Entire positive solution to the system of nonlinear elliptic equations

ENTIRE POSITIVE SOLUTION TO THE SYSTEM OF

NONLINEAR ELLIPTIC EQUATIONS

LINGYUN QIU AND MIAOXIN YAO

Received 8 November 2005; Revised 12 May 2006; Accepted 15 May 2006

The second-order nonlinear elliptic system −Δu= f1(x)uα_+g1(x)u−β_+h1(x)uγ_P(v),

−Δv= f2(x)vα_+g2(x)v−β_+h2(x)vγ_{P(u) with 0< α,β,γ <1 , is considered inR}N_.

Un-der suitable hypotheses on functionsfi,gi,hi(i=1, 2), andP, it is shown that this system

possesses an entire positive solution (u,v)∈C2,θ

loc(RN)×C 2,θ

loc(RN) (0< θ <1) such that bothuandvare bounded below and above by positive constant multiples of|x|2−N_for

all|x| ≥1.

Copyright © 2006 L. Qiu and M. Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

This paper is concerned with the second-order nonlinear elliptic system

−Δu= f1(x)uα_+g1(x)u−β_+h1(x)uγ_P(v),

−Δv= f2(x)vα+g2(x)v−β+h2(x)vγP(u),

x∈RN_{(N≥3), (1.1)}

whereΔis the Laplacian operator, 0< α,β,γ <1 are constants, the functions fi,gi,hi(i=

1, 2) are nonnegative and locally H¨older continuous with exponentθ∈(0, 1) inRN_{, and} P:R+→R+is a continuous differentiable function, whereR+=(0, +∞),R+=[0, +∞).

We are interested in the study of the existence of entire positive solutions (u(x),v(x)) to (1.1) which satisfy the condition that each of its elements decays between two positive multiples of|x|2−N_{asxtends to infinity. By an entire solution of (1.1) is meant a pair of}

functions (u,v)∈C2,θ

loc(RN)×C 2,θ

loc(RN) which satisfies (1.1) at every pointxinRN. The existence of entire positive solutions of the equation

Δu+f(x,u)=0, x∈RN_{,N≥3, (1.2)}

Hindawi Publishing Corporation Boundary Value Problems

has been proved under various hypotheses, see [6,7,10–12,14,17]. In Particular, for the generalized Emden-Fowler equation

Δu+K(x)uλ=0, x∈RN_{,N≥3, (1.3)}

whereλis a constant, andK is a positive locallyθ-H¨older continuous function inRN_,

Fukagai [7] has proved forλ∈(0, 1) that if +∞

1 s

N−1−λ(N−2)_K∗_{(s)ds <+∞, K}∗_(s)=max

|x|=sK(x), (1.4)

then there is an entire positive solution of (1.3) that is minimal, that is, bounded below and above, respectively, by a positive constant times|x|2−N_{asxtends to infinity.}

Equation (1.3) withλ∈(0, 1) is said to be of sublinear type; ifλis negative, then (1.3) is said to be of singular type, and such equations arise from the boundary layer theory of viscous fluids, see [3,13]. In this paper, we focus on elliptic systems of mixed type.

It is well known that some reaction-diffusion equations have been investigated in con-nection with models of population dynamics [2,5,9,15]. To mention some, in [15], the equation∂u/∂t−dΔum_{=f(x,u) is studied. For some mutualistic symbiosis population}

models of two species, it may be necessary to study equation systems such as

∂u ∂t −d1Δu

m_{= f1(x)u}ρ_+g1(x)uσ_+h1(x)uμ_P(v),

∂v ∂t −d2Δv

m_{= f2(x)v}ρ_+g2(x)vσ_+h2(x)vμ_P(u),

x∈RN_{, (1.5)}

where 0< ρ,μ < m,−m < σ <0, andd1,d2>0. Obviously, the positive equilibrium solu-tions to system (1.5) inRN_{are corresponding to the entire positive solutions of a system}

in the form of (1.1).

Some existence results of elliptic system

Δu+F1(x,u,v)=0,

Δv+F2(x,u,v)=0 (1.6)

have been established in [4,11,16,19–21]. In particular, in [20], the existence of the equilibrium solutions is established for the Volterra-Lotka mutualistic symbiosis model in the case of equal linear birth rates, using the method of upper and lower solutions. However, for so-called mixed type in whichF1andF2involve both singular and sublinear terms, results regarding the existence of positive entire solutions cannot be derived from those in the literature.

as

−Δu= f1(x)uα_+g1(x)u−β_+h1(x)uγ_vδ_,

−Δv= f2(x)vα+g2(x)v−β+h2(x)vγuδ,

x∈RN_{, (1.7)}

or

−Δu=f1(x)uα_+g1(x)u−β_+h1(x)uγ_c

0+v −δ

,

−Δv=f2(x)vα_+g2(x)v−β_+h2(x)vγ_c

0+u −δ

,

x∈RN_{, (1.8)}

with 0< α,β,γ <1,c0≥0, 0< δ <1−γ, and some other kinds of systems even more general (seeRemark 2.2). Moreover, our method can be used to deal with similar systems on a bounded domain.

2. Main results

First, we denote byφthe function defined onR:

φ(t)=1, if 0≤t <1; φ(t)=t2−N_{, ift≥1. (2.1)}

A solution (u(x),v(x)) for equation system (1.1) is usually called a minimal positive entire solution if bothu(x) andv(x) are between two positive constant multiples of function

φ(|x|) in wholeRN_{. This term comes from the fact that no positive solution ofΔu≤0 in}

an exterior domain can decay more rapidly than a constant multiple of|x|2−N_{, see [18].}

Theorem2.1. Suppose that0< α,β,γ <1are constants and the functionsgi,hi(i=1, 2), andPsatisfy the following conditions:

(T) fi,gi,hiare locally H¨older continuous with exponentθ∈(0, 1)inRN_and

+∞

1 s

N−1−α(N−2)_f∗

i (s)ds <+∞, fi∗(s)=max_|x|=sfi(x),

+∞

1 s

N−1+β(N−2)_g∗

i (s)ds <+∞, gi∗(s)=max_| x|=sgi(x),

+∞

1 s

N−1−γ(N−2)_h∗

i (s)P∗(s)ds <+∞, h∗i(s)=max_| x|=shi(x), P

∗_(s)=max |x|=sP

φ|x|,

fi_∗(s) +gi_∗(s) +hi_∗(s)≡0 fors≥0,

fi_∗(s)=min

|x|=sfi(x), gi∗(s)=min|x|=sgi(x); hi∗(s)=min|x|=shi(x);

(2.2)

(P)P:R+→R+is a continuous differentiable function satisfying that there exists aλ∈ (0, 1−γ)such that for allk≥1andc∈[k−1_,k],

Then the system (1.1) possesses a positive entire solution(u,v)∈C2,θ

loc(RN)×C 2,θ

loc(RN)such that each ofuandvdecays between two positive constant multiples ofφ(|x|)asxtends to infinity, that is, the solution is minimal.

Remark 2.2. Examples of functionP(s) satisfying the condition (P) are

P(s)=c0+s −δ

, c0>0, 0< δ≤λ

(2.4)

as suggested in (1.8),

P(s)=sδ+s−σ, 0< δ≤λ, 0< σ≤λ (2.5)

or

P(s)= sδ

c0+sσ

, c0>0,δ >0,σ >0, 0< δ+σ≤λ (2.6)

and so on.

3. Proof of results

Lemma3.1. Consider the equation

−Δu=f(x)uα+g(x)u−β+h(x)uγ. (3.1)

Suppose that f,g,hare nonnegative functions defined onRN_{, and0< α,β,γ <1are}

con-stants. If f,g,hare locally H¨older continuous with exponentθ∈(0, 1)inRN_and

(T)

+∞

1 s

N−1−α(N−2)_f∗_{(s)ds <+∞, f}∗_(s)=max |x|=sf(x),

+∞

1 s

N−1+β(N−2)_g∗_{(s)ds <+∞, g}∗_(s)=max |x|=sg(x),

+∞

1 s

N−1−γ(N−2)_h∗_{(s)ds <+∞, h}∗_(s)=max |x|=sh(x),

f∗(s) +g∗(s) +h∗(s)≡0, fors≥0,

f∗(s)=min

|x|=sf(x), g∗(s)=min|x|=sg(x), h∗(s)=min|x|=sh(x),

(3.2)

then (3.1) possesses a unique positive entire solutionu∈C2,θ

loc(RN)such thatudecays be-tween two positive constant multiples ofφ(|x|)asxtends to infinity, that is, the solution is minimal.

Lemma3.2. Suppose that f :RN_×R+_{→Ris a continuous function such that one of the} following assumptions is satisfied:

(F2)s−1_{f(x,s)is strictly decreasing insfor eachxin a subsetΩ}

0ofRNand both f(x,s) ands−1_{f(x,s)are nonincreasing insfor allxin the remainder partR}N_−Ω

0. Letw,v∈C2_(RN_)satisfy

(a)Δw+ f(x,w)≤0≤Δv+f(x,v)inRN_,

(b)w,v >0inRN_{andlim inf}

|x|→∞(w(x)−v(x))≥0,

(c)ΔvinL1_(RN_).

Thenw≥v∈RN_.

The proof of Lemma 3.1 is given for completeness in the appendix of this article.

Lemma 3.2is an extension of [17, Lemma 1], so the proof is omitted here for briefness. Proof ofTheorem 2.1. Consider the equation

−Δu= f1(x)uα+g1(x)u−β+h1(x)uγPφ|x|, x∈RN_{. (3.3)}

In view of (T) andLemma 3.1, we find that there exists, for (3.3), a unique entire positive solutionu0(x)∈C2,θ

loc(RN). With the same argument, for the equation

−Δv=f2(x)vα+g2(x)v−β+h2(x)vγPφ|x|, x∈RN_{, (3.4)}

there exists a unique entire positive solutionv0(x)∈C2,θ

loc(RN). Moreover, it is obvious that there is a constantc0>1 such that for anyx∈RN,

c−01φ

|x|≤u0(x)≤c0φ

|x|,

c−1 0 φ

|x|≤v0(x)≤c0φ

|x|. (3.5)

For any constantE≥1, denote

UE≡

u∈C0,θ

loc

RN_|E−1_{u0(x)≤u(x)≤Eu0(x), x∈R}N_,

VE≡

v∈C0,θ

loc

RN_|E−1_{v0(x)≤v(x)≤Ev0(x),x∈R}N_,

Q≡UE×VE.

(3.6)

Obviously,Qis closed and convex. For each (u,v)∈UE×VE, by Poisson equations theory

and (T), the problem

−Δu= f1(x)uα+g1(x)u−β+h1(x)uγP(v), x∈RN_{, (3.7)}

has a unique solutionu∈C2,θ

loc(RN)⊂C 0,θ

loc(RN), and the problem

−Δv=f2(x)vα_+g2(x)v−β_+h2(x)vγ_{P(u), x∈R}N_{, (3.8)}

has a unique solutionv∈C2,θ

loc(RN)⊂C 0,θ

loc(RN). Defining the mappingsA1:Q→C0,locθ(RN) byA1(u,v)=uandA2:Q→C0,locθ(RN) byA2(u,v)=v, we have thatAi(u,v)∈C

2,θ

loc(RN),

i=1, 2, and hence

Φ(Q)⊂C2,θ

loc

RN_×C2,θ

loc

We claim that ifEis a positive constant large enough, then

E−1_{u0(x)≤u(x)≤Eu0(x), x∈R}N_,

E−1v0(x)≤v(x)≤Ev0(x), x∈RN_, (3.10)

hence we haveA1(Q)⊂UEandA2(Q)⊂VE. In fact, we have

Ec0 −1

φ|x|≤E−1u0(x)≤u(x)≤Eu0(x)≤Ec0

φ|x|,

−Δu= f1(x)uα_+g1(x)u−β_+h1(x)uγ_P(v)

≤ f1(x)Eαu0α+g1(x)Eβu

−β

0 +h1(x)u

γ

0EγP _v(x)

φ|x|φ

|x|

≤ f1(x)Euα

0+g1(x)Eu−

β

0 +h1(x)u

γ

0Eγ

Ec0 λ

Pφ|x|,

(3.11)

while on the other hand, we have

−ΔEu0

=f1(x)Euα0+g1(x)Eu

−β

0 +h1(x)u

γ

0E P

φ|x|. (3.12)

Thus, ifEis so large thatE(1−r−λ)/λ_{≥c0, then we haveΔu≥Δ(Eu0). It follows from the}

maximum principle for the operator−Δthat

u(x)≤Eu0(x), x∈RN_{. (3.13)}

Similarly, we have

u(x)≥E−1u(x), x∈RN_{. (3.14)}

With the same argument, we conclude that

E−1_{v0(x)≤v(x)≤Ev0(x), x∈R}N_{. (3.15)}

Fix thisEand defineΦ:Q→Qby

Φ(u,v)=A1(u,v),A2(u,v), ∀(u,v)∈Q, (3.16)

and now we only need to prove thatΦhas a fixed point inQ.

In order to use the Schauder-Tychonofffixed point theorem, we will prove that the operatorΦsatisfies the conditions through three steps.

(1)Φ(Q)⊂Q. This is a direct conclusion of (3.10).

(2)Φ:Q→Qis continuous. Obviously, it suffices to prove thatA1 andA2are both continuous in the sense that forwn→w in Q, it holds true thatAiwn−Aiw0,θ→0, n→ ∞,i=1, 2, here, for any sequence{un} ⊂C0,θ

loc(RN), by writingun0,θ→0,n→ ∞,

we mean that, for any closed bounded domainG⊂RN_,un

Denote

F(x)≡ f1(x)uα+g1(x)u−β+h1(x)uγP(v),

Fn(x)≡f1(x)uα_n+g1(x)u−nβ+h1(x)uγnP(vn).

(3.17)

We have obviously that _Fn−F

0,θ−→0, asun−u0,θ+vn−v0,θ−→0, n−→ ∞. (3.18)

ByLemma 3.1, we may letube the unique solution of the equationΔu=F(x) and let

unbe the unique solution of the equationΔun =Fn(x). Then by the Schauder estimation theory, we know that for any bounded domainG⊂RN_{, there exists a constantC>0 such}

that

un−u_C2,θ(G)≤CFn−FC0,θ(G), (3.19) and hence

un−uC0,θ_(G)≤CFn−FC0,θ_(G). (3.20)

Therefore,

un−u0,θ−→0, asun−u0,θ+vn−v0,θ−→0, n−→ ∞, (3.21)

that is,A1 is a continuous mapping fromQ toUE. Similarly, A2 is also a continuous mapping fromQtoVE.

(3)Φ(Q) is relatively compact inC0,θ

loc(RN)×C 0,θ

loc(RN).

We first recall the gradient estimates for Poisson’s equation (see [8]). For any bounded domainΩ⊂RN_{, ifΔu= f inΩ, then}

sup Ω dx

_Du(x)≤Csup

Ω |u|+ supΩ d 2

xf(x)

, (3.22)

wheredx=dist(x,∂Ω) andC=C(N).

DenoteBm≡ {x∈RN;x< m},m=1, 2,. . . .For eachu∈A1(Q), we have by (3.22)

that

sup

Bm

_Du(x)≤sup

Bm

dxDu(x)≤sup

Bm+1

dxDu(x)

≤C

sup

Bm+1

_{u(x)+ sup}

Bm+1

d2xF(x)

≤C

sup

Bm+1

_{Eφ(x)+ (m+ 1)}2_sup

Bm+1

_F(x)≤Km,

(3.23)

Furthermore, by (3.23), we know that _u(x)−u(y)

|x−y| ≤Du

t0x+1−t0y≤Km, ∀x,y∈Bm. (3.24)

This shows thatA1(Q), restricted onBm, is a bounded subset ofC0,1(Bm). By the compact

embedding result (see [1]);C0,1_{(Ω) C}0,θ_{(Ω), for any bounded domainΩ⊂R}N_{, it is}

seen thatA1(Q), restricted onBm, is a relative compact subset ofC0,θ_{(Bm). Therefore, for}

any arbitrary sequence{un}n≥1⊂A1(Q), there exists a subsequence{u(nm)}n≥1⊂A1(Q) which is convergent onBm in the sense of the norm · C0,θ_(B

m). The case forA2(Q) is

similar.

Considering∞_m=1Bm=RN, by the diagonal method, we conclude, fori=1 andi=2, respectively, that for an arbitrary sequence{un}n≥1⊂Ai(Q), there exists a subsequence,

say,{u(nn)}n≥1⊂Ai(Q), which is convergent in the sense of the norm · C0,θ_(K) on any

compact subset K of RN_{, that is, Ai(Q) is relatively compact inC}0,θ

loc(RN). Therefore, Φ(Q)=A1(Q)×A2(Q) is a relatively compact subset ofC0,θ

loc(RN)×C 0,θ

loc(RN).

Therefore, by the Schauder-Tychonofffixed point theorem, there exists an element (u,v)∈Qsuch thatΦ(u,v)=(u,v), that is, (u,v) satisfies the system (1.1). This completes

the proof ofTheorem 2.1.

Appendix

Proof ofLemma 3.1. Let

F(x,u)=f(x)uα_+g(x)u−β_+h(x)uγ_,

G(t,u)=f∗(t)uα+g∗(t)u−β+h∗(t)uγ,

g(t,u)=f∗(t)uα+g∗(t)u−β+h∗(t)uγ,

(A.1)

theng(|x|,u)≤F(x,u)≤G(|x|,u),x∈RN_{,u >0. It follows from (T) that}

0<

+∞

0 s

N−1_{gs,φ(s)ds <} +∞

0 s

N−1_{Gs,φ(s)ds <+∞. (A.2)}

Then we define two functions by

y(t)=᏶Gt,φ(t), z(t)=᏶gt,φ(t), t >0, (A.3)

where᏶is the integral operator defined by

᏶[E](t)= 1

N−2 t

0 _s

t N−2

sE(s)ds+ +∞

t sE(s)ds

By the simple calculation, we have

y+N−1

t y

_{= −Gt,φ(t), z+}N−1

t z

_{= −gt,φ(t), t >0, (A.5)}

l1φ(t)≤y(t), z(t)≤l2φ(t), t >0, (A.6)

for some positive constantsl1andl2.

Takeλ=max{α,β,γ}, then for anyk≥1, ifk−1_{≤c≤k, then}

G(t,cu)≤kλ_{G(t,u), t≥0,u >0,}

g(t,cu)≥k−λ_{g(t,u), t≥0,u >0.} (A.7)

Moreover, lettingy∗(x)=k1λy(x), andk1is a number such thatl1kλ1≥1, we have

y_∗+N−1

t y

∗= −kλ1G

t,φ(t), t >0,

φ(t)≤y∗(t)≤kλ1l2φ(t), t >0,

(A.8)

by (A.5) and (A.6). Hence,

Gt,y∗=f∗(t)y∗α+g∗(t)y−∗β+h∗(t)y∗γ

≤f∗(t)l2kλ1 α

φα_+g∗_(t)φ−β_+h∗_(t)l

2kλ1 γ

φγ

≤kλ1

f∗(t)φα+g∗(t)φ−β+h∗(t)φγ=kλ1G(t,φ),

(A.9)

where we take k1 so big that k1≥l2α/λ(1−α) and k1≥lγ/λ (1−γ)

2 , that is, (l2k1λ)α≤kλ1 and (l2kλ1)γ≤kλ1.

Therefore, it follows that

y_∗+N−1

t y

∗≤ −G

t,y∗. (A.10)

Similarly, lettingz∗(x)=k−2λz(x) with constantk2−λl2≤1, we obtain

z_∗+N−1

t z

∗≥ −g

t,z∗, (A.11)

such thatz∗(t)< δfor anyt >0 and

z∗(t)≤y∗(t), t >0. (A.12)

Definey∗(0) andz∗(0) by continuity with (A.3), and letv(x)=y∗(|x|) andw(x)= z∗(|x|) forx∈RN, then from (A.10) and (A.11)vandware, respectively, a supersolution

and a subsolution of (3.1), withv(x)≥w(x) satisfied. Therefore, by super-subsolution principle, (3.1) has a positive entire solutionusuch that

For proving the uniqueness of such solutions, we suppose thatvandware two positive entire solutions of (3.1), then it is easily seen that the conditions (a) and (b) inLemma 3.2

are satisfied even whenvinterchanges withw.

Moreover, since|Δv|and|Δw|are, respectively, given by f(x,v) andf(x,w), and both

vandware between two constant multiples ofφ(|x|), we have, by (A.5) for somek >0,

_{Δv(x),Δw(x)≤k}λ_{Gx,φ|x|, (A.14)}

hence it follows from (A.2) that bothΔvandΔware inL1_(RN_).

Therefore, usingLemma 3.2, we havev≥was well asw≥vinRN_{, and hencev≡w.}

References

[1] R. A. Adams,Sobolev Spaces, Academic Press, New York, 1975.

[2] D. Aronson, M. G. Crandall, and L. A. Peletier,Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Analysis6(1982), no. 10, 1001–1022.

[3] A. Callegari and A. Nachman,Some singular, nonlinear differential equations arising in boundary layer theory, Journal of Mathematical Analysis and Applications64(1978), no. 1, 96–105. [4] D. G. de Figueiredo and J. Yang,Decay, symmetry and existence of solutions of semilinear elliptic

systems, Nonlinear Analysis, Theory, Methods & Applications33(1998), no. 3, 211–234. [5] P. de Mottoni, A. Schiaffino, and A. Tesei,Attractivity properties of nonnegative solutions for a

class of nonlinear degenerate parabolic problems, Annali di Matematica Pura ed Applicata, Serie Quarta136(1984), 35–48.

[6] A. L. Edelson,Entire solutions of singular elliptic equations, Journal of Mathematical Analysis and Applications139(1989), no. 2, 523–532.

[7] N. Fukagai,On decaying entire solutions of second order sublinear elliptic equations, Hiroshima Mathematical Journal14(1985), no. 3, 551–562.

[8] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd ed., Fundamental Principles of Mathematical Sciences, vol. 224, Springer, Berlin, 1983.

[9] M. E. Gurtin and R. C. MacCamy,On the diffusion of biological populations, Mathematical Bio-sciences33(1977), no. 1-2, 35–49.

[10] N. Kawano,On bounded entire solutions of semilinear elliptic equations, Hiroshima Mathematical Journal14(1984), no. 1, 125–158.

[11] N. Kawano and T. Kusano,On positive entire solutions of a class of second order semilinear elliptic systems, Mathematische Zeitschrift186(1984), no. 3, 287–297.

[12] O. A. Ladyzhenskaya and N. N. Ural’tseva,Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

[13] A. Nachman and A. Callegari,A nonlinear singular boundary value problem in the theory of pseu-doplastic fluids, SIAM Journal on Applied Mathematics38(1980), no. 2, 275–281.

[14] M. Naito,A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Mathematical Journal14(1984), no. 1, 211–214.

[15] A. Okubo,Diffusion and Ecological Problems: Mathematical Models, Biomathematics, vol. 10, Springer, Berlin, 1980.

[17] J. Shi and M. Yao,On a singular nonlinear semilinear elliptic problem, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics128(1998), no. 6, 1389–1401.

[18] C. A. Swanson,Extremal positive solutions of semilinear Schr¨odinger equations, Canadian Mathe-matical Bulletin26(1983), no. 2, 171–178.

[19] T. Teramoto,Existence and nonexistence of positive entire solutions of second order semilinear el-liptic systems, Funkcialaj Ekvacioj42(1999), no. 2, 241–260.

[20] L. Z. Wang and A. X. Huang,Existence of positive solutions of a kind of reaction-diffusion system, Journal of Xi’an Jiaotong University36(2002), no. 2, 211–213.

[21] C. S. Yarur,Existence of continuous and singular ground states for semilinear elliptic systems, Elec-tronic Journal of Differential Equations1998(1998), no. 1, 1–27.

Lingyun Qiu: Department of Mathematics, Tianjin University, Tianjin 300072, China

Current address: Liu Hui Center for Applied Mathematics, Nankai University and Tianjin University, Tianjin 300072, China

E-mail address:[email protected]

Miaoxin Yao: Department of Mathematics, Tianjin University, Tianjin 300072, China

Current address: Liu Hui Center for Applied Mathematics, Nankai University and Tianjin University, Tianjin 300072, China