数理解析研究所講究録
900 巻 1995 年 28-41

28

2 階巳 Yn41。\mbox{\boldmath $\gamma$}--:i.-c\vee / \sim \rightar ow 聖方桟$\lambda_{\perp}$$\sqrt$広11.

$\mathrm{n}\mathrm{t}$$\ovalbox{\tt\small REJECT}$

roduc

夫秀 j 合$\mathrm{t}\mathrm{i}$$\ovalbox{\tt\small REJECT}\tau_{J\mathrm{J}\mathrm{i}}+^{\wedge}\ovalbox{\tt\small REJECT}$on and Resul

呆

広

,

$\mathit{0}_{\rangle^{+}}arrow.\mathrm{b}\mathfrak{l}\ovalbox{\tt\small REJECT} x,,\mathrm{I}\check{\Xi}_{j}^{\perp}$

問(Hiroyuki USAMI)$\ovalbox{\tt\small REJECT}$$\mathrm{t}\mathrm{s}$This is a joint work with Professor Manabu Naito (Hiroshima
Univer$s\mathrm{i}\mathrm{t}\mathrm{y}$).

Let us consider the following binary elliptic system of the
Emden-Fowler type in an exterior domain$\Delta u_{1}+p_{1}(I)|u_{22}|^{\sigma_{1^{-}}1}u=$$\Omega\subset \mathrm{R}^{N}$

$0$

:

,
(S)

$\Delta u_{2}+p_{2}(X)|u|^{\sigma_{2^{-}}1}u11$

$=$

$0$

,

we always assume the next conditions:where$(\mathrm{A}_{1})$$(\mathrm{A}_{2})$$\sigma_{1}$$p_{i}$

System (S)solutions,$\sigma_{2}>$$0$;$\in C(\Omega;[0, \infty))$$\mathrm{i}\mathrm{s}$, and$\mathrm{i}\mathrm{s}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}p_{i}$unbounded,called oscillatory if , for any $R>$$(u_{1} , u_{2})$sati sfying$u_{1}u_{2}>0$in$\Omega\cap$

$0$

$i$

,

$\{x .

$=$

1 , 2.

it has no|x| >R\}$.

For the single elliptic equation$\Delta u$

useful oscillationsee$\mathrm{e}$.

$\mathrm{g}$

.

[3 ,

$+p(x)|u|^{\sigma-1}u$$\mathrm{c}\mathrm{r}\mathrm{i}$4 , 10 ,

$=$

$0$

,

$\sigma>$

$0$

(0)

,

teria have been obtained by many authors;

111.

Of course,(0)$\mathrm{i}\mathrm{s}$called$\mathrm{o}\mathrm{s}\mathrm{c}$illatory

if it has no positive solutions which are of constant sign near29$\infty$.

The next oscillation criteria are well-known:

Theorem

sa

$0$

.

Le

$t$

$tisf_{\mathcal{Y}}ing0\leq\hat{p}(\gamma)$

(i)

and

$\sigma\neq 1$$\hat{p}$be a $conti$ nuous$\leq\min_{1x1=r}P(x)$Le $tN=2$ .

Then

(0)

Le $tN\geq 3$ .

Then

(0)

,

$l$

arge

$|x|$

$Ti$

on

.

$tor\mathrm{y}if$$= \min\{1, \sigma\}$$\sigma_{*}$.

$isoscill$ a $toryif$$\int^{\infty}r^{N-1-\sigma^{*}}\hat{p}(N-2)(_{\mathcal{T}})dr=\infty$

By considering theor$isoscill$ a$\int^{\infty}r(\log r)^{\sigma_{*}}\hat{p}(r)dr=\infty$

(\"u)

$f$

func

,

$= \max\{1, \sigma\}$$\sigma^{*}$.

case where $p(x)$ has radial symmetry, we findthat this theorem characterizes the oscillation situation of (0)in some sense.

However , turning our attention to system (S) , werealize that there exist few results which give effective$\mathrm{c}\mathrm{r}\mathrm{i}$teria for oscillation of system

Motivated by thi(S).

$\mathrm{s}$fact

we make an attempt to give a contribution to this problem.
Other related results of asymptotic theory for elliptic systems
like (S) are found in [1 , 2 , 5, 6 , 7 , 12 , 13].

First we introduce some notation.

Let$\hat{p}_{i}$,$i$$=1$ ,

2 , becontinuous functions such that$0\leq\hat{p}_{i}(r)$$\leq$$\min$

1 $x|=r$
Define the functions$P_{i}(T)$$P_{i}(r)$

$P$
$i$

$p_{i}(x)$$(r)$ ,$= \int_{T}^{\infty}s\hat{p}_{i}(S)ds$

$i$

$=$

for

$|x|$

large.

1 , 2 , by
if $N=2$ ; and$= \int_{T}s\infty-\sigma_{i}(N-2)-3\hat{p}(S)d_{S}i$

if $N\geq 3$ .

It will be seen below that system (S) is oscillatory if$P_{i}$$\equiv\infty$

30

for some$i$.

Hence as uming the exi stence of

$P$

$s$

$=$

$i$

’

$i$

$1$

2 , loses,

no generality.
are as follows:

Our oscillation criteria for (S)Le

Theorem 1.

cons $tants\lambda$$\lambda_{1}+\lambda_{2}=1$$\sigma_{1}$,$\sigma_{2}>1$.

Supposethere are$that$$(i =1 , 2)$ and $\epsilon>0$ sa $tisf\mathrm{y}ing$

$>0$

$i$

$t$

,

$\epsilon>0$$-\lambda_{j}-$$\lambda_{i}\sigma_{i}$

for

$i$

,

$j\in\{1,2\}$ ,$i$$\neq j$,and$\int^{\infty}r^{N-3\lambda_{1}\lambda_{2}}[P_{1}(r)1[P(2r)]$.
Then ,$( \int^{r_{SP}}N-31(_{S)d\mathrm{I}^{\lambda\sigma-\lambda T}}S221^{-}\epsilon(\int sd_{S})N-3_{P(_{S)}}12r\lambda_{1}\sigma-\lambda-\epsilon d=\infty$(S)$oscill$ a $tory$ .

$is$Theorem 2.

some

$i$

(i)

$j\in\{1 , 2\}$ ,

,

$i$

Le

$<1$

$t\sigma\sigma$

1 2$\neq j$and $N=2$ .

Suppose $thatf$ or,$\int^{\infty}r(\log r)\sigma_{i}(\sigma_{j}+1)\hat{p}(ri)[P(r)1dTj\sigma_{i}=\infty$

Then,

(S)

(\"u)

{1,2},$t\sigma\sigma$
$1$

$2$

$<1$

and $N\geq 3$.

Suppose $thatf$ or some(S),$j\in$.

$isoscill$ a $tory$ .

Theorem 3.
{1,2},$i$$i\neq j$ ,$\int^{\infty}r^{\sigma_{1}\sigma_{2}(}N-2)-3\hat{p}_{i}(r)[P_{j}(T)]d\sigma_{i}r=\infty$

Then ,.

$oscill$ at $ory$ .

$is$

Le

.

Le

$t\sigma\sigma$

$12$

$>1$

.

Suppose $thatf$ or some$i$,$j\in$$i\neq j$ ,$\sigma.+1$$J$$\int^{\infty}r^{N-3}P_{i}(r)dr=\infty$

and$\lim$

inf $\frac{P.(r)}{P(r)}[\int^{r}s^{N^{-}3}Pi(S)ds]$$rarrow\infty$

Then

(S)

$is$

$oscill$ a$t_{Or}\mathrm{y}$

.

$i$

$>0$

.

31

Theorem 4.

Let$\sigma\sigma$

1 2

$=1$

.

Suppose that$\int^{\infty}r^{N-3}\min\{P_{1}(r) , P_{2}(r)\}dr=\infty$

SupposemoreoveT($\lim\sup$

.

that

$\int^{T}s^{-1_{\min}}\{P_{1}(r) , P_{2}(r)\}dr$$\sigma^{1/(\sigma+1)}\log 1o\mathrm{g}r$)$>-\infty$$(N-2)\sigma^{1/(\sigma+1)}\log r$)$>-\infty$-$rarrow\infty$

if $N=2$ ,$\lim_{rarrow}\sup_{\infty}$$ifN\geq 3$ ,

Thi

$s$

$oT$

(

$\int^{T}s^{N-3}\min\{P_{1}(r) , P_{2}(T)\}dr$

.

-

Then

(S)

$isoscill$ atory.

report proceeds as follows.

In\S 2 we give awhere$\sigma=\max\{\sigma_{1} , \sigma_{2}\}$comparison principle to the effect that the existence ofa positive

positive solution of (S) guarantees the existence ofsolution of an ordinary differential system

as sociated to$\mathrm{i}\mathrm{s}$(Here and(S).

def ined to be pos$\mathrm{i}\mathrm{t}$the problem ofHence ,

ive if$\mathrm{f}$$(\hat{\mathrm{S}})$

a

\S 2

in

in the sequel, a vector function

bo.th component

inding oscillation

are pos tive. )$\mathrm{i}$$\mathrm{s}$$\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}$ia for (S)reduces to the problem of finding oscillation criteria for
one-dimens ional problem\S 3.

$(\hat{\mathrm{S}})$.

Thi$\mathrm{s}$problem$\mathrm{i}s$fully di scussed in

The author believes that the results in this section is ofindependent intere

omitted in thi$s\mathrm{t}$.

The proofs of Theorems 1-4

report , because they can be eas ily carr ed out$\mathrm{i}$$\mathrm{s}$by comb ining Propos tion 2$\mathrm{i}$

2.

Reduc

are actually$\mathrm{t}\mathrm{i}$$\mathrm{w}\mathrm{i}$th Propos tions 3-6 in$\mathrm{i}$on to One-D imens onal Probl ems$\mathrm{i}$We begin with the following proposition.

\S 3.

32

Propos$(u_{1} , u_{2})f$or$\mathrm{i}\mathrm{t}\mathrm{i}$

on 1.

$|x|$

Suppose that$\geq R,$a $p_{oS}i\tau ive$ so $luti$ on$(r^{N^{-}1_{w_{1}}}’)$$(r^{N-1}w_{2^{)}}’$$wi$

th $Rsuffici$ en$tly\mathit{1}$arge.

$+r^{N-1_{\hat{p}_{1}}}(r)|w_{2}|^{\sigma_{1^{-1}}}w_{2}=0$.

Then ,

thereof the $ordi$ nary $dif$ feren $ti$ a$(w_{1} , w_{2})$’has a $pQsiiive$ so $luti$ on

(S)

$is$

$ls\mathrm{y}_{S\mathrm{r}}em$,$(\hat{\mathrm{S}})$$+r^{N-1}\hat{p}_{2}(T)|w_{1}|^{\sigma_{2^{-1}}}w_{1}=0$,

for $r\geq R$ , such that$\min$$0<w_{i}(r)\leq$1$u_{i}(x)$$r\geq R$ ,,1 , 2.

$i$$x|=r$

Proposition1 yields the following simple comparison

principle

on which our results are heavily based:

Propos$\mathrm{i}\mathrm{t}\mathrm{i}$on 2.

one-dimens onal$i$$(\hat{\mathrm{S}})$$s\mathrm{y}_{St}em$$isoscill$ atory.

The proof of Proposition 1

Theorem 2. 1].

Lemma 1.
de $fined$ on $R\leq$$(w_{1}, w_{2})$$\mathrm{i}s$$(u_{1}, u_{2})$

.

there

$|x|\leq b$$\hat{u}_{i}(r)$

Proo

$(r)$

Then,

.
$\leq$$i$$w_{i}(b)$

\^u

$(T)$
$i$

$= \min_{1_{X}1=r^{u}}i(x)$ ,$\mathrm{f}\mathrm{o}\mathrm{f}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}*$The

Proposa $positive$ so $luti$ on of
$is$

(S)

a $positive$ so $luti$ on

on $R\leq r\leq b$ such that$(\hat{\mathrm{S}})$$=\hat{u}_{i}(R)$

$0<w$
where

o.f theneeded to prove Proposition1.

Le $tb>R$ , andof system$w_{i}(R)$similar to that of [8 ,$\mathrm{i}\mathrm{s}$We give only the sketch

following lemma$isoscill$ atory $if$ the(S)$ElliP\zeta icsy_{St}em$$\mathrm{i}\mathrm{t}\mathrm{i}$

on 1.

,

$=\hat{u}_{i}(b)$. $i=1$$R\leq r\leq b$ ,$r\geq R$ ,

Let

$i$

$\{b_{m}\}$$i$$=1$ ,

be a

$=$

2;

,

$1$

,

and,

2,

2.

$s$

equence such that

33

$R<b_{1}<b_{2}<$we obtain a sequence

By Lemma 1$(r^{N-1_{w_{1m}}})|$ ’,$<b_{m}<$and$\mathrm{l}\mathrm{i}\mathrm{m}marrow\infty b_{m}=\infty$.

such that$\{(w_{1m}, w_{2m})\}^{\infty}m=1$$+r^{N-1}\hat{p}_{1}(r)w^{\sigma}2m1=0$ ,$R\leq r\leq b_{m}$ ;$(r^{N-1}w2m|)$ ’$w_{im}(R)$

$0$

We

$=\hat{u}_{i}(R)$,$<w_{im}(r)$$+r^{N-1}\hat{p}_{2}(r)w^{\sigma}1m2=0$ ,$w_{im}(b_{m})$$\leq\hat{u}_{i}(r)$can choo se a subsequence$\{(w_{1\mu} , w_{2\mu})\}$,. $i=1$$R\leq r\leq b_{m}$ ,$\{(w_{1\mu}, w_{2\mu})\}$

of

$=$

$i$

a desired solution of$(\hat{\mathrm{S}})$

as$[R, \infty)$

.

2;

,

$1$

,

and,

2.

$\{(w_{1m}, w_{2m})\}$

converges to a positive function

on each compact sub interval of
gives$=\hat{u}_{i}(b_{m})$$\{(w_{1} , w_{2})\}$$\muarrow\infty$.

Thi$\mathrm{s}$such thatuniformly$(w_{1} , w_{2})$For the detailed argument werefer the reader to [8] .

3.

Osc llat on Theorems for One-D imens onal Probl ems$\mathrm{i}$$\mathrm{i}$$\mathrm{i}$Instead of dealing with system$(\hat{\mathrm{S}})$

directly,

we shalltransform it into the simple system of the form$+$

$y_{1}^{||}$$a_{1}(t)|y_{2}|^{\sigma_{1^{-}}1}\mathcal{Y}_{2}$

$=$

$0$

,
$(\mathrm{S}_{0})$’$+$$y_{2}^{1}$

When $N=$$w$$i$$(et)$ ,$2$$i$$a_{2}(t)|y_{1}|^{\sigma_{2^{-}}1}\mathcal{Y}_{1}$

$=$

$0$

.

,

consider the change of variables$=$

1 , 2.

We then$\mathrm{f}$ind that$(\hat{\mathrm{S}})$

$t$

.
$+$

$e^{2\mathrm{r}_{)}}t_{\hat{p}_{1}(e}|z_{2}|^{\sigma 1}1^{-}z2$$=$$\log r$ ,$z_{i}(t)$equivalent to the$\mathrm{i}\mathrm{s}$system$.z_{1}$

$=$

$0$

,

$=$

34

$e\hat{p}_{2}(e^{T})2t|z_{1}|^{\sigma_{2^{-}}1}z_{1}=0$$.z_{2}.+$where. $=d/dt$ .

$rN-2$ ,$z_{i}(t)$,we make the change of valuables

When $N\geq 3$ ,$tw_{i}(t^{1/(N-2)})$ ,

Then

2.

1,

$i$

$=$

$t$

reduces to the$(\hat{\mathrm{S}})$

system$.z_{1}.+$.
$+$$.z_{2}$$(N-2)^{-2}$$t^{-\sigma_{1}-N/}(N^{-}2)_{\hat{p}_{1}(}1/(tN^{-}2)_{)}|z_{2}|^{\sigma_{1^{-}}1}z_{2}=0$,$(N-2)^{-2}$$t^{-\sigma_{2}-N/(}\hat{p}_{2}N-2)(t^{1/()}N-2)|z_{1}|^{\sigma_{2^{-}}1}z_{1}$

,

where

.

the

cillatory$\mathrm{o}s$$=$.

$d/dt$

let$0$

Note that the transformations used here keeP

property.

consider system
Now,

$=$

$(\mathrm{S}_{0})$For

our purpose , it$\mathrm{i}\mathrm{s}$sufficient to.

us consider system$(\mathrm{s}_{0})$under the following basic

assump tions:$(\mathrm{B}_{1})$$(\mathrm{B}_{2})$$\sigma_{1}$$a_{i}$,$\sigma_{2}>0$;$\in C([t_{0} , \infty)$

Define the functions$A_{i}(t)$Asseen below, if; $[0 , \infty))$ , and$(t)$ ,

$A$

$=$

$i$

$1$

$\mathrm{i}\mathrm{s}$$a$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}$unbounded,

$=$

$i$

$1$

’

$i$

2 , by,$i$$= \int_{t}^{\infty}a_{i}(s)ds$$\equiv\infty$$A_{i}$

for

$s$

,

ome

$t$

$i$

$\geq$$t_{0}$, then.
$(\mathrm{s}_{0})$

has

no

$\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}$tivesolut ions.

Proposition 3.

cons $tants\lambda_{i}$$>0$$\lambda_{1}+\lambda_{2}=1$,

Let$\sigma_{1}$,$\sigma_{2}>1$.

Suppose that there$(i =1 , 2)$ and $\epsilon>0$ sa$\lambda_{i}\sigma_{i}$and$\int^{\infty}[A_{1}(t)][\lambda_{1}\lambda_{2}A(t)2]$$\lambda_{j}-\epsilon>0$

$f$

$t$

or

are

$isf_{\mathcal{Y}}ing$$i$,$j\in\{1,2\}$ ,$i\neq j$ ,

2.

35

.
Then ,$( \int^{t\lambda_{22^{-}1}}A_{1}(s)d_{S})\sigma\lambda-\epsilon(\int t)^{\lambda-\lambda-}A(_{S})dSt21\sigma 12\epsilon_{d}$$isoscillator\mathcal{Y}$$(\mathrm{s}_{0})$Propos{1, 2},

ion 4.

$\mathrm{i}\mathrm{t}$

Le

.

.

$t\sigma\sigma$

$12$

$<1$

.

Suppose $thatf$ or some$i$.

$j\in$$i\neq j$ ,a $i(t)[A_{j}(t)]dt\sigma i=\infty$ .

$\int^{\infty}t\sigma_{i}(\sigma_{j}\star 1)$

Then.

$=\infty$$isoscill$ at $ory$ .

$(\mathrm{S}_{0})$Propos{1, 2},$i$$\mathrm{i}\mathrm{t}\mathrm{i}$Le$t\sigma\sigma$

$12$

$>1$

.

Suppose $thatf$ or some$i$.

$j\in$,$j$$\neq$

on 5.

$\sigma.+1$
$J$

and

$\int^{\infty}A_{i}(t)dt=\infty$$\lim$

inf$is$$(\mathrm{s}_{0})$.

$i$$tarrow\infty$

Then

$>0$

$\frac{A.(t)}{A(t)}[\int^{t}A_{i}(S)ds]$$oscill$ a $tory$ .

Proposition 6.

Let$\sigma\sigma$

1 2

$=1$

$\int^{\infty}\min\{A_{1}(t), A_{2}(t)\}dt$.

Suppose that(1)$=\infty$and$( \int^{t}\min\{A_{1}(s), A_{2}(s)\}ds-\sigma^{1/(}\mathrm{l}\mathrm{o}\mathrm{g}1+\sigma)t)$$\lim\sup$$>-\infty$(2),$tarrow\infty$

where$\sigma=\max\{\sigma_{1} , \sigma_{2}\}$

Proo

Propos$\mathrm{f}\mathrm{o}\mathrm{f}$has a solution$\geq 0$

.

,

$-$

$(\infty)$$y_{i}$

for

$i$

.

$j$

$t$

$\in$

Then,

$\mathrm{i}\mathrm{t}\mathrm{i}$$\geq T$$\mathrm{y}_{1}(t)$,$y_{i}^{1}(t)${1 , 2}.

$k$$=$$+ \int_{t}^{\infty}$

$i$

on 3.

$(\mathrm{s}_{0})$$>0$ and$1$,$isoscill$ a $tory$ .

Suppose to the contrary thatsuch that$(\mathcal{Y}_{1}, y_{2})$may assume that$y_{\dot{k}}(t)$.

$y_{2}(t)$

$>0$

.

for

Then by$\geq \mathcal{T}$$t$$(\mathrm{S}_{0})$.

We

we have2, anda $i(s)[y_{j}(s)]ds\sigma_{i}$$\neq j$

.

$0$

$>$

$y_{1}(t)y_{2}(t)$$(\mathrm{S}_{0})$In view of

$=$

$0$

$y_{k}(\infty)$

,

$t$

$>$

$0$

,

$\geq \mathcal{T}$

thi

(3)

,

$\mathrm{s}$$\mathrm{i}\mathrm{m}\mathrm{P}^{\mathrm{l}\mathrm{i}}\mathrm{e}\mathrm{S}$36

that the functions$A_{k}(t)$

,

$k$

$=$

$1$

are well2,,$\mathrm{d}\mathrm{e}\mathrm{f}$

and

ined,

$\sigma_{i}$$y_{i}(t)$Put $w(t)$$=y_{1}(t)y_{2}(t)$$w’(t)$$t$$\geq \mathcal{T}$

.

,

for

$i$

,

$j$

{1, 2},$\in$

Then $w(t)$ $>0$ ,$t\geq \mathcal{T}$$\geq A_{1}(t)[y_{2}(t)]^{\sigma_{1}+1}+A_{2}(t)[y_{1}(t)]^{\sigma_{2}+1}$

in theNow,.

$\geq T$$t$$\geq A_{i}(t)[y_{j}(t)]$$\mathrm{r}\mathrm{i}$$\neq j$

$i$

,

.

(4)

and

$t\geq$T.

(5)

we apply the well-known

ght hand side of (5) ,inequality$+\chi_{2}\geq C$$\chi_{1}$where$w’(t)$

.

$0$

$>$

$C(\lambda_{1}, \lambda_{2})$for$(\lambda_{1} , \lambda_{2})X_{1}^{\lambda_{1}}X_{2}\lambda_{2}$a constant.

$\mathrm{i}\mathrm{s}$$\chi_{1}$,$\chi_{2}\geq 0$

We then obtain$\geq C_{1}[A_{1}(t)]^{\lambda_{1}}[A_{2}(t)]^{\lambda_{2}}$$[y(21t)]^{\lambda_{1}}1^{+1)-1\lambda_{2^{()+}}}-\epsilon[y(\sigma(\mathrm{r})]2^{+}-1-\epsilon[\sigma 1w(t)]^{1\epsilon}$

where

,

$C_{1}>0$$y_{i}(t)$$\mathrm{i}\mathrm{s}$a constant.

$[y_{j}(\mathcal{T})]\sigma_{i}$$\geq$(6)$t\geq T$ ,Since (4) shows that$\int_{T}^{t}A_{i}(s)ds$,$t$$\geq T$

;

$i$

,

$j\in$$\{1 , 2\}$,$\neq j$$i$,substituting these inequalities into (6) and integrating theresulting inequality,$\mathrm{R}^{\epsilon}wT\epsilon$

we have$\geq c_{2}\int_{T}^{t}[A(1s)]^{\lambda_{1}}[A_{2}(s)]^{\lambda_{2}}$$( \int_{\tau^{A_{1}}}^{s}(r)dr)^{\lambda(\sigma}22+1)-1-\epsilon(\int_{\tau^{A_{2}}}^{S}(r)d_{T}\mathrm{I}^{\lambda_{1}}(\sigma_{1}+1)-1-\epsilon dS$

where

contradiction.

Proo
of

$>0$

$c_{2}=c_{2}(T)$$\mathrm{t}\mathrm{s}_{0}$)$\mathrm{f}\mathrm{o}\mathrm{f}$

for

$t$

is a constant.

Letting$tarrow\infty$

,

,

$t$

$\geq T$,

we have a

The proof is complete.

Propo$\geq T$.

$s\mathrm{i}\mathrm{t}\mathrm{i}$

on 4.

Let

$(\mathcal{Y}_{1} , y_{2})$bea positive solution

We then obtain (3) , and an integration

37

implies that$y_{i}(t)$where$\geq c_{i}t\int_{t}^{\infty}a_{i}(s)[y_{j}(s)]ds\sigma_{i}$

$0$

$>$

$c_{i}$$[y_{i}(t)]\sigma_{j}$.

a constant.

$\mathrm{i}\mathrm{s}$$\geq T$

$t$

.

Sub

$\mathrm{s}\mathrm{t}\mathrm{i}$,Thi$\geq T$$t$$\mathrm{s}\mathrm{y}\mathrm{i}$tuting thi

;

$i$

elds

$\mathrm{s}$

,

$j$

$\in$

$y_{j}(t)${1, 2},

inequality in (7) ,$\hat{C}>$$0$$\mathrm{i}\mathrm{s}$of the above.

a constant.

$t$.

we have$\geq T$ght hand$\mathrm{r}\mathrm{i}$,$\mathrm{s}\mathrm{i}$

de

Then,
$\geq\hat{C}t\sigma_{i}(1+\sigma_{j})$$-w’(t)$Define $w(t)$ by the

,

(7)

,

$\neq j$$\geq c_{j}tA_{j}(t)$$\frac{y_{i}(t)}{t}\geq\hat{C}\int_{t}^{\infty}S\sigma_{i}(1+\sigma)ja_{i}(s)[A_{j}(s)]\sigma_{i}[\frac{y_{i}(s)}{s}]^{\sigma_{1}\sigma_{2}}d_{S}$

where

$i$

a

$\dot{\mathrm{t}}(t)[A_{j}(t)]\sigma_{i}[w(t)]^{\sigma_{1}\sigma_{2}}$,$t$$\geq T$,$t$$\geq T$.

from which, by an integration,$[w(T1-\sigma)]1-\sigma 1\sigma 1\sigma_{2}2$

Letting$t$$arrow\infty$,$\geq\hat{C}\int_{T}^{T}s\sigma_{i}(1+\sigma_{j})$a $i(s)[A_{j}(s)]ds\sigma_{i}$ ,we get a contradiction.

Hence the proof iscomplete.

In proving Proposition3 we need the follwing lemma.
proof of thi$\mathrm{s}$Lemma 2.

lemma

found in [91.

$\mathrm{i}s$

Le $t\alpha<\beta$ be $p_{oSi\ddagger}ive$ cons $tants$ , and $q\in C$

a $posi\mathrm{r}ivefuncti$ on suchinf$\lim$

The

$[$

$t$

1

’

$\infty)$$tha\ell$$\mathrm{r}^{1+\beta_{q}}(t)$

$>0$

.

$tarrow\infty$Then,

the $re$ $exist$ no $positive$$w’(t)$

Proo$>0$ ,$\mathrm{f}\mathrm{o}\mathrm{f}$solut ion of$([w’(t)]^{\alpha})$Propos$(\mathrm{S}_{0})$.

$\mathrm{i}\mathrm{t}\mathrm{i}$Asfunc $tionsw(t)$$\geq q(t)[w(t)]^{\beta}$

on 5,

Let

$(y_{1} , y_{2})$$f$,seen before, we have$which$sa $tisfy$

or a $ll$ 1 arge$t$$\geq I^{\backslash }$

(4).

,

be a

$t$

.

$\mathrm{p}\mathrm{o}\mathrm{s}$itive

An integrat ion

38

give

$\mathrm{s}$$y_{i}(t)$We denote the$\mathrm{r}\mathrm{i}$$w_{i}(t)$Then,ght hand$\mathrm{s}\mathrm{i}$de of the above by$+ \int_{T}^{t}A_{i}(s)[y_{j}(s)]dS\sigma_{i}$$\equiv y_{i}(T)$$\geq T$$t$$+ \int_{\mathcal{T}}^{i}A_{i}(s)[\mathrm{y}_{j}(s)1ds\sigma_{i},$$\geq y_{i}(T)$

$(t)$

$w$
$i$

,

:

$\geq T$

$t$

.

.

it is easy to see that

.

’

,

$0$

$>$

$w_{i}$

The change of$\geq A_{j}(t)w_{i}\sigma_{j}$$(( \frac{w_{i}}{A_{i}(t)}.)^{1/\sigma}i)$$\mathrm{v}\mathrm{a}\mathrm{r}$iable$\int_{T}^{t}A_{i}(s)ds$$=$$\tau$$\geq T$$t$transforms.
inequality$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}$into$\frac{d}{d\tau}[(\frac{dw_{i}}{d\tau})^{1/\sigma}i]$$\geq\frac{A.(t)}{A(t)}w_{i}\sigma_{j}$,$\tau\geq 0$.

$i$Observe that$dw_{i}/d\tau>$$0$,$\mathrm{f}\mathrm{o}\mathrm{f}$Propos$\min\{A_{1}(t) , A_{2}(t)\}$ ,(1)

implies that$t$$\mathrm{i}\mathrm{t}\mathrm{i}$on 6.

$\geq$$t_{0}$.

To Prove the theorem,

shows that$\infty$.

Thi$\mathrm{s}$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{P}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}$in view of Lemma 2 , wecomplete.

For simplicity$=\infty$,$s\mathrm{u}\mathrm{P}\mathrm{p}\mathrm{o}$,$t$se that

we have$\geq T$

we put$\sigma=$$B$$\sigma_{1}$$\int^{\infty}t^{\sigma_{1}}a_{1}(i)dt$$( t)$.

$=$$\mathrm{S}$$\infty$

$=$

ince

.

to the contrary that there is,$\lim y_{2}^{1}(t)$

of
$=0$

$(\mathrm{S}_{0})$.

We$\mathrm{f}\mathrm{i}\mathrm{r}$st show that.

(8)$tarrow\infty$$tarrow\infty$$y_{1}’(t)$$=$$\mathrm{i}\mathrm{s}$$\mathrm{s}\mathrm{u}\mathrm{P}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{e}$$(y_{1} , y_{2})$$\mathrm{l}\mathrm{i}\mathrm{m}y_{2}(t)/t$

In fact , if thiThen,

We may$\int^{\infty}ta_{1}(t)dt$

a pos itive solution.

The proof

reach a contradi ction.

Proo$\tau\geq 0$$\mathrm{i}\mathrm{s}$$\mathrm{s}$not the case, the identity$-y_{1}^{1}(T)$$+ \int_{\tau^{a_{1}(_{S}}}^{i})[y_{2}(s)]^{\sigma}1dS$$\int^{\infty}a_{1}(t)[y_{2}(t)]^{\sigma_{1}}dt$

contradiction provesion 5, we$\mathrm{f}$$<\infty$

,

(8).

$=$

$0$

,

implying thatExactly

ind that the function$t$$\geq T$,$\int^{\infty}t^{\sigma_{1}}a_{1}(t)dt$

as in the Proof of$w\mathrm{d}\mathrm{e}\mathrm{f}$ined by$<$39$w(t)$(9)$t\geq T$ ,,$(\leq y_{2}(t))$$+ \int_{\tau^{B}}^{t}(_{S)}[y_{1}(s)]ds\sigma_{2}$$\equiv y_{2}(T)$satisfies’$\geq B(t)w^{\sigma}$$[( \frac{w^{1}}{B(t)})^{\sigma}]$$t\geq$(10)

T.

Notice that (8) and (9) implies that$\lim w(t)/t$

.

(11)

$=$

$0$

$=$

$\int_{T^{B(s}}^{t}$$tarrow\infty$

The change of variable$\tau$$\geq w^{\sigma}$$\frac{d}{d\tau}[(\frac{dw}{d\tau})^{\sigma}]$$=C\exp(\sigma^{-1/(+1}\tau\sigma)\mathrm{I}$$v(\tau)$

th $C>$a constant.

$0$

It

$\mathrm{i}\mathrm{s}$$\mathrm{e}\mathrm{a}\mathrm{s}$

(10)

into

.

$\tau\geq 0$

introduce the auxiliary functionNow,$\mathrm{w}\mathrm{i}$transforms) $ds$

defined by

$=$

$v$

$v_{C}$,$\tau\geq 0$,

seen that, for any $C>0$ ,

ily

$v$

solves the half-linear equation$\frac{d}{d\tau}[(\frac{dv}{d\tau})^{\sigma}]$$\mathrm{S}$ince$v(0)$

$C$

$=$

,

,

and

$v$

sufficiently small $C>$

(0)
$0$

$C\sigma^{-1/(}\sigma+1)$$=$

so thatThen , by the well-known compar$v(\tau)$,$\geq 0$$\tau$$\tau\geq 0$$v^{\sigma}$$=$$\mathrm{i}$$narrow\infty$,$>$

chooseand $w’(0)$$v(0)$

$>$

(0).

’
$v$

son principle , we have $w(\tau)$$\geq C\exp(\sigma^{-1/}\int^{t}(\sigma+1)B(_{S})dsT)$ ,On the other hand, condition (2)$>$$c_{1}$. we can$\geq$

namely ,,$w(t)$

constant$w(\mathrm{O})$.

$0$and a sequence$t$$\geq T$.

assures the existence of a$\subset$$\{ \mathrm{r}_{n}\}$$[T, \infty)$

such that$t_{n}$\dag er$\infty$and$\frac{1}{t_{n}}\exp(\sigma\int_{\tau}^{n_{B}}-1/(\sigma+1)st()ds)$$\geq c_{1}$From thi$\mathrm{s}$,we have$\mathrm{l}\mathrm{i}\mathrm{m}\inf_{narrow\infty}w(t_{n})/t_{n}>0$

for. .. Thi$\mathrm{s}$$n\in$

N.

contradicts

as

40

(11),

and hence the proof

complete.

$\mathrm{i}s$

References[1] Y.

Furusho,Existence of positive entire solutions forweakly coupleds emilinear elliptic systems,
120A (1992). 79-91.

Soc. Edinburgh ,[2]

Proc. Royal

N. Kawano & T. Kusano,

Positive solutions of a class of

second order semilinear elliptic systems in exterior
domains ,[3]

Math. Nachr. , 121 (1985), 11-23.
An oscillation theorem for

Y. Kitamura & T. Kusano,

sublinear Schrodinger equat ions,(1978) ,[4]

T. Kura,Utilitas Math. , 14

171-175.
Oscillation criteria for a class of sublinear

elliptic equat ions of the second order,(1982) ,[5]

Nonoscillation criteria forstrongly ellipt ic systems,[6]

Bollettino U. M. I.

G. S. Ladde , V. Lakshmikantham & A. S. Vatsala,

Pitman, Boston,11

E. Mitidier i,

1985.
A Rellich type identity and applications,

M. Naito , Y. Naito & H. Usami ,18 (1993) , 125-151.

0 s cillation theory for

semilinear elliptic equations with arbitrary
nonlinearities,

Mono t one

f or Non li near Differen ti a l Equa tionS,

Comm. Partial Differential Equations,
[8]

(4) ,

166-173.

ft era rive Techn i ques

[7]

Ut ilitas Math. , 22335-341.

T. Kusano & N. Yoshida,(1975) ,a

preprint.

41

[9]

Y. Naito & H. Usami ,Nonexi stence results of pos itive

entire solut ions for quas ilinear elliptic inequalities,in

preparat ion.
[10]

E. S. Nous sair & C. A. Swanson ,Oscillation theory for

semilinear Schrodinger equations and inequalities,

[11]

Proc.

75A (1976) 67-81.

Royal Soc.

Edinburgh ,H. Onose ,Oscillation cr iteria for the sublinear
Proc. Amer. Math. Soc. , 85 (1982),

Schr0dinger equat ion,

69-72.
[12]

L. A. Peletier & R. C. A. M. van der Vorst,

Existence andnon-existence of positive solutions of non-linear elliptic
systems and the biharmonic equation,
Integral Equations ,
[13]

[14]

5

(1992) ,Differential and

747-767.

W. T. Reid,

S turmi an Theory f or OrdinaTy Differenti a 1

Equa

Springer-Verlag, New York ,$ti_{\mathit{0}}ns$,

C. A. Swanson,

1980.

Compar i son and Oscill a ti on Theory

Diff eren ti a l Equa tions ,of Li near

Academi c Pres s, New York , 1968.