\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 86, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/86\hfil Entire solutions] {Entire solutions of semilinear elliptic equations} \author[A. Gladkov, N. Slepchenkov\hfil EJDE-2004/86\hfilneg] {Alexander Gladkov, Nickolai Slepchenkov} % in alphabetical order \address{Alexander Gladkov \hfill\break Mathematics Department, Vitebsk State University, Moskovskii pr. 33, 210038 Vitebsk, Belarus} \email{gladkov@vsu.by, gladkoval@mail.ru} \address{Nickolai Slepchenkov \hfill\break Mathematics Department, Vitebsk State University, Moskovskii Pr. 33, 210038 Vitebsk, Belarus} \email{slnick@tut.by} \date{} \thanks{Submitted November 3, 2003. Published June 23, 2004.} \subjclass[2000]{35J60} \keywords{Semilinear elliptic equation; entire solutions; nonexistence} \begin{abstract} We consider existence of entire solutions of a semilinear elliptic equation $\Delta u= k(x) f(u)$ for $x \in \mathbb{R}^n$, $n\ge3$. Conditions of the existence of entire solutions have been obtained by different authors. We prove a certain optimality of these results and new sufficient conditions for the nonexistence of entire solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lem}[theorem]{Lemma} \newtheorem{cor}[theorem]{Corollary} \newtheorem{prop}[theorem]{Proposition} \newtheorem{rmk}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction}\label{in} In this paper we study the existence of entire solutions of the semilinear elliptic equation $$\label{1.1} \Delta u= k(x) f(u), \quad x \in \mathbb{R}^n, n\ge3,$$ where $k(x)$ is a nonnegative continuous function in $\mathbb{R}^n$, $f(u)$ is a positive continuous function which is defined either in $\mathbb{R}$ or $\mathbb{R}_+$. We denote here $\mathbb{R}_+ = (0,+\infty)$. By an entire solution of equation (\ref{1.1}) we mean a function $u \in C^2(\mathbb{R}^n)$ which satisfies (\ref{1.1}) at every point of $\mathbb{R}^n$. The important particular cases of (\ref{1.1}) are the equations $$\label{1.2} \Delta u= k(x) u^\sigma, \, \sigma>1, \quad \Delta u= k(x) \exp {(2u)}.$$ The existence and the nonexistence of entire solutions for (\ref{1.2}) have been investigated by many authors (see, for example, \cite{Lin} -- \cite{O} and the references therein). Equations (\ref{1.2}) arise in physics and geometry, as stated in \cite{Cheng98,Ni,Ni82}. Equation (\ref{1.1}) has also been studied in papers such as \cite{Usami87,Usami92,W}, where it is shown the existence of entire solutions. It has also been known \cite{Ku,Usami87} that for some classes functions $f(u)$ under the condition $$\label{1.3} \int_0^\infty s\overline k (s) \, ds < \infty,$$ where $\overline k (s)=\sup_{|x|=s} k(x)$, equation (\ref{1.1}) possesses infinitely many entire solutions if $\mathop{\rm dom} f =\mathbb{R}$ and infinitely many positive entire solutions if $\mathop{\rm dom}f = \mathbb{R}_+$. We shall use in this paper the following nonexistence statement of entire solutions of (\ref{1.1}). \begin{theorem}\label{Th2} Let $f(u)$ satisfy the following conditions: \begin{gather} f(u) \, \textrm {is convex}, \label{1.4} \\ \int_{1}^\infty\Big(\int_{0}^v f(u)\, du\Big)^{-1/2} \, dv< \infty, \label{1.5} \end{gather} and there exists nonnegative non-increasing continuous function $k_\star (r)$ such that \begin{gather} k_\star (|x|)\leq k(x),\quad \int_0^{+\infty}s\, k_\star (s)\,ds=+\infty, \label{1.6} \\ \limsup_{r\to+\infty} k_\star (r)\,r^2>0. \label{1.7} \end{gather} Then (\ref{1.1}) has no entire solutions if $\, \mathop{\rm dom} f = \mathbb{R}$ and has no positive entire solutions if $\, \mathop{\rm dom} f = \mathbb{R}_+$. \end{theorem} Theorem~\ref{Th2} is a little more general assertion than \cite[Corollary 2.1]{Usami92} and can be easily obtained from that paper. The main purpose of the present paper is to present new sufficient conditions for nonexistence of entire solutions of (\ref{1.1}), and to show a certain optimality of (\ref{1.3}) for the existence of entire solutions of (\ref{1.1}). The distribution of this paper is as follows. We show an optimality of the condition (\ref{1.3}) for the existence of entire solutions of (\ref{1.1}) for some class functions $f(u)$ in Section 2. In Section 3 we construct example of (\ref{1.1}) with radially symmetric function $k(x)$ which demonstrates that the condition (\ref{1.3}) is not necessary for the existence of entire solutions. In Section 4, we give new sufficient conditions for the nonexistence of entire solutions of (\ref{1.1}). In particular it is shown that Theorem~\ref{Th2} is valid without assumption (\ref{1.7}). \section{Optimality of existence condition }\label{uec} \noindent The aim of this section is to show a certain optimality of the condition (\ref{1.3}) for the existence of entire solutions of (\ref{1.1}). The similar result for ordinary differential equation of second order with $f(u) =u^\lambda$, $\lambda>1$, has been obtained in \cite{Iz} and we shall use here some ideas of that paper. \begin{theorem}\label{Th3} Let $f(u)$ satisfy \eqref{1.4}, \eqref{1.5} and $\varphi (r)$ be any positive continuous function such that $\varphi (r) \to \infty$ as $r \to \infty$. Then there exist radially symmetric positive continuous function $k(x)=\overline k (|x|)$ such that $$\label{2.1} \int_0^\infty \frac{s\overline k (s)}{\varphi (s)} \, ds < \infty,$$ and the equation (\ref{1.1}) has no entire solutions if $\mathop{\rm dom} f = \mathbb{R}$ and has no positive entire solutions if $\mathop{\rm dom} f = \mathbb{R}_+$. \end{theorem} \begin{proof} Without lose of generality we can suppose that $\varphi(r)\geq 1$ for $r\geq 0$. We shall construct positive locally H{\"o}lder continuous function $\overline \varphi(r)$ such that $$\label{2.2}\begin{split} 1\leq\bar\varphi(r)\leq\sqrt{\varphi(r)},\quad \bar\varphi(r) \textrm{ does not decrease, }\\ \frac{\bar\varphi(r)}{r}\to 0 \textrm { as } r\to\infty \textrm{ and does not increase for } r\geq R_0, \end{split}$$ where $R_0>0$. Let $r_0=0$ and $\varphi_0=\inf_{r\geq r_0}\sqrt{\varphi(r)}\geq 1$. We put $\varphi_2=\varphi_0+1$ and choose $r_1$ such that $r_1\geq \max\{r_0+1, \exp({\varphi_0})\}$ and $\inf_{r\geq r_1}\sqrt{ \varphi(r)} \geq \varphi_2$. Denote $r_2=r_1\,\exp (1)$. We define $\bar\varphi(r)$ on the interval $[r_0,r_2)$ in the following way \begin{equation*} \bar{\varphi}(r)=\begin{cases} \varphi_0, &r\in\left[r_0,r_1\right),\\ \varphi_0+\ln(r/r_1),&r\in\left[r_1,r_2 \right). \end{cases}\end{equation*} Then $\bar{\varphi}(r_1)=\varphi_0$, $\bar{\varphi}(r_2)= \varphi_0+1= \varphi_2$. It is easy to see that $\bar{ \varphi}(r) \leq \ln r$ for $r\in [r_1,r_2)$. For $k=2,3,\dots$ we put $\varphi_{2k}=\varphi_{2k-2}+1$ and $r_{2k-1}$ choose such that $r_{2k-1}\geq \max \{r_{2k-2}+1,\exp( \varphi_{ 2k-2})\}$ and $\inf_{r\geq r_{2k-1}} \sqrt{\varphi(r)}\geq \varphi_{2k}$. Now set $r_{2k}=r_{2k-1}\,\exp(1)$ and $\bar{\varphi}(r)=\begin{cases} \varphi_{2k-2}, &r\in\left[r_{2k-2},r_{2k-1}\right),\\ \varphi_{2k-2}+\ln(r/r_{2k-1}),&r\in\left[r_{2k-1},r_{2k} \right). \end{cases}$ It is not difficult to verify that $\bar\varphi(r_{2k-1})=\varphi_{2k-2}$, $\bar\varphi(r_{2k})= \varphi_{2k}$ and $\bar\varphi(r)\leq\ln r$ for $r\in[r_{2k-1},r_{2k})$. Constructed function $\bar\varphi(r)$ is locally H{\"o}lder continuous for $r \geq 0$ and satisfies (\ref{2.2}). We define now a sequence $\tau_p$, $p=0,1,\dots$ as follows: $\tau_0=0,\quad 1\leq \tau_{p+1}-\tau_p\leq \tau_{p+2}-\tau_{p+1},\quad 2\tau_p\leq\tau_{p+1},\quad (p+1)^2\leq\bar{\varphi}(\tau_p)$ and introduce for $r\geq R_0$ the function $\overline k (r)=\frac{\bar{\varphi}(r)\,\psi(r)}{r},$ where $\psi(r)$ is positive locally H{\"o}lder continuous function such that \begin{equation*} \psi(r)=\begin{cases} 1/\delta_p,&r\in[\tau_p,\tau_{p+1}-\delta_p/10),\\ a_p\,r+b_p,&r\in[\tau_{p+1}-\delta_p/10,\tau_{p+1}). \end{cases} \end{equation*} Here $p=0,1,\dots$, $\delta_p=\tau_{p+1}-\tau_p$, and coefficients $a_p$ and $b_p$ we choose to join points $(\tau_{p+1}-\delta_p/10, 1/\delta_p)$ and $(\tau_{p+1},1/\delta_{p+1})$. For $0\leq r0,\quad 1-\Bigl(\frac{r_p}{r_{p}+4(n-2)r_{p}[g(r_{p})]^{-1}} \Bigr)^{n-2}\leq\frac12\frac{a_p}{f(\bar{a}_p) },\\ g(r_{p}) \geq 4 (n-2), \quad r_{p}+4(n-2)r_{p}[g(r_{p})]^{-1}0$. Hence we are able to apply the Schauder-Tychonoff fixed point theorem and conclude that $T$ has a fixed point $u$ in $U$. This fixed point satisfies (\ref{3.8}), and so we obtain a solution $u(|x|)$ of (\ref{1.1}). \end{proof} \section{ Nonexistence of entire solutions}\label{ne} The main purpose of this section is to get new sufficient conditions for nonexistence of entire solutions of (\ref{1.1}). We introduce an auxiliary function $I(\beta)=\int_\beta^\infty\Big(\int_\beta^vf(u)\,du\Big)^{-1/2}\,dv< \infty,\ \beta>0.$ Let $\bar u(r)$ denote the mean value of $u(x)$ over the sphere $|x|=r$, that is, $\bar u(r)=\frac{1}{\omega_nr^{n-1}}\int_{|x|=r}u(x)\,dS,$ where $\omega_n$ is the surface area of the unit sphere in $\mathbb{R}^n$, $dS$ is the volume element in the surface integral. We shall use two lemmas which have been proved in \cite{Usami92}. \begin{lem}\label{L2} Let $f(u)$ be convex function and there exists nonnegative continuous function $k_\star(r)$ such that $k_\star(|x|)\leq k(x)$. If $u(x)$ is a solution of \eqref{1.1} then $\bar u(r)$ satisfies the following conditions $$\begin{gathered} \label{4.1} \bar u''(r)+\dfrac{n-1}{r}\bar u'(r)\geq k_\star(r)\,f(\bar u(r)),\\ \bar u'(0)=0,\quad \bar u(0)=u(0). \end{gathered}$$ \end{lem} \begin{lem}\label{L3} Let $f(u)$ satisfy \eqref{1.4} and \eqref{1.5}. Then function $I(\beta)$ does not increase for sufficiently large values of $\beta$ and $\lim_{\beta \to \infty} I(\beta) = 0$. \end{lem} Now we prove an auxiliary statement which has independent interest. \begin{theorem}\label{Th5} Let $f(u)$ satisfy \eqref{1.4}, \eqref{1.5} and $k_\star(r)$ be nonnegative continuous function possessing the properties \eqref{1.6} and $$\label{4.2} (s/r)^\delta\leq \int_{R_0}^s t\, k_\star(t)\,dt/\int_{R_0}^r t\, k_\star(t)\,dt$$ for $r\geq s\geq R_0^*> R_0$, where $\delta$, $R_0^*$ and $R_0$ are some positive constants. Then the equation (\ref{1.1}) has no entire solutions if $\mathop{\rm dom} f = \mathbb{R}$ and has no positive entire solutions if $\mathop{\rm dom} f = \mathbb{R}_+$. \end{theorem} \begin{proof} Let $u(x)$ be any entire solution of (\ref{1.1}). Then by Lemma~\ref{L2} $\bar u(r)$ satisfies (\ref{4.1}) which imply the following integral inequality with $\alpha =u(0)$ $$\label{4.3} \bar u(r)\geq\alpha+\frac{1}{n-2}\int_0^r\big(1-(\frac{s}{r} )^{n-2}\big)s k_\star(s)f(\bar u(s))\,ds.$$ Moreover, $\bar u(r)$ is nondecreasing and $\bar u(r)\to\infty$ as $r\to\infty$. Since $k_\star(r)$ is nonnegative continuous function then sets $A(R,r) \equiv \{s \in (R,r): k_\star(s)>0 \}$ and $A(R,\infty) \equiv \{s \in (R,\infty): k_\star(s)>0 \}$ are union of finite or countable number of intervals. By sets $A(R,r) = \bigcup_i (a_i,b_i)$ and $A(R,\infty) = \bigcup_i (\overline a_i,\overline b_i)$ we introduce the auxiliary sets in the following way $A[R,r) = \bigcup_i [a_i,b_i)$ and $A[R,\infty) = \bigcup_i [\overline a_i,\overline b_i).$ For $r \in A[R_0,\infty)$, we put $$\label{4.4} h(r)=\int_{A[R_0,r)} s k_\star (s)\,ds.$$ By virtue of (\ref{1.6}) and (\ref{4.4}) $h$ maps in a one-to-one manner $A[R_0,\infty)$ on $[0,\infty)$. Hence there exists inverse for $h$ function $g$. We denote $$\label{4.5} t=h(r),\ \tau=h(s),\ \bar u(g(t))=w(t).$$ Due to (\ref{1.4}), (\ref{1.5}) function $f(u)$ is increasing for sufficiently large values of $u$. Therefore $f(\bar u(r))$ is nondecreasing for $r>R_1$ for some $R_1>0$. We take $R_2$ such that $R_2 \geq \max\{R_1, R_0^*\}$, $k_\star (R_2) \neq 0$. Then by (\ref{4.3}) -- (\ref{4.5}) for $t > h(R_2)$ we get \label{4.6} \begin{aligned} w(t)&\geq \alpha + \frac{1}{n-2}\int_{A[R_2,g(t))} \Big(1-\big(\frac{s}{g(t)} \big)^{n-2}\Big)s\, k_\star (s)\, f(\bar u(s))\,ds\\ &= \alpha+\frac{1}{n-2} \int_{h(R_2)}^{t} \Big(1-\big(\frac{ g(\tau)}{g(t)}\big)^{n-2}\Big)f(w(\tau))\,d\tau. \end{aligned} It follows from (\ref{4.2}) that $$\label{4.7} g(\tau)/g(t)\leq (\tau/t)^{1/\delta}.$$ From (\ref{4.6}) and (\ref{4.7}) we deduce $$\label{4.8} w(t)\geq\alpha+\frac{1}{n-2}\int_{h(R_2)}^t\big(1-(\frac{\tau}{t} )^{(n-2)/\delta}\big)f(w(\tau))\,d\tau.$$ Let $T>h(R_2)$ and $T\leq\tau \leq t\leq 2T$. Using (\ref{4.8}) and the inequality \begin{gather*} 1-(\frac{\tau}{t})^{(n-2)/\delta} \geq (n-2)C(\delta) \frac{t-\tau}{\tau}, \end{gather*} where $C(\delta)=\min\{1/2,1/2^{(n-2)/\delta}\}/\delta$, we obtain $w(t)\geq \beta+C(\delta)\int_T^t\frac{t-\tau}{\tau} f(w(\tau))\,d\tau.$ Here we denote $\beta=\alpha+\frac{1}{n-2}\int_{h(R_2)}^T\big(1-(\frac{\tau}{t} )^{(n-2)/\delta}\big)f(w(\tau))\,d\tau.$ It is obvious $\beta\to\infty$ as $T\to\infty$. Put $z(t)=\beta+C(\delta)\int_T^t\frac{t-\tau}{\tau} f(w(\tau))\,d\tau.$ Then we have $$\label{4.9} z''(t)=C(\delta)\frac{1}{t}f(w(t))\geq C(\delta)\frac{1}{t}f(z(t))$$ and $z(T)=\beta,$$\, z'(T)=0$. If we multiply (\ref{4.9}) by $z^\prime (t)$ and then integrate over $[T,t]$, we get $(z'(t))^2\geq 2C(\delta)\frac{1}{t}\int_{\beta}^{z(t)}f(u)\,du.$ Elementary calculations shows that $\Big(\int_{\beta}^{z(t)}f(u)\,du\Big)^{-1/2}z'(t)\geq \sqrt\frac{2C(\delta)}{t}.$ Integrating the above inequality over $[T,t]$, we infer $$\label{4.10} I(\beta)\geq\int_{\beta}^{z(t)}\Big(\int_{\beta}^vf(u)\,du \Big)^{-1/2}dv \geq 2\sqrt{2C(\delta)}(\sqrt t- \sqrt T).$$ We put now $t=2T$ in (\ref{4.10}) and pass to the limit $T \to \infty$. Then left hand side of (\ref{4.10}) tends to zero due to Lemma~\ref{L3}, on the other hand right hand side of (\ref{4.10}) tends to infinity. Obtained contradiction proves theorem. \end{proof} \begin{cor}\label{C1} Let function $f(u)$ satisfy the conditions \eqref{1.4}, \eqref{1.5} and $k_\star (r)$ be nonnegative continuous function possessing the properties \eqref{1.6} and $$\label{4.11} k_\star (r)\leq \frac{C}{r^2} \,\, \textrm{ for } \,\, r\geq R_3 > 0$$ for some values of $R_3$ and $C>0$. Then \eqref{1.1} has no entire solutions if $\mathop{\rm dom} f = \mathbb{R}$ and has no positive entire solutions if $\mathop{\rm dom} f = \mathbb{R}_+$. \end{cor} \begin{proof} We show that (\ref{4.2}) is valid with $\delta=1$. Really, it is easy to verify that $\frac{d}{dr}\Big(\frac{\int_{R_0}^rt\, k_\star(t)\,dt}{r} \Big)= \frac{r^2\, k_\star(r)-\int_{R_0}^rt\, k_\star(t)\,dt}{r^2}\leq \frac{C-\int_{R_0}^rt\, k_\star(t)\,dt}{r^2}<0$ for sufficiently large values of $r$. Now by Theorem~\ref{Th5} the conclusion of corollary follows. \end{proof} \begin{rmk}\label{R0} {\rm We constructed in Section 3 the function $k(x)=\overline k(|x|)$ such that $\int_0^\infty s \overline k(s) \, ds = \infty,\$ $\overline k(r) \leq g(r)/r^2$ for $r \geq r_1 >0$, where $g(r)$ is any positive nondecreasing continuous function with properties: $g(r) \to \infty$ and $g(r)/r \to 0$ as $r \to \infty$, and the equation (\ref{1.1}) has infinitely many positive entire solutions. Hence the upper bound in (\ref{4.11}) is optimal.} \end{rmk} \begin{rmk}\label{R1} {\rm For the equations (\ref{1.2}) similar to Theorem~\ref{Th5} and Corollary~\ref{C1} statements have been proved in \cite{Cheng87} under the additional assumption $\int_{0}^r s k_\star (s) \, ds \,\, \textrm{is strictly increasing in} \,\, [0,\infty).$ } \end{rmk} Using Corollary~\ref{C1} and Theorem~\ref{Th2} it is not difficult to establish the following assertion. \begin{cor}\label{C2} Let function $f(u)$ satisfy the conditions \eqref{1.4}, \eqref{1.5} and $k_\star (r)$ be nonnegative continuous non-increasing for large values of $r$ function satisfying \eqref{1.6}. Then the equation \eqref{1.1} has no entire solutions if $\mathop{\rm dom} f = \mathbb{R}$ and has no positive entire solutions if $\mathop{\rm dom} f = \mathbb{R}_+$. \end{cor} \begin{rmk}\label{R10} {\rm Corollary~\ref{C2} gives new nonexistence criterion for (\ref{1.1}) and this statement is more general than any one in \cite{Usami92}. In particular Theorem~\ref{Th2} is true without assumption (\ref{1.7}).} \end{rmk} \begin{rmk}\label{R2} {\rm All results of this section are valid for more general equation $\Delta u = p(x,u)$ where $p(x,u)$ is nonnegative continuous function satisfying the inequality $p(x,u) \geq k(x) f(u).$ Here the functions $k(x)$ and $f(u)$ possess the same properties as in our statements. In particular the equation (\ref{1.1}) with function $f(u)$ satisfying the conditions (\ref{1.4}), (\ref{1.5}) and function $k(x)$ satisfying the inequality $k(x) \geq \{ c|x|^2 (\ln |x|) (\ln\ln |x|)\dots (\ln \dots \ln |x|) \}^{-1},$ where $c>0$ and $|x| \geq r_\star >0$, has no entire solutions if $\mathop{\rm dom} f = \mathbb{R}$ and has no positive entire solutions if $\mathop{\rm dom} f = \mathbb{R}_+$.} \end{rmk} \begin{thebibliography}{99} \bibitem{A}{L. V. Ahlfors}, An extension of Schwarz's lemma, {\sl Trans. Amer. Math. Soc.,} {\bf 43}(1938), 359-364. \bibitem{Cheng87} {K.-S. Cheng and J.-T. Lin}, On the elliptic equations $\Delta u=k(x)\,u^\sigma$ and $\Delta u=k(x)\,e^{2u}$, {\sl Trans. Amer. Math. Soc.,} {\bf 304}(1987), 639-668. \bibitem{Cheng98} {K.-S. Cheng and J.-N. Wang}, On the conformal Gaussian curvature equation in $R^2$, {\sl J. Diff. Equat.,} {\bf 146}(1998), 226-250. \bibitem{Ku} {T. Kusano and S. 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