Nonexistence and Radial Symmetry of Positive Solutions of Semilinear Elliptic Systems

Abstract

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

1. Introduction

In this paper we consider the more general semilinear elliptic system

−Δ𝑢=𝑘₁𝑢^𝑝1+𝑘₂𝑣^𝑝2+𝑘₃𝑢^𝑝3𝑣^𝑝4,−Δ𝑣=𝑙₁𝑢^𝑞1+𝑙₂𝑣^𝑞2+𝑙₃𝑢^𝑞3𝑣^𝑞4,inℝ^𝑁(𝑁≥3), (1.1) where 𝑘_𝑖 and 𝑙_𝑖 (𝑖=1,2,3) are nonnegative constants. The question is to determine for which values of the exponents 𝑝_𝑖 and 𝑞_𝑖 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(ℝ^𝑁). In the case of the Emden-Fowler equation

Δ𝑢+𝑢^𝑘=0,𝑢≥0inℝ^𝑁. (1.2) 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

𝑐𝑢(𝑥)=(𝑑+|𝑥−𝑥|²)^(𝑁−2)/2, (1.3) where 𝑐=[𝑁(𝑁−2)𝑑]^1/2 with 𝑑>0 and 𝑥∈ℝ^𝑁. 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(ℝ^𝑁) 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 that 𝑘_𝑖,𝑙_𝑖≥0(𝑖=1,2,3), but 𝑘_𝑖 and 𝑙_𝑖 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(ℝ^𝑁).

Theorem 1.2. Suppose that 𝑘_𝑖,𝑙_𝑖>0(𝑖=1,2,3), 𝑝_𝑗=𝑞_𝑗=(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∈ℝ^𝑁, 𝑐𝑢(𝑥)=(¹𝑑+|𝑥−𝑥|²)^(𝑁−2)/2𝑐,𝑣(𝑥)=(²𝑑+|𝑥−𝑥|²)^(𝑁−2)/2, (1.4) where 𝑐₁,𝑐₂>0 and satisfy the following equalities: 𝑁(𝑁−2)𝑑𝑐₁^(𝑁−2)/2=𝑘₁𝑐₁^(𝑁+2)/2+𝑘₂𝑐₂^(𝑁+2)/2+𝑘₃𝑐^{((𝑁−2)/2)𝑝3}₁𝑐^{((𝑁−2)/2)𝑝4}₂,𝑁(𝑁−2)𝑑𝑐₁^(𝑁−2)/2=𝑙₁𝑐₁^(𝑁+2)/2+𝑙₂𝑐₂^(𝑁+2)/2+𝑙₃𝑐^{((𝑁−2)/2)𝑞3}₁𝑐^{((𝑁−2)/2)𝑞4}₂. (1.5)

Remark 1.3. Obviously Theorem 1.1 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(ℝ^𝑁).

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 Theorems 1.1 and 1.2, we will use the method of moving spheres. We first prove a number of lemmas as follows. For 𝑥∈ℝ^𝑁 and 𝜆>0, let us introduce the Kelvin transforms

𝑢_𝑥,𝜆𝜆(𝑦)=^𝑁−2|𝑦−𝑥|^𝑁−2𝜆𝑢(𝑥+²(𝑦−𝑥)|𝑦−𝑥|²),𝑣_𝑥,𝜆𝜆(𝑦)=^𝑁−2|𝑦−𝑥|^𝑁−2𝜆𝑣(𝑥+²(𝑦−𝑥)|𝑦−𝑥|²), (2.1) which are defined for 𝑦∈ℝ^𝑁⧵{𝑥}. For any 𝑦∈ℝ^𝑁⧵{𝑥}, one verifies that 𝑢_𝑥,𝜆 and 𝑣_𝑥,𝜆 satisfy the system

−Δ𝑢_𝑥,𝜆=𝑘₁(𝜆)|𝑦−𝑥|^{𝑁+2−𝑝1(𝑁−2)}𝑢^𝑝1_𝑥,𝜆+𝑘₂(𝜆)|𝑦−𝑥|^{𝑁+2−𝑝2(𝑁−2)}𝑣^𝑝2_𝑥,𝜆+𝑘₃(𝜆)|𝑦−𝑥|^{𝑁+2−(𝑝3+𝑝4)(𝑁−2)}𝑢^𝑝3_𝑥,𝜆𝑣^𝑝4_𝑥,𝜆,−Δ𝑣_𝑥,𝜆=𝑙₁(𝜆)|𝑦−𝑥|^{𝑁+2−𝑞1(𝑁−2)}𝑢^𝑞1_𝑥,𝜆+𝑙₂(𝜆)|𝑦−𝑥|^{𝑁+2−𝑞2(𝑁−2)}𝑣^𝑞2_𝑥,𝜆+𝑙₃(𝜆)|𝑦−𝑥|^{𝑁+2−(𝑞3+𝑞4)(𝑁−2)}𝑢^𝑞3_𝑥,𝜆𝑣^𝑞4_𝑥,𝜆. (2.2) Our first lemma says that the method of moving spheres can get started.

Lemma 2.1. For every 𝑥∈ℝ^𝑁, there exists 𝜆₀(𝑥)>0 such that 𝑢_𝑥,𝜆(𝑦)≤𝑢(𝑦) and 𝑣_𝑥,𝜆(𝑦)≤𝑣(𝑦), for all 0<𝜆<𝜆₀(𝑥) and |𝑦−𝑥|≥𝜆.

Proof. Without loss of generality we may take 𝑥=0. We use 𝑢_𝜆 and 𝑣_𝜆 to denote 𝑢_0,𝜆, and 𝑣_0,𝜆, respectively. Clearly, there exists 𝑟₀>0 such that 𝑑𝑑𝑟(𝑟^(𝑁−2)/2𝑢(𝑟,𝜃))>0,∀0<𝑟<𝑟₀,𝜃∈𝕊^𝑁−1. (2.3) Consequently, 𝑢_𝜆(𝑦)≤𝑢(𝑦),∀0<𝜆≤|𝑦|<𝑟₀. (2.4) By the superharmonicity of 𝑢 and the maximum principle (see [4, Corollary 1.1]), 𝑢(𝑦)≥(min_{𝜕𝐵𝑟0}𝑢)𝑟₀^𝑁−2|𝑦|^2−𝑁,∀|𝑦|≥𝑟₀. (2.5) Let 𝜆₀=𝑟₀(min_{𝜕𝐵𝑟0}𝑢min_𝐵𝑟0𝑢)^1/(𝑁−2)≤𝑟₀. (2.6) Then for every 0<𝜆<𝜆₀, and |𝑦|≥𝑟₀, we have 𝑢_𝜆𝜆(𝑦)≤^0𝑁−2|𝑦|^𝑁−2max_𝐵𝑟0𝑟𝑢≤^0𝑁−2min_{𝜕𝐵𝑟0}𝑢|𝑦|^𝑁−2. (2.7) It follows from (2.4), (2.5), and (2.7) that for every 0<𝜆<𝜆₀, 𝑢_𝜆(𝑦)≤𝑢(𝑦),|𝑦|≥𝜆. (2.8) Similarly, there exists 𝜆₀>0 such that for every 0<𝜆<𝜆₀, we obtain 𝑣_𝜆(𝑦)≤𝑣(𝑦),|𝑦|≥𝜆. (2.9) We can choose 𝜆₀=min{𝜆₀,𝜆₀}.

Set, for 𝑥∈ℝ^𝑁,

𝜆_𝑢(𝑥)=sup{𝜇>0∣𝑢_𝑥,𝜆(𝑦)≤𝑢(𝑦),∀|𝑦−𝑥|≥𝜆,0<𝜆≤𝜇},𝜆_𝑣(𝑥)=sup{𝜇>0∣𝑣_𝑥,𝜆(𝑦)≤𝑣(𝑦),∀|𝑦−𝑥|≥𝜆,0<𝜆≤𝜇}. (2.10) By Lemma 2.1, 𝜆_𝑢(𝑥) and 𝜆_𝑣(𝑥) are well defined and 0<𝜆_𝑢(𝑥),𝜆_𝑣(𝑥)≤∞ for 𝑥∈ℝ^𝑁. Let 𝜆=min{𝜆_𝑢,𝜆_𝑣}, then we have the following

Lemma 2.2. If 𝜆(𝑥)<∞ for some 𝑥∈ℝ^𝑁, then 𝑢_{𝑥,𝜆(𝑥)}≡𝑢 and 𝑣_{𝑥,𝜆(𝑥)}≡𝑣 on ℝ^𝑁⧵{𝑥}.

Lemma 2.3. If 𝜆(𝑥)=∞ for some 𝑥∈ℝ^𝑁, then 𝜆(𝑥)=∞ for all 𝑥∈ℝ^𝑁.

Proof of Lemma 2.2. Without loss of generality, we assume that 𝜆=𝜆_𝑢 and take 𝑥=0 and let 𝜆=𝜆(0),𝑢_𝜆=𝑢_0,𝜆 and 𝑣_𝜆=𝑣_0,𝜆, and Σ_𝜆={𝑦;|𝑦|>𝜆}. We wish to show 𝑢_𝜆≡𝑢 and 𝑣_𝜆≡𝑣 in ℝ^𝑁⧵{0}. Clearly, it suffices to show 𝑢_𝜆≡𝑢,𝑣_𝜆≡𝑣onΣ_𝜆. (2.11) We first prove 𝑢_𝜆≡𝑢. We know from the definition of 𝜆 that 𝑢_𝜆≤𝑢,𝑣_𝜆≤𝑣onΣ_𝜆. (2.12) In view of (1.1), a simple calculation yields −Δ𝑢_𝜆=𝑘₁(𝜆)|𝑦|^{𝑁+2−𝑝1(𝑁−2)}𝑢^𝑝1_𝜆+𝑘₂(𝜆)|𝑦|^{𝑁+2−𝑝2(𝑁−2)}𝑣^𝑝2_𝜆+𝑘₃(𝜆)|𝑦|^{𝑁+2−(𝑝3+𝑝4)(𝑁−2)}𝑢^𝑝3_𝜆𝑣^𝑝4_𝜆,𝜆>0. (2.13) Therefore, −Δ(𝑢−𝑢_𝜆)=𝑘₁𝑢^𝑝1+𝑘₂𝑣^𝑝2+𝑘₃𝑢^𝑝3𝑣^𝑝4−𝑘₁(𝜆)|𝑦|^{𝑁+2−𝑝1(𝑁−2)}𝑢^𝑝1_𝜆−𝑘₂(𝜆)|𝑦|^{𝑁+2−𝑝2(𝑁−2)}𝑣^𝑝2_𝜆−𝑘₃(𝜆)|𝑦|^{𝑁+2−(𝑝3+𝑝4)(𝑁−2)}𝑢^𝑝3_𝜆𝑣^𝑝4_𝜆≥𝑘₁(𝜆)|𝑦|^{𝑁+2−𝑝1(𝑁−2)}(𝑢^𝑝1−𝑢^𝑝1_𝜆)+𝑘₂(𝜆)|𝑦|^{𝑁+2−𝑝2(𝑁−2)}(𝑣^𝑝2−𝑣^𝑝2_𝜆)+𝑘₃(𝜆)|𝑦|^{𝑁+2−(𝑝3+𝑝4)(𝑁−2)}(𝑢^𝑝3𝑣^𝑝4−𝑢^𝑝3_𝜆𝑣^𝑝4_𝜆)≥0onΣ_𝜆. (2.14) If 𝑢−𝑢_𝜆≡0 on Σ_𝜆, we stop. Otherwise, by the Hopf lemma and the compactness of 𝜕𝐵_𝜆, we have 𝑑𝑑𝑟(𝑢−𝑢_𝜆)|_𝜕𝐵𝜆≥𝐶>0. (2.15) By the continuity of ∇𝑢, there exists 𝑅>𝜆 such that 𝑑𝑑𝑟(𝑢−𝑢_𝜆𝐶)≥2>0,for𝜆≤𝜆≤𝑅,𝜆≤𝑟≤𝑅. (2.16) Consequently, since 𝑢−𝑢_𝜆=0 on 𝜕𝐵_𝜆, we have 𝑢(𝑦)−𝑢_𝜆(𝑦)>0,for𝜆≤𝜆<𝑅,𝜆<|𝑦|≤𝑅. (2.17)
Set 𝑐=min_𝜕𝐵𝑅(𝑢−𝑢_𝜆)>0. It follows from the superharmonicity of 𝑢−𝑢_𝜆 that
𝑢−𝑢_𝜆≥𝑐𝑅^𝑁−2|𝑦|^𝑁−2,∀|𝑦|≥𝑅. (2.18) Therefore, 𝑢(𝑦)−𝑢_𝜆(𝑦)≥𝑐𝑅^𝑁−2|𝑦|^𝑁−2−(𝑢_𝜆(𝑦)−𝑢_𝜆(𝑦)),∀|𝑦|≥𝑅. (2.19)
By the uniform continuity of 𝑢 on 𝐵_𝑅, there exists 0<𝜖<𝑅−𝜆 such that for all 𝜆≤𝜆≤𝜆+𝜖,
|𝜆^𝑁−2𝜆𝑢(²𝑦|𝑦|²)−𝜆^𝑁−2𝑢(𝜆²𝑦|𝑦|²)|<𝑐𝑅^𝑁−22,∀|𝑦|≥𝑅. (2.20) It follows from (2.19) and the above inequality that 𝑢(𝑦)−𝑢_𝜆(𝑦)>0,for𝜆≤𝜆≤𝜆+𝜖,|𝑦|≥𝑅. (2.21) Estimates (2.17) and (2.21) violate the definition of 𝜆.
From 𝑢_𝜆≡𝑢 and (2.14), we easily know that 𝑣_𝜆≡𝑣 in Σ_𝜆. Lemma 2.2 is proved.

Proof of Lemma 2.3. Since 𝜆(𝑥)=∞, we have 𝑢_𝑥,𝜆(𝑦)≤𝑢(𝑦),𝑣_𝑥,𝜆(𝑦)≤𝑣(𝑦),∀𝜆>0,|𝑦−𝑥|≥𝜆. (2.22) It follows that lim_|𝑦|→∞|𝑦|^𝑁−2𝑢(𝑦)=∞. (2.23) On the other hand, if 𝜆(𝑥)<∞ for some 𝑥∈ℝ^𝑁, then, by Lemma 2.2, lim_|𝑦|→∞|𝑦|^𝑁−2𝑢(𝑦)=lim_|𝑦|→∞|𝑦|^𝑁−2𝑢_{𝑥,𝜆(𝑥)}(𝑦)=𝜆^𝑁−2(𝑥)𝑢(𝑥)<∞, (2.24) which is a contradiction. Similarly, we also obtain a contradiction for 𝑣.

3. Proofs of Theorems 1.1 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]). Let 𝑓∈𝐶¹(ℝ^𝑁),𝑁≥1,𝜈>0. Suppose that for every 𝑥∈ℝ^𝑁, there exists 𝜆(𝑥)>0 such that (𝜆(𝑥))|𝑦−𝑥|^𝜈𝜆𝑓(𝑥+²(𝑥)(𝑦−𝑥)|𝑦−𝑥|²)=𝑓(𝑦),𝑦∈ℝ^𝑁⧵{𝑥}, (3.1) Then for some 𝑐≥0,𝑑>0,𝑥∈ℝ^𝑁,𝑐𝑓(𝑥)=±(𝑑+|𝑥−𝑥|²)^𝜈/2. (3.2)

Lemma 3.2 (See [8, Lemma 11.2]). Let 𝑓∈𝐶¹(ℝ^𝑁),𝑁≥1,𝜈>0. Assume that (𝜆)|𝑦−𝑥|^𝜈𝜆𝑓(𝑥+²(𝑦−𝑥)|𝑦−𝑥|²)≤𝑓(𝑦),∀𝜆>0,𝑥∈ℝ^𝑁,|𝑦−𝑥|≥𝜆, (3.3) Then 𝑓≡𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.

Proof of Theorem 1.1. We first claim that 𝜆(𝑥)=∞ for all 𝑥∈ℝ^𝑁. We prove it by contradiction argument. If 𝜆(𝑥)<∞ for some 𝑥, then by Lemma 2.2, 𝑢_{𝑥,𝜆(𝑥)}≡𝑢 and 𝑣_{𝑥,𝜆(𝑥)}≡𝑣 on ℝ^𝑁⧵{𝑥}. But looking at equations in system (2.2) we realize that this is impossible. Therefore, 𝑢_𝑥,𝜆(𝑦)≤𝑢(𝑦),𝑣_𝑥,𝜆(𝑦)≤𝑣(𝑦),∀𝜆>0,𝑥∈ℝ^𝑁,|𝑦−𝑥|≥𝜆. (3.4) This, by Lemma 3.2, implies that 𝑢,𝑣≡𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. From system (1.1) we know that it is also impossible.

Proof of Theorem 1.2. We first claim that 𝜆(𝑥)<∞ for all 𝑥∈ℝ^𝑁. We prove it by contradiction argument. If 𝜆(𝑥)=∞ for some 𝑥, then by Lemma 2.3, 𝜆(𝑥)=∞ for all 𝑥, that is, 𝑢_𝑥,𝜆(𝑦)≤𝑢(𝑦),𝑣_𝑥,𝜆(𝑦)≤𝑣(𝑦),∀𝜆>0,𝑥∈ℝ^𝑁,|𝑦−𝑥|≥𝜆. (3.5) 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∈ℝ^𝑁, there exists 𝜆(𝑥)>0 such that 𝑢_{𝑥,𝜆(𝑥)}≡𝑢 and 𝑣_{𝑥,𝜆(𝑥)}≡𝑣. Then by Lemma 3.1, for some 𝑐_𝑖,𝑑>0(𝑖=1,2) and some 𝑥∈ℝ^𝑁, 𝑐𝑢(𝑥)≡(¹𝑑+|𝑥−𝑥|²)^(𝑁−2)/2𝑐,𝑣(𝑥)≡(²𝑑+|𝑥−𝑥|²)^(𝑁−2)/2. (3.6) Theorem 1.2 follows from the above and the fact that (𝑢,𝑣) is a solution of (1.1).

Acknowledgment

This paper is supported by Youth Foundation of NSFC (no. 10701061).