Funkcialaj Ekvacioj, 37 (1994) 345-361
Existence and Asymptotic Behavior of Positive Solutions of
Second Order Quasilinear Differential Equations
By
KUSANO Takasi and Akio OGATA
(Hiroshima University and Miyazaki University, Japan)
0. Introduction
We consider second order quasilinear ordinary differential equations of
the type
(A)
,
$(r(t)(y^{¥prime})^{¥alpha*})^{¥prime}+f(t, y, y^{¥prime})=0$
where
$¥alpha>0$
,
$(y^{¥prime})^{¥alpha*}=|y^{¥prime}|^{¥alpha}¥mathrm{s}¥mathrm{g}¥mathrm{n}y^{¥prime}$
and
$r:[a,$
$¥infty$
)
$¥rightarrow R_{+}$
and $f:[a,$
$¥infty$
)
,
$¥times¥overline{R}_{+}¥times R¥rightarrow R$
are continuous,
$R_{+}=(0, ¥infty)$
,
.
,
Our objective is to study the problem of existence and asymptotic behavior
of positive solutions of (A), showing under appropriate hypotheses on $r(t)$ and
that (A) possesses positive solutions with various asymptotic behaviors
which is
. By a solution of (A) we mean a function $y:[a,$
)
as
.
and satisfies (A) on
continuously differentiate together with
We distinguish the two cases
$¥overline{R}_{+}=[0,$
$¥infty)$
$a¥geqq 0$
$f(t, y, y^{¥prime})$
$¥infty$
$ t¥rightarrow¥infty$
$[a,$
$r(t)(y^{¥prime})^{a*}$
$¥int_{a}^{¥infty}(r(t))^{-1/a}dt=¥infty$
and
$¥rightarrow R$
$¥int_{a}^{¥infty}(r(t))^{-1/¥alpha}dt<¥infty$
$¥infty)$
,
and for each of these cases we construct three kinds of positive solutions with
different asymptotic properties. A standard fixed point technique is applied
for this purpose. The main results for (A) are presented in Sections 1 and
2. These results are applied in Section 3 to obtain criteria for the existence
of positive symmetric solutions, either bounded or unbounded, of particular
classes of quasilinear partial differential equations of the forms
$¥sum_{i=1}^{N}D_{i}$
( Du
$|$
$|^{p-2}D_{i}u$
)
$+g$
(
$¥sum_{i=1}^{N}D_{i}(|D_{i}u|^{p-2}D_{i}u)+g$
$¥mathrm{x}, u, $
(
Du)=0,
$x, ¥mathrm{u}, $
Du)=0
346
KUSANO Takasi and Akio OGATA
in exterior domains in
, where $p>1$ , $N¥geqq 2$ ,
,
, $i=1,¥cdots,N$ , $Du=(D_{1}u,¥cdots,D_{N}u)$ .
For closely related results we refer to the papers [2, 5] in which oscillation
theory is developed for less general equations of the form
$R^{N}$
$¥mathrm{x}$
$=(¥mathrm{x}_{1},¥cdots,¥mathrm{x}_{N})¥in R^{N}$
$D_{i}=¥partial/¥partial x_{i}$
.
$(r(t)(y^{¥prime})^{a*})^{¥prime}+f(t, y)=0$
1. The case
$¥int_{a}^{¥infty}(r(t))^{-1/¥alpha}dt=¥infty$
We first examine the equation (A) in which
(1.2)
$r(t)$
satisfies
.
$¥int_{a}^{¥infty}(r(t))^{-1/¥alpha}dt=¥infty$
Extensive use will be made of the function
(1.2)
$R_{a}(t)=¥int_{a}^{t}(r(s))^{-1/¥alpha}ds$
,
$t¥geqq a$
.
Noting that the unperturbed equation
has the solutions
, we intend to construct positive solutions of (A) having the following
asymptotic properties:
$(r(t)(y^{¥prime})^{¥alpha*})^{¥prime}=0$
$¥{1, R_{¥alpha}(t)¥}$
(I)
(II)
(III)
$¥lim_{t¥rightarrow¥infty}[y(t)/R_{¥alpha}(t)]=$
$¥lim_{t¥rightarrow¥infty}y(t)=$
const
const
$>0$
$>0$
;
;
$¥lim_{r¥rightarrow¥infty}[y(t)/R_{¥alpha}(t)]=0,¥lim_{t¥rightarrow¥infty}y(t)=¥infty$
.
To formulate theorems guaranteeing the existence of these three types of
solutions of (A) we need the following structure hypotheses:
(Majorization) There is a continuous function $F:[a,$
which is nondecreasing in the second and third variables and satisfies
$¥infty)¥times¥overline{R}_{+}^{2}¥rightarrow¥overline{R}_{+}$
$(¥mathrm{F}_{1})$
$|f(t, y, z)|¥leqq F(t, y, |z|)$
,
$(t, y, z)¥in[a,$
(Superlinearity) For any $(t, u, v)¥in[a,$
is nondecreasing in $¥lambda>0$ and satisfies
$(¥mathrm{F}_{2})$
$¥lambda u$
,
$¥lambda v)$
$¥infty$
$¥infty)¥times¥overline{R}_{+}¥times R$
)
$¥lim_{¥lambda¥rightarrow+0}¥lambda^{-a}F(t, ¥lambda u, ¥lambda v)=0$
For any $(t, u, v)¥in[a,$
is nonincreasing in ¥
and satisfies
$(¥mathrm{F}_{3})$
$¥lambda u$
,
$¥mathrm{k}¥mathrm{v})$
(Sublinearity)
$ lambda>0$
$¥infty$
)
$¥times¥overline{R}_{+}^{2}$
.
the function
$¥lambda^{-¥alpha}F(t$
,
$¥lambda^{-¥alpha}F(t$
,
.
$¥times¥overline{R}_{+}^{2}$
the function
Differential
Second Order Quasilinear
Equations
$¥lim_{¥lambda¥rightarrow+¥infty}¥lambda^{-¥alpha}F(t, ¥lambda u, ¥lambda v)=0$
347
.
Criteria for the existence of solutions of the types (I) and (II) now follow.
Theorem 1.1.
holds.
In addition to (1.1) suppose that either
If there is a constant $c>0$ such that
$¥{(¥mathrm{F}_{1}), (¥mathrm{F}_{3})¥}$
(1.3)
$¥int_{a}^{¥infty}F(t, cR_{¥alpha}(t), c(r(t))^{-1/¥alpha})dt<¥infty$
of
the type (I).
In addition to (1.1) suppose that either
If there is a constant $c>0$ such that
$¥{(¥mathrm{F}_{1}), (¥mathrm{F}_{3})¥}$
(1.4)
or
,
then the equation (A) has infinitely many positive solutions
Theorem 1.2.
holds.
$¥{(¥mathrm{F}_{1}), (¥mathrm{F}_{2})¥}$
$¥{(¥mathrm{F}_{1}), (¥mathrm{F}_{2})¥}$
or
,
$¥int_{a}^{¥infty}(¥frac{1}{r(t)}¥int_{t}^{¥infty}F(s, c, c(r(s))^{-1/¥alpha})ds)^{1/¥alpha}dt<¥infty$
then the equation (A) has infinitely many positive solutions
Proof of
of
Theorem 1. 1. In view of (1.3) and
or
aid of the Lebesgue dominated convergence theorem that
$(¥mathrm{F}_{2})$
$(¥mathrm{F}_{3})$
the type (II).
we see with the
,
$¥lim_{k¥rightarrow*}k^{-1}¥int_{a}^{¥infty}.F(t, (2k)^{1/¥alpha}R_{¥alpha}(t), (2k)^{1/a}(r(t))^{-1/¥alpha})dt=0$
where $*=0$ if
chosen so that
$(¥mathrm{F}_{2})$
(1.5)
holds and
$*=¥infty$
if
$(¥mathrm{F}_{3})$
holds.
Therefore $k>0$ can be
.
$¥int_{a}^{¥infty}F(t, (2k)^{1/¥alpha}R_{a}(t), (2k)^{1/¥alpha}(r(t))^{-1/¥alpha})dt¥leqq k/2$
Note that there are infinitely many such positive constants .
We define
to be the set of functions $y¥in C^{1}[a,$ ) satisfying
$k$
$¥mathrm{Y}$
$¥infty$
(1.6)
$(k/2)^{1/¥alpha}R_{¥alpha}(t)¥leqq y(t)¥leqq(2k)^{1/a}R_{¥alpha}(t)$
for
$t¥geqq a$
(1.7)
.
Let
$¥swarrow¥Gamma$
,
$(k/2)^{1/¥alpha}(r(t))^{-1/¥alpha}¥leqq y^{¥prime}(t)¥leqq(2k)^{1/¥alpha}(r(t))^{-1/¥alpha}$
denote the mapping (integro-differential operator) defined by
,
$¥swarrow^{¥varpi}y(t)=¥int_{a}^{t}(¥frac{1}{r(s)}(k+¥int_{s}^{¥infty}f(¥sigma, y(¥sigma), y^{¥prime}(¥sigma))d¥sigma))^{1/¥alpha}ds$
maps
It can be shown that (i)
is relatively compact in $C^{1}[a,$
Y. If $y¥in 7$, then by
(i)
$¥swarrow¥Gamma$
$¥swarrow^{¥varpi}(¥mathrm{Y})$
$¥swarrow^{¥varpi}(¥mathrm{Y})¥subset$
into
$¥mathrm{Y}$
$¥infty)$
$¥mathrm{Y};(¥mathrm{i}¥mathrm{i})¥swarrow¥varpi$
.
$(¥mathrm{F}_{1})$
and (1.5)
$t$
$¥geqq a$
.
is continuous; and (iii)
348
KUSANO Takasi and Akio OGATA
$|¥int_{s}^{¥infty}f(¥sigma, y(¥sigma), y^{¥prime}(¥sigma))d¥sigma|¥leqq¥int_{a}^{¥infty}F(¥sigma, y(¥sigma), y^{¥prime}(¥sigma))d¥sigma$
$¥leqq¥int_{a}^{¥infty}F(¥sigma, (2k)^{1/¥alpha}R_{¥alpha}(¥sigma), (2k)^{1/¥alpha}(r(¥sigma))^{-1/¥alpha})d¥sigma¥leqq k/2$
,
$s¥geqq a$
,
and so we have from (1.7)
(1.8)
$(k/2)^{1/¥alpha}R_{¥alpha}(t)¥leqq¥swarrow^{¥varpi}y(t),.¥leqq(2k)^{1/a}R_{¥alpha}(t)$
,
$t¥geqq a$
,
and
(1.9)
$(¥frac{k}{2r(t)})^{1/¥alpha}¥leqq(d^{¥Gamma}y)^{¥prime}(t)=(¥frac{1}{r(t)}(k+¥int_{t}^{¥infty}f(s, y(s), y^{¥prime}(s))ds))^{1/¥alpha}¥leqq(¥frac{2k}{r(t)})^{1/a}$
$t$
$¥geqq a$
.
maps
It follows that
, which implies that
into itself.
sequence
Let
be
a
is
continuous.
of
elements
of converging
(ii)
in the topology of $C^{1}[a,$ ), which is the uniform convergence of
to
functions and their first derivatives on compact subintervals of [ ,
. In
converge,
and
view of (1.7) it is easy to see that
respectively, to
and
on any compact subinterval of $[a,$
,
converges to
in $C^{1}[a,$ ). This proves the
which means that
continuity of .
is relatively compact. Noting that (1.8) and (1.9) hold for
(iii)
every
, it suffices to prove the equicontinuity of the set
:
on every compact interval of the form $[a, T]$ , $T>a$ .
Let $T>a$ be fixed and let $a¥leqq t_{1}<t_{2}¥leqq T$. Then,
$¥swarrow¥Gamma$
$¥swarrow^{¥varpi}y¥in ¥mathrm{Y}$
$¥swarrow¥varpi$
$¥mathrm{Y}$
$¥mathrm{Y}$
$¥{y_{¥mathrm{v}}¥}$
$y¥in ¥mathrm{Y}$
$¥infty$
$a$
$¥{(l^{¥Gamma}y_{¥mathrm{v}})^{¥prime}(t)¥}$
$¥{¥ovalbox{¥tt¥small REJECT} y_{¥mathrm{v}}(t)¥}$
$¥ovalbox{¥tt¥small REJECT} y(t)$
$¥infty)$
$(J^{¥varpi}y)^{¥prime}(t)$
$¥infty)$
$l^{¥Gamma}y$
$¥{¥ovalbox{¥tt¥small REJECT} y_{v}¥}$
$¥infty$
$¥swarrow c^{j^{-}}$
$J^{¥varpi}(¥mathrm{Y})$
$J^{¥Gamma^{l}}(¥mathrm{Y})=¥{(¥swarrow^{¥Gamma}y)^{¥prime}$
$y¥in ¥mathrm{Y}$
$y¥in ¥mathrm{Y}¥}$
(1.10)
$(¥swarrow^{¥Gamma}y)^{¥prime}(t_{2})-$
$(¥swarrow^{¥Gamma}y)^{¥prime}(t_{1})=(g(t_{2}))^{1/¥alpha}-(g(t_{1}))^{1/¥alpha}$
,
where
$g(t)=¥frac{1}{r(t)}(k+¥int_{t}^{¥infty}f(s, y(s), y^{¥prime}(s))ds)$
.
Combining (1.10) with the inequalities
$|(g(t_{2}))^{1/¥alpha}-(g(t_{1}))^{1/¥alpha}|¥leqq|g(t_{2})-g(t_{1})|^{1/¥alpha}$
for
$¥alpha¥geqq 1$
$|(g(t_{2}))^{1/¥alpha}-(g(t_{1}))^{1/¥alpha}|¥leqq(1/¥alpha)|g(¥tau)|^{(1-¥alpha)/¥alpha}|g(t_{2})-g(t_{1})|$
where
$¥tau¥in[t_{1}, t_{2}]$
, and
$|g(t_{2})-g(t_{1})|¥leqq¥frac{1}{r(t_{1})}|¥int_{¥mathrm{r}_{1}}^{¥mathrm{r}_{2}}f(s, y(s), y^{¥prime}(s))ds|$
,
for
$0<¥alpha<1$ ,
Second Order Quasilinear
Differential
349
Equations
$+|¥frac{1}{r(t_{2})}-¥frac{1}{r(t_{1})}|(k+¥int_{¥mathrm{r}_{2}}^{¥infty}f(s, y(s), y^{¥prime}(s))ds)$
$¥leqq¥frac{1}{r(t_{1})}¥int_{t_{1}}^{t_{2}}F(s,$
$+2k|¥frac{1}{r(t_{2})}-¥frac{1}{r(t_{1})}|$
we conclude that for any given
$t_{1}$
,
$t_{2}¥in[a, T]$
, implies
equicontinuous on
$¥epsilon>0$
$|(¥ovalbox{¥tt¥small REJECT} y)^{¥prime}(t_{2})-$
$[a, T]$
.
,
$(2k)^{1/¥alpha}R_{¥alpha}(s)$
,
there exists
such that $|t_{2}-t_{1}|<¥delta$ ,
is
, showing that
is relatively compact in the
$¥delta>0$
$(¥swarrow^{¥Gamma}y)^{¥prime}(t_{1})|<¥epsilon$
It follows that
$(2k)^{1/¥alpha}(r(s)^{-1/¥alpha})ds$
$¥ovalbox{¥tt¥small REJECT}(¥mathrm{Y})$
$¥ovalbox{¥tt¥small REJECT}^{¥prime}(¥mathrm{Y})$
)-topology.
Therefore the Schauder-Tychonoff fixed point theorem (see
such that
, and there is an element
applicable to
$C^{1}[a,$
$y(t)=¥int_{a}^{¥mathrm{r}}(¥frac{1}{r(s)}(k+¥int_{s}^{¥infty}f(¥sigma, y(¥sigma), y^{¥prime}(¥sigma))d¥sigma))^{1/¥alpha}ds$
Differentiating this equation, we see that
. We also see that
equation (A) on [ ,
$a$
$¥infty)$
L’Hospital’s rule we find
(I). This completes the proof.
$¥mathrm{e}.¥mathrm{g}$
$y=¥swarrow^{¥varpi}y$
$y¥in ¥mathrm{Y}$
$¥swarrow¥varpi$
,
.
,
$t¥geqq a$
[1, 4]) is
,
$¥mathrm{i}.¥mathrm{e}.$
.
is a positive solution of the
, and hence by
$y(t)$
$¥lim_{t¥rightarrow¥infty}(r(t))^{1/¥alpha}y^{¥prime}(t)=k$
.
$¥lim_{t¥rightarrow¥infty}[y(t)/R_{¥alpha}(t)]=k$
Thus
is of the type
$y(t)$
Theorem 1.2. As in the beginning of the proof of Theorem 1.1
one can see the existence of infinitely many positive constants such that
Proof of
$k$
(1.11)
$¥int_{a}^{¥infty}F(t, ¥mathit{2}k, 2k(r(t))^{-1/¥alpha})dt¥leqq(k/2)^{¥alpha}$
and
(1.12)
$¥int_{a}^{¥infty}(¥frac{1}{r(t)}¥int_{t}^{¥infty}F(s, ¥mathit{2}k, 2k(r(s))^{-1/¥alpha})ds)^{1/a}dt¥leqq k/2$
define the set
For such a constant
) as follows:
$k$
$¥ovalbox{¥tt¥small REJECT}:Z¥rightarrow C^{1}[a,$
(1.13)
$Z¥subset C^{1}[a,$
$¥infty$
.
) and the mapping
$¥infty$
$Z=¥{z¥in C^{1}[a, ¥infty):k/2¥leqq z(t)¥leqq 2k, |z^{¥prime}(t)|¥leqq 2k(r(t))^{-1/¥alpha}, t¥geqq a¥}$
(1.14)
,
$¥ovalbox{¥tt¥small REJECT} z(t)=k-¥int_{t}^{¥infty}(¥frac{1}{r(t)}¥int_{s}^{¥infty}f(¥sigma, y(¥sigma), y^{¥prime}(¥sigma))d¥sigma)^{1/a*}ds$
where
Let
$¥xi^{1/¥alpha*}=|¥xi|^{1/¥alpha}¥mathrm{s}¥mathrm{g}¥mathrm{n}¥xi$
$z¥in Z$
.
Since
.
$t¥geqq a$
,
,
KUSANO Takasi and Akio OGATA
350
$|¥int_{t}^{¥infty}(¥frac{1}{r(t)}¥int_{s}^{¥infty}f(¥sigma, z(¥sigma), z^{¥prime}(¥sigma))d¥sigma)^{1/¥alpha*}ds|$
$¥leqq¥int_{a}^{¥infty}(¥frac{1}{r(s)}¥int_{s}^{¥infty}F(¥sigma, ¥mathit{2}k, 2k(r(¥sigma))^{-1/¥alpha})ds)^{1/¥alpha}ds$
we see from (1.14) and (1.12) that
(1.14) and using (1.11), we have
$k/2¥leqq¥ovalbox{¥tt¥small REJECT} z(t)¥leqq 2k$
for
,
$t¥geqq a$
$|(¥ovalbox{¥tt¥small REJECT} z)^{¥prime}|¥leqq(¥frac{1}{r(t)}¥int_{t}^{¥infty}F(s, ¥mathit{2}k, 2k(r(s))^{-1/¥alpha})ds)^{1/¥alpha}¥leqq¥frac{k}{2}(r(t))^{-1/¥alpha}$
Differentiating
.
,
$t¥geqq a$
.
. The continuity of
and the relative
It follows therefore that
$C^{1}[a,$
in
compactness of
) can be verified as in the proof of Theorem
1.1. The Schauder-Tychonoff theorem then ensures the existence of a fixed
element $z¥in Z$ of , which satisfies
$¥ovalbox{¥tt¥small REJECT}(Z)¥subset Z$
$¥ovalbox{¥tt¥small REJECT}$
$¥infty$
$¥ovalbox{¥tt¥small REJECT}(Z)$
$¥ovalbox{¥tt¥small REJECT}$
$z(t)=k-¥int_{t}^{¥infty}(¥frac{1}{r(s)}¥int_{s}^{¥infty}f(¥sigma, z(¥sigma), z^{¥prime}(¥sigma))d¥sigma)^{1/¥alpha*}ds$
From this equation we infer that
and satisfies
defined on [ ,
,
$t¥geqq a$
.
is a positive solution of (A) which is
. This completes the proof.
In order to establish the existence of type (III) solutions of (A) we need
the following conditions on $f(t, y, z)$ .
is continuous, and $f(t, y, z)$ is nondecreaing in
, )
and .
is nonthe function
For any $(t, y, z)¥in[a,$ )
increasing in
and satisfies
$¥infty)$
$a$
$(¥mathrm{F}_{4})f:[a$
$¥infty$
$z(t)$
$¥lim_{r¥rightarrow¥infty}z(t)=k$
$¥times¥overline{R}_{+}^{2}¥rightarrow¥overline{R}_{+}$
$z$
$y$
$¥infty$
$(¥mathrm{F}_{5})$
$¥lambda^{-¥alpha}f(t, ¥lambda y, ¥lambda z)$
$¥times¥overline{R}_{+}^{2}$
$¥lambda>0$
.
$¥lim_{¥lambda¥rightarrow+¥infty}¥lambda^{-¥alpha}f(t, ¥lambda y, ¥lambda z)=0$
Theorem 1.3. In addition to (1.1) suppose that
and
hofd
equation (A) has infinitely many positive solutions of the type (III) if
$(¥mathrm{F}_{4})$
(1.15)
for some
(1.16)
$(¥mathrm{F}_{5})$
$¥int_{a}^{¥infty}f(t, cR_{¥alpha}(t), c(r(t))^{-1/¥alpha})dt<¥infty$
constant $c>0$ , and
$¥int_{a}^{¥infty}(¥frac{1}{r(t)}¥int_{t}^{¥infty}f(s, d, 0)ds)^{1/a}dt=¥infty$
for every constant $d>0$ .
Proof. There are infinitely many
constants $k>0$ such that
The
Second Order Quasilinear Differential Equations
(1.17)
351
$¥int_{a}^{¥infty}f(t, k(1+R_{¥alpha}(t)), k(r(t))^{-1/¥alpha})dt¥leqq k^{¥alpha}$
Take one such
$k$
.
and define
(1.18)
$W=¥{w¥in C^{1}[a, ¥infty):k¥leqq w(t)¥leqq k(1+R_{¥alpha}(t)), 0¥leqq w^{¥prime}(t)¥leqq k(r(t))^{-1/¥alpha}, t¥geqq a¥}$
(1.19)
,
$¥ovalbox{¥tt¥small REJECT} w(t)=k+¥int_{a}^{t}(¥frac{1}{r(s)}¥int_{s}^{¥infty}f(¥sigma, w(¥sigma), w^{¥prime}(¥sigma))d¥sigma)^{1/¥alpha}ds$
$t¥geqq a$
,
.
is a coutinuous mapping which
It is not a difficult task to verify that
sends $W$ into a compact subset of $W$. Therefore there exists a fixed point
$w¥in W$ of
,
,
$¥ovalbox{¥tt¥small REJECT}$
$¥ovalbox{¥tt¥small REJECT}$
$¥mathrm{i}.¥mathrm{e}.$
,
$t¥geqq a$
$w(t)=k+¥int_{a}^{t}(¥frac{1}{r(s)}¥int_{s}^{¥infty}f(¥sigma, w(¥sigma), w^{¥prime}(¥sigma))d¥sigma)^{1/¥alpha}ds$
Differentiation of the above equation shows that
. From the relation
[ ,
$a$
.
is a solution of (A) on
$w(t)$
$¥infty)$
$w^{¥prime}(t)=(¥frac{1}{r(t)}¥int_{t}^{¥infty}f(s, w(s), w^{¥prime}(s))ds)^{1/¥alpha}$
$¥geqq a$
$t$
,
we obtain by L’Hospital’s rule
$¥lim_{t¥rightarrow¥infty}[w(t)/R_{¥alpha}(t)]=¥lim_{t¥rightarrow¥infty}(r(t))^{1/¥alpha}w^{¥prime}(t)=0$
.
On the other hand, we have
$w(t)¥geqq k+¥int_{a}^{t}(¥frac{1}{r(s)}¥int_{s}^{¥infty}f(¥sigma, k, 0)d¥sigma)^{1/¥alpha}ds$
from which, using (1.16), we see that
$¥lim_{¥mathrm{r}¥rightarrow¥infty}w(t)=¥infty$
.
,
$t¥geqq a$
,
This shows that
$w(t)$
is
a solution of the type (III), and proof is complete.
Example 1.4.
(1.20)
Consider the equation
$((y^{¥prime})^{¥alpha*})^{¥prime}+t^{¥lambda}y^{¥beta*}+t^{¥mu}(y^{¥prime})^{¥gamma*}=0$
,
$t¥geqq 1$
,
where $¥alpha>0$ , $¥beta>0$ , $¥gamma>0$ , and are constants. This equation is a special
. The condition
case of (A) with $r(t)=1$ and
the conditions
, and with this
holds with the choice
. It
or
or
holds according to whether
$
¥
beta+
¥
lambda<-1$
and $¥mu<-1$ , and
is easily seen that (1.3) holds if and only if
$¥lambda$
$¥mu$
$f(t, y, z)=t^{¥lambda}y^{¥beta*}+t^{¥mu}z^{¥gamma*}$
$F(t, u, v)=t^{¥lambda}u^{¥beta}+t^{¥mu}v^{¥gamma}$
$(¥mathrm{F}_{2})$
$(¥mathrm{F}_{3})$
$¥{¥alpha<¥beta, ¥alpha<¥gamma¥}$
$(¥mathrm{F}_{1})$
$F$
$¥{¥alpha>¥beta, ¥alpha>¥gamma¥}$
352
KUSANO Takasi and Akio OGATA
(1.4) holds if and only if
and $¥alpha+¥mu<-1$ .
From Theorems 1.1 and 1.2 we conclude under the condition
,
or
that:
(i) if $¥beta+¥lambda<-1$ and $¥mu<-1$ , then (1.20) possesses infinitely many
positive solutions $y(t)$ on [1, ) such that
const $>0$ ;
(ii) if $¥alpha+¥lambda<-1$ and $¥alpha+¥mu<-1$ , then (1.20) possesses infinitely many
positive solutions $y(t)$ on [1, ) such that
const $>0$ .
It is clear that
holds for (1.20) and that
is satisfied if
and
. Theorem 1.3 applied to (1.20) yields the following statement.
$1-¥alpha¥leqq¥lambda<-1$
. If
and
and $¥mu<-1$ , then
(iii) Let
$y(t)$
on [1, ) satisfying
(1.20) possesses infinitely many positive solutions
and
.
$¥alpha+¥lambda<-1$
$¥{¥alpha<¥beta$
$¥{¥alpha>¥beta, ¥alpha>¥gamma¥}$
$¥alpha<¥gamma¥}$
$¥infty$
$¥lim_{t¥rightarrow¥infty}[y(t)/t]=$
$¥infty$
$¥lim_{t¥rightarrow¥infty}y(t)=$
$(¥mathrm{F}_{4})$
$¥alpha>¥beta$
$(¥mathrm{F}_{5})$
$¥alpha>¥gamma$
$¥alpha>¥beta$
$¥alpha>¥gamma$
$-¥beta$
?
$¥infty$
$¥lim_{t¥rightarrow¥infty}[y(t)/t]=0$
2. The case
$¥lim_{t¥rightarrow¥infty}y(t)=¥infty$
$¥int_{a}^{¥infty}(r(t))^{-1/¥alpha}dt<¥infty$
We now turn to the case where the function
(2. 1)
$¥int_{a}^{¥infty}(r(t))^{-1/¥alpha}dt<¥infty$
We define the function
(2.2)
$¥rho_{¥alpha}(t)$
$r(t)$
in (A) satisfies
.
by
$¥rho_{¥alpha}(t)=¥int_{t}^{¥infty}(r(s))^{-1/¥alpha}ds$
,
$t$
$¥geqq a$
.
From the fact that the unperturbed equation
has the solutions
it is natural to expect that (A) possibly possesses positive solutions
having the following three types of asymptotic behavior:
$(r(t)(y^{¥prime})^{¥alpha*})^{¥prime}=0$
$¥{1, ¥rho_{¥alpha}(t)¥}$
[I]
[II]
[III]
$¥lim_{t¥rightarrow¥infty}y(t)=$
const
$>0$
$¥lim_{t¥rightarrow¥infty}[y(t)/¥rho_{¥alpha}(t)]=$
;
const
$>0$
;
$¥lim_{t¥rightarrow¥infty}y(t)=0,¥lim_{t¥rightarrow¥infty}[y(t)/¥rho_{a}(t)]=¥infty$
.
That positive solutions of these types do exist under suitable conditions
will be shown in the theorems to follow.
Theorem 2.1.
holds.
$¥{(¥mathrm{F}_{1}), (¥mathrm{F}_{3})¥}$
In addition to (2.1) suppose that either
If there is a constant $c>0$ such that
$¥{(¥mathrm{F}_{1}), (¥mathrm{F}_{2})¥}$
or
Differential
Second Order Quasilinear
(2.3)
353
Equations
$¥int_{a}^{¥infty}F(t, c, c(r(t))^{-1/¥alpha})dt<¥infty$
,
then the equation (A) has infinitefy many positive solutions
of
In addition to (2.1) suppose that either
If there is a constant $c>0$ such that
Theorem 2.2.
holds.
$¥{(¥mathrm{F}_{1}), (¥mathrm{F}_{3})¥}$
the type [I].
or
$¥{(¥mathrm{F}_{1}), (¥mathrm{F}_{2})¥}$
,
(2.4)
$¥int_{a}^{¥infty}F(t, c¥rho_{a}(t), c(r(t))^{-1/¥alpha})dt<¥infty$
then the equation (A) has infinitefy many positive solutions
of
the type [II].
The ideas and techniques to be used to prove these theorems are
essentially the same as the ones developed in the proofs of Theorems 1.1 and
1.2. So, we give only a sketch of the proof, leaving the verification of the
details to the reader.
Proof of
Theorem 2. 1.
(2.5)
Choose $k>0$ so that
$¥int_{a}^{¥infty}F(t, ¥mathit{2}k, 2k(r(t))^{-1/¥alpha})dt¥leqq k^{¥alpha}$
and
(2.6)
,
$¥int_{a}^{¥infty}(¥frac{1}{r(t)}¥int_{a}^{t}F(s, ¥mathit{2}k, 2k(r(s))^{-1/¥alpha})ds)^{1/¥alpha}dt¥leqq k/2$
and define the set
$¥mathrm{Y}$
and the mapping
$d^{¥varpi}$
by
,
(2.7)
$¥mathrm{Y}=¥{y¥in C^{1}[a, ¥infty):k/2¥leqq y(t)¥leqq 2k, |y^{¥prime}(t)|¥leqq 2k(r(t))^{-1/¥alpha}, t¥geqq a¥}$
and
(2.8)
,
$J^{¥varpi}y(t)=k-¥int_{a}^{t}(¥frac{1}{r(s)}¥int_{a}^{s}f(¥sigma, y(¥sigma), y^{¥prime}(¥sigma))d¥sigma)^{1/a*}ds$
Then, by the Schauder-Tychonoff theorem,
gives rise to a type [I] solution of (A).
Proof of
(2.9)
has a fixed element
.
, which
$y¥in ¥mathrm{Y}$
Choose $k>0$ so that
$¥int_{a}^{¥infty}F(t, (2k)^{1/a}¥rho_{¥alpha}(t), (2k)^{1/a}(r(t))^{-1/¥alpha})dt¥leqq k/2$
and define
(2.10)
Theorem 2.2.
$¥swarrow¥varpi$
$t¥geqq a$
$Z$
to be the set of functions
$(k/2)^{1/¥alpha}¥rho_{a}(t)¥leqq z(t)¥leqq(2k)^{1/¥alpha}¥rho_{a}(t)$
$z¥in C^{1}[a,$
,
$¥infty$
) satisfying
$|z^{¥prime}(t)|¥leqq(2k)^{1/¥alpha}(r(t))^{-1/a}$
,
$t¥geqq a$
.
354
KUSANO Takasi and Akio OGATA
The desired solution of the type [II] of (A) is obtained as a fixed point
of the mapping
defined by
$z¥in Z$
$¥ovalbox{¥tt¥small REJECT}$
(2.11)
,
$t¥geqq a$
$¥ovalbox{¥tt¥small REJECT} z(t)=¥int_{t}^{¥infty}(¥frac{1}{r(s)}(k-¥int_{s}^{¥infty}f(¥sigma, z(¥sigma), z^{¥prime}(¥sigma))d¥sigma))^{1/¥alpha}ds$
.
Finally we discuss the existence of solutions of the type [III] by requiring
the following sign and sublinearity conditions on $f(t, y, z)$ which are similar
to
and
used in Theorem 1.3.
¥
¥
,
is continuous, ¥
[
], and
$f(t, y, z)$ is nondecreasing in
and
.
For any $(t, y, z)¥in[a,$ )
the function
is
¥
nonincreasing in
and satisfies
$(¥mathrm{F}_{4})$
$(¥mathrm{F}_{5})$
$(¥mathrm{F}_{4}^{¥prime})f:$
$a$
$ overline{R} _=(- infty,$
$¥infty)¥times¥overline{R}_{+}¥times¥overline{R}¥_¥rightarrow¥overline{R}_{+}$
$|z|$
$y$
$¥infty$
$(¥mathrm{F}_{5}^{¥prime})$
$0$
$¥times¥overline{R}_{+}¥times¥overline{R}¥_$
$¥lambda^{-¥alpha}f(t, ¥lambda y, ¥lambda z)$
$ lambda>0$
.
$¥lim_{¥lambda¥rightarrow+¥infty}¥lambda^{-¥alpha}f(t, ¥lambda y, ¥lambda z)=0$
Theorem 2.3.
addition that
Suppose that (2.1),
(2.12)
and
$(¥mathrm{F}_{5}^{¥prime})$
are satisfied.
$¥lim_{¥mathrm{r}¥rightarrow¥infty}¥frac{1}{r(t)}¥int_{a}^{t}f(s, 1, -1)ds=0$
(2.13)
for some
$(¥mathrm{F}_{4}^{¥prime})$
Suppose in
,
$¥int_{a}^{¥infty}(¥frac{1}{r(t)}¥int_{a}^{t}f(s, c, -c)ds)^{1/¥alpha}dt<¥infty$
constant $c>0$ , and
(2. 14)
$¥int_{a}^{¥infty}f(t, d¥rho_{¥alpha}(t), 0)dt=¥infty$
for every
constant $d>0$ . Then $fAe$ equation (A) has infinitely many positive
solutions of the type [III] which are defined on an interval [ $T$,
with $T>a$
$¥infty)$
sufficientfy large.
Proof.
Because of (2.12) there is $T>a$ such that
(2.15)
Put
$¥sup_{t¥geqq T}¥frac{1}{r(t)}(1+¥int_{T}^{t}f(s, 1, -1)ds)¥leqq 1$
$¥theta=2^{1/¥alpha}(1+¥rho_{¥alpha}(T))$
(2.11)
and define
.
.
Choose $k>0$ in such a way that
,
$¥int_{T}^{¥infty}(¥frac{1}{r(t)}¥int_{T}^{t}f(s, ¥theta k, -¥theta k)ds)^{1/¥alpha}dt¥leqq k$
$k¥theta¥geqq 1$
and
Second Order Quasilinear
Differential
355
Equations
(2.17)
$W=¥{w¥in C^{1}[T, ¥infty):k¥rho_{a}(t)¥leqq w(t)¥leqq¥theta k, -¥theta k¥leqq w^{¥prime}(t)¥leqq 0, t ¥geqq T¥}$
(2.18)
$¥ovalbox{¥tt¥small REJECT} w(t)=¥int_{t}^{¥infty}(¥frac{1}{r(s)}(k^{¥alpha}+¥int_{T}^{s}f(¥sigma, w(¥sigma), w^{¥prime}(¥sigma))d¥sigma))^{1/¥alpha}ds$
Let
$w¥in W$.
,
,
.
$t¥geqq T$
Then from (2.18) and (2.16) we see that
$k¥int_{t}^{¥infty}¥frac{ds}{(r(s))^{1/¥alpha}}¥leqq¥ovalbox{¥tt¥small REJECT} w(t)$
$¥leqq¥int_{t}^{¥infty}¥frac{1}{(r(s))^{1/a}}((2k^{¥alpha})^{1/¥alpha}+(2¥int_{T}^{s}f(¥sigma, w(¥sigma), w^{¥prime}(¥sigma))d¥sigma)^{1/¥alpha})ds$
$¥leqq¥int_{t}^{¥infty}¥frac{2^{1/¥alpha}k}{(r(s))^{1/¥alpha}}ds+2^{1/a}¥int_{t}^{¥infty}(¥frac{1}{r(s)}¥int_{T}^{s}f(¥sigma, ¥theta k, -¥theta k)d¥sigma)^{1/¥alpha}ds$
$¥leqq 2^{1/¥alpha}k(¥rho_{¥alpha}(T)+1)=¥theta k$
,
.
$t¥geqq T$
Furthermore, using the inequality
,
$¥sigma¥geqq T$
$(¥ovalbox{¥tt¥small REJECT} w)^{¥prime}(t)=-(¥frac{1}{r(t)}(k^{¥alpha}+¥int_{T}^{t}f(¥sigma, w(¥sigma), w^{¥prime}(¥sigma))d¥sigma))^{1/¥alpha}$
$t¥geqq T$
$f(¥sigma, w(¥sigma), w^{¥prime}(¥sigma))¥leqq f(¥sigma, ¥theta k, -¥theta k)¥leqq(¥theta k)^{¥alpha}f(¥sigma, 1, -1)$
which is implied by
$(¥mathrm{F}_{4}^{¥prime})$
and
$(¥mathrm{F}_{5}^{¥prime})$
,
, in the equation
,
we obtain
$0¥geqq(¥ovalbox{¥tt¥small REJECT} w)^{¥prime}(t)¥geqq-(¥frac{1}{r(t)}(k^{¥alpha}+(¥theta k)^{¥alpha}¥int_{T}^{t}f(¥sigma, 1, -1)d¥sigma))^{1/¥alpha}$
,
.
$¥geqq-¥theta k(¥frac{1}{r(t)}(1+¥int_{T}^{t}f(¥sigma, 1, -1)d¥sigma))^{1/¥alpha}¥geqq-¥theta k$
The above computations show that
maps
of
and the relative compactness of
there exists a function $w¥in W$ such that
$¥ovalbox{¥tt¥small REJECT}$
$¥ovalbox{¥tt¥small REJECT}$
$t¥geqq T$
into itself. Since the continuity
are proved in a routine manner,
$W$
$¥ovalbox{¥tt¥small REJECT}(W)$
,
,
$w(t)=¥int_{t}^{¥infty}(¥frac{1}{r(s)}(k^{¥alpha}+¥int_{T}^{s}f(¥sigma, w(¥sigma), w^{¥prime}(¥sigma))d¥sigma))^{1/a}ds$
$t¥geqq T$
and
$w^{¥prime}(t)=-(¥frac{1}{r(t)}(k^{¥alpha}+¥int_{T}^{s}f(¥sigma, w(¥sigma), w^{¥prime}(¥sigma))d¥sigma))^{1/¥alpha}$
It is obvious that
$w(t)$
satisfies (A) for
, and
$t¥geqq T$
$w(t)¥rightarrow 0$
.
$t¥geqq T$
as
$ t¥rightarrow¥infty$
.
That
KUSANO Takasi and Akio OGATA
356
$¥lim_{t¥rightarrow¥infty}[w(t)/¥rho_{a}(t)]=¥infty$
?
follows from the inequality
,
$t¥geqq T$
$(r(t))^{1/¥alpha}w^{¥prime}(t)¥geqq(k^{¥alpha}+¥int_{T}^{t}f(s, k¥rho_{¥alpha}(s), 0)ds)^{1/¥alpha}$
This completes the proof.
combined with the condition (2.14).
Consider the equations
Example 2.4.
$(2.19_{¥pm})$
$(e^{¥lambda t}(y^{¥prime})^{¥alpha*})^{¥prime}+e^{¥mu t}y^{¥beta*}¥pm e^{¥mathrm{v}t}(y^{¥prime})^{¥gamma*}=0$
where $¥alpha>0$ , $¥beta>0$ , ¥
special case of (A) with
$ lambda>0$
,
$¥mu$
and
and
$r(t)=e^{¥lambda t}$
$v$
,
$t¥geqq 0$
,
are constants. These equations are a
$f(t, y, z)=e^{¥mu t}y^{¥beta*}¥pm e^{vt}z^{¥gamma*}$
.
holds with $F(t, u, v)=e^{¥mu t}u^{¥beta}+e^{vt}v^{¥gamma}$ , and
The condition
and
are
and
guaranteed to hold if
, respectively.
¥
2.2
to
Applying Theorems 2.1 and
, we have the following
$(¥mathrm{F}_{1})$
$(¥mathrm{F}_{2})$
$¥{¥alpha<¥beta, ¥alpha<¥gamma¥}$
$(¥mathrm{F}_{3})$
$¥{¥alpha>¥beta, ¥alpha>¥gamma¥}$
$(2.19_{ pm})$
statements.
or
(i) If either
then $(2.19_{¥pm})$ , have positive solutions
(ii) If either
$ v/¥lambda<¥gamma/¥alpha$
const
, then
$y(t)$
$¥mu<0$
$¥lim_{t¥rightarrow¥infty}y(t)=$
or
, have positive solutions
$¥{¥alpha>¥beta, ¥alpha>¥gamma¥}$
$¥{¥alpha<¥beta, ¥alpha<¥gamma¥}$
$(2.19_{¥pm})$
and if
such that
$¥{¥alpha>¥beta, ¥alpha>¥gamma¥}$
$¥{¥alpha<¥beta, ¥alpha<¥gamma¥}$
$y(t)$
and if
such that
and
$ v/¥lambda<¥gamma/¥alpha$
const
,
$>0$ .
$¥mu/¥lambda<¥beta/¥alpha$
and
$¥lim_{t¥rightarrow¥infty}e^{(¥lambda/¥alpha)t}y(t)=$
$>0$ .
Theorem 2.3 is applicable to the equation $(2.19¥_)$ , for which
holds
trivially and
is satisfied if
and
.
. If
and
and $v/¥lambda<1$ , then
(iii) Suppose that
$(2.19¥_)$ has positive solutions $y(t)$ satisfying
and
.
$(¥mathrm{F}_{4}^{¥prime})$
$¥alpha>¥beta$
$(¥mathrm{F}_{5}^{¥prime})$
$¥alpha>¥beta$
$¥alpha>¥gamma$
$¥beta/¥alpha¥leqq¥mu/¥lambda<1$
$¥alpha>¥gamma$
$¥lim_{t¥rightarrow¥infty}e^{(¥lambda/¥alpha)t}y(t)=¥infty$
$¥lim_{t¥rightarrow¥infty}y(t)=0$
3.
Application to partial differential equations
Let us consider the partial differential equations
(B)
$¥sum_{i=1}^{N}D_{i}$
( Du
(C)
$¥sum_{i=1}^{N}D_{i}(|D_{i}u|^{p-2}D_{i}u)+g$
$|$
$|^{p-2}D_{i}u$
where $p>1$ is a constant,
$Du=(D_{1}u,¥cdots,D_{N}u)$ ,
$x$
)
$+g$
(
$|¥mathrm{x}|$
(
, , Du )
$u$
$|x|_{p*},$
$|$
$|$
$u$
$=0$ ,
, Du )
$|$
$=(¥mathrm{x}_{1},¥cdots,¥mathrm{x}_{N})¥in R^{N}$
$|_{p}$
,
$¥mathrm{x}¥in¥Omega_{a}$
$=0$ ,
$N¥geqq 2$
,
$¥mathrm{x}¥in¥Omega_{a}^{*}$
$D_{i}=¥partial/¥partial ¥mathrm{x}_{i}$
,
,
,
$ i=1,¥cdots$
, $N$ ,
Second Order Quasilinear
$|¥mathrm{x}|=(¥sum_{i=1}^{N}|¥mathrm{x}_{i}|^{2})^{1/2}$
$|$
,
|¥mathrm{x}|¥geqq a¥}$
357
Equations
,
$|¥mathrm{x}|_{p^{*}}=(¥sum_{i=1}^{N}|¥mathrm{x}_{i}|^{p/(p-1)})^{(p-1)/p}$
Du $|=(¥sum_{i=1}^{N}|D_{i}u|^{2})^{1/2}$ ,
$¥Omega_{a}=¥{¥mathrm{x}¥in R^{N} :
Differential
$|$
,
Du
$|_{p}=$
$(¥sum_{i=1}^{N}|D_{i}u|^{p})^{1/p}$
$¥Omega_{a}^{*}=¥{¥mathrm{x}¥in R^{N} :
,
|¥mathrm{x}|_{p^{*}}¥geqq a¥}$
,
$a>0$ ,
is a continuous function. Because of their importance
and $g:[a,$ )
in various situations, these equations have been the object of intensive
. [3, 6]).
investigations in recent years (see
We are interested in the existence and asymptotic behavior of positive
[resp.
]; such a
solutions of (B) [resp. (C)] which depend only on
solution is referred to as a symmetric solution of (B) or (C).
Basic to the subsequent discussions is the observation that a symmetric
function $u=y(|¥mathrm{x}|)$ [resp.
] is a solution of (B) [resp. (C)] if and
$y(t)$
equation
ordinary
differential
satisfies the
only if
$¥infty$
$¥times¥overline{R}_{+}^{2}¥rightarrow R$
$¥mathrm{e}.¥mathrm{g}$
$|x|$
$|¥mathrm{x}|_{p^{*}}$
$u=y(|¥mathrm{x}|_{p^{*}})$
(3.1)
$(t^{N-1}|y^{¥prime}|^{p-2}y^{¥prime})^{¥prime}+t^{N-1}g(t, y, |y^{¥prime}|)=0$
,
$¥geqq a$
,
$t¥geqq a$
.
$t$
which is rewritten as
,
$(t^{N-1}(y^{¥prime})^{p-1^{*}})^{¥prime}+t^{N-1}g(ty, |y^{¥prime}|)=0$
The equation (3.1) is a special case of (A) in which
(3.2)
$¥alpha=p-1$ , $r(t)=t^{N-1}$ , $f(t, y, z)=t^{N-1}g(t, y, z)$
,
and so the results for (A) developed in §§1, 2 can be applied to (3.1) to derive
information about the existence and asymptotic behavior of positive symmetric
solutions of (B) and (C). We note that the function $r(t)=t^{N-1}$ satisfies (1.1)
or (2.1) according to whether $N¥leqq p$ or $N>p$ . If $N¥leqq p$ , the function
given by (1.2) can be taken to be
$R_{¥alpha}(t)$
(3.3)
$R_{N,p}(t)=t^{(p-N)/(p-1)}$
while if $N>p$, the function
(3.4)
for $N<p;R_{N,p}(t)=¥log(t/a)$ for $N=p$ ,
$¥rho_{¥alpha}(t)$
defined by (2.2) can be taken to be
$¥rho_{N,p}(t)=t^{(p-N)/(p-1)}$
.
Hypotheses on the structure of (B) and (C) will be chosen from the list
below.
There is a positive continuous function $G(t, y, z)$ on
which is nondecreasing in and and satisfies
$[a,$
$(¥mathrm{G}_{1})$
$y$
$z$
$|g(t, y, z)|¥leqq G(t, y, z)$
For any
nondecreasing in ¥
$(¥mathrm{G}_{2})$
$(t, y, z)¥in[a,$
$ lambda>0$
$¥infty)¥times¥overline{R}_{+}^{2}$
$¥infty$
)
and satisfies
,
$(t, y, z)¥in[a,$
$¥times¥overline{R}_{+}^{2}$
$¥infty)¥times¥overline{R}_{+}^{2}$
the function
.
$¥lambda^{1-p}G(t, ¥lambda y, ¥lambda z)$
is
KUSANO Takasi and Akio OGATA
358
$¥lim_{¥lambda¥rightarrow+0}¥lambda^{1-p}G(t, ¥lambda y, ¥lambda z)=0$
For any
$(¥mathrm{G}_{3})$
nonincreasing in
$(t, y, z)¥in[a,$
$¥lambda>0$
$¥infty$
)
$¥mathrm{x}¥overline{R}_{+}^{2}$
.
the function
$g:[a,$
in
$¥infty$
)
and .
For any
nonincreasing in ¥
is
and satisfies
$¥lim_{¥lambda¥rightarrow¥dagger¥infty}¥lambda^{1-p}G(t, ¥lambda y, ¥lambda z)=0$
$(¥mathrm{G}_{4})$
$¥lambda^{1-p}G(t, ¥lambda y, ¥lambda z)$
$¥times¥overline{R}_{+}^{2}¥rightarrow¥overline{R}_{+}$
.
is continuous, and
$g(t, y, z)$
is nondecreasing
$z$
$y$
$(¥mathrm{G}_{5})$
$(t, y, z)¥in[a,$
$ lambda>0$
$¥infty$
)
$¥times¥overline{R}_{+}^{2}$
the function
$¥lambda^{1-p}g(t, ¥lambda y, ¥lambda z)$
is
and satisfies
,
$¥lim_{¥lambda¥rightarrow+¥infty}¥lambda^{1-p}g(t, ¥lambda y, ¥lambda z)=0$
Now we will formulate some existence results for (B) and (C) which follow
directly from the theorems presented in the preceding sections. For simplicity
.
intead of
we use therein the notation
$|x|_{*}$
$|¥mathrm{x}|_{p^{*}}$
Theorem 3.1. Suppose that $N=p$ and either
holds. If there is a constant $c>0$ such that
(3.5)
$¥int_{a}^{¥infty}t^{N-1}G$
( , clog $(t/a)$ ,
$t$
$ct^{-1}$
$¥{(¥mathrm{G}_{1}), (¥mathrm{G}_{2})¥}$
)
$dt$ $<¥infty$
or
$¥{(¥mathrm{G}_{1}), (¥mathrm{G}_{3})¥}$
,
then the equation (B) [resp. (C)] has infinitefy many positive symmetric solutions
$u(|x|)$ [resp.
] with the property
$u(|¥mathrm{x}|_{*})$
(3.6)
$¥lim_{|x|¥rightarrow¥infty}[u(|x|)/¥log|x|]=$
const
$>0$
[resp.
$¥lim_{|x|_{*}¥rightarrow¥infty}[u(|¥mathrm{x}|_{*})/¥log|¥mathrm{x}|_{*}]=$
Theorem 3.2. Suppose that $N¥neq p$ and either
holds. If there is a constant $c>0$ such that
(3.7)
$¥{(¥mathrm{G}_{1}), (¥mathrm{G}_{2})¥}$
$¥int_{a}^{¥infty}t^{N-1}G(t, ct^{(p-N)/(p-1)}, ct^{(1-N)/(p-1)})dt<¥infty$
or
const
$>0$ ].
$¥{(¥mathrm{G}_{1}), (¥mathrm{G}_{3})¥}$
,
then the equation (B) [resp. (C)] has infinitefy many positive symmetric solutions
$u(|x|)$ [resp.
] with the property
$u(|¥mathrm{x}|_{*})$
(3.8)
$¥lim_{|x|¥rightarrow¥infty}[u(|¥mathrm{x}|)/|¥mathrm{x}|^{(p-N)/(p-1)}]=$
[resp.
const
$>0$
$¥lim_{|x|_{*}¥rightarrow¥infty}[u(|¥mathrm{x}|_{*})/|¥mathrm{x}|_{*}^{(p-N)/(p-1)}]=$
const
$>0$
].
Second Order Quasilinear
Theorem 3.3.
(3.9)
$N¥leqq p$
for some
Suppose that either
and
359
Equations
or
$¥{(¥mathrm{G}_{1}), (¥mathrm{G}_{2})¥}$
$¥{(G_{1}), (¥mathrm{G}_{3})¥}$
holds.
If
$¥int_{a}^{¥infty}(t^{1-N}¥int_{t}^{¥infty}s^{N-1}G(s, c, cs^{(1-N)/(p-1)})ds)^{1/(p-1)}dt<¥infty$
constant $c>0$ , or
(3.10)
Differential
if
$N>p$ and
$¥int_{a}^{¥infty}t^{N-1}G(t, c, ct^{(1-N)/(p-1)})dt<¥infty$
for some
constant $c>0$ , then the equation (B) [resp. (C)] has infinitefy many
[resp. $u(|x|_{*})$ ] with the property
positive symmetric solutions
$u(|¥mathrm{x}|)$
(3.11)
$¥lim_{|x|¥rightarrow¥infty}u(|¥mathrm{x}|)=$
Theorem 3.4.
const
$>0$
[resp.
const
$¥lim_{|x|_{*}¥rightarrow¥infty}u(|¥mathrm{x}|_{*})=$
Suppose that $N>p$ and
$¥{(¥mathrm{G}_{4}), (¥mathrm{G}_{5})¥}$
holds.
(3.12)
$¥lim_{t¥rightarrow¥infty}t^{1-N}¥int_{a}^{t}s^{N-1}g(s, 1, -1)ds=0$
(3.13)
$¥int_{a}^{¥infty}(t^{1-N}¥int_{a}^{t}s^{N-1}g(s, c, -c)ds)^{1/(p-1)}dt<¥infty$
$>0$ ].
If
,
for some
constant $c>0$ , and
(3.14)
$¥int_{a}^{¥infty}t^{N-1}g(t, dt^{(p-N)/(p-1)}, 0)dt=¥infty$
for
every constant $d>0$ , then the equation (B) [resp. (C)] has infinitely many
positive solutions which are defined in a small neighborhood of infinity and satisfy
(3. 15)
$¥lim_{|x|¥rightarrow¥infty}u(|¥mathrm{x}|)=0,¥lim_{|x|¥rightarrow¥infty}[u(|¥mathrm{x}|)/|¥mathrm{x}|^{(p-N)/(p-1)}]=¥infty$
[resp.
$¥lim_{|x|_{*}¥rightarrow¥infty}u(|¥mathrm{x}|_{*})=0,¥lim_{|x|_{*}¥rightarrow¥infty}[u(|¥mathrm{x}|_{*})/|¥mathrm{x}|_{*}^{(p-N)/(p-1)}]=¥infty.$
]
Theorem 3.1 follows from Theorem 1.1. Theorems 1.1 and 2.2 together
imply Theorem 3.2. Theorem 3.3 is derived from Theorems 1.2 and
2.1. Application of Theorem 2.3 yields Theorem 3.4.
Example 3.5.
(3.16)
$p>1$ ,
Specializing the above theorems to the equation
$¥sum_{i=1}^{N}D_{i}$
$¥gamma>0$
,
$¥delta>0$
,
$¥lambda$
( Du
$|$
and
$¥mu$
$|^{p-2}D_{i}u$
)
$+|¥mathrm{x}|^{¥lambda}u^{¥gamma}+|x|^{¥mu}|$
Du
$|^{¥delta}=0$
,
$¥mathrm{x}¥in¥Omega_{a}$
,
being constants, we have the following statements.
360
KUSANO Takasi and Akio OGATA
(i) Let $N=p$ and suppose that either
or
$¥{¥gamma>p-1, ¥delta>p-1¥}$
$¥{¥gamma<p-1_{i}$
If $¥lambda<-N$ and $¥mu<-N+¥delta$ , then (3.16) possesses infinitely many
positive symmetric solutions
which are asymptotic to constant multiples
of
.
as
$N
¥
neq
p$
Let
and suppose that either $¥{¥gamma>p-1, ¥delta>p-1¥}$ or
(ii)
$¥{¥gamma<p-1, ¥delta<p-1¥}$ .
If
$¥delta<p-1¥}$
.
$u(|¥mathrm{x}|)$
$¥log|¥mathrm{x}|$
$|¥mathrm{x}|¥rightarrow¥infty$
$¥lambda<-N-¥frac{¥gamma(p-N)}{p-1}$
and
$¥mu<-N+¥frac{¥delta(N-1)}{p-1}$
,
then (3.16) possesses infinitely many positive symmetric solutions $u(|x|)$ which
are asymptotic to constant multiples of
as
.
$
¥
{
¥
gamma>p-1,
¥
delta>p-1
¥
}$
$
¥
{
¥
gamma<p-1,
¥delta<p-1¥}$ .
Suppose
that
either
or
(iii)
If
$|¥mathrm{x}|^{(p-N)/(p-1)}$
$¥lambda<-¥max¥{N, p¥}$
and
$|¥mathrm{x}|¥rightarrow¥infty$
$¥mu<-¥max¥{N, p¥}+¥frac{¥delta(N-1)}{p-1}$
then (3.16) possesses infinitely many positive symmetric solutions
to positive constants as
regardless of the relation between
$
¥gamma<p-1$ and $¥delta<p-1$ .
Suppose
$N>p$
that
,
If
(iv)
$|¥mathrm{x}|¥rightarrow¥infty$
?
$N+¥frac{¥gamma(N-p)}{p-1}¥leqq¥lambda<-p$
and
,
tending
and .
$u(|¥mathrm{x}|)$
$N$
$p$
$¥mu<-p$ ,
then (3.16) possesses infinitely many positive symmetric solutions
defined
in a neighborhood of infinity and having the asymptotic proerty (3.15). These
solutions decay to zero more slowly than
as
.
$u(|¥mathrm{x}|)$
$|¥mathrm{x}|^{-(N-p)/(p-1)}$
$|x|¥rightarrow¥infty$
References
[1]
[2]
[3]
[4]
[5]
[6]
R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston,
New York, 1965.
A. Elbert and T. Kusano, Oscillation and nonoscillation theorems for a class of second
order quasilinear differential equations, Acta Math. Hungarica 56 (1990), 325-336.
J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. I, Elliptic
Equations, Pitman, Boston-London-Melbourne, 1985.
M. Hukuhara, Sur l’existence des points invariants d’une transformation dans l’espace
fonctionnel, Japan. J. Math. 20 (1950), 1-4.
T. Kusano, A. Ogata and H. Usami, Oscillation theory for a class of second order
quasilinear ordinary differential equations with application to partial differential equations,
Japan. J. Math. (N.S.) 19 (1993), 131-147.
J.-L. Lions, Quelques Methodes de Resolution des Problems aux Limites Nonlineaires,
Dunod Gauthier-Villars, Paris, 1969.
Second Order Quasilinear
Differential
Equations
nuna adreso:
Kusano Takasi
Department of Mathematics
Faculty of Science
Hiroshima University
Akio Ogata
Department of Mathematics
Faculty of Education
Miyazaki University
(Ricevita la 25-an de majo, 1992)