# Nonexistence and Radial Symmetry of Positive Solutions of Semilinear Elliptic Systems

Authors: Zhengce Zhang; Liping Zhu
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

Nonexistence and radial symmetry of positive solutions for a class of semilinear elliptic systems are considered via the method of moving spheres.


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## Body

## 1. Introduction

In this paper we consider the more general semilinear elliptic system(1.1)-Δu=k1up1+k2vp2+k3up3vp4,-Δv=l1uq1+l2vq2+l3uq3vq4,inℝN(N≥3),
where ki and li (i=1,2,3) are nonnegative constants. The question is to determine for which values of the exponents pi and qi the only nonnegative solution (u,v) of (1.1) is (u,v)=(0,0). The solution here is taken in the classical sense, that is, u,v∈C2(ℝN). In the case of the Emden-Fowler equation(1.2)Δu+uk=0,u≥0inℝN.
When 1≤k<(N+2)/(N-2)(N≥3), it has been proved in [1] that the only solution of (1.2) is u=0. In dimension N=2, a similar conclusion holds for 0≤k<∞. It is also well known that in the critical case, k=(N+2)/(N-2), problem (1.2) has a two-parameter family of solutions given by(1.3)u(x)=(cd+|x-x¯|2)(N-2)/2,
where c=[N(N-2)d]1/2 with d>0 and x¯∈ℝN. If k1=k2=l1=l2=0,k3,l3>0, p3,q4>1,p4,q3≥0 and min{p3+2p4,q4+2q3}≤(N+2)/(N-2), using Pokhozhaev's second identity, Chen and Lu ([2, Theorem 2]) have proved that problem (1.1) has no positive radial solutions with u(x)=u(|x|). Suppose that p3,p4,q3, and q4 satisfy 0≤p3,q4≤1,p4,q3>1 and other related conditions, using the method of integral relations, Mitidieri ([3, Theorem 1]) has proved that problem (1.1) has no positive solutions of C2(ℝN) with k3=l3=1. In present paper, we study problem (1.1) by virtue of the method of moving spheres and obtain the following theorems of nonexistence and radial symmetry of positive solutions.Theorem 1.1.
Suppose thatki,li≥0(i=1,2,3), but ki and li are not equal to zero at the same time. Moreover, max{p1,p2,p3+p4},max{q1,q2,q3+q4}≤(N+2)/(N-2) with p1,p3,q2,q4≥0,p2,p4,q1,q3>0, but p1,p2,p3+p4 and q1,q2,q3+q4 are not both equal to (N+2)/(N-2), then Problem (1.1) has no positive solution of C2(ℝN).Theorem 1.2.
Suppose thatki,li>0(i=1,2,3), pj=qj=(N+2)/(N-2)(j=1,2), and p3+p4=q3+q4=(N-2)/(N+2), then the positive C2 solution of (1.1) is of the form (1.3), that is, for some d>0,x¯∈ℝN,
(1.4)u(x)=(c1d+|x-x¯|2)(N-2)/2,v(x)=(c2d+|x-x¯|2)(N-2)/2,
where c1,c2>0 and satisfy the following equalities:
(1.5)N(N-2)dc1(N-2)/2=k1c1(N+2)/2+k2c2(N+2)/2+k3c1((N-2)/2)p3c2((N-2)/2)p4,N(N-2)dc1(N-2)/2=l1c1(N+2)/2+l2c2(N+2)/2+l3c1((N-2)/2)q3c2((N-2)/2)q4.Remark 1.3.
Obviously Theorem1.1 contains new region of k,t,p, and q which can not be covered by [2, Theorem 2] and [3, Theorem 1]. Moreover, Theorem 1.2 gives the exact forms of positive solutions of C2(ℝN).There are some related works about problem (1.1). For k2=l1=1 and k1=k3=l2=l3=0, Figueiredo and Felmer (see [4]) proved Theorem 1.1 using the moving plane method and a special form of the maximum principle for elliptic systems. Busca and Manásevich obtained a new result (see [5, Theorem 2.1]) using the same method as in [4]. It allows p2 and q1 to reach regions where one of the two exponents is supercritical. In [6], Zhang et al. first introduced the Kelvin transforms and gave a different proof of Theorem 1.1 in [4] using the method of moving spheres. This approach was suggested in [7], while Li and Zhang who had made significant simplifications prove some Liouville theorems for a single equation in [8]. In this paper, we consider the general case of nonlinearities and do not need the maximum principle for elliptic systems. Moreover, the exact form of positive solution is proved in Theorem 1.2. If we can find a proper transforms instead of the Kelvin transforms, we suspect that [5, Theorem 2.1] can also be proved via the method of moving spheres. We leave this to the interested readers.Let us emphasize that considerable attention has been drawn to Liouville-type results and existence of positive solutions for general nonlinear elliptic equations and systems, and that numerous related works are devoted to some of its variants, such as more general quasilinear operators and domains. We refer the interested reader to [9–15], and some of the references therein. We refer the interested reader to [16, 17].


## 2. Preliminaries and Moving Spheres

To prove Theorems1.1 and 1.2, we will use the method of moving spheres. We first prove a number of lemmas as follows. For x∈ℝN and λ>0, let us introduce the Kelvin transforms(2.1)ux,λ(y)=λN-2|y-x|N-2u(x+λ2(y-x)|y-x|2),vx,λ(y)=λN-2|y-x|N-2v(x+λ2(y-x)|y-x|2),
which are defined for y∈ℝN∖{x}. For any y∈ℝN∖{x}, one verifies that ux,λ and vx,λ satisfy the system(2.2)-Δux,λ=k1(λ|y-x|)N+2-p1(N-2)ux,λp1+k2(λ|y-x|)N+2-p2(N-2)vx,λp2+k3(λ|y-x|)N+2-(p3+p4)(N-2)ux,λp3vx,λp4,-Δvx,λ=l1(λ|y-x|)N+2-q1(N-2)ux,λq1+l2(λ|y-x|)N+2-q2(N-2)vx,λq2+l3(λ|y-x|)N+2-(q3+q4)(N-2)ux,λq3vx,λq4.
Our first lemma says that the method of moving spheres can get started.Lemma 2.1.
For everyx∈ℝN, there exists λ0(x)>0 such that ux,λ(y)≤u(y) and vx,λ(y)≤v(y), for all 0<λ<λ0(x) and |y-x|≥λ.Proof.
Without loss of generality we may takex=0. We use uλ and vλ to denote u0,λ, and v0,λ, respectively. Clearly, there exists r0>0 such that
(2.3)ddr(r(N-2)/2u(r,θ))>0,∀0<r<r0,θ∈𝕊N-1.
Consequently,
(2.4)uλ(y)≤u(y),∀0<λ≤|y|<r0.
By the superharmonicity of u and the maximum principle (see [4, Corollary 1.1]),
(2.5)u(y)≥(min∂Br0u)r0N-2|y|2-N,∀|y|≥r0.
Let
(2.6)λ̂0=r0(min∂Br0uminB¯r0u)1/(N-2)≤r0.
Then for every 0<λ<λ̂0, and |y|≥r0, we have
(2.7)uλ(y)≤λ̂0N-2|y|N-2maxB¯r0u≤r0N-2min∂Br0u|y|N-2.
It follows from (2.4), (2.5), and (2.7) that for every 0<λ<λ̂0,
(2.8)uλ(y)≤u(y),|y|≥λ.
Similarly, there exists λ̃0>0 such that for every 0<λ<λ̃0, we obtain
(2.9)vλ(y)≤v(y),|y|≥λ.
We can choose λ0=min{λ̂0,λ̃0}.Set, forx∈ℝN,(2.10)λ¯u(x)=sup{μ>0∣ux,λ(y)≤u(y),∀|y-x|≥λ,0<λ≤μ},λ¯v(x)=sup{μ>0∣vx,λ(y)≤v(y),∀|y-x|≥λ,0<λ≤μ}.
By Lemma 2.1, λ¯u(x) and λ¯v(x) are well defined and 0<λ¯u(x),λ¯v(x)≤∞ for x∈ℝN. Let λ¯=min{λ¯u,λ¯v}, then we have the followingLemma 2.2.
Ifλ¯(x)<∞ for some x∈ℝN, then ux,λ¯(x)≡u and vx,λ¯(x)≡v on ℝN∖{x}.Lemma 2.3.
Ifλ¯(x¯)=∞ for some x¯∈ℝN, then λ¯(x)=∞ for all x∈ℝN.Proof of Lemma2.2.
Without loss of generality, we assume thatλ¯=λ¯u and take x=0 and let λ¯=λ¯(0),uλ=u0,λ and vλ=v0,λ, and Σλ={y;|y|>λ}. We wish to show uλ¯≡u and vλ¯≡v in ℝN∖{0}. Clearly, it suffices to show
(2.11)uλ¯≡u,vλ¯≡vonΣλ¯.
We first prove uλ¯≡u. We know from the definition of λ¯ that
(2.12)uλ¯≤u,vλ¯≤vonΣλ¯.
In view of (1.1), a simple calculation yields
(2.13)-Δuλ=k1(λ|y|)N+2-p1(N-2)uλp1+k2(λ|y|)N+2-p2(N-2)vλp2+k3(λ|y|)N+2-(p3+p4)(N-2)uλp3vλp4,λ>0.
Therefore,
(2.14)-Δ(u-uλ¯)=k1up1+k2vp2+k3up3vp4-k1(λ¯|y|)N+2-p1(N-2)uλ¯p1-k2(λ¯|y|)N+2-p2(N-2)vλ¯p2-k3(λ¯|y|)N+2-(p3+p4)(N-2)uλ¯p3vλ¯p4≥k1(λ¯|y|)N+2-p1(N-2)(up1-uλ¯p1)+k2(λ¯|y|)N+2-p2(N-2)(vp2-vλ¯p2)+k3(λ¯|y|)N+2-(p3+p4)(N-2)(up3vp4-uλ¯p3vλ¯p4)≥0onΣλ¯.
If u-uλ¯≡0 on Σλ¯, we stop. Otherwise, by the Hopf lemma and the compactness of ∂Bλ¯, we have
(2.15)ddr(u-uλ¯)|∂Bλ¯≥C>0.
By the continuity of ∇u, there exists R>λ¯ such that
(2.16)ddr(u-uλ)≥C2>0,forλ¯≤λ≤R,λ≤r≤R.
Consequently, since u-uλ=0 on ∂Bλ, we have
(2.17)u(y)-uλ(y)>0,forλ¯≤λ<R,λ<|y|≤R.
Setc=min∂BR(u-uλ¯)>0. It follows from the superharmonicity of u-uλ¯ that
(2.18)u-uλ¯≥cRN-2|y|N-2,∀|y|≥R.
Therefore,
(2.19)u(y)-uλ(y)≥cRN-2|y|N-2-(uλ(y)-uλ¯(y)),∀|y|≥R.
By the uniform continuity ofu on B¯R, there exists 0<ϵ<R-λ¯ such that for all λ¯≤λ≤λ¯+ϵ,
(2.20)|λN-2u(λ2y|y|2)-λ¯N-2u(λ¯2y|y|2)|<cRN-22,∀|y|≥R.
It follows from (2.19) and the above inequality that
(2.21)u(y)-uλ(y)>0,forλ¯≤λ≤λ¯+ϵ,|y|≥R.
Estimates (2.17) and (2.21) violate the definition of λ¯.
Fromuλ¯≡u and (2.14), we easily know that vλ¯≡v in Σλ¯. Lemma 2.2 is proved.Proof of Lemma2.3.
Sinceλ¯(x¯)=∞, we have
(2.22)ux¯,λ(y)≤u(y),vx¯,λ(y)≤v(y),∀λ>0,|y-x¯|≥λ.
It follows that
(2.23)lim|y|→∞|y|N-2u(y)=∞.
On the other hand, if λ¯(x)<∞ for some x∈ℝN, then, by Lemma 2.2,
(2.24)lim|y|→∞|y|N-2u(y)=lim|y|→∞|y|N-2ux,λ¯(x)(y)=λ¯N-2(x)u(x)<∞,
which is a contradiction. Similarly, we also obtain a contradiction for v.


## 3. Proofs of Theorems1.1 and 1.2

In this section we first present two calculus lemmas taken from [8] (see also [7]).Lemma 3.1 (See [8, Lemma 11.1]).
Letf∈C1(ℝN),N≥1,ν>0. Suppose that for every x∈ℝN, there exists λ(x)>0 such that
(3.1)(λ(x)|y-x|)νf(x+λ2(x)(y-x)|y-x|2)=f(y),y∈ℝN∖{x},
Then for some c≥0,d>0,x¯∈ℝN,(3.2)f(x)=±(cd+|x-x¯|2)ν/2.Lemma 3.2 (See [8, Lemma 11.2]).
Letf∈C1(ℝN),N≥1,ν>0. Assume that
(3.3)(λ|y-x|)νf(x+λ2(y-x)|y-x|2)≤f(y),∀λ>0,x∈ℝN,|y-x|≥λ,
Then f≡constant.Proof of Theorem1.1.
We first claim thatλ¯(x)=∞ for all x∈ℝN. We prove it by contradiction argument. If λ¯(x¯)<∞ for some x¯, then by Lemma 2.2, ux¯,λ¯(x¯)≡u and vx¯,λ¯(x¯)≡v on ℝN∖{x¯}. But looking at equations in system (2.2) we realize that this is impossible. Therefore,
(3.4)ux,λ(y)≤u(y),vx,λ(y)≤v(y),∀λ>0,x∈ℝN,|y-x|≥λ.
This, by Lemma 3.2, implies that u,v≡constant. From system (1.1) we know that it is also impossible.Proof of Theorem1.2.
We first claim thatλ¯(x)<∞ for all x∈ℝN. We prove it by contradiction argument. If λ¯(x¯)=∞ for some x¯, then by Lemma 2.3, λ¯(x)=∞ for all x, that is,
(3.5)ux,λ(y)≤u(y),vx,λ(y)≤v(y),∀λ>0,x∈ℝN,|y-x|≥λ.
This, by Lemma 3.2, implies that u,v≡constant, a contradiction to (1.1). Therefore, it follows from Lemma 2.2 that for every x∈ℝN, there exists λ¯(x)>0 such that ux,λ¯(x)≡u and vx,λ¯(x)≡v. Then by Lemma 3.1, for some ci,d>0(i=1,2) and some x¯∈ℝN,
(3.6)u(x)≡(c1d+|x-x¯|2)(N-2)/2,v(x)≡(c2d+|x-x¯|2)(N-2)/2.
Theorem 1.2 follows from the above and the fact that (u,v) is a solution of (1.1).



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*Source: 629749-2009-10-13.xml*