Funkcialaj Ekvacioj, 29 (1986) 189-195
On Bounded Positive Entire Solutions of
Semilinear Elliptic Equations
By
Hiroyuki USAMI
(Hiroshima University, Japan)
Introduction
§1.
In this paper we consider the elliptic equation
(1)
$¥Delta u+f(x, u, ¥nabla u)=0$
,
$x$
$¥in R^{n}(n¥geq 3)$
,
¥
and
is locally Holder continuous (with exponent $¥theta¥in(0,1)$ ). Our objective is to obtain conditions guaranteeing the existence of positive entire solutions of
. By an entire
(1) which have prescribed positive limits as
solution of (1) we mean a function $u¥in C^{2}(R^{n})$ which satisfies (1) at every point of .
Recently Naito [3] has considered the case $f(x, u,p)=a(x)g(u)$ , where $g:(0, ¥infty)$
is positive and nondecreasing and obtained some results under the condition
where
$x=(x_{1^{ }},--, x_{n})$
,
$¥Delta=¥partial^{2}/¥partial x_{1}^{2}+$
?
$+¥partial^{2}/¥partial x_{n}^{2}$
,
$¥nabla=(¥partial/¥partial x_{1^{ }},--, ¥partial/¥partial x_{n})$
$ f:R^{n} times$
$(0, ¥infty)¥chi R^{n}¥rightarrow R$
$|x|=(x_{1}^{2}+ --+x_{n}^{2})^{1/2}¥rightarrow¥infty$
$R^{n}$
$¥rightarrow R$
$¥int_{0}^{¥infty}$
$ ta^{*}’(t)dt<¥infty$
,
where
$a^{*}(t)=¥max_{|x|=t}|a(x)|$
.
Moreover he has obtained a possible range of the limit values at infinity of such
positive entire solutions. On the other hand, Kusano and Oharu [2] have established
sufficient conditions for equation (1) to have infinitely many entire solutions which
are bounded from above and below by positive constants. In this paper an attempt
is made to improve and strengthen considerably both of the above results.
Our method, essentially based on Naito’s [3], is to construct explicitly supersolutions and subsolutions having the same prescribed limits at infinitly as entire solutions
.
of suitable Poisson equations in
$R^{n}$
§2.
Main result
With regard to (1) the following conditions are assumed to hold without further
mention:
, there exists a
(A) (Nagumo condition) For any bounded domain
, ) such that
:
continuous function
$¥Omega¥subset R^{n}$
$¥rho_{¥Omega}$
$(0, ¥infty)¥rightarrow[0$
$¥infty$
$|f(x, u,p)|¥leq¥rho_{¥Omega}(u)(1+|p|^{2})$
190
H. USAMI
.
on [0, )
(B) There exists a nonnegative function
which is locally Holder continuous (with exponent ), monotone in
decreasing in , and satisfies
for
$(x, u,p)¥in¥Omega¥times(0, ¥infty)¥chi R^{n}$
$F(t, u, ¥mathrm{v})$
$¥infty$
$¥times(0, ¥infty)¥times[0,$
$¥theta$
$u$
$¥infty)$
and non-
$v$
$|f(x, u,p)|¥leq F(|x|, u, |p|)$
for
.
In what follows we employ the notation:
$(x, u,p)¥in R^{n}¥times(0, ¥infty)¥times R^{n}$
$d(t, u, u;¥lambda)=¥frac{F(t,¥lambda u,¥lambda U)}{¥lambda}$
.
Hypotheses on the nonlinearity of $F$ are formulated below in terms of
for each fixed $(t, u, u)¥in[0,$
considered as a function of
.
,
Our result is the following theorem.
$¥infty)¥chi(0, ¥infty)¥times$
$¥lambda¥in(0, ¥infty)$
$[0$
$d(t, u, u;¥lambda)$
$¥infty)$
Theorem. Suppose that one of the following conditions is satisfied:
$F$ is nondecreasing in
, and for any fixed $(t, u, v)$ , $d(t, u, v;¥lambda)$ is a non;
and satisfies
decreasing function of
$d(t,
u, v;¥lambda)$ is a non$F$ is nondecreasing in
$(t,
u,
v)$
,
, and for any fixed
;
and satisfies
increasing function of
$F$ is nonincreasing in
is a non, and for any fixed $(t, u, v)$ ,
.
increasing function of
Suppose moreover that
$u$
$(¥mathrm{C}_{1})$
$¥lim_{¥lambda¥rightarrow 0}d(t, u, v;¥lambda)=0$
$¥lambda¥in(0, ¥infty)$
$u$
$(¥mathrm{C}_{2})$
$¥lim_{¥lambda¥rightarrow¥infty}d(t, u, u;¥lambda)=0$
$¥lambda¥in(0, ¥infty)$
$¥lambda d(t, u, v, ¥lambda)$
$u$
$(¥mathrm{C}_{3})$
$¥lambda¥in(0, ¥infty)$
(2)
$¥int_{1}^{¥infty}tF(t, c, c/t)dt<¥infty$
for some $c>0$ in case
$(¥mathrm{C}_{1})$
(3)
$(¥mathrm{C}_{2})$
holds, or
$¥int_{1}^{¥infty}tF(t, c, ¥sigma c/t)dt<¥infty$
holds.
in case
holds, or any sufficiently large
sufficiently small $l>0$ in case
holds, there exists a positive entire solution $u(x)$ of (I) satisfying
or
for some $c>0$ and
Then, for any
$l>0$
or
in case
$(¥mathrm{C}_{2})$
$¥sigma>1$
$(¥mathrm{C}_{3})$
$(¥mathrm{C}_{1})$
$(¥mathrm{C}_{3})$
$¥lim_{|x|¥sim¥infty}u(x)=l$
.
We use the supersolution-subsolution method (see Noussair and Swanson [4] which is by now well-known. We put $k(t)=¥max¥{1, t¥}$ for ¥ .
: From the conditions imposed on $F(t, u, v)$ it follows that
Case
Proof.
$t geq 0$
$)$
$(¥mathrm{C}_{1})$
191
Semilinear Elliptic Equations
(4)
if
$k(t)F(t, ¥lambda, ¥lambda/k(t))/¥lambda¥leq k(t)F(t, c, c/k(t))/c$
$0<¥lambda¥leq c$
,
$c$
,
$t¥in[0,$
being the constant appearing in (2), and that for
as
$k(t)F(t, ¥lambda, ¥lambda/k(t))/¥lambda¥rightarrow 0$
$¥infty)$
$t¥in[0,$
$¥lambda¥rightarrow 0$
,
$¥infty)$
,
.
The Lebesgue dominated convergence theorem then shows that
as
$¥int_{0}^{¥infty}k(t)F(t, ¥lambda, ¥lambda/k(t))dt/¥lambda¥rightarrow 0$
Therefore two constants ,
$¥eta$
$¥zeta>0$
$¥lambda¥rightarrow 0$
.
can be chosen small enough so that
,
$¥int_{0}^{¥infty}k(t)F(t, ¥eta, ¥eta/k(t))dt¥leq¥eta$
(5)
,
$¥int_{0}^{¥infty}k(t)F(t, ¥zeta, ¥zeta/k(t))dt<¥zeta$
$¥zeta=¥eta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t, ¥eta, ¥eta/k(t))dt$
With this choice of , , we define functions
$¥zeta$
$¥eta$
(6)
$z$
and
$y$
.
on [0,
$¥infty$
) by
$tt=0>0,$
$¥mathrm{f}¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{o}¥mathrm{r}$
$z(t)=¥left¥{¥begin{array}{l}¥zeta-t^{2-n}¥int_{0}^{t}s^{n-3}(¥int_{s}^{¥infty}rF(r,¥zeta,¥zeta/k(r))dr)ds¥¥¥zeta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t,¥zeta,¥zeta/k(t))dt¥end{array}¥right.$
.
(7)
$y(t)=¥{_{¥eta}¥eta-t^{2-n}¥int_{0}^{t}s^{n-3}(¥int_{0}^{s}rF(r, ¥eta, ¥eta/k(r))dr)ds$
$tt=0>0,$
$¥mathrm{f}¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{o}¥mathrm{r}$
.
It is easily seen that
as
$ z(t)¥rightarrow¥zeta$
$ t¥rightarrow¥infty$
$z^{¥prime}(t)=¥int_{0}^{t}(s/t)^{n-1}F(s, ¥zeta, ¥zeta/k(s))ds¥geq 0$
for
$t¥geq 0$
,
$(t^{n-1}z^{¥prime}(t))^{¥prime}=t^{n-1}F(t, ¥zeta, ¥zeta/k(t))$
for
$t¥geq 0$
.
From the second equation we have
$0¥leq t¥leq 1$ , we
have
$z^{¥prime}(0)=0^{-},$
and
$|z^{¥prime}(t)|¥leq¥zeta/k(t)$
for
,
$t¥geq 0$
$|z^{¥prime}(t)|¥leq¥int_{0}^{1}F(s, ¥zeta, ¥zeta/k(s))ds¥leq¥int_{0}^{¥infty}k(s)F(s, ¥zeta, ¥zeta/k(s))ds<¥zeta$
and if
$t¥geq 1$
, we have
,
.
In fact, if
192
H. USAMI
$|z^{¥prime}(t)|¥leq(¥int_{0}^{1}+¥int_{1}^{t})(s/t)F(s, ¥zeta, ¥zeta/k(s))ds$
$¥leq(1/t)(¥int_{0}^{1}F(s, ¥zeta, ¥zeta/k(s))ds+¥int_{1}^{¥infty}sF(s, ¥zeta, ¥zeta/k(s))ds)$
$=(1/t)¥int_{0}^{¥infty}k(s)F(s, ¥zeta, ¥zeta/k(s))ds<¥zeta/t$
Then the function
$w(x)=z(|x|)¥in C_{1¥mathrm{o}¥mathrm{c}}^{2+¥theta}(R^{n})$
.
satisfies the following inequalities:
$ 0<¥zeta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t, ¥zeta, ¥zeta/k(t))dt¥leq w(x)¥leq¥zeta$
,
$¥Delta w(x)=F(|x|, ¥zeta, ¥zeta/k(|x|))$
$¥geq F(|x|, w(x), |¥nabla w(x)|)¥geq-f(x, w(x), ¥nabla w(x))$
,
$X¥in R^{n}$
.
Therefore $w(x)$ is a subsolution of (1) in
.
$y(t)$
In the same way we can show that
satisfies the following:
$R^{n}$
$ y(t)¥rightarrow¥eta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t, ¥eta, ¥eta/k(t))dt=¥zeta$
$y^{¥prime}(t)=-¥int_{0}^{t}(s/t)^{n-1}F(s, ¥eta, ¥eta/k(s))ds¥leq 0$
$|y^{¥prime}(t)|¥leq¥eta/k(t)$
$v(x)=y(|x|)¥in C_{1¥mathrm{o}¥mathrm{c}}^{2+¥theta}(R^{n})$
$¥zeta¥leq v(x)¥leq¥eta$
$ t¥rightarrow¥infty$
,
,
,
$(t^{n-1}y^{¥prime}(t))^{¥prime}=-t^{n-1}F(t, ¥eta, ¥eta/k(t))$
So, the function
as
,
for
$t¥geq 0$
.
satisfies the following inequalities:
,
$¥Delta v(x)=-F(|x|, ¥eta, ¥eta/k(|x|))$
,
$¥leq-F(|x|, v(x), |¥nabla v(x)|)¥leq-f(x, v(x), ¥nabla v(x))$
$X¥in R^{n}$
.
Therefore $v(x)$ is a supersolution of (1) in . Since $w(x)¥leq v(x)$ in , applying the
supersolution-subsolution method [4, p. 125], we see that (1) has an entire solution
$u(x)$ such that $w(x)¥leq u(x)¥leq v(x)$ in
. Since
, we
conclude that
, as desired.
Case
: The conditions imposed on $F(t, u, v)$ imply that (4) holds for
,
and for each $t¥in[0,$ ),
as
. It follows that
$R^{n}$
$R^{n}$
$R^{n}$
$¥lim_{|x|¥rightarrow¥infty}w(x)=¥lim_{|x|¥rightarrow¥infty}v(x)=¥zeta$
$¥lim_{|x|¥rightarrow¥infty}u(x)=¥zeta$
$¥lambda¥geq c$
$(¥mathrm{C}_{2})$
$¥infty$
$k(t)F(t, ¥lambda, ¥lambda/k(t))/¥lambda¥rightarrow 0$
$¥int_{0}^{¥infty}k(t)F(t, ¥lambda, ¥lambda/k(t))dt/¥lambda¥rightarrow 0$
$¥lambda¥rightarrow¥infty$
as
$¥lambda¥rightarrow¥infty$
.
We now choose positive constants and so large that (5) is satisfied, and with these
by (6) and (7), respectively. The same argument
and , define the functions
yields the existence of an entire solution $u(x)$ of (1) such
as in the proof of case
that
.
$¥eta$
$¥eta$
$¥mathrm{z}¥dot{¥mathrm{a}}¥mathrm{n}¥mathrm{d}$
$¥zeta$
$(¥mathrm{C}_{1})$
$¥lim_{|x|¥rightarrow¥infty}u(x)=¥zeta$
$y$
$¥zeta$
193
Semilinear Elliptic Equations
: From the conditions imposed on
Case
can be chosen large enough so that
$F(t, u, v)$
$(¥mathrm{C}_{3})$
$¥zeta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t, ¥zeta/¥sigma, ¥zeta/k(t))dt¥geq¥zeta/¥sigma$
$¥int_{0}^{¥infty}k(t)F(t, ¥zeta/¥sigma, ¥zeta/k(t))dt¥leq¥zeta$
$¥zeta$
$¥eta$
and
$z$
$¥eta$
$¥zeta>0$
,
,
$¥eta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t, ¥zeta/¥sigma, ¥zeta/k(t))dt=¥zeta$
With this choice of , , we define the functions
, two constants ,
$y$
.
by
$¥mathrm{f}¥mathrm{f}¥mathrm{o}¥mathrm{r}t=0¥mathrm{o}¥mathrm{r}t>0,$
$z(t)=¥left¥{¥begin{array}{l}¥zeta-t^{2- n}¥int^{tt}s^{n-3}(¥int_{s}^{¥infty}rF(r,¥zeta/¥sigma,¥zeta/k(r))dr)ds¥¥¥zeta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t,¥zeta/¥sigma,¥zeta/k(t))dt¥end{array}¥right.$
.
for
$t>0$ ,
for
$t=0$ .
$y(t)=¥{_{¥eta}¥eta-t^{2-n}¥int_{0}^{t}s^{n-3}(¥int_{0}^{s}rF(r, ¥zeta/¥sigma, ¥zeta/k(r))dr)ds$
Arguing as in the preceding cases, we obtain
as
$z(t)¥rightarrow¥zeta=¥eta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t, ¥zeta/¥sigma, ¥zeta/k(t))dt$
$z^{¥prime}(t)=¥int_{0}^{t}(s/t)^{n-1}F(s, ¥zeta/¥sigma, ¥zeta/k(s))ds¥geq 0$
$|z^{¥prime}(t)|¥leq¥zeta/k(t)$
$ t¥rightarrow¥infty$
,
,
,
$(t^{n-1}z^{¥prime}(t))^{¥prime}=t^{n-1}F(t, ¥zeta/¥sigma, ¥zeta/k(t))$
,
for
as
$ y(t)¥rightarrow¥eta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t, ¥zeta/¥sigma, ¥zeta/k(t))dt=¥zeta$
$t¥geq 0$
,
$ t¥rightarrow¥infty$
,
,
$y^{¥prime}(t)=-¥int_{0}^{t}(s/t)^{n-1}F(s, ¥zeta/¥sigma, ¥zeta/k(s))ds¥leq 0$
$|y^{¥prime}(t)|¥leq¥zeta/k(t)$
,
$(t^{n-1}y^{¥prime}(t))^{¥prime}=-t^{n-1}F(t, ¥zeta/¥sigma, ¥zeta/k(t))$
Therefore the functions
for
,
$v(x)=y(|x|)$ , $w(x)=z(|x|)$
$t¥geq 0$
.
satisfy the following relations:
,
$ 0<¥zeta-¥frac{1}{n-2}¥int_{0}^{¥infty}tF(t, ¥zeta/¥sigma, ¥zeta/k(t))dt¥leq w(x)¥leq¥zeta¥leq v(x)¥leq¥eta$
,
$¥Delta w(x)¥geq F(|x|, w(x), |¥nabla w(x)|)¥geq-f(x, w(x), ¥nabla w(x))$
,
$¥Delta v(x)¥leq-F(|x|, v(x), |¥nabla v(x)|)¥leq-f(x, v(x), ¥nabla v(x))$
$x$ $¥in R^{n}$
,
194
H. USAMI
.
$¥lim_{|x|¥rightarrow¥infty}w(x)=¥lim_{|x|¥rightarrow¥infty}v(x)=¥zeta$
Thus we can conclude that there exists a positive entire solution
. This completes the proof.
$u(x)$
of (1) satisfying
$¥lim_{|x|¥rightarrow¥infty}u(x)=¥zeta$
Remark.
Consider the equation
(8)
$¥Delta u+a(x)¥phi(u)=0$
,
$x¥in R^{n}$
where $a(x)¥phi(u)$ is locally Holder continuous, and
Let $a^{*}(t)=¥max_{|x|=t}|a(x)|$ , and suppose that
$¥int_{0}^{¥infty}$
(9)
$¥phi:(0, ¥infty)¥rightarrow(0, ¥infty)$
$ ta^{*}(t)dt=A^{*}<¥infty$
In this case, the situation is simpler.
large that
$¥eta-¥frac{A^{*}}{n-2}¥phi(¥zeta)=¥zeta$
$¥eta$
$¥zeta$
is nonincreasing.
.
We can choose positive constants
,
$¥zeta-¥frac{A^{*}}{n-2}¥phi(¥xi)¥geq¥xi$
With this choice of , , , we define the functions
$¥xi$
,
$y$
and
$z$
$¥xi$
,
$¥eta,¥check{¥mathrm{t}_{¥ni}}$
so
.
by
$y(t)=¥{_{¥eta}¥eta-¥phi(¥zeta)t^{2-n}¥int_{0}^{t}s^{n-3}(¥int_{0}^{s}ra^{*}(r)dr)ds$
for
$t>0$ ,
for
$t=0$ .
for
$t>0_{y}$
for
$t=0$ .
$z(t)=¥{¥zeta^{-n}-¥int_{¥zeta-¥frac{¥phi(A^{*}¥xi)t^{2}}{n-2}¥phi(¥xi)}0ts^{n-3}(¥int_{s}^{¥infty}ra^{*}(r)dr)ds$
In the same manner as in the proof of the theorem, we can conclude that there exists
. If in particular
a positive entire solution $u(x)$ of (8) satisfying
as
$¥phi(u)=u^{-¥gamma}(¥gamma>0)$ , then from (9) we see that
(8) has a positive entire solution $u(x)$
having the prescribed limit at infinity provided
.
$ u(x)¥rightarrow¥zeta$
$¥zeta$
$¥zeta¥geq¥frac{¥mathcal{T}+1}{¥gamma}$
$|x|¥rightarrow¥infty$
$(¥mathcal{T}A^{*}/(n-2))^{1/(1+¥mathcal{T})}$
Acknowledgement. The author wishes to express his sincere thanks to Professor
Takasi Kusano for a number of helpful suggestions and stimulating discussions.
Semilinear Elliptic Equations
195
References
[1]
[2]
[3]
[4]
Kawano, N., On bounded entire solutions of semilinear elliptic equations, Hiroshima
Math. J., 14 (1984), 125-158.
Kusano, T. and Oharu, S., On entire solutions of second order semilinear elliptic equations, J. Math. Anal. Appl., 113 (1986), 123-135.
Naito, M., A note on bounded positive entire solutions of semilinear elliptic equations,
Hiroshima Math. J., 14 (1984), 211-214.
Noussair, E. S. and Swanson, C. A., Positive solutions of quasilinear elliptic equations
in exterior domains, J. Math. Anal. Appl., 75 (1980), 121-133.
nuna adreso:
Department of Mathematics
Faculty of Science
Hiroshima University
Hiroshima, Japan
(Ricevita la 27-an de augusto, 1984)