ON POSITIVE SOLUTIONS OF ONE CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

1996,16(4):469-476

ON POSITIVE SOLUTIONS OF ONE CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS 1 Yo.ng Ch'unpeng ( Dept. of

Sy.~.

;fh~_

)

Sci. and Math., Zhen.fJzholl, University, Zhc1t!lzholl.

J,5005~,

China..

Abstract In this paper, we establish a 1-1 correspondence between positive solutions of one class of nonlinear differential equations and a. class of harmonic functions. Thc8C results give au explicit description of E.B.Dynkiu's class H a of positive harmonic fnnc-

tions.

Key words Brownian motion, Super-Brownian motion, Regular harmonic function,

1 Introd uction 1.1 We consider positive solutions of a nonlinear differential equation

(1) where A is the usual Laplacian operator in R d , D is a domain in It is well know that, the Laplace equation ~7l,(x)

tc',

1< u ~ 2.

= 0

can be studied probabilistically by using pa.th of the Brownia.n motion ~ = {~t, II:,: }. Analogously, the nonlinear differential equation (1) can be investigated by using the superBrownian rnotion (see [1, 4]). Let G(3;, y) be Green's function and K(:l;, y) be the poisson kernel for ~ in D, E.B .Dynkin proved that the following formula

(2) establishs a 1-1 correspondence between the class Un of all positive solutions dominated by harmonic functions and a class of H o of positive harmonic functions (see Sect. 4.7 in [4]). We certainly want to know what kind of harrnonic function belong to Hoo In this paper, we shall give an explicit description of the class H n . We prove that: Ho

= {h: h is positive harmonic in D,

1 Received

lirnh(et) exists a,s II:,;, and tlr

Nov.21, 1994; revised Nov. 10,1995

470

ACTA MATHEMATICA SCIENTIA

et

where 'T = {t > 0, rt D} is the first exit time of 1.2 Consider the following real functions

<,O(x, z) = a(x) z + b(x)z2

+

1

Vol. 16

efrom D.

00

(e- U Z

-

1 + u z)n",(du),

where n x (du) is a kernel from R d to (0,00), a(x), b(x) and A( x) = ~rooo (u 2 /\ u )n;,; (d'u) are positive bounded Borel functions. The superprocesses can be constructed in a very general set tinge Here we only briefly describe the result. Assurning that = {et, II;r.} is a standard Brownian motion in Rd. Denote by B the Borel a-algebra in R d , by M the set of all finite rneasure on B and by M the a-algebra in M generated by the following functions

e

= JL(B), B E B. [3,5], there exists a Markov process X = {Xt, PJl} in (M, M) such that: !B(J-L)

Then by 1) If ! is a bounded continuous function, then R+

= [0,(0), where < V(x),J-L >=

JRd

< !, X; > is right continuous in t

on

V(x)JL(dx). 2) For every J-L EM, and! is a positive Borel fountion, PJlexp

where

Vi! is

< -!,Xt >= exp < -vt!,J-L >,

(3)

the unique solution of the following integral equation

Vd + II",

r <,O(e., vt-.!(e.))ds = II",!(et).

./0

(4)

Moreover, to every set DEB, there corresponds a random measure X T on (R d , B) associated with the first exit time 'T

=

inf {t,

et rt D}

which is determined by the following formula

PJlexp < -!,XT >= exp < -VT!,J-L >,

(5)

where (6)

=

We call the collection X (Xt,XT,PJl) the super-Brownian rnotion with parameters (e,lp) and lp(x, z) the branching mechanism function of X. The heuristic meaning of randorn measure X; and X T can be explained in terms of branching particle systems (see [3,4]). In this paper, we always assume that

where 1 <

0: :::;

2.

Yang: SOLUTIONS OF ONE CLASS OF NONLINEAR EQUATIONS

No.4

471

2 Preliminaries Definition 2.1 A harmonic function h in D is said to be regular in D, if for every 3; ED 1) the following limit liU1tjT h(f.t) exists and is finite a.s IT;I:; 2) the above limit is integrable with respect to IT x and 11,(3;) = IT:,: (lin1tjT h(f.t)). We use H to denote the class-of all harmonic functions in D, H r the class of all positive regular harmonic functions in D, H 1 the class of all positive bounded harmonic functions in D. Let H 2 = {h E H; 3 b« E HI such that h.", T 11, < oo}.

Then we have the following interesting result. Lemma 2.1 For the proof of this result, see TheorerH 2 in [9]. Lemma 2.2 Let l be a bounded Borel function in a Green domain D and

F(x) =

no:

r f(f..)ds.

io

Then 1) F (x) is hounded and F (x) is continuously differentiable on D. 2) If / is Hijldcr continuous, then F(:I;) is twice continuously differentiable and

!:1F

= -2/

in

D.

For the proof of this result, see Theorern 4.6.6 in [8]. Proposition 2.1 Let D be an open set in Rd.Then 1) for every J..L E MD

P,,(Xr E MaD) 2) for every

J..L

= 1.

E MDc

where MDc, MaD denote the finite measure on R(I which supported by DC, BI), Proof Let 1" E bpB (bounded positive B measurable function), for every II, EM, it follows frOIH formula (5.9) in [1]

- Let /(3;) = ID(x), then for every JL E M

By the Markov property of

f.,

for all xED

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ACTA MATHEMATICA SCIENTIA

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So that for all J.L E MD

PIl(XT E MaD) == 1. Clearly for x

ft

D,

II:r,( T == 0) == 1. If J.L E MDc, then by (5) (6), It is clear

Proposition 2.2 Let D be an open set in R d , then u(x) is a positive solution of (1)

if and only if for every relative compact open subset Do C D exp(-u(x)) == P~:J: exp < -u(x),XT o >

x E Do,

for

e

where TO is the first exit time of from Do. Proof Suppose that u(x) is a positive solution of (1) in D. Let Do be a relative cornpact open set in D, then u( x) is continuous in D and by Theorem 1.1 ill [1]

V(x) = -logP~:J: exp < -u(x), X T o > is a solution of (1) in Do, and under the boundary condition

V(x)

-+

u(a), as

x

-+

a E 8r D o ,

:/; E Do,

the equation (1) has a unique solution in Do (where 81·D o denotes the regular point of Do). So that for any x E Do

V(x)

= u(x).

Hence for any x E Do

On the other hand, if for every relative compac t open set Do in D, and for any x E Do

Then by Theorem 1.1 in [1]

u(x) == -logP~:J: exp < -u(x), X T o > is a solution of (1) in Do. By the arbitrary of Do, we conclude that u(:J;) satisfies (1) in D. Proposition 2.3 If IUn(x) is a sequence of positive solutions of (1) and un(J:) 1 u(J;) pointwisely, then u( x) is a solution of (1). Proof By Proposition 2.2, for every relatively compact open set Do C D "Ix E Do.

By the Dominated Convergence Theorem, let n then we obtain that

-+

(X)

in both side of the above equation,

No.4

Yang: SOLUTIONS OF ONE CLASS OF NONLINEAR EQUATIONS

473

Hence 'u( x) is a solution of (1) by Proposition 2.2. Remark The proof of this result seemed sirnple by probabilistic rnethod. Proposition 2.4 Suppose D is an open set in R d . Let D n be a sequence of relative

i D, D n C D n +1 . If 'u(x)

compact open sets in D such that D n

is a positive solution of (1),

then for every J-L E MD forms a rnartingale, where

Tn

is the first exit time from D n

,

MD

IS

the finite rneasure

supported in D. Pr-oof By Proposition 2.2, for every x E D n exp( -u(x)) By Proposition 2.1, for every x E

= P6

;c

exp

< --u(a;), X

;c

exp

< -u(x),X

Tn

>.

D~

So that for every xED exp(--u(x))

= P6

Tn

>.

By (4), for every 11, E MD Pp'exp

< -u(x),X

Tn

>= exp < -VT n U( X), I1, >.

Therefore for every xED

This conclude that for every IL E MD

Pp. exp < -'u(x), X T n >= exp < -u(x), J-L > . Let m. < n. Apply the special Markov property of super-Brownian rnotion (see Theorem 1.4 in [2]), then

< -u(x),XT n >1 :F~Tm) = Px exp < -'ll(x), X n > = exp < -u(x),XT m > a.s Pit. P/-l(exp

T

~n

So that for every IL E MD

forrns a martingale. Remark By Proposition 2.4, we know that lim exp

n--+oo

< -u(x), X

Tn

>

exists

a.s

P/-l'

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3 Main Result In this section, we always assurne that D is a bounded open set in Rd. Theorem 3.1 Suppose that U is a positive solution of (1), then exist a sequence of bounded positive solutions Urn of (1) such that Urn (x) i u(x). Proof Choose a sequence of relative compact open subsets D n of D such that D 1£ i D, D n C D n + 1 - Set Ul1~(X)

=-

log P6:z: exp( -( lim < 1£-+00

tL(X),

X r n > Arn))

for

xED.

Then 'U l1t (

x) ::; tn,

Urn ( x)

I u(x ).

Next we shall prove that for every m, Urn (x) is a positive solution of (1). Let Do be an arbitrary relative cornpact open subset of D. Apply the special Markov property of the super-Brownian motion, for every x E Do

P6:z:'exp < -urn(x),X r o >

= P6:z: P X ro exp( -(lirn1£ -+ oo < u(x), X r n > Arn))

= P6:z:(P6:z:(exp(-(lim

n

= P6:z: exp(-(limn -+ oo

< u(x),Xr n > Am))I;:~r(.)) < u(x),X r n > Am)) -+ 00

= exp( -Ul1t (x )). Hence

'lLl1t (x)

is a positive solution of (1) by Proposition 2.2.

Theorem 3.2 Suuppose that the boundary of D is of class C 2 , then E.q. (2) establish a 1-1 correspondence between the class of all bounded positive solutions of (1) and the class of all bounded positive harmonic functions H t . Proof Suppose that u is a bounded positive solution of (1), then for every sequence of relative compact open subsets D 1£ of D such that D n i D, D n C D n + t , and for every J.L E M P/lexp < -u(x),X r n >= exp < -lL(X),j.t > by Proposition 2.2. It follows from (6) that (7)

It is clear that

Tn

i

T.

Let n

- t 00

in (7), then

Set h(J;) == linln -+ oo U(~Tn). It has been proved that 11. is a harrnonic function in there exists h E H t satisfies E.q. (2). On the other hand, suppose that h E H t , then for

J;

E D,

D[S],

i.c.,

Yang: SOLUTIONS OF ONE CLASS OF NONLINEAR EQUATIONS

No.4

and

h(x)

475

= nx{limh{~t)). tTT

As aD is Lipchitz, the Martin boundary of D is identified with aD according to Hunt and Wheeden[7l. Hence by Theorern 9.1 in [6], there is a bounded positive Borel function g 011 aD such that a.s

for

fix

3~

E D.

Set

u(x) = - log

P6;e

exp

< -y, X > . T

Then by Theorern 1.1 in [1] u is a bounded positive solution of (1) and

i.e., there exists a bounded positive solution of (1) u satisfies the E.q. (2) with h.

Theorem 3.3 Suppose the boundary of D is of class C 2 , then E.q. (2) establish a 1-1 correspondence between the class of all positive solutions of (1) such that ~rD G( 3~, Y)tt/.l' (y)dy < 00 and the class of all positive regular harmonic functions H; in D. Proof Let u(x) be a positive solution of (1) such that G{x, y)uO:{y)dy < 00. By Theorem 3.1, there exists a sequence of bounded positive solutions urn(x) of (1) such that um{x) i u(x). Apply Theorem 3.2, for every unl(x) there exists a hounded positive harmonic function hnJ, (x) such that

In

Let m.

~ 00

in both side of above formula, then

U(x)+j G{x,y)uO:{y)dy= lirn hrn(x). D

By Harnack Theorern h( x)

=

rn--+oo

lim n --+ oo hT", (x) is a harrnonic function In D and h( 3;)

limn --+ oo hm{x) E H; by Lemma 2.1, i.e., there exists li E H; satisfies (2). On the other hand, suppose h( x) E Hi , then for xED

exis ts

. lirn h(~t) tTT

a.s

II:z:

and h(J~)

=

I1:r.(1inlh(~t)). tiT

Set V = lirntTT h(~t). Let h n = IIx(V /\ n) in D. Then 0 < h n :S nand h n is harmonic in D. As BI) is Lipchitz, the Martin boundary of D is identified with aD according to Hunt and Wheeden[7l. Hence by Theorem 9.1 in [oJ, there is a bounded positive Borel function gn on aD such that a.s

II:z:

for

xED.

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ACTA MATHEMATICA SCIENTIA

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But by the rnartingale convergence theorem lirn hn(eTk)

k--+oo

where

Tk

= V /\ n

a.s

.is.the first exit time of efrom Dk and

subsets such that o, 1 D. Let Y SUPnYn. Then V :::;

=

g(~T)

Let Vm = g /\ m, and 'urn.(x) and satisfies

=-

Il;

a.s

log p~a! exp

Um(X) + iT

II x Dk

for

<

for

is a sequence of relative compact open

xED. Hence for :/; E D,

-~rt,XT

u~.(e..)ds =

> . Then u,n(:t;)

n.,vm(eT).

~

1'n,u n t ( x )

1,

(8)

By Theorem 1.1 in [1] um(x) is a bounded positive solution of(l). Set n(3:) = Ihun--+ex> um(x). Then by Proposition 2.3 u(x) is a positive solution 'of (1). Let m --+ 00 in both side of (8), then

i.e., there exists a positive solution of (1) 1J, satisfies the E.q. (2) with h. Acknowledgements I am indebted to my advisor professor Wu Rong for her encour.agernentsand helpful suggestions, References 1 Dynkin E B. A probabilistic approach to one class of nonlinear differential equations. Probab. Th. R.el. Fields. 1991,89: 89-115. 2 Dynkin E B. Path processes and historical processes, Probab, Th. ReI. Fields, 1991, 90: 1-36. 3 Dynkin E B. Branching parbicle systems and Superprocesses. AIUl. Probab., 1991,90: 1157-1194. 4 Dynkin E B. Superprocesses and partial differential equation. AIU1. Probab., 1993,21 = 1185-1262. 5 Fitzswu11.011S P J. Construction and regularity of measure-valued Markov processes, Israel.iJ .Math., 1988, 64: 337-361., 6 Doob J L. Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc.

Math. Frall.ee., 1957,85: 431-458. 7 Hunt R A, Wheedell R L. Positive harmonic functions on Lipschitz domains, Trans.Amer.Math.Soc., 1970,1 '14: 507-527. 8 Prot S C, Stone C J. Brownian motion and classical potential theory. 1978. 9 Chan Li- Zhi. Decomposition, Representation and extension of the harmonic function. Aet.a. Math.

Scientia (Chinese), 1987,1: 75-83.