ON POSITIVE SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL-EQUATIONS - A PROBABILISTIC APPROACH

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ELSEVIER Stochastic Processes and their Applications 59 (1995) 43-53

stochastic

processes

and their

applications

On positive solutions o f s o m e nonlinear differential

equations - A probabilistic approach

Yuan-Chung Sheu *

Department of Applied Mathematics, National Chiao-Tung University, Hsinchu, Taiwan Received September 1994; revised March 1995

Abstract

By using connections between superdiffusions and partial differential equations (established recently by Dynkin, 1991), we study the structure of the set of all positive (bounded or unbounded) solutions for a class of nonlinear elliptic equations. We obtain a complete classification of all bounded solutions. Under more restrictive assumptions, we prove the uniqueness property of unbounded solutions, which was observed earlier by Cheng and Ni (1992).

Keywords: Branching particle systems; Measure-valued processes; Nonlinear elliptic equation; Range; Superdiffusions.

1. Introduction

Throughout this paper we consider positive solutions o f the following nonlinear differential equation :

Lu(x) = k(x)u~(x), x E •d, (1)

where 1 < ct~<2, k is a bounded strictly positive continuous function on R d satisfying condition (6) below, and L is a differential operator in R d o f the form

d O ~

L = E a,J ~xi Oxj + ~i bi O X i (2)

i,j=l

such that it satisfies the following:

(1.a) The functions aij = aji and bi are bounded smooth functions in R d. (1.b) There exists a constant c > 0 such that

d

i j= 1

for all x E ~" and all Ul,U2 . . . u d. * Email: sheu@math.nctu.edu.tw.

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44 Y.-C. SheulStochastic Processes and their Applications 59 (1995) 43-53

If k - 0, then Eq. (1) becomes a linear equation and it can be studied probabilis- tically by using paths o f a diffusion ~ = (~t,l-Ix) with the generator L. By using the superdiffusion X = (Xt,X~, Pu) with parameters (L, ~b), where ~k(x, z) = k ( x ) z ~, we shall investigate the structure of the set of all positive solutions of Eq. (1).

I f L corresponds to a recurrent diffusion, then there is no nontrivial bounded position solution to (1). ( The following argument is provided by an anonymous referee. Assume u is such a solution and choose x0 E R d and a ball B such that u(xo) > SUpyeBU(y). By Ito's formula, IIxou(~t^~B)>~u(xo), where zB --- inf{t~>0, ~t E B}. By recurrence,

Ilxo[ZB < c~] ---- 1. Thus letting t ~ o¢ gives u(xo) > HxoU(~B)>>.u(xo), a contradic-

tion.) Therefore, we assume further that L corresponds to a transient diffusion. The superdiffusion X = ( X t , X ~ , P , ) is a branching measure-valued Markov process describing the evolution of a random cloud. It can be obtained as a limit of branching particle systems by speeding up the branching rate, decreasing the mass o f particles, and increasing the number of particles. For every t > 0, the random measure Xt is a limit of mass distribution of branching particle systems X # at time t, as fl ~ 0. For every z, the first exit time of ~ from a domain D C ~d, the corresponding random measure X~ can be obtained as a limit, as fl ~ 0, of the mass distribution of the particle systems X p at the first exit time from D (see Section 2 for more detail). Let zn be the first exit time o f ~ from the Euclidean ball of radius n, centered at 0 and let H stand for the set o f all bounded positive functions h on R d with L h ( x ) = 0 for all x E R d. Denote by (f,/~) the integral of f with respect to p. We write P Y for the expected value o f a random variable Y on a probability space ( f 2 , ~ - , ~ ) . We obtain the following in Section 3.

Theorem I. (a) I f h E H, then Zh = limn--.oo(h,X~,) exists a.s. (which means Pu-a.s.

f o r all #) a n d the function vh(x) = - l o g & exp{--Zh}

is the unique positive solution o f (1) with u = h at oc. ( I f u and v are two functions on R d, we write u = v at c~ i f u(x ) - v(x ) --~ 0 as Ilxll ~ ~ . )

(b) I f u is a bounded solution o f ( I ) , then u = vh f o r some h E H.

Therefore we characterize all bounded solutions of (1). (A similar result was established earlier by Cheng and Ni (1992) under more restrictive assumptions.)

Denote by E the set of all unbounded solutions u(x) o f (1) with u(x) ~ oo as IIxll ~ e~. The following theorem implies that E is not empty.

Theorem II. (a) The function

l ( x ) --- - log Prx[Xr, = 0 f o r n sufficiently large ], is the largest element o f E.

(b) The function

J ( x ) - - - - l o g P a x [ X ~ , ~ O a s n ~ o o ], x E ~ d

x E R d

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Y.-C. Sheu/ Stochastic Processes and their Applications 59 (1995) 43-53 45 We will prove Theorem II in Section 4, and give equivalence conditions in Theorem 4.2, for the uniqueness of unbounded solutions u o f (1) with u ( x ) ~ oo as Ilxll ~ o~.

As an application of Theorem 4.2, we give an alternative probabilistic proof of Cheng and Ni's result (el. Cheng and Ni, 1992, Theorem II):

Theorem III. I f L = A, where A = ~a=l O~ Ox---7, and k(x) ,- Ilxll - t , l > 2, a s Ilxll ~ o~,

then there is only one unbounded solution u o f ( I ) with u(x) ~ oo, as IIxH ~ o~. (Writing u ~ v as Ilxll ~ oo means there exist two constants cl,c2 > 0 such that cl v(x ) <<, u(x ) <<. c2v(x ) f o r x sufficiently large. )

In Section 5, as an application of Theorem III, we evaluate the probability for the range 9t of X to be compact.

In this paper c always denotes a constant and it may have different values in different lines. The notation Bn stands for the open ball with radius n, centered at 0.

2. Preliminaries

2.1 Let L be a differential operator in R d o f the form (2) satisfying conditions (1.a) and (1.b). Then there exists a Markov process ~ = (~t, l l x ) in R d with continuous

paths such that for every bounded continuous function f on R d,

ut(x) = H x f ( ~ t )

is the unique solution of the equation ~u

- - ---- L u

~t

with the property ut(x) ~ f ( x ) as t ~ 0 (see, e.g. Stroock and Varadhan, 1979).

We call ~ the diffusion with the generator L.

2.2 Denote by 9~ the Borel a-algebra in •d and by M the set of all finite measures on ~ . Write ~¢ for the a-algebra in M generated by the functions fs(/~) = I~(B),B E ~ .

For every positive bounded Borel function k(x) in R d and 1 < ~ < 2 , there exists

a Markov process X = (Xt,Pu) in (M,~t') such that the following conditions are

satisfied.

(2.2.a) If f is a bounded continuous function, then ( f , Xt) is right continuous in t

on ~+.

(2.2.b) For every # C M and for every positive bounded Borel function f , P~ e x p { - ( f , Xt)} = e x p { - ( v t , # ) } ,

where v is the unique solution o f the integral equation

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46 Y.-C Sheul Stochastic Processes and their Applications 59 (1995) 43-53

Moreover, to every D E 9~, there corresponds a random measure X~ on (~d, 9~) asso- ciated with the first exit time z = inf{t, it ¢~ D} from D by the formula

Pu exp{-(f,X~)} = exp{-(v, p), } (3)

where v satisfies the integral equation

[/0

]

v(x) + IIx k(¢s)V~(~s)ds = H x f ( ~ ) . (4)

(See Dawson (1993) or Dynkin (1994).) Note that for every p E M with supp(p)C D, X~ concentrates on dD, P~,-a.s.

We call X = (Xt,X~,P~) the superdiffusion with parameters (L,~), where ~ ( x , z ) = k(x)z c'. We explain the heuristic meaning of Xt and X~ in terms of branching particle systems. Consider a system of particles which undergo random motion and branching on R a according to the following rules.

a. Particles are distributed at time 0 according to the Poisson point process with intensity # E M.

h. Each particle survives with probability e x p { - f o k(~s)ds} at time t.

e. At the end of its lifetime, a dying particle gives birth to n offsprings at its own site, with probability Pn, where if 1 < c¢ < 2,

{ i ( ( : ) i f n = l , Pn = - 1 ) n i f n ¢ 1,

1

and i f ~ - - 2 , P 2 = p 0 = i .

d. During its lifetime, the motion of each particle is governed by the process 4. e. All particle lifetimes, motions, and branching are independent of one another. The historical path H f of a particle consists of its own trajectory and the trajectories of all its ancestors. If each particle has mass r, then

Xt~(B) = fl E 1B(Hta)

a

is the mass distribution at time t. (The sum is taken over all particles which are alive at time t.) Set

X~(B) : fl E IB(H~.),

a

where za = inf{t, H f ¢~ D}. (Here we identify particles a and b if Z a = T b and H a = H b

for all s ~ Za.) If k# = ~ and p/~ = ~, then Xt ~ and Xfl converge weakly to Xt and X~ as fl ~ 0. Let z be the first exit time of ~ from a domain D C Na. A boundary point y of D is called regular if lly[z = 0] = 1. We quote two theorems from Dynkin (1991).

Theorem 2.1. Let D be a bounded domain in R a. For every positive bounded Borel function f on dD, the function

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Y.-C Sheu I Stochastic Processes and their Applications 59 (1995) 43-53 47 satisfies the equation

Lv(x ) = k(x )v~(x ) (5)

for x • D. Moreover if D is reoular and i f f is continuous, then v = f on dD. (We write v = f on K C OD if for every y • K, limx~O,x-.y V(x) = f ( y ) . )

Theorem 2.2. Let D be an arbitrary domain in R a. Choose a sequence of bounded

regular domains {Dn}n with Dn T D and let Zn be the first exit time of ~ from D,. I f u is a solution of (5) in D, then Z~ = l i m , ~ ( u , X ~ . ) exists a.s. and u(x) =

- log Pax exp{-Zu} for all x • D.

3. Bounded solutions of L u = k u ~

Throughout this paper we consider a strictly positive bounded continuous function

k(x) on R a satisfying the condition :

l i m r - ~ SUpx~Ra f 9(x, y ) k ( y ) d y = 0, (6)

JII yJl>r

where 9(x, y ) is the Green function of the operator L on R a.

Example 1. If

k(x)<~h(llxll)

for x sufficiently large, and f o~ sh(s)ds < e~ for some

a > 0, then k satisfies condition (6) for L = A and d = 3.

Proof. It follows from Eq. (13.74) in Dynkin (1965) that

g(x,y)<~cllx-yl[

2-d for all

x , y • R d. Our conclusion follows from Zhao (1993, Propositions 1 and 2). []

As before, Zn stands for the first exit time of the diffusion ~ from the domain Bn and X = (Xt,X~,Pu) is the superdiffusion with parameters (L, qJ), where ~k(x,z) = k(x)z ~. Put Zh,, = (h,X~,) if h • H.

Lemma 3.1. For every ~ • M and h • H, P~-a.s., Zh = lim,~o~ Zh,, exists and

Zh < 00. Proof. Let ~ , and Eqs. (3) and (4),

Pu[exp{-Zh,.}l~m] = Px,. exp{-Zh, n} = exp{-(wn,Xr.) },

where Wn satisfies the equation

w.(x) + nx w ~ ( G ) k ( G ) d s = h(x), x • 8 . . Clearly wn(x)<~h(x) and we have, by (7),

= a{X, k, 1 <<.k<<.n}. If m < n, then, by the strong Markov property

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48 Y.-C Sheul Stochastic Processes and their Applications 59 (1995) 43-53

Therefore, (exp{-Zh, n},~n,Pu) is a bounded submartingale and Zh = l i m n ~ Z h , n exists Pu-a.s.

It follows from (3) and (4) that PuZh,~ = l I , h ( ~ , ) = (h,l~). B y Fatou's lemma, PuZ <~ liminf,~o~P~Zh, n = (h,t~) < cx~,

which implies that Zh < c~,Pu-a.s. []

W e will write Z for Zh if h(x) = 1 for all x E •d. Theorem 3.2. For every h E H, the function

vh(x) = - log P~x e x p { - Z h } , x E ~a,

is the unique solution o f (1) with vh = h at e~. Moreover vh satisfies the integral equation

v(x) + fna O(x' Y)k(y)v~(Y) dy = h(x) (8)

in ~u.

Proof. Set Vh,,(x) = -- log P,~x exp{--Zh, n}, x E Bn. It follows from Eqs. (3) and (4) that vh,~ satisfies the equation

Vh, n(X) + ~_ 9n(X, y)k(y)v~m(y)dy =-- h(x), x E Bn, (9)

d I S n

where 9n(X,Y) is the G r e e n ' s function o f L on B,. Set 9n(x,y) = 0 i f x ~ Bn or y f[ Bn, and put vh,~(x) = h(x) if x ~/B,. Then equation (9) becomes

+ fu~ 9n(X'y)k(y)V~'n(Y)dY = h(x), Vx E R d. (10) Vh, n(X)

Note that gn(X,y) T g ( x , y ) as n ~ cx~, and, by L e m m a 3.1, Vh(X) = lim,~oovh, n(x) for all x E ~d. Therefore, for every x E R d,

gn(x, y)vh,,(y) ~ g(x, y)v~,(y) as n --* cx~. Since vh,,(y)~<[Ihll, we have vh(y)<~llhll and so

g~(x, y)v~,~(y)k(y) <~ cg(x, y)k(y).

For every x E ~a, condition (6) implies that 9(x, y ) k ( y ) is integrable. Letting n --~ in (10), the dominated convergence theorem implies that vh satisfies Eq. (8).

For n > m, both the functions Vh, n and - log Pa~ exp{--(vh~,XT.)} are solutions o f (5) in Bin, and they have the same boundary values o n OB m. The m a x i m u m principle (see, e.g. Dynkin (1991, Theorem 0.5)) implies that

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E-C. Sheul Stochastic Processes and their Applications 59 (1995) 43-53 49

Letting n ---, oo, we get oh(x) = - - l o g P6~ exp{--(vh,X~,)} in Bin. Theorem 2.1 implies that vh is a solution o f (5). To check Vh = h at c~, it suffices, by (8), to show that

faa g ( x , y ) k ( y ) d y ---, 0 as Ilxll ~ ~ . w r i t e fn~ O(x, y ) k ( y ) dy = A(x, r) + B(x, r), ( 11 ) where £ A(x,r) = / 9 ( x , y ) k ( y ) d y JII yll<~r and / - B(x, r) = ] 9(x, y ) k ( y ) dy. JII yll>~r

Since k is bounded and g(x,y) --~ 0 as Ilx - yll ~ c ~ , for every r > 0, A(x,r) goes to 0 as Ilxll ~ oo. Clearly condition (6) implies that SUPxeR~B(x,r ) ~ 0 as r ---, c~. Letting Ilxll ~ ~ and then r ---, c~ in (11), we observe that f g ( x , y ) k ( y ) d y ~ 0 as Let u be a solution o f ( l ) with u = h at c~. Since Z < oo and u = h at oo,

(u,X~,) ---* Zh. By Theorem 2.2, u(x) = - log P ~ e x p { - Z h } -- vh(x). []

R e m a r k . (a) I f L = A and k is radial, then Vc, n is radial for every constant function h = c a n d s o is Vc.

(b) Under the same assumptions on k(x) as in Example 1, Kawano (1984) and Cheng and Ni (1992) obtained similar results for L = A. By using Brownian path integration and potential theory, Zhao (1993) studied related problems for L = A. Theorem 3.3. I f u is a bounded solution o f (1), then v = h at co for some h E H and v(x) = Vh(X).

Proof. Note that if u satisfies Eq. (8) for some h E H , our conclusions follow from the same arguments as in the p r o o f o f Theorem 3.2. Since u(x) = - l o g P ~ x exp{-(u,X~,)},

(3) and (4) imply that u satisfies the equation

u(x) + ~ gn(x,y)k(y)u~(y)dy = hn(x), x E nn, (12)

n

w h e r e hn(x) = IIx[u(~,)]. C l e a r l y Lhn = 0 and hn(x) =/-/xu(¢~n)~llull for all n. By passing n to the limit in (12), h(x) = limn--.~ hn(x) exists for all x E R d. Therefore, h is bounded and Lh = 0 in R a. By passing to the limit in (12) again, u satisfies Eq. (8). []

Remark. The special case o f Theorem 3.2, where L = A and k satisfies the conditions in Example 1, was observed earlier by Cheng and Ni (1992).

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5 0 Y.-C Sheul Stochastic Processes and their Applications 59 (1995) 43-53

4. Unbounded solutions o f L u = k u ~'

L e m m a 4.1. (a) L e t B = {x; [ I x - x°ll < R} and u(x) = 2(R 2 - r 2) ,-g-'

where 2 is a positive constant and r = [Ix - xol[. W e have

lim u(x) = c~

x---~a, xEB

f o r all a E OB, and Lu - ku~ ~ O in B

~ a ~

f o r some 2 dependin9 only on ~, the dimension d and the upper bounds f o r aij = --

k

and [~i = ~ in B.

(b) I f B c D f o r some open set D, and v is a solution o f ( 5 ) in D, then v ~ u in B.

Proof. (a) is quoted from Dynkin (1991, Lemma 3.1) and (b) follows easily from the maximum principle. []

P r o o f o f Theorem II. (a) For x E B n and m > 0, set In, m(X) = - logP6~ e x p { - m Z I , n } . Theorem 2.1 implies that In, m satisfies (5) in Bn and In, m = m on 0Bn. By the maximum principle, In, m is increasing in m. Therefore, for every x E Bn,

In(x) = limm~o~ ln, m(X )

exists. Clearly In(x) = - l o g P 6 x [ Z l , n -- 0] and In = ~ on OBn. Let B be an arbitrary bounded open ball and let ~ be the first exit time o f ~ from B. I f / ~ C Bn for some n, Theorem 2.1 and the maximum principle imply that

~ , m ( X ) = - l o g P 6 ~ exp{-(~,m,X~)}, x E B. (13) Note that, by L e m m a 4.1, [ ~ , m [ ~ c in B, for all m. Leaing m go to c~ in (13), we have

In(x) = - l o g P~x e x p { - ( I n , X ~ ) } . (14) Clearly l ( x ) = limn--.ooln(x) and, letting n ~ c~ in (14), we observe I ( x ) = -

log Prx e x p { - ( I , X ~ ) } . By Theorem 2.1, I satisfies Eq. (1).

Assume u is a solution o f (1). Since In = cx~ on c3B,, the maximum principle implies that u ~< In in Bn and so u ~< I.

(b) Write Vc for vh in Theorem 3.2 if h is a constant function c. Clearly Vc(X) T J ( x )

as c T ~ and J ( x ) ~ o0 as IIxll - - ' oo. Since vc(x) -- - log t'61 e x p { - ( V c , X ~ , ) } in Bn, we have J ( x ) = - log Prx e x p { - ( J , X ~ , ) } in Bn. By Lemma 4.1, J is bounded on aBn, and so Theorem 2.1 implies that J is a solution o f (1).

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Y.-C Sheul Stochastic Processes and their Applications 59 (1995) 43-53 51 Assume u E E. For every c > O, (u,X~.)>~cZl,n for sufficiently large n. For any c > O ,

u ( x ) = l i m n ~ - l o g P~x e x p { - ( u , X ~ , ) } ~> limn~o~ - log P6~ e x p { - c Z l , n } = v~(x).

Letting e T oo, u ( x ) > ~ J ( x ) for all x E R a. Therefore J is the minimal element in E. []

Remark. (a) Assume L = A and k is radial. Then both Into and In are radial. Therefore I is radial. Clearly J is radial.

(b) Let kl,k2 be two bounded strictly positive continuous functions on R d which both satisfy condition (6). Assume further that kl(x)<<.k2(x) for all x. For s = 1,2, let

Is, nan,Is, and Js denote In, m,1, and J respectively, with k replaced by ks. For x E Bn, we have

: - ~ >~ - k2I~,n, m" L12,n,m - k2I~,n,m : 0 LIl,n,m klll,n, m ~-Llkn, m

The M a x i m u m principle implies that Iz, n,m<<.Ii,n,m on Bn. Therefore 12<<.11. Similar arguments imply that J2 ~<J1.

Denote by

IEI

the cardinality o f E. B y Theorem II,

IEI

>/1.

Theorem 4.2. The f o l l o w i n g three s t a t e m e n t s are equivalent.

(a)

IEI--

1.

(b) F o r every m e a s u r e p E M with c o m p a c t support, we have Pu[Zl,n --* O] = Pu[Z1,n = 0 f o r sufficiently large n ].

(c) T h e r e exists a constant c such that I ( x ) < , c J ( x ) f o r x sufficiently large,

where I a n d J are f u n c t i o n s in T h e o r e m II.

Proof. Note that for every # E M with compact support, we have

P u [ Z = O] = e x p { - ( J , # ) } and

Pu[Zl,n = 0 for sufficiently large n ] = e x p { - (/, #) }.

Therefore (a) and (b) are equivalent. Clearly (b) implies (c). Assume that (c) holds. To prove (a), it suffices to show that I = J . Fix x E R a. By Theorem 2.2, both ZI = limn~o~(/,X~,) and Z j = limn--.~(I,X~,) exist Px-a.s. Since I ( x ) ~ c~ and

J ( x ) ~ c~, ZI = J j = cx~ on {Z > 0}. Combining with Theorem 2.2, we have - l o g P6x[Z = O] = J ( x ) = - l o g P6, e x p { - Z j }

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52 Y.-C. Sheul Stochastic Processes and their Applications 59 (1995) 43-53

Therefore Zj = 0 on {Z = 0}Px-a.s. By assumption, we have (I,X~,) <<.c(J,X~,) for n sufficiently large, which implies that Zt = 0 on {Z = 0}, Px-a.s. Therefore

I(x) = - log P6~ e x p { - ZI}

= - l o g P6x[exp{-Zl},Z = 0] -- - log P6x[Z = O] = J(x). []

I f L = A and if k is radial with k(x) = clxl -t, l > 2, for large x, then the first part o f Theorem 4.3 o f Cheng and Ni (1992), implies that for every radial solution u o f (1),

I - - 2

we have u(x) ~ Ixl 7~-~ as Hxll ~ c~. By using this observation and Theorem 4.2, we prove Theorem III.

P r o o f o f Theorem I I L By assumption, there exist two constants el, c: and two radial functions k~ and k2 with kl(x) = Ix] - l = k2(x) for x sufficiently large and

clkl(x)<~k(x)<~c2kz(x) for all x E R a.

For s = 1,2, let Is,Js denote I and J respectively with k replaced by Gks. By Remark (b) following the proof o f Theorem II, we have, for all x,

Iz(x) <~I(x) <~II (x)

and

J2(x) <~J(x) <~Jl (x).

/ - - 2 --2

Since Is(x) ~ Ixl ~-' and J ~ ( x ) ~ Ix ,---~,s = 1,2, as Ilxll ~ c~, we get

I(x) .< I i ( x ) ~<c

J ( x ) "~" J2(x)

for x sufficiently large. Our result follows from Theorem 4.2. []

R e m a r k . In general we do not know if condition (6) is a sufficient condition for

l e l - - 1.

5. A p p l i c a t i o n

The range ~ o f the superdiffusion X is the smallest closed subset o f R d such that contains supports o f Xt for all t~>0. For constant k, Iscoe (1986) proved that ~ is compact a.s. for L = A and Dynkin (1991) observed this for general L.

T h e o r e m 5.1. I f L = A and k(x) ,,~

Ixl -z, l

> 2, as

Ilxl[

~ ~ , then for every # c M

with compact support,

P~[ g8 is compact] = e x p { - ( / , p / } ,

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Y.-C. Sheul Stochastic Processes and their Applications 59 (1995) 43-53 53 Proof. If/~ E M has compact support, we have, P~-a.s.,

{:~ is compact} =

U{Z,,n

= 0}.

n = l

(See Dynkin, 1991). Our statement follows from Theorem II and Theorem III. []

Acknowledgements

The author is indebted to a referee who read the paper carefully and made some invaluable suggestions.

References

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