Positive Solutions to Some Semilinear Elliptic Equations in L1(Rn)

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 357-367 (1986)

Positive Solutions to Some Semilinear Elliptic Equations in L1(R") By Masaharu A R A Ia n d Akira NAKAOKA Abstract.

A semilinear elliptic equation

Au=&)l.(u)

-m

in the whole space R" is treated. The coefficient a(x)is assumed to be positive and bounded and p(u) is assumed to satisfy the conditions (@1)-(@3) in the text. We give a necessary and sufficient condition on a(x) for this equation to have a unique non-negative solution in L1(Rn)for any non-negative f ( x ) in L1(R"). The result is extended to the equation

.

Au=@(x, u ) - ~ ( x )

Key words: semilinear elliptic equation, positive solution 1. Introduction

We shall study a non-negative solution in L 1 ( R n to ) a semilinear equation ( E)

Au=a(x)cp(u)-f(x)

(xE

R")

where we assume that O
y(u)= 1-e-"

as a n auxiliary tool of the study of the asymptotic behavior of some chemical reaction system. (This equation will be denoted by (E,).) They gave a sufficient condition on a ( x ) for the equation (E,) to have a unique non-negative solution in L 1 ( R n f) o r any positivef(x) in L 1 ( R * ) :

Theorem 0. If the Lebesgue measure of { x ;a ( x ) < c } with some positive constant c isfinite, then the equation (E,) has a unique non-negative solution in L 1 ( R n ) for any non-negative f in L 1 ( R n ) . They also gave a sufficient condition on the pair { a , f}for the non-existReceived March 29, 1985. Revised December 25, 1985.

M. ARAI and A. NAKAOKA

358

ence of positive solutions in L1(Rn). But there is a large gap between the two conditions. On the other hand, BBnilan, Brezis and Crandall [ l ] study a positive soiution in Maricinkiewicz space iW'(n-2)(Rn) and in L1(Rn) to the equation (E) with constant a. We note that they show, among other things, that if cp(u) satisfies the conditions (0-1) and (0-2) below and a is constant then the equation (E) has a unique non-negative solution in L1(Rn)for any non-negative f in L1(R"). Their study in Mn/("-*)(Rn) is extended by Gallouet and Morel [2] to the case where the non-linear term depends also on x , but the solutions are not necessarily in L1(Rn). The aim of this paper is to give a necessary and sufficient condition on a ( x ) for the equation (E) with ~ ( usatisfying ) the conditions (0-1)-(0-3) below to have a unique non-negative solution in L1(Rn) for any non-negative f in L'( R"). In what follows we assume that

(A-1)

U(X)€

Lm(Rn),

(A-2) f ( x ) is non-negative and of class L1(Rn), and that

(0-1) y(u) a is monotone increasing function of class c~(R,) , (a-2) cp(O)=O and c p ' ( O ) f O , (0-3) y'(u) is bounded in

E+.

We mean a solution u(x) to be non-negative and of class L1(Rn)satisfying (E) in the sense of distribution. and K , be the Let k , ( x ) be the inverse Fourier transform of convolution operator kc*,which is the inverse of (c2-A). Our main theorem is Theorem 1. Assume (A-l), (A-2), ( 0 - l ) , (0-2) and (0-3). The equation (E) has a solution for any f ( x ) if and only if inf (K,a)(x)>O

(A)

X

for some

c>O

.

Remark 1. If the condition (A) is valid for some positive c, then it is so for any positive c . We use the notations I/ 11 and 11 I(_ to denote the L1 norm and L" norm, respectively. For a condition P ( x ) of x , E(P) denotes the totality of x satisfying P ( x ) , e.g. E ( u > E ) = { x E R " ;a ( x ) > & } . B ( z ; r ) and B ( z ) denote the closed balls in Rn with radius r and center at z and at the origin, respectively. For the measurable set S in Rn, mS denotes its Lebesgue measure. Remark 2.

As for the problem in a bounded open set, see the review

Semilinear Elliptic Equation in L1(R")

3 59

article by P. L. Lions [3] and the abundant list of references there. One would be able to regard the equation (E) as a model of the following situation: a system of a reaction of a n enzyme of consistency a(x) and a substance of consistency u(x), which is constantly supplied by the amount of f ( x ) per unit volume per unit time. If the reaction obeys the MichaelisMenten law, the substance is diffusible and is not a n enzyme and the system reaches a stable state, then the stable state would be given by the equation (E) with

y(u)=u/(u+K)

( K : positive constant) .

2. Preliminaries

As is well known, the function k,(x) has the following properties: Lemma 2.1. (K-1) k,(x) depends only on 1x1 and k,(xj>O (K-2)

\

.

k,(x)dx=l/c2 .

(K-4) k,(x)lM(n, c) exp [ - c l x l / l / 2 ] I x l 1 - " , where M(n, c ) is a constant depending only on n and c. To study the equation (E) we use the method of iteration: (2.1)

Au,

- C'U,

= ~ ( x ) P ( u , _ ~-)c ~ u , --J(x) ~ ,

u,(x)=O

,

which is equivalent to (2.2)

u, =Ke(c2u,-1-ay(u,- 1 ) ) +KJ , uo=O

.

In what follows we choose c so large that c 2 u - a ( x ) ~ ( u ) is a monotone increasing function of u, which is possible by (A.l) and (0-3). Then it is easy to see

Lemma 2.2. 0 < u,(x)

U , + ~ ( X ) E L'(R").

Arguments similar to those in [4] show

Lemma 2.3. The equation (E) has a solution and in this case the solution is given by (2.3)

u(x)=lim u,(x)

.

if and onZy if { Ilu,l\}

is bounded

M. ARAI and A. NAKAOKA

360

Lemma 2.4.

The solution to (E) is unique if it exists.

Lemma 2.5. Let (pl and ‘pz satisfy the assumptions (@-1)-(@-3) and p l l y z . If the equation (E) with v=pI has the solution then so does equation (E) with p=pz.

Proof. Let {uj,rn}be the sequence defined by (2.2) with p replaced by pj ( j = l ,2). Then we have ul,rn-uZ,rn=K,{a(pz(uz,rn-l)-pl(uz,m-~)}

+ ~ e ~ l ~ 2 ~ ~ , m - t - ~ ~ ~ ( ~ ~ , m - ~ ~ ~ -9 ~ ~ 2 ~ ~ , m - ~ - ~

which is positive by the assumption of the induction so that Lemma 2.3 yields the result.

3. Proof of the “if” Part By virtue of assumptions (0-1)and (0-2), ‘p(u) is bounded from below by a function p1 satisfying

(0-4) p(u)=C,u for small u and p(u)=C, for large u (Cl, Cz: positive constants) with p replaced by

‘pl and

(0-5) monotone concave and smooth.

By virtue of Lemma 2.5 we may assume (0-4) and (0-5) without loss of generality in this section. Note that these assumptions imply

(3.1)

[cp’(O)-’p’(u)]/y(u)

is bounded in

and

(3.2)

0 5 ‘p’ ( u ) uIp(u) I p’(0)u

I

Define K , by (2.2) and u by (2.3), which is not yet known to be in L‘ or not. By virtue of Lemma 2.3 it suffices to show that { ~ ~ K , I I } is bounded. Lemma 3.1.

and

I t holds that

Semilinear EIliptic Equation in L1(Rn)

361

(3.5)

Proof. Integrating (2.2) over R”,we have

c-“

\

\

15

f ( x ) d x- a(x)p(u,-,(x))dx = u,(x)dx-

s

u,-,(x)dx

,

which is non-negative by Lemma 2.2 so that we have (3.3). The identity (2.2), Lemma 2.2 and (3.2) show (3.4). Integration of (3.4) over R n gives (3.5). Q.E.D.

Lemma 3.2. Zf g E L”(R”) then ( K , g ) ( x )is continuous. If g E L1n L” then (K,g)(x)tends to 0 as x tends to the infinity. Proof. I (K,g)(x’)-(K,g)(x)I x since k , is in L’. For any positive E , we have (3.6)

I(K,g)(x)/S

1

5

Ik,(x’)-k,(x)ldxllgl1, tends to zero as x’+

k(x-y)lg(y)ldy+

E(l&lZE)

s

E(ISli8)

k,(x-Y)Ig(Y)ldY

The second term in the right hand side of (3.6) is estimated by term of the right hand side of (3.6) is estimated by

s

k,(x -Y) I g(Y)I dY

B(z;IXI/Z)~

+

s

E/c~.

*

The first

~ c ( X - Y ) ~ Y l l ~ I l 9. a

B ( X ; I z 112) n E i 181Z E )

whose first term is bounded by the value of k , ( y ) at lyl = lxli2 times llgll which tends to zero as x tends to the infinity and whose second term equal to 5 B ( 1 2 1 , 2 ) ,,~l,i,,,k,(y)dyllg1I, which tends to zero as x tends to the infinity since mE( l g l 2 E ) is finite. Q.E.D.

Lemma 3.3.

If (A) holds, then q-sup (Kc(c2-ap’(u))(x)<1 z

.

Proof. Note that q l l . We put g=c2-up’(u), g,=c2-p’(0)a(x) and Now we have g2(x)=a(x)(cp’(0)--cp’(u(x)). Then g , , g 2E L” and g = g , + g , . sup ( K , g , ) ( x = ) 1-y’(O) inf (K,a)(x)< 1 z

X

by the assumption (A). Next, since g , E L 1 n L - by virtue of (3.1) and (3.3), Lemma 3.2 shows that lim (K,g,)(x)=O .

M. ARAIand A. NAKAOKA

362

Hence, if 9-1, it must be the maximum, say at x = x o . Then

so that a(x)p’(u(x))=O for almost all x, which with (3.1) and (3.3) shows a ( x )€ L 1 . Then Lemma 3.2 and the assumption (A) contradict each other.

Q.E.D.

Proof of the

p(u) implies

“if”

part of Theorem 1. Lemma 2.2 and the concavity of c2- a(x)p’( u,(

x)) Ic2-a(x)p’( u(x)) .

Multiply (3.4) with this inequality side by side and integrate it to obtain

4.

Condition (A)

The condition (A) describes a global property of a(x). In this section we give local versions of the condition (A), which will be convenient to prove the “only if” part of Theorem 1. We introduce the following two propositions (B) and (C) on a ( x ) :

(B) For any E > 0, there exists a triplet of sequences ({xp}, b p } ,

{ell})

such that ( i ) x p E R n + m as p + m , ( i i ) ap>O and ap+m as p-’m, (iii) e p is a measurable subset of B ( x p ;a p )whose Lebesgue measure tends to zero as p tends to the infinity and satisfies

Semilinear Elliptic Equation in L’(R”)

(4.1)

B(x,; a,)\e,cE(a
( C ) There exist positive constants (4.2)

E,

363

.

and ro such that

m ( B ( x ;r,) n E(a > e,)) 2 1

for any x € R”

.

We denote by -(B) the negation of (B).

Theorem 2.

The propositions (A), -(B) and ( C ) are mutudy equivalent.

Proof will be decomposed into a series of lemmas. Lemma 4.1.

(A) implies -(B).

Proof. Assume (B). Then we have

which is estimated by 2e for large p . Lemma 4.2.

-(B) implies ( C ) .

Proof. Assume that (C) does not hold. sequence { x , } in Rn such that

Then for any e > O , there exists a

m(B(x,; p ) n E(a > 4 )< 1

.

Now consider the balls with radius p‘ (Oe)) tends to zero. If the sequence {x,’} accumulates at a finite point, say x o , then B ( x o ; p u / 2 ) c B ( x , ’ ; p ufor ) large p . Let a,=p/4, x , be a point satisfying Ix,-x,I E ) . Then a triplet of sequences ( { x , } , {a,}, {e,}) satisfies the condition (B). If a sequence {x,’} has n o accumulation point, (B) is already valid. Lemma 4.3.

( C ) implies (A).

Proof. Assume ( C ) . Then (K,u)(x)is estimated from below by k , ( x - y ) d y 2Eok0(X)II I I =r 5.

*

Proof of the “only if” Part of Theorem 1 By virtue of the assumptions ( 0 - 2 ) and (0-3), ~ ( uis) estimated by C,u with

M. ARAI and A. NAKAOKA

364

some positive constant C, so that by Lemma 2.4 it is sufficient to show the existence of somef to which the equation (E) with p ( u ) = C , u has n o solution. Replacing C,a with a , it is sufficient to show

Lemma 5.1.

Assume inf (K,a)(x)= O

(5.1)

z

.

Then there exists (not necessarily non-negative) f ( x )E L1(Rn)such that the equation (5.2)

(-A+a(x))~(x)=f(x)

has no solution in L 1 ( R n ) . Indeed, let f be as in the above lemma and f*=max (kf, 0). Then at least one of the equations (5.2) with f replaced b y f , has no solution so that we have the "only if" part of Theorem 1. Proof of Lemma 5.1. Assume (5.1). Then, by Theorem 2 and the diagonal argument, (B) holds with E replaced by l/p. Let v,(x) be the characteristic function of the ball B(x,; a,) multiplied by the inverse of its measure so that I1upll=l. Then the L' norm o f f , ~ [ Z - ( ~ ~ - a ) K , ]isu ,estimated by (5.3)

s

B I Z p ; a,)

s

I v,(x) --c2(K,v,)(x)Idx+

+(UP)

5

B ( z p :& , ) C

I.,

(Kcv,)(4dx+c2 B ( z p ;a,)

(K,u,)(x)dx ,

f C Z j

l~,(X)-CZ(K,~,)(X)l~X

(K,up)(x)dx

B(zp;apC)

where we choose c so large that c 2 2[lullm. Since v , ~ c 2 ( K c u , ) ( xin) B(x,; a,), the first term of (5.3) is equal to

=cz\

(K,v,)(x)dx B(zp;ap)C

so that the right hand side of (5.3) is estimated by (5.4)

3c2

\

+

+

(K,v,)(x)dx ( l/p)c-2 rne,/mB(a,)

BIZ,; u p C )

.

Lemma 5.2 below shows that the first term of (5.4)tends to zero as p tends to infinity. Thus j p tends to zero while IIvpl/=l so that the operator [ I (c2-a)K,] from L1 into L1 is not bijective and so the operator

Semilinear Elliptic Equation in L1(R”)

-A

+u =c2

-

A - (c’- a ) = [ I -

365

(cZ- u)K,..(c~ - A)

from {u € L’; Au E L 1 }into L1 is not bijective. Let u be a solution of the equation

(-A+u)u=O.

(5.5)

Then we have (cz-A)u=(c?-a)u and (5.6)

for c2>ess supa(x). Integration of (5.6) shows jlaull=O, which with (5.5) shows Au=O and so u=O. Thus -A+a is injective so that it is not surjective, which completes the proof. Lemma 5.2.

For any z € R n and R > 0, we have

Z(z, R ) = \

B(z;RIC

dx\

k,(x-y)dy
B ( z : n)

where a(R) is independent of z and lim,,,o(R)=O. Proof. The proof of the case of n = l is similar to and simpler than that of the case of n 2 2 . So we prove only the case of n 2 2 . By virtue of (K-4), we have

I( z, R) I Const.,R n+ I

I -yll-ndy.

e-clz-Yl/J2

Change the variables to x’=c(x-y)/(&-R), y’=c(x-y)/(z/TR) and omit the primes to obtain (5.7)

sB(.f;c

Z(z, R ) ~ C o n s tRn+’ .

where we put c ‘ = c / d / T .

e-Rlvllyll-”dy , dx \ B ( % ; c f )

Since the second integral in the right hand side of 0 , x ) and take the polar coordi-

(5.7) depends only on 1x1, we put x=(O,

nates in x and y to obtain (5.8)

.-.,

I ( z , R)
ede\8+e-Rcdr, 8-

where s + = r cos 6 + l / d z - r z sin213 and sin Bo=c’/r. The last integral in the right hand side of (5.8) is estimated by R - l c R a - . Change the valuable 0 to s=s-. Then sin 0de=r-1[(r2-c’z)/(2s2)- 1/2]ds< (r2-c’2)/(2rs2)ds

M. ARAIand A. NAKAOKA

366

6.

Additional Remark The arguments in the preceding sections can be applicable to the equation

Au=@(x, u ) - ~ ( x ) ,

(El'

where @(x,u) is a function on R n x K such that (6.1)

(6.2)

,

u)

and

@(x,O)=O

0 I@Jx, u)

and

@,Ax,u) I0 ,

OI@(X,

(6.3)

a(x)=@,(x, O ) € L " ,

and there exists a positive constant C such that

(6.4) Theorem 3. by ( E)' holds.

[ @ A x ,O ) - @ J X , u)l/@(x,u) Ic

.

Under the assumptions (6.1)-(6.4)Theorem 1 with (E) replaced

Note that (6.4)implies

(6.5)

@ ( x ,u) >a(x)(1- e - c u ) / C

.

Replace @(x,u ) in (E)' with the right hand side of (6.5) and then replace Cu and Cf with u and f, respectively, to obtain (Eo). Hence, by Lemma 2.4, as for the existence of the solution, it suffices to consider (Eo).

Examples. Let a ( x ) and b ( x ) be positive bounded functions. ( i ) @ ( x ,u ) = a ( x ) ( l - e e - b ( z ) usatisfies ) the conditions (6.1)-(6.4) with C= sup b ( x ) . ( i i ) @ ( x ,u)=a(x)u/(l+b(x)u)satisfies (6.1)-(6.4)with C = 2 .

Semilinear Elliptic Equation in L'(R")

367

References [ 1 ] P. BCnilan, H. Brezis and M. G. Crandall, A semilinear equation in LL(RN), Ann. Scoula Norm. Sup. Pisa, 2 (1975), 523-555. [ 2 ] T. Gallouet and J.-M. Morel, Une equation semi-IinCaire elliptique dans L1(RN), C.R. Acad. Sci. Paris, 296 (1983), 493-496. [ 3 ] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. [ 4 ] M. Mimura and A. Nakaoka, On some degenerate diffusion system related with a certain reaction system, J. Math. Kyoto Univ., 12 (1972), 95-121.

Masaharu ARAI Faculty of Economics Ritumeikan University Kyoto 603, Japan Akira NAKAOKA Department of Mathematics Kyoto Institute of Technology Kyoto 606, Japan