\documentclass[twoside]{article} \usepackage{amssymb} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Solutions to the Kohn--Laplace equation\hfil EJDE--1999/12} {EJDE--1999/12\hfil Stepan Tersian \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 1999}(1999), No.~12, pp. 1--11. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Nontrivial solutions to the semilinear Kohn--Laplace equation on ${\mathbb R}^3$ \thanks{ {\em 1991 Mathematics Subject Classifications:} 35J20, 35J25. \hfil\break\indent {\em Key words and phrases:} Kohn--Laplace equation, homoclinic type solutions, \hfil\break\indent homogeneous spaces, mountain-pass theorem. \hfil\break\indent \copyright 1999 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted November 26, 1998. Published April 15, 1999.} } \date{} % \author{Stepan Tersian} \maketitle \begin{abstract} The existence of nontrivial solutions to the semilinear Kohn--Laplace equation $$-\Delta _Hu+V(P)u=f(u)$$ is considered under appropriate assumptions on $V(P)$ and $f(u)$. Results are obtained using a variational method and a compact-embedding lemma. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Variational methods have been used for obtaining homoclinic solutions of second-order semilinear ordinary differential equations, and homoclinic type solutions of semilinear elliptic equations on the whole space. See for example Ding and Ni \cite{DN}, Rabinowitz \cite{R}, Omana and Willem \cite{OW}, and Korman and Lazer \cite{KL}. In most of the references, the linear part of the equation is assumed strongly elliptic, which motivates us to consider the degenerate semilinear elliptic case. In particular, we study the existence of a nontrivial solution, $u\in W_H^{1,2}({\mathbb R}^3)$, to the semilinear Kohn--Laplace or Heisenberg equation $$-\Delta _Hu+V(P)u=f(u),\quad P( x,y,z) \in {\mathbb R}^3\,, \label{k1}$$ where$\Delta _H=\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}% +4( ( x^2+y^2) \frac{\partial ^2}{\partial z^2}+y\frac{% \partial ^2}{\partial z\partial x}-x\frac{\partial ^2}{\partial x\partial y}% )$ is the Kohn--Laplacian. We assume that $V(P)\in C({\mathbb R}^n{\mathbb R})$, \begin{eqnarray} &V(P)>0,\quad \forall P\in {\mathbb R}^n\,,\quad\mbox{and}& \label{k2} \\ &V(P)\to +\infty \quad \mbox{as}\quad |P|\to +\infty \,,& \label{k3} \end{eqnarray} where $|P|=\sqrt{x^2+y^2+z^2}$ is the norm in ${\mathbb R}^3$. We assume that $f(t)\in C({\mathbb R})$ satisfying \begin{eqnarray} &f(0)=0,\quad f(t)=o(|t|)\quad \mbox{as}\quad t\to 0\,, &\label{k4}\\ &f(t)=o(|t|^3)\quad \mbox{as}\quad t\to \infty \,, &\label{k5}\\ &0<\mu F(t)\equiv \mu \int_0^tf(s)\,ds\le tf(t)\quad \mbox{with $\mu >2$.} &\label{k6} \end{eqnarray} Let $W_H^{1,2}( {\mathbb R}^3)$ be the Sobolev space associated with the Kohn--Laplacian $\Delta _H$, and $X=\{u(P)\in W_H^{1,2}({\mathbb R}^3):\int_{{\mathbb R}^3}(|\nabla _Hu(P)|^2+V(x)|u(P)|^2)\,dP<\infty \}.$ Hereafter, $\int$ means integration in ${\mathbb R}^3$. Our main result is as follows \begin{theorem} \label{T1} Let (\ref{k2})--(\ref{k6}) hold for $V$ and $f$. Then the equation $-\Delta _Hu+V(P)u=f(u)$ has a nontrivial solution $u\in W_H^{1,2}({\mathbb R}^3)$. \end{theorem} This paper is organized as follows: In the first section, we state some preliminary results on the Kohn--Laplace operator and prove embedding lemmata which are essential for the forthcoming considerations. We use properties of ${\mathbb R}^3$ as a homogeneous space, with homogeneous dimension 4 with respect to the intrinsic distance, using the Sobolev and Poincare inequalities considered in Biroli and Mosco \cite{BM}, Biroli, Mosco and Tchou \cite{BMT}, and in Jerison \cite{J}. In the second section, we prove an embedding lemma and an existence result for (\ref{k1}) using the mountain-pass theorem. This idea comes from Omana and Willem \cite{OW}, where homoclinic solutions of Hamiltonian systems are considered. There are difficulties involved in checking that the corresponding functional satisfies the Palais--Smale condition (as a difference with the one-dimensional case). To overcome these difficulties, we prove a proposition that is analogous to the one in P.L. Lions \cite{PL}. In a forthcoming paper by Biroli and Tersian, an extension to semilinear equations related to a general Dirichlet's form will be given. \section{Preliminary results} First, we recall the definition of the Kohn--Laplace operator $\Delta _H$. Let \begin{eqnarray*} &\xi =\frac \partial {\partial x}+2y\frac \partial {\partial z},\qquad \eta =\frac \partial {\partial y}-2x\frac \partial {\partial z}\,, \nabla _H=( \xi ,\eta ) =\sigma \nabla \,, &\\ &\sigma =\left[ \begin{array}{ccc} 1 & 0 & 2y \\ 0 & 1 & -2x \end{array} \right] ,\quad \nabla =( \frac \partial {\partial x},\frac \partial {\partial y},\frac \partial {\partial z}) , &\\ &\Delta _H=\nabla _H^2={\rm div}\,( \sigma ^T\sigma \nabla ) \,,&\\ &\Delta _H=\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2} +4( ( x^2+y^2) \frac{\partial ^2}{\partial z^2}+y\frac{ \partial ^2}{\partial z\partial x}-x\frac{\partial ^2}{\partial x\partial y})\,. &\end{eqnarray*} The operator $\Delta _H$ is elliptic, $P^T\sigma ^T\sigma P\geq 0$ for every $P\in {\mathbb R}^3$, but not necessarily strongly elliptic, because the eigenvalues of the matrix$\sigma ^T\sigma =\left[ \begin{array}{ccc} 1 & 0 & 2y \\ 0 & 1 & -2x \\ 2y & -2x & 4y^2+4x^2 \end{array} \right]$ are $0, 1, 1+4y^2+4x^2$ and its rank is $2$. The intrinsic distance $\rho ( P,P')$ between $P( x,y,z)$ and $P'( x',y',z^{\prime })$ associated with the operator $\Delta _H$ is defined as$\rho ( P,P') =\left( ( ( x-x')^2+( y-y') ^2) ^2+( z-z'-2( x'y-xy') ) ^2\right) ^{1/4}.$ Under the distance $\rho$ the intrinsic ball $B_\rho ( P_0,r)$ is defined as$B_\rho ( P_0,r) =\{P:\rho ( P_0,P) \leq r\}.$ For vectors $P( x,y,z)$ and $P'( x',y',z')$, we define the $P'$ right translation as $P\oplus P'=( x+x',y+y',z+z'+2(x'y-xy') ) .$ Let $m( B)$ denote the volume of the Euclidean ball with radius $r$, $B(P_0,r)\subset {\mathbb R}^3$. \begin{proposition} \label{P1} \begin{description} \item{(i)} $B( 0,r) \subset B_\rho ( 0,r) \subset B( 0,r^2)$ for $r\geq 1$; and $B( 0,r^2) \subset B_\rho ( 0,r) \subset B( 0,r)$ for $00.$ \end{lemma} \paragraph{Proof.} Let $\{u_k\}$ be a bounded sequence in $W_{H,0}^{1,2}(B_\rho ( 0,R) )$, with $\|u_k\|_{W_H^{1,2}}\leq A$, and $u_k\rightarrow u$ weakly in $W_{H,0}^{1,2}(B_\rho ( 0,R) )$ and $L^2(B_\rho ( 0,R) )$. We also denote by $u_k$ the extension of $u_k$ to ${\mathbb R}^3$ by $0$, which belongs to $W_H^{1,2}({\mathbb R}^3)$. Let $\varepsilon$ be an arbitrary positive number and $\{B_j\}$, $B_j=B_\rho (P_j,r)$, be the covering of $B_R=B_\rho ( 0,R)$ by intrinsic balls with radius $r=( \frac \varepsilon {32AC}) ^{1/2}$, such that every point of $B_R$ belongs to at most 4 balls $B_j$. By a result of Jerison \cite{J}$\int_{B_j}|u(P)-\bar{u}_j|^2\,dP\leq Cr^2\int_{B_j}|\nabla _Hu(P)|^2\,dP\,,$ where $C$ is a constant independent of $u$ and $j$, and$\bar{u}_j=\frac 1{m(B_j)}\int_{B_j}u(P)\,dP\,.$ We have \begin{eqnarray} \int_{B_R}u(P)^2\,dP &\leq &2(\sum_j\int_{B_j}|u(P)-\bar{u% }_j|^2\,dP+\sum_j\frac 1{m(B_j)}(\int_{B_j}u(P)\,dP)^2) \nonumber \label{c1} \\ &\leq &\frac \varepsilon {16A}\sum_j\int_{B_j}|\nabla _Hu(P)|^2\,dP+\frac{C_1}{\varepsilon ^2}\sum_j(\int% \limits_{B_j}u(P)\,dP)^2 \nonumber \\ &\leq &\frac \varepsilon {4A}\int_{B_R}|\nabla _Hu(P)|^2\,dP+\frac{C_1}{% \varepsilon ^2}\sum_j(\int_{B_j}u(P)\,dP)^2, \label{k7} \end{eqnarray} where $C_1=2(32AC/\pi)^2$. By (\ref{k7}) for $w_{k,n}=u_k-u_n$, \begin{eqnarray} \int_{B_R}w_{k,n}^2\,dP &\leq &\frac \varepsilon {4A}\int_{B_R}|\nabla _Hw_{k,n}|^2\,dP+\frac{C_1}{\varepsilon ^2}% \sum_j(\int_{B_j}w_{k,n}\,dP)^2 \nonumber \label{c2} \\ &\leq &\frac \varepsilon 2+\frac{C_1}{\varepsilon ^2}\sum_j (\int_{B_j}w_{k,n}\,dP)^2. \label{k8} \end{eqnarray} Since $u_k\rightarrow u$ weakly in $L^2(B_R)$ we have that there exists $N$ such that $\int_{B_R}w_{k,n}dP \leq \frac{\varepsilon ^3}{2C_1} \quad\mbox{for all } k,n>N\,.$ Then by (\ref{k8}) $\int_{B_R}w_{k,n}^2dP \leq \varepsilon \quad\mbox{for all } k,n>N\,;$ therefore, $\{u_k\}$ converges in $L^2(B_R)$. Using the extension by zero of $u\in W_H^{1,2}(B_\rho ( 0,R) )$ on $B_\rho ( 0,R( 1+\delta ) ) \backslash B_\rho ( 0,R)$, with $\delta >0$, we deduce that $W_H^{1,2}(B_\rho ( 0,R) )\subset W_{H,0}^{1,2}(B_\rho ( 0,R( 1+\delta ) ) .$ By the first part, it follows that the inclusion $W_H^{1,2}(B_\rho ( 0,R) )\subset L^2(B_\rho ( 0,R( 1+\delta ) )$ is compact.\hfil$\diamondsuit$ \begin{lemma} \label{L2} The embedding $W_H^{1,2}({\mathbb R}^3)\subset L^4({\mathbb R}^3)$ is continuous. \end{lemma} \paragraph{Proof.} As the homogeneous dimension of $( {\mathbb R}^3,\rho ,m)$ is 4, by the Sobolev inequality \cite{BM}, \begin{eqnarray*} \lefteqn{ \left(\int_{B_\rho ( P_0,r) }|u(P)|^4\,dP\right)^{1/4} } \\ &\leq& C_2\left(r^2\int_{B_\rho ( P_0,r) }|\nabla _Hu(P)|^2\,dP+\int_{B_\rho ( P_0,r) }|u(P)|^2\,dP\right)^{1/2}, \end{eqnarray*} for $u\in C_0^\infty (B_\rho ( P_0,r) )$. Covering the Euclidean space ${\mathbb R}^3$ by intrinsic balls $B_j=B_\rho ( P_j,r)$ such that each point of ${\mathbb R}^3$ is covered by at most 4 balls for $u\in C_0^\infty ({\mathbb R}^3)$, we have \begin{eqnarray*} \int_{{\mathbb R}^3}|u(P)|^4\,dP &\leq &\sum_jC_2(r^2\int_{B_j}|\nabla _Hu(P)|^2\,dP+\int_{B_j}|u(P)|^2\,dP)^2 \\ &\leq &2C_2( \sum_j(\int_{B_j}|\nabla _Hu(P)|^2\,dP)^2+(\int_{B_j}|u(P)|^2\,dP))^2) \\ &\leq &8C_2( \int_{{\mathbb R}^3}|\nabla _Hu(P)|^2\,dP+\int_{% {\mathbb R}^3}|u(P)|^2\,dP) ^2. \end{eqnarray*} Then $\|u\|_{L^4}\leq C_3\|u\|_{W_H^{1,2}}$. \hfil$\diamondsuit$ \section{Existence result} We consider the existence of a nontrivial solutions $u\in W_H^{1,2}({\mathbb R}^3$) to the semilinear Kohn--Laplace equation $-\Delta _Hu+V(P)u=f(u).$ Suppose that $V(P)\in C({\mathbb R}^n,{\mathbb R})$ and $f(t)\in C({\mathbb R})$ satisfy assumptions (\ref{k2})--(\ref{k6}). By (\ref{k6}) it follows that there exists $m>0$ such that $$|F(t)\geq m|t|^\mu ,\quad\mbox{for} |t|\geq 1\,. \label{k9}$$ Let us consider the operator $L=-\Delta _H+V(x)$ in the space $E=L^2({\mathbb R}^3)$ under assumptions (\ref{k2}) and (\ref{k3}). Let $X$ be the domain of operator $L$ in $E$, $X=\{u(P)\in W_H^{1,2}({\mathbb R}^3):\int (|\nabla _Hu(P)|^2+V(x)|u(P)|^2)\,dP<\infty \}.$ It follows that $V(P)$ is uniformly positive; i.e., there exists $a>0$ such that $$V(P)\geq a>0,\quad \forall P\in {\mathbb R}^3. \label{k10}$$ Notice that $L$ is a positive selfadjoint operator in $E$. Then the graph norm of $L$ in $X$, $\|u\|_Y^2=\|u\|_{L^2}^2+\|\nabla _Hu\|_{L^2}^2\,,$ is equivalent to the norm $\|u\|^2=\int (|\nabla _Hu(P)|^2+V(x)|u(P)|^2)\,dP=\langle Lu,u\rangle \,.$ Also notice that $X$ is a Hilbert space with the scalar product $(u_1,u_2)=\int (\nabla _Hu_1\nabla _Hu_2+V(P)u_1u_2)\,dP\,.$ \begin{lemma} \label{L3} Suppose $V(x)$ satisfies (\ref{k2}) and (\ref{k3}). Then the embedding of $X$ in $E$ is compact. \end{lemma} \paragraph{Proof.} Let $\{u_k( P) \}$ be a bounded sequence in $X$, with $\|u_k\|\leq A$, and $u_k\rightarrow u$ weakly in $X$. We shall show that $u_k\rightarrow u$ strongly in $E$. Assuming that $u=0$, we prove that $$\int u_k^2( P) \,dP\rightarrow 0,\quad \mbox{as}\quad k\rightarrow \infty \,. \label{k11}$$ Let $\varepsilon >0$, $\delta >0$ and $R>0$ be such that $$V( P) \geq \frac{1+A}\varepsilon \quad \mbox{if} \quad |P|\geq R( 1+\delta ) \,. \label{k12}$$ The operator $S:X\rightarrow W_H^{1,2}( B( 0,R) )$, $Su=u\mid _{B( 0,R) }$ is linear and continuous. By Lemma \ref{L1} the inclusion $W_H^{1,2}( B( 0,R) ) \subset L^2( B( 0,R( 1+\delta ) ) )$ is compact and therefore $\int_{B( 0,R( 1+\delta ) ) }u_k^2( P) \,dP\rightarrow 0,\quad \mbox{as}\quad k\rightarrow \infty \,.$ Let $k_0$ be such that for $k\geq k_0$ $\int_{B( 0,R( 1+\delta ) ) }u_k^2( P) \,dP\leq \frac \varepsilon {1+A}\,.$ Then for $k\geq k_0$, \begin{eqnarray*} \int u_k^2( P) \,dP &=&\int_{|P|\geq R( 1+\delta ) }u_k^2( P) \,dP+\int_{B( 0,R( 1+\delta) ) }u_k^2( P) \,dP \\ &\leq &\frac \varepsilon {1+A}(1+\int_{|P|\geq R( 1+\delta ) }V( P) u_k^2( P) \,dP) \\ &\leq &\frac \varepsilon {1+A}(1+\|u_k\|^2) \\ &\leq &\varepsilon \,. \end{eqnarray*} \hfill$\diamondsuit$ \medskip Since $W_H^{1,2}({\mathbb R}^3)$ is not included in $L^\infty ({\mathbb R}^3)$ the approach by Omana and Willem \cite{OW} does not work in the case. By Lemmas \ref{L1} and \ref{L2} we have the embeddings \begin{eqnarray*} X &\subset &L^2({\mathbb R}^3)=E\quad \mbox{compactly}, \\ X &\subset &W_H^{1,2}({\mathbb R}^3)\subset L^4({\mathbb R}^3)\,. \end{eqnarray*} Now we prove the following proposition that is analogous to the one in P.L. Lions \cite{PL}. \begin{lemma}\label{L4} Suppose $g\in C({\mathbb R})$ satisfies \begin{eqnarray*} &g(t)=o(|t|)\quad \mbox{as}\quad |t|\rightarrow 0\,&\\ &g(t)=o(|t|^3)\quad \mbox{as}\quad |t|\rightarrow +\infty \,.& \end{eqnarray*} If $\{u_k\}$ is a bounded sequence in $L^4({\mathbb R}^3)$ and $u_k\rightarrow u$ in $L^2({\mathbb R}^3)$, then $\int |g(u_k)(u_k-u)|\,dP\rightarrow 0\,\quad\mbox{as}\quad k\rightarrow +\infty \,.$ \end{lemma} \paragraph{Proof.} From the assumptions, for every $\varepsilon >0$ there exists $\rho >0$ such that $g(t)\leq \varepsilon |t|^3$ for $|t|\geq \rho$, and that there exists $\delta >0$ such that $|g(t)|\leq \varepsilon |t|$ when $|t|\leq \delta <\rho$, we have $|g(t)|\leq \varepsilon (|t|+|t|^3)+C_\varepsilon |t|,$ where $C_\varepsilon =\delta ^{-1}\max_{\delta \leq |t|\leq \rho }|g(t)|$. Then \begin{eqnarray*} \int |g(u_k)(u_k-u)|\,dP &\leq& \varepsilon \int ( |u_k|(|u_k|+|u|)+|u_k|^3(|u_k|+|u|)) \,dP \\ &&+C_\varepsilon \int |u_k||u_k-u|\,dP \\ &\leq &C_1\varepsilon \int (|u_k|^2+|u|^2+|u_k|^4+|u|^4)\,dP \\ &&+C_\varepsilon \int |u_k||u_k-u|\,dP\,. \end{eqnarray*} Then the result follows by the convergence in $L^2({\mathbb R}^3)$ and the boundedness in $L^4({\mathbb R}^3)$ of $\{u_k\}$. \hfil$\diamondsuit$\medskip Let us consider the functional $$\varphi (u)={\frac 12}\|u\|^2-\int F(u(x))\,dP\,. \label{k13}$$ It can be proved that $\varphi \in C^1(X,{\mathbb R})$ and $$\langle \varphi '(u),v\rangle =(u,v)-\int f(x,u(x))v(x)dP ,\quad \forall v\in X\,. \label{k14}$$ The critical points of $\varphi$ are weak solutions of the equation (\ref{k1}) in the function space $W_H^{1,2}({\mathbb R}^3)$. We are looking for nontrivial solutions, $u\neq 0$, in $W_H^{1,2}({\mathbb R}^3)$ of the equation (\ref{k1}). To prove the existence of nontrivial critical points, $u\neq 0$, we apply the mountain-pass theorem of Ambrosetti and Rabinowitz to the functional $\varphi$. \begin{lemma} \label{L5} If $f\in C( {\mathbb R})$ satisfies (\ref {k3})--(\ref{k5}), then the functional $\varphi$ satisfies the Palais--Smale condition in $X$. \end{lemma} \paragraph{Proof.} Let $\{u_k\}$ be a sequence in $X$ such that $$|\varphi (u_k)|\leq C,\ \varphi '(u_k)\rightarrow 0\mbox{ in }X^{*} \label{k15}$$ as $k\rightarrow +\infty$. Then there exists $k_0$ such that for $k\geq k_0$ $|\langle \varphi '(u_k),u_k \rangle |\leq \mu \|u_k\|.$ Then \begin{eqnarray*} C+\|u_k\| &\geq &\varphi (u_k)-\frac 1\mu \langle\varphi '(u_k),u_k\rangle \\ &=&(\frac 12-\frac 1\mu )\|u_k\|^2+\frac 1\mu \int (f(u_k)u_k-\mu F(u_k))\,dP \\ &\geq &(\frac 12-\frac 1\mu )\|u_k\|^2\,; \end{eqnarray*} so $\{u_k\}$ is bounded in $X$. By Lemmas \ref{L2}, \ref{L3} and \ref{L4} there exists a subsequence denoted again by $\{u_k\}$, such that $u_k\rightarrow u\in X$ weakly, $u_k\rightarrow u$ in $L^2({\mathbb R}^3)$ strongly, and $$\lim_{k\rightarrow +\infty }\int \left| f(u_k)(u_k-u)\right| \,dP=0\,. \label{k16}$$ By (\ref{k15}) we have that $$|\langle\varphi '(u_k),v\rangle|\leq \varepsilon _k\|v\|\,. \label{k17}$$ where $\varepsilon _k\rightarrow 0$ as $k\rightarrow +\infty$. Then by (\ref {k16}) it follows $\lim_{k\rightarrow +\infty }(\|u_k\|^2-\langle u,u_k\rangle) =\lim_{k\rightarrow +\infty}\int f(u_k)(u_k-u)\,dP=0\,.$ Then $\{u_k\}$ converges to $u$ strongly in $X$. \hfil$\diamondsuit$ \paragraph{Proof of Theorem 1.} The existence of a solution follows from the the mountain-pass theorem. It remains to show its geometric conditions. There exists a constant $C$ such that $\|u\|_E\leq C\|u\|,\quad \|u\|_{L^4}\leq C\|u\|\,.$ Let $0<\varepsilon < 1/(2C^2)$. Then \begin{eqnarray*} \int F( u) \,dP &\leq &\varepsilon ( \|u\|_E^2+\|u\|_{L^4}^4) +C_\varepsilon \|u\|_{L^4}^4 \\ &\leq &\varepsilon C^2\|u\|^2+(\varepsilon +C_\varepsilon )C^4\|u\|^4 \end{eqnarray*} and $\varphi (u)\geq (\frac 12-\varepsilon C^2)\|u\|^2-(\varepsilon +C_\varepsilon )C^4\|u\|^4>0$ for small enough $\|u\|=r>0$. Let us take a $u_0\in C_0^\infty ( {\mathbb R}^3)$ such that $u_0>0$, supp\,$u_0\subset B(0,2)$ and $\left. u_0\right| _{B(0,1)}=1$. Then $\|u_0\|>a|B(0,1)|>\rho$. By (\ref{k9}), for $x\in B(0,1)$, we have $F(u_0(x))\geq m|u_0(x)|^\mu \,.$ For $\gamma >1$, we have \begin{eqnarray*} \varphi (\gamma u_0) &=&\frac 12\gamma ^2\|u_0\|^2-\int F(\gamma u_0)\,dP \\ &\leq &\frac 12\gamma ^2\|u_0\|^2-\int_{B(0,1)}F(\gamma u_0)\,dP \\ &\leq &\frac 12\gamma ^2\|u_0\|^2-\gamma ^\mu |B(0,1)|<0\,, \end{eqnarray*} for sufficiently large $\gamma$, because $\mu >2$. \hfil$\diamondsuit$ \paragraph{Acknowledgment.} The author thanks to Professor Biroli for his helpful discussions and comments. \begin{thebibliography}{9} \bibitem{BM} M. Biroli, U. Mosco, Sobolev inequalities on homogeneous spaces. {\it Proceedings Potential theory and degenerate P.D. operators''.} Parma Feb. 1994, Kluwer 1995. \bibitem{BMT} M. Biroli, U. Mosco and N.A.\ Tchou, Homogenization by the Heisenberg group. {\it Publications du Laboratoire D'Analyse Numerique. R 94032, Univ. Piere et Marie Curie.} \bibitem{DN} W.Y. Ding and W.M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation. {\it Arch. Rational Mech. Anal.} {\bf 91} , 283- 308 (1986). \bibitem{J} D. Jerison, The Poincare inequality for vector fields satisfying H\"{o}rmander's condition. {\it Duke Math. J.} {\bf 53}, 503-523 (1986). \bibitem{PL} P.L. Lions, Symmetrie et compacite dans les espaces de Sobolev. {\it J. Funct. Anal.} {\bf 49}, 315- 334 (1982). \bibitem{OW} W. Omana and M. Willem, Homoclinics orbits for a class of Hamiltonian systems. {\it Diff. Int. Eq}. {\bf 5}, No 5, 1115-1120 (1992). \bibitem{KL} P. Korman and A.C. Lazer. Homoclinic orbits for a class of symmetric Hamiltonian systems. {\it Electron. Journal of Diff. Eq.} v. 1994, N 01, 1-10 (1994). \bibitem{R} P. Rabinowitz, A note on semilinear elliptic equation on ${\mathbb R}^n$. {\it Nonlinear Analysis, Scuola Norm. Superiore, Pisa,} 307-317 (1991). \end{thebibliography} \medskip \noindent{\sc Stepan Tersian }\\ Center of Applied Mathematics and Informatics \\ University of Rousse \\ 8, Studentska \\ 7017 Rousse, Bulgaria \\ E-mail: tersian@ami.ru.acad.bg \end{document}