\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 $10$, 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 $10$ such that $G(\zeta)>0$, where $G(u)=\int_0^{u}g(t)dt$. \end{itemize} \smallskip \noindent\textbf{Example.} Let $10$ 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 $10$ and $u\in W^{1,N}(\mathbb{R}^N)$, then $$\int_{\mathbb{R}^N}\Big[ \exp \big( \alpha | u | ^{\frac N{N-1}}\big) -S_{N-2}\big( \alpha ,u\big) \Big]\,dx <\infty . \label{TM1}$$ 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 $$\int_{\mathbb{R}^N}\big[ \exp \big( \alpha | u| ^{\frac N{N-1}}\big) -S_{N-2}( \alpha ,u) \big] \,dx \leq C( N,M,\alpha ) . \label{TM2}$$ \end{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: $10$. 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}, $$(N-p)\int_{\mathbb{R}^N}|\nabla w|^p\,dx=Np\int_{\mathbb{R}^N}G(w)\,dx.$$ \end{remark} Let $$\label{m} m:=\inf \big\{I(u) :u\in W^{1,p}(\mathbb{R}^N)\backslash\{0\} \mbox{ and u is a solution of \eqref{eq:1.1}} \big\}.$$ 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 $10$, Ding and Ni \cite{DN} obtained the characterization $$c=m=\inf_{v\in W^{1,p}(\mathbb{R}^N)\setminus\{0\}}\max_{t>0} I(tv) \label{eq:c}.$$ \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 $10$ 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 $$|D^\alpha w(x)|\leq Ce^{-\delta|x|}, \quad \mbox{if } |\alpha|\leq1 \label{eq:D.1}$$ \item[(iii)] Moreover, $w$ is a solution with \textit{minimal energy}, i.e., $00$, 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 : $10$ there exists $C_\epsilon>0$ such that $$\label{morena} g(s)\leq(\epsilon-\nu)s^{p-1}+C_\epsilon s^{p^*-1}, \quad \mbox{for } s\geq0.$$ 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) $$\int_{\mathbb{R}^N}| u| ^q\big[ \exp ( \beta | u| ^{\frac N{N-1}}) -S_{N-2}( \beta ,u) \big]\,dx \leq C(\beta ,N)\| u\| _{W^{1,N}(\mathbb{R}^N)}^q, \label{M1}$$ 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 $$\int_{\mathbb{R}^N}R(\beta ,u^{*})| u^{*}| ^q \,dx=\int_{| x| \leq \sigma }R(\beta ,u^{*})| u^{*}| ^q \,dx+\int_{| x| \geq \sigma }R(\beta ,u^{*})| u^{*}| ^q \,dx, \label{eu}$$ 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 $$( u+v) ^{\frac N{N-1}}\leq u^{\frac N{N-1}}+Au^{\frac 1{N-1}}v+v^{\frac N{N-1}},\quad \forall u,v\geq 0. \label{TM3}$$ If $\gamma$ and $\gamma '$ are positive real numbers such that $% \gamma +\gamma '=1$, then for all $\varepsilon >0$, we have $$u^\gamma v^{\gamma \prime }\leq \varepsilon u+\varepsilon ^{-\frac \gamma {\gamma '}}v,\quad \forall u,v\geq 0, \label{TM4}$$ 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, $$\int_{| x| \leq r}\exp \big( \alpha | u^{*}| ^{\frac N{N-1}}\big) <\infty ,\quad \forall u\in W^{1,N}(\mathbb{R}^N) ,\quad \forall \alpha >0. \label{TM5}$$ Furthermore, taking $\epsilon >0$ such that $( 1+\varepsilon) \alpha <\alpha _N$, we obtain \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} 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 $$\int_{| x| \leq \sigma }R(\beta ,u^{*})| u^{*}| ^q \,dx \leq C(\beta ,N)\|u\|_{W^{1,N}(\mathbb{R}^N)}^q. \label{M2}$$ 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, $$\int_{| x| \geq\sigma}R_N(\beta ,u^{*})| u^{*}| ^q \,dx\leq C(N,M)\exp (\beta )\| u\| _{W^{1,N}(\mathbb{R}^N)}^q. \label{M3}$$ 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 $$w\in\gamma([0,1])\quad \mbox{and}\quad \max_{t\in[0,1]}I(\gamma(t))=m,$$ 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 $10,\\ 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, $$\label{Mari} \int_{\mathbb{R}^N}\varphi(\theta w)w^N\,dx>0\quad \mbox{for all } \theta\in [1, \theta_1].$$ 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*} Choosingt_0\in(0,1)$sufficiently small, we have $$\label{Maria2} \|\nabla w_t\|_{L^N(\mathbb{R}^N)}^N-t_0^N\int_{\mathbb{R}^N}\varphi(\theta w)w^N\,dx>0\quad \mbox{for all } \theta\in [1, \theta_1].$$ By$(\ref{Mari})$, we can also choose$t_1>1$such that for all$\theta\in [1,\theta_1]$, $$\label{Maria3} \|\nabla w\|_{L^N(\mathbb{R}^N)}^N-t_1^N\int_{\mathbb{R}^N}\varphi(\theta w)w^2\,dx\leq -\frac{1}{\theta_1-1}\|\nabla w\|^N_{L^N(\mathbb{R}^N)}\,.$$ 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} Withc$and$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 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 $$\label{F} P(u)>0 \mbox{ if } 0<\|u\|_\infty\leq\rho_0.$$ 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 $$\int_{r_1}^\infty r^{N-1}|w'|^{p-2}w'\varphi'\,dr =\int_{r_1}^\infty r^{N-1}g(u(r))\varphi\,dr<0 \label{eq:1.r}$$ 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 $$\varphi(r)= \begin{cases} 0 & \mbox{if } 0\leq r\leq r'-\delta,\\ \frac{w(r'+\delta)}{2\delta}(r-r'+\delta) & \mbox{if } r'-\delta0 such that w'(r)\leq0 for r\geq r_o. Next, we show that w' has exponential decay. Since w is radial, $$\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),$$ where h(r)=r^{N-1}g(w(r)). If u(r)=\int_r^\infty h(s)ds we have u'(r)=-h(r). Consequently, if v(r)=r^{N-1}|w'(r)|^{p-2}w'-u(r) we have$$ \int_0^\infty v(s)\varphi'(s)ds=0\ \ \ \ \ \forall\ \ \varphi\in W_r^{1,p}(0,\infty). $$Therefore, by \cite[Lemma VIII.1]{HB}, there exists a constant C such that $$r^{N-1}|w'|^{p-2}w'=C+u(r)\,. \label{eq:1.6R}$$ We claim that C=0. Indeed, suppose that C\neq0. By the exponential decay of u and (\ref{eq:1.6R}), there exists a constant C_1>0 such that for r sufficiently large$$ r^{N-1}|w'(r)|^{p-1}\geq C-ce^{-\theta r}\geq C_1/r, $$that is, $$|w'(r)|\geq\frac{C_1}{r^\alpha}\label{eq:1.7},$$ where \alpha=\frac{N}{p-1}. Since p\leq N we have \alpha>1. Integrating (\ref{eq:1.7}) from R to r and using the fact of that w'(r)\leq0 for r\geq r_o, we obtain $$-w(r)+w(R)\geq \frac{C_1}{1-\alpha}(\frac{1}{r^{\alpha-1}}-\frac{1}{R^{\alpha-1}}) \label{eq:1.8}.$$ Letting r\to\infty in (\ref{eq:1.8}) we have$$ w(R)\geq\frac{C_1}{(\alpha-1)}\frac{1}{R^{\alpha-1}},$$for$R$sufficiently large. This contradicts the exponential decay of$w$. Therefore, $$r^{N-1}|w'|^{p-2}w'=u(r). \label{eq:1.9}$$ It follows from (\ref{eq:1.9}) that$w'$has exponential decay. Moreover,$w\in C^2(r_o,\infty)$. 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