\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
\begin{equation}\label{1.1}
\Delta u= k(x) f(u), \quad x \in \mathbb{R}^n, n\ge3,
\end{equation}
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
\begin{equation}\label{1.2}
\Delta u= k(x) u^\sigma, \, \sigma>1, \quad \Delta u= k(x) \exp
{(2u)}.
\end{equation}

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
\begin{equation}\label{1.3}
\int_0^\infty s\overline k (s) \, ds < \infty,
\end{equation}
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
\begin{equation}\label{2.1}
\int_0^\infty \frac{s\overline k (s)}{\varphi (s)} \, ds < \infty,
\end{equation}
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
   \begin{equation}\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}\end{equation}
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 $0\leq r<R_0$
we can define $\overline k(r)$ in any way to get positive non-increasing
locally H{\"o}lder continuous function.

Let $R_0\in[\tau_i,\tau_{i+1})$.  Using the definitions of
$\bar\varphi(r)$, $\overline k(r)$ and $\psi(r)$, we verify the validity
of (\ref{2.1}),
    \begin{align*}
 \int_{\tau_{i+1}}^\infty\frac{\overline k(s)\,s}{\varphi(s)}ds
&= \sum_{p=i+1}^\infty\int_{\tau_p}^{\tau_{p+1}}\frac{\bar\varphi(s)\,
    \psi(s)}{\varphi(s)}ds\\
&\leq \sum_{p=i+1}^\infty\int_{\tau_p}^{\tau_{p+1}}\frac{\psi(s)\,ds}{\sqrt{
    \varphi(s)}} \\
&\leq\sum_{p=i+1}^{\infty}\frac{1}{\bar{\varphi}(\tau_p)}\int_{\tau_p}^{
    \tau_{p+1}}\frac{ds}{\tau_{p+1}-\tau_p}\\
&\leq  \sum_{p=i+1}^\infty\frac{1}{(p+1)^2}<\infty.
    \end{align*}
Now we show that $\overline k(r)$ satisfies (\ref{1.6}) and (\ref{1.7}).
Indeed, we have
  \begin{align*}
  \int_{R_0}^\infty s \overline k(s)\,ds
&\geq\sum_{p=i+1}^\infty
  \int_{\tau_p}^{\tau_{p+1}} \bar{\varphi}(s)\,\psi(s)\,ds\\
&\geq \sum_{p=i+1}^\infty \int_{\tau_p}^{\tau_{p+1}-\delta_p/10}
  \frac{\bar{\varphi}(s)}{\tau_{p+1}-\tau_p}ds\\
&\geq\sum_{p=i+1}^\infty\frac{9}{10}\bar{\varphi}(\tau_p)=
   \frac{9}{10}\sum_{p=i+1}^\infty(p+1)^2=\infty.
\end{align*}
Put $r_p=\dfrac{\tau_{p+1}+\tau_p}{2}$. Then for $p\geq i+1$ we
get
    \begin{equation*}
    \begin{split}
    \overline k(r_p)\,r_p^2&=\bar{\varphi}(r_p)\,\psi(r_p)\,r_p=
    \frac{\bar{\varphi}(r_p)\,r_p}{\tau_{p+1}-\tau_p}
    \geq\frac{\bar{\varphi}(r_p)(\tau_{p+1}+\tau_p)}{2\tau_{p+1}}=\\
    &=\frac{1}{2}\,\bar{\varphi}(r_p)\big(1+\frac{\tau_p}{\tau_{p+1}}\big)
    \geq\frac{1}{2}\bar{\varphi}(\tau_{p})\geq\frac{1}{2}\,(p+1)^2.
    \end{split}
    \end{equation*}
According to Theorem~\ref{Th2} the equation  (\ref{1.1}) with
function $k(x)=\overline k(|x|)$ 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{proof}

\section{Counterexample to necessity of (\ref{1.3})} \label{cn}

The condition (\ref{1.3}) is not necessary for the existence of
entire solutions of the equation (\ref{1.1}). To show this we give
an explicit $k(x)=\overline k(|x|)$ which satisfies $\int_0^\infty
s \overline k(s) \, ds = \infty$ and we construct a solution of
(\ref{1.1}) with this $k(x)$. Note that analogous examples of
entire solutions for the equations (\ref{1.2}) have been
constructed in \cite{Lin87}. We modify that construction.
Constructed solution will also demonstrate in Section 4 an
optimality additional to (\ref{1.3}) condition for the
nonexistence of entire solutions of the equation (\ref{1.1}).

We suppose that $g(r)$ be any positive nondecreasing continuous
function such that $g(r) \to \infty$ and $g(r)/r \to 0$ as
$r \to \infty$. Let $\{a_p\}_{p=1}^\infty$ and  $\{r_p\}_{p=1}^\infty$
are sequences which have the following properties:
  \begin{equation}\label{3.1}
  \begin{gathered}
  a_1=2\alpha,\quad a_{p+1}=a_{p}+2f(\bar{a}_p),\quad
  f(\bar{a}_p)=\max_{\alpha \leq a\leq a_p}f(a),\\
 r_1>0,\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}<r_{p+1},
  \end{gathered}
  \end{equation}
where $\alpha$ is some positive constant. We put $\overline r_{p}=
r_{p}+4(n-2)r_{p}[g(r_{p})]^{-1}$ and denote $\overline k(r)$ a smooth
function which satisfies the following relations:
  \begin{gather}
  \label{3.2}
  0\leq\bar{k}(r)\leq\frac{g(r)}{r^2}\quad\text{for } r_p  \leq r<\overline r_{p},
  \; p=1,2,\dots,\\
  \label{3.3}
  \bar{k}(r)=0\quad \text{for }0\leq r<r_1,\;
  \overline r_{p}\leq r< r_{p+1},\; p=1,2,\dots, \\
  \label{3.4}
  \frac{1}{n-2}\int_{r_p}^{r_{p+1}}r\,\bar{k}(r)dr=1,\quad
  p=1,2,\dots.
  \end{gather}
It is not difficult to show the existence of $\bar{k}(r)$ with
  properties (\ref{3.2}) -- (\ref{3.4}). Indeed
\begin{align*}
\int_{r_p}^{\bar{r_{p}}} \frac{g(r)}{r}~dr &\geq g(r_p)
\int_{r_p}^{r_p+\frac{4(n-2)r_p}{g(r_p)}} \frac{dr}{r} = g(r_p)
\ln \big(1+\frac{4(n-2)}{g(r_p)}\big) \\
& \geq g(r_p)\frac{2(n-2)}{g(r_p)} = 2(n-2).
\end{align*}
We used here that $g(r)$ is a nondecreasing function, $g(r_p) \geq
4(n-2)$ and the inequality
$$
\ln (1+x) \geq \frac{x}{2}, \quad 0\leq x \leq 1.
$$
Note also that we can choose $\int_{r_p}^{r_{p+1}} r\bar{k}(r)~dr$
any between $0$ and its upper bound
$\int_{r_p}^{\overline{r}_{p}}\frac{g(r)}{r}~dr$.
Let $\bar{w}(r)$ be the piecewise continuous function  defined as
  \begin{equation}
  \label{3.5}
  \bar{w}(r)=\begin{cases}
  \frac12a_1&\text{for }  0\leq r<r_1,\\
  a_p&\text{for }  r_p\leq r<\overline r_p,\; p=1,2,\dots,\\
  \frac12a_{p+1}&\text{for } \overline r_p\leq r<r_{p+1},\; p=1,2,\dots.
  \end{cases}
  \end{equation}
We put
  \begin{equation}
  \label{3.6}
  T\bar{u}=\alpha+\frac{1}{n-2}\int_0^r\big(1-\left(\frac{s}{r}\big)^{n-2}
  \right) s\,\bar{k}(s)\,f(\bar{u}(s))\,ds.
  \end{equation}

\begin{lem}\label{L1}
Let  $\bar{u}(r)$ satisfy the inequalities
  $ \alpha\leq \bar{u}(r)\leq   \bar{w}(r)$.
Then  $T\bar u(r)\leq \bar{w}(r)$.
\end{lem}

\begin{proof} At first we suppose that $0\leq r\leq r_1$. Due to
(\ref{3.3}), (\ref{3.5}) and (\ref{3.6})
  \begin{equation*}
  T\bar{u}=\alpha\leq\bar{w}(r).
  \end{equation*}

Assume now that $r_p\leq r<\overline r_{p}$. Using  (\ref{3.1}) --
(\ref{3.6}) we get
  \begin{align*}
  T\bar{u}&=
  \frac12a_1+\frac{1}{n-2}\sum_{j=1}^{j=p-1}\int_{r_j}^{\overline r_j}
  \big(1-(\frac{s}{r})^{n-2} \big)s\,\bar{k}(s)\,f(\bar{u}(s))\,ds \\
&\quad +\frac{1}{n-2}\int_{r_p}^r\big(1-(\frac{s}{r})^{n-2}\big)
s\,\bar{k}(s)\,f(\bar{u}(s))\,ds\\
&\leq \frac12a_1+\sum_{j=1}^{j=p-1}f(\bar{a}_j)
  +f(\bar{a}_p)\,\frac{1}{n-2}\Bigl(1-\bigl(\frac{r_p}{\overline r_{p}}
  \bigr)^{n-2}\Bigr)\int_{r_p}^{\overline r_p}s\,\bar{k}(s)\,ds \\
&= \frac12a_p+f(\bar{a}_p)\,\Bigl(1-\bigl(\frac{r_p}{\overline r_{p}}
  \bigr)^{n-2}\Bigr)\\
&\leq   \frac12a_p+f(\bar{a}_p)\frac12\frac{a_p}{f(\bar{a}_p)}=a_p=
  \bar{w}(r).
  \end{align*}
For $\overline r_{p}\leq r<r_{p+1}$ we have
  \begin{equation*}
  \begin{split}
   T\bar{u}&=
   \frac12a_1+\frac{1}{n-2}\sum_{j=1}^{j=p}\int_{r_j}^{\overline r_j} s\,\bar{k}(s)\,f(\bar{u}(s))\,ds\\
      &\leq\frac12a_1+\sum_{j=1}^{j=p}f(\bar{a}_j)=\frac12a_{p+1}=\bar{w}(r).
   \end{split}
   \end{equation*}
\end{proof}

Now we can prove the main result of this section.

\begin{theorem}\label{Th4}
Let $\bar k(r)$be a smooth function
satisfying  \eqref{3.2} -- \eqref{3.4}. Then
    \[ \int_0^\infty s\,\bar k(s)\,ds=\infty,\]
and the equation (\ref{1.1}) with  $k(x)=\overline k(|x|)$ has
infinitely many positive entire solutions.
\end{theorem}

\begin{proof} We consider the problem
  \begin{equation}
  \label{3.7}
  \begin{gathered}
  \bar{u}''(r)+\dfrac{n-1}{r}\bar u'(r)=\bar{k}(r)\,f(\bar{u}(r)),\\
  \bar{u}(0)=\alpha,\quad \bar{u}'(0)=0,
  \end{gathered}
  \end{equation}
or equivalently the integral equation
  \begin{equation}
  \label{3.8}
  \bar{u}(r)=\alpha+\frac{1}{n-2}\int_0^r \big(1-(\frac{s}{r}
  )^{n-2}\big)s\,\bar{k}(s)\,f(\bar{u}(s)).
  \end{equation}
We shall prove that (\ref{3.8}) has a solution for each
$0<\alpha\leq1$, and therefore the equation (\ref{1.1}) with
$k(x)=\overline k(|x|)$ has infinitely many positive solutions.

Let $C[0,\infty)$ denote the locally convex space of all
continuous function on $[0,\infty)$ with the topology of uniform
convergence on every compact set of $[0,\infty)$. Let $U$ be the
set
  \[
  U=\left\{ \bar{u}(r)\in C[0,\infty),\;\alpha\leq\bar{u}(r)\leq\bar{w}(r)
\text{ for } r \geq 0 \right\},
  \]
where $0<\alpha\leq1$ and $\bar{w}(r)$ was defined in (\ref{3.5}).
Clearly, $U$ is a closed convex subset of $C[0,\infty)$. Now we
consider the mapping $T$ which was defined in (\ref{3.6}).  It is
obvious
\[
T\bar u(r)\geq\alpha.
\]
Due to Lemma~\ref{L1}
\[ T\bar u\leq\bar w(r).\]
Thus $T$ maps $U$ into itself. It is easy to see that $U$ is
continuous. To prove that $T$ is also compact, we just compute
    \begin{equation*}
     0 \leq (T\bar u)'(r)=\int_0^r\left(\frac{s}{r}\right)^{n-1}\bar{k}(s)\,
     f(\bar u(s))\,ds \equiv M(r),
     \end{equation*}
where $M(r)$ is a bonded function on any segment $[0,R], R>0$. 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{equation}
  \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{equation}
\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
  \begin{equation}\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
  \end{equation}
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 $\alpha =u(0)$
  \begin{equation}
  \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.
  \end{equation}
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
  \begin{equation}
  \label{4.4}
  h(r)=\int_{A[R_0,r)} s k_\star (s)\,ds.
  \end{equation}
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
 \begin{equation}
  \label{4.5}
  t=h(r),\ \tau=h(s),\ \bar u(g(t))=w(t).
  \end{equation}
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
\begin{equation}\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}
\end{equation}
It follows from (\ref{4.2}) that
  \begin{equation}\label{4.7}
  g(\tau)/g(t)\leq (\tau/t)^{1/\delta}.
  \end{equation}
 From  (\ref{4.6}) and (\ref{4.7}) we deduce
  \begin{equation}\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.
  \end{equation}
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
  \begin{equation}\label{4.9}
  z''(t)=C(\delta)\frac{1}{t}f(w(t))\geq C(\delta)\frac{1}{t}f(z(t))
  \end{equation}
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
  \begin{equation} \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).
  \end{equation}
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
 \begin{equation} \label{4.11}
  k_\star (r)\leq \frac{C}{r^2} \,\, \textrm{ for } \,\, r\geq R_3 > 0
 \end{equation}
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}


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\end{document}