N $.} We consider the functional
\[
P(u)=\frac{N-p}{p}\|\nabla u\|_{L^p(\mathbb{R}^N)}^p-\int_{\mathbb{R}^N}G(u)\,dx
=NI(u)-\|\nabla u\|_{L^p(\mathbb{R}^N)}^p,
\]
defined in $ W^{1,p}(\mathbb{R}^N)$. Using (\ref{morena}), it is
not difficult to see that there exists $ \rho_0 > 0 $ such that
\[
P(u)>0 \quad \mbox{ for } 0 < \|u\|_{W^{1,p}(\mathbb{R}^N)}\leq \rho_0 .
\]
For each $\gamma\in \Gamma$ we have
$P(\gamma(1))=NI(\gamma(1))-\|\nabla\gamma(1)\|_{L^p(\mathbb{R}^N)}^p\leq NI(\gamma(1))< 0$
and $\gamma(0)=0$. Thus
there exists $t_0\in [0,1]$ such that
$$
\|\gamma(t_0)\|_{W^{1,p}(\mathbb{R}^N)}>\rho_0,
\quad \mbox{and}\quad  P(\gamma(t_0))=0.
$$
Therefore, $\gamma(t_0)\in \gamma([0,1])\cap{\mathcal{P}}$.

\noindent \textbf{Case: $p=N$.}  We consider $ \rho\in
C_0^\infty(\mathbb{R}^N , [0,\infty))$ such that
$\int_{\mathbb{R}^N}\rho(x)\,dx=1$.
For $\gamma\in\Gamma$ and $\epsilon>0$, we define
$\gamma_\epsilon:[0,1]\to W^{1,N}(\mathbb{R}^N)$ given by
$$
\gamma_\epsilon(t)(x)=\int_{\mathbb{R}^N}\rho\big(\frac{x-y}{\epsilon}\big)
\gamma(t)(y)dy.
$$
It is easy to see that the function $\gamma_\epsilon$ satisfies
the following three properties:
\begin{itemize}
\item[(i)] $\gamma_\epsilon(t)\in
    L^{\infty}(\mathbb{R}^N)$, for all $t\in [0,1]$
\item[(ii)] $\gamma_\epsilon \in C([0,1], L^{\infty}(\mathbb{R}^N))$
\item[(iii)]
$\max_{t\in[0,1]}\|\gamma_\epsilon(t)-\gamma(t)\|_{W^{1,N}(\mathbb{R}^N)}\to0$
    as $\epsilon\to0$.
\end{itemize}
Now, using assumption (G3) there exists $\rho_0 >0$ such that
\begin{equation}\label{F}
P(u)>0 \mbox{ if } 0<\|u\|_\infty\leq\rho_0.
\end{equation}
 By (iii), we have $P(\gamma(1))\leq N I(\gamma_\epsilon(1))<0$
and $\gamma(0)=0$ for all $\epsilon>0$. Thus, using (\ref{F}) and
(ii) we obtain that $P(\gamma_\epsilon(t))>0$ for $t>0$
sufficiently small. Therefore, we can find $t_\epsilon\in[0,1]$
such that
\[
\|\gamma_\epsilon(t_\epsilon)\|_\infty>\rho_0,\quad
P(\gamma_\epsilon(t_\epsilon))=0.
\]
That is, $\gamma_\epsilon(t_\epsilon)\in \mathcal{P}$.
We extract a subsequence $\epsilon_n\to 0$ such that
$t_{\epsilon_n}\to t_0$. From (ii)-(iii) it follows that
$$
\|\gamma_\epsilon(t_{\epsilon_n})-\gamma(t_0)\|_{W^{1,N}(\mathbb{R}^N)}\to0,\quad
P(\gamma(t_0))=0\,.
$$
Now we claim that $\gamma(t_0)\neq0$. Indeed, by Theorem~\ref{p=N},
$$
\inf_{u\in\mathcal{P}}\|\nabla u\|_{L^N(\mathbb{R}^N)}^N=2m>0.
$$
Therefore, $\|u\|_{W^{1,N}(\mathbb{R}^N)}\geq(Nm)^{1/N}$ for all
$u\in\mathcal{P}$. In particular,
$$
\|\gamma_\epsilon(t_{\epsilon_n})\|_{W^{1,N}(\mathbb{R}^N)}\geq(Nm)^{1/N}.
$$
Consequently,
$\|\gamma(t_0)\|_{W^{1,N}(\mathbb{R}^N)}\geq(Nm)^{1/N}>0$. Thus
$\gamma(t_0)\in\gamma([0,1])\cap\mathcal{P}$ and
$\gamma([0,1])\cap\mathcal{P}\neq\emptyset$. This, show the Lemma in
the case $p=N$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{jean}]
By Corollary \ref{jean1} we have $c\leq m$.
On the outer hand,  Lemmas \ref{lima} and \ref{I} imply
$$ m=\inf_{u\in\mathcal{P}}I(u)\leq c.
$$
Thus, the proof of Theorem is complete.
\end{proof}

\section{Asymptotic Behavior}

In this section we show the decay at infinity of the weak solution
and its derivatives.

\begin{proof}[Proof of Theorem \ref{theo:1}]
The exponential decay of $w$ at infinity  is already known \cite[Theorem 2.3]{Su}.
We show first that there exists $r_o>0$
such that $w'(r)\leq0$ for $r\geq r_o$. Indeed, since $w$ has
exponential decay at infinity, it follows form (G1) that there
exists $r_1>0$ such that
\begin{equation}
\int_{r_1}^\infty r^{N-1}|w'|^{p-2}w'\varphi'\,dr
=\int_{r_1}^\infty r^{N-1}g(u(r))\varphi\,dr<0
\label{eq:1.r}
\end{equation}
for all $ 0\leq\varphi\in W_r^{1,p}(0,+\infty)$ with
$\mathop{\rm supp}\varphi\subset(r_1,\infty)$. The result then follows by
contradiction. Take $r_o>r_1+1$ and suppose that exists $r'\geq
r_o$ such that, $w'(r')>0$. Since $w'$ is continuous, there exists
$\delta>0$ such that $w'(r)>0$ for $r\in (r'-\delta, r'+\delta)$.
Choosing the test function
$$
\varphi(r)= \begin{cases}
0 & \mbox{if } 0\leq r\leq r'-\delta,\\
\frac{w(r'+\delta)}{2\delta}(r-r'+\delta)
 &  \mbox{if }  r'-\delta<r\leq r'+\delta, \\
w(r) & \mbox{if } r\geq r'+\delta
\end{cases}
$$
in (\ref{eq:1.r}) we have
$$
\int_{r'-\delta}^{r'+\delta}r^{N-1}|w'|^{p-2}w'(r)\,dr<0.
$$
This is a contradiction. Therefore, there exists $r_o>0$ such that
$w'(r)\leq0$ for $r\geq r_o$. Next, we show that $w'$ has
exponential decay. Since $w$ is radial,
\begin{equation}
\int_0^\infty r^{N-1}|w'|^{p-2}w'\varphi'\,dr=\int_0^\infty
h(r)\varphi\,dr\quad  \forall \, \varphi\in W_r^{1,p}(\mathbb{R}^N),
\end{equation}
where $h(r)=r^{N-1}g(w(r))$. If $u(r)=\int_r^\infty h(s)ds$ we
have $u'(r)=-h(r)$. Consequently, if
$v(r)=r^{N-1}|w'(r)|^{p-2}w'-u(r)$ we have
$$
\int_0^\infty v(s)\varphi'(s)ds=0\ \ \ \ \ \forall\ \ \varphi\in
W_r^{1,p}(0,\infty).
$$
Therefore, by \cite[Lemma VIII.1]{HB}, there exists a constant
$C$ such that
\begin{equation} r^{N-1}|w'|^{p-2}w'=C+u(r)\,.
\label{eq:1.6R}
\end{equation}
We claim that $C=0$. Indeed, suppose that $C\neq0$. By the
exponential decay of $u$ and (\ref{eq:1.6R}), there exists a
constant $C_1>0$ such that for $r$ sufficiently large
$$
r^{N-1}|w'(r)|^{p-1}\geq C-ce^{-\theta r}\geq C_1/r,
$$
that is,
\begin{equation}
|w'(r)|\geq\frac{C_1}{r^\alpha}\label{eq:1.7},
\end{equation}
where $\alpha=\frac{N}{p-1}$. Since $p\leq N$ we have $\alpha>1$.
Integrating (\ref{eq:1.7}) from $R$ to $r$ and using the fact of
that $w'(r)\leq0$ for $r\geq r_o$, we obtain
\begin{equation}
-w(r)+w(R)\geq
\frac{C_1}{1-\alpha}(\frac{1}{r^{\alpha-1}}-\frac{1}{R^{\alpha-1}})
\label{eq:1.8}.
\end{equation}
Letting $r\to\infty$ in (\ref{eq:1.8}) we have $$
w(R)\geq\frac{C_1}{(\alpha-1)}\frac{1}{R^{\alpha-1}},
$$
for $R$ sufficiently large. This contradicts the exponential decay
of $w$. Therefore,
\begin{equation}
r^{N-1}|w'|^{p-2}w'=u(r). \label{eq:1.9}
\end{equation}
It follows from (\ref{eq:1.9}) that $w'$ has exponential decay.
Moreover, $w\in C^2(r_o,\infty)$. This completes the proof of
Theorem~\ref{theo:1}.
\end{proof}

\subsection*{Acknowledgments}
This was work partially supported by  CNPq, PRONEX-MCT/Brazil and Millennium
Institute for the Global Advancement of Brazilian Mathematics -
IM-AGIMB.

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