Necessary Conditions for Existence Results of Some Integral System

Abstract

In this paper, we give some necessary conditions for the existence of positive solutions for integral systems.

1. Introduction

In this paper, we study the necessary condition for the existence of positive solutions for the following integral system: 𝑢(𝑥)=_ℝ𝑁𝑥−𝑦^𝜆𝑣𝑦^𝑝𝑑𝑦inℝ^𝑁,𝑣(𝑥)=_ℝ𝑁𝑥−𝑦^𝜆𝑢𝑦^𝑞𝑑𝑦inℝ^𝑁, (1) where 𝜆, 𝑝,  and  𝑞 are real parameters.

As for one single equation 𝑢(𝑥)=_ℝ𝑁𝑥−𝑦^𝜆𝑢𝑦^𝑝𝑑𝑦inℝ^𝑁, (2) there are a lot of results of this problem. If 𝜆=𝛼−𝑁 with 0<𝛼<𝑁, then problem (2) is equivalent to the following differential equation: (−Δ)^𝛼/2𝑢(𝑥)=𝑢(𝑥)^𝑝inℝ^𝑁. (3) This problem has been widely studied in the past few years. For example, in order to answer a question raised by Lieb in [1], the authors studied the symmetric property and the uniqueness of solutions for problem (2) in [2]. Later, they studied the integral system (1) in [3]. Also, after the work of [2], Li studied the general form of (2) in [4]. For the case 𝜆<0, he obtained similar results to [2] but with less regularity requirement. For the case 𝜆>0, he shows that if problem (2) has a nonnegative solution in ℝ^𝑁 and 𝜆(𝑝+1)+2𝑁≥0, then 𝑞=−1−2𝑁/𝜆. The main ingredients in these papers are the moving plane method and moving sphere method based on the maximum principle of integral forms. This method has been widely used in other works. For example, inspired by these works, the author studied the Liouville-type theorems for problems (1) and (2) with general nonlinearities in [5, 6]. For further results of this type of integral equations, see [7–18], and so forth. We note that all these results concern the cases 𝜆<0 and 𝑝>0. A natural question is whether similar results hold for 𝜆>0 or 𝑝<0. We note that the case 𝜆<0 and 𝑝>0 is quite different from the case 𝜆>0 or 𝑝<0. Generally speaking, the moving plane method or the moving sphere method does not work in the latter case, so we have to look for other methods. In a recent paper [19], the author give a sufficient and necessary condition for the existence of positive solutions for problem (2) with 𝜆>0. Based on some integral estimates, the author proved that problem (2) possesses a positive solution if and only if 𝜆𝑝=−(𝜆+2𝑁). Inspired by the work of [19], we first study the integral system (1) with 𝜆>0. Our main result is the following theorem.

Theorem 1. Suppose that 𝜆>0 and problem (1) possesses a 𝐶¹ positive solution; then 1+11+𝑝𝜆1+𝑞=−𝑁. (4)

As for 𝜆<0, we have the following nonexistence result.

Theorem 2. If −𝑁<𝜆<0, then problem (1) possesses no 𝐶¹ positive solution provided that 𝑝<0 or 𝑞<0.

This paper is organized as follows. We prove Theorem 1 in Section 2. The proof of Theorem 2 is completed in Section 3.

2. Proof of Theorem 1

We first claim that 𝑢(𝑥)∈𝐿^𝑞+1(ℝ^𝑁) and 𝑣(𝑥)∈𝐿^𝑝+1(ℝ^𝑁).

In fact, we infer from 𝑣(0)=_ℝ𝑁𝑦^𝜆𝑢𝑦^𝑞≥𝑑𝑦_{ℝ𝑁⧵𝐵1(0)}𝑦^𝜆𝑢𝑦^𝑞𝑑𝑦≥_{ℝ𝑁⧵𝐵1(0)}𝑢𝑦^𝑞𝑑𝑦 (5) that 𝑢(𝑥)∈𝐿^𝑞(ℝ^𝑁). Also, it follows from (1) that |𝑥|^𝜆𝑢𝑥|𝑥|²=_ℝ𝑁|𝑥|^𝜆𝑥|𝑥|²−𝑦^𝜆𝑣𝑦^𝑝=𝑑𝑦_ℝ𝑁𝑦^𝜆𝑦𝑥−𝑦^2𝜆𝑣^𝑝𝑦𝑑𝑦. (6) Now taking limit in (6) by letting 𝑥→0, we obtain lim_|𝑥|→0|𝑥|^𝜆𝑢𝑥|𝑥|²=_ℝ𝑁𝑣^𝑝𝑦𝑑𝑦<∞. (7) We point out that we can take the limit under the integral sign because of the dominated convergence theorem. In fact, we note that when 𝜆>0 and |𝑥|≤1, we have 𝑦𝑥−|𝑦|^2𝜆≤|𝑥|+1𝑦^𝜆≤11+𝑦^𝜆. (8) It is easy to check that |𝑦|^𝜆(1+1/|𝑦|)^𝜆𝑣^𝑝(𝑦)∈𝐿¹(ℝ^𝑁).

It follows from (7) that there exist 𝑅>0 and 𝐶>0 such that 𝐶⁻¹|𝑥|^𝜆≤𝑢(𝑥)≤𝐶|𝑥|^𝜆 (9) for |𝑥|≥𝑅. Finally, we have _{ℝ𝑁⧵𝐵𝑅(0)}𝑢^𝑞+1(𝑥)𝑑𝑥=_{ℝ𝑁⧵𝐵𝑅(0)}𝑢^𝑞𝑢𝑑𝑥≤𝐶_ℝ𝑁|𝑥|^𝜆𝑢^𝑞(0),𝑑𝑥=𝐶𝑣 (10) which further implies that 𝑢(𝑥)∈𝐿^𝑞+1(ℝ^𝑁). Similarly, we have 𝑣(𝑥)∈𝐿^𝑝+1(ℝ^𝑁).

Next, we can prove as in [19] that ∇𝑢(𝑥)=𝜆_ℝ𝑁𝑥−𝑦^𝜆−2𝑥−𝑦𝑣^𝑝𝑦(𝑥)𝑑𝑦,∇𝑣=𝜆_ℝ𝑁𝑥−𝑦^𝜆−2𝑥−𝑦𝑢^𝑞𝑦𝑑𝑦 (11) in the sense of distribution. Hence, we infer from (11) that ∇𝑢^1+𝑞(𝑥)=1+𝑞𝑢^𝑞(𝑥)𝜆⋅_ℝ𝑁𝑥−𝑦^𝜆−2𝑥−𝑦𝑣^𝑝𝑦𝑑𝑦,∇𝑣^1+𝑝(𝑥)=1+𝑝𝑣^𝑝(𝑥)𝜆⋅_ℝ𝑁𝑥−𝑦^𝜆−2𝑥−𝑦𝑢^𝑞𝑦𝑑𝑦. (12)

Now we choose a cut-off function 𝜂∈𝐶^∞([0,+∞)) satisfying 0≤𝜂≤1, 0≤|𝜂^′|≤2,  𝜂(𝑡)=1 for 𝑡≤1 and 𝜂(𝑡)=0 for 𝑡≥2. For any 𝑅>0, if we multiply (12) by 𝜂(|𝑥|/𝑅)𝑥 and integrate over ℝ^𝑁, then we get _ℝ𝑁𝜂|𝑥|𝑅𝑣^𝑝(𝑥(𝑥=)⟨𝑥,∇𝑣)⟩𝑑𝑥_ℝ𝑁𝜂|𝑥|𝑅𝑣^𝑝_ℝ𝑁𝜆𝑥−𝑦^𝜆−2𝑥,𝑥−𝑦𝑢𝑦^𝑞𝑑𝑦𝑑𝑥,_ℝ𝑁𝜂|𝑥|𝑅𝑢^𝑞(𝑥)(𝑥)=⟨𝑥,∇𝑢⟩𝑑𝑥_ℝ𝑁𝜂|𝑥|𝑅𝑢^𝑞(𝑥)_ℝ𝑁𝜆𝑥−𝑦^𝜆−2⟨𝑥,𝑥−𝑦⟩𝑣𝑦^𝑝𝑑𝑦𝑑𝑥. (13)(14) While the left-hand side of (13) equals _ℝ𝑁𝜂|𝑥|𝑅𝑣^𝑝(𝑥(𝑥=1)⟨𝑥,∇𝑣)⟩𝑑𝑥1+𝑝_ℝ^𝑁𝜂|𝑥|𝑅𝑥,∇𝑣^1+𝑝(𝑥𝑁)𝑑𝑥=−1+𝑝_ℝ^𝑁𝜂|𝑥|𝑅𝑣^1+𝑝(𝑥)−1𝑑𝑥1+𝑝_ℝ^𝑁∇𝜂|𝑥|𝑅,𝑥𝑣^1+𝑝𝑑𝑥, (15) it follows from ∇𝜂|𝑥|𝑅≤2|𝑥|,𝑥𝑅 (16) that _ℝ𝑁∇𝜂|𝑥|𝑅,𝑥𝑣^1+𝑝𝑑𝑥≤4_{𝑅≤|𝑥|≤2𝑅}𝑣^1+𝑝𝑑𝑥⟶0 (17) as 𝑅→∞ by 𝑣∈𝐿^1+𝑝(ℝ^𝑁). Thus we conclude that lim_{𝑅→∞ℝ𝑁}𝜂|𝑥|𝑅𝑣^𝑝(𝑥(𝑥𝑁)⟨𝑥,∇𝑣)⟩𝑑𝑥=−1+𝑝_ℝ^𝑁𝑣^1+𝑝(𝑥)𝑑𝑥. (18) While the right-hand side of (13) equals 𝜆_ℝ𝑁𝜂|𝑥|𝑅𝑣^𝑝(𝑥)⋅_ℝ𝑁𝑥−𝑦^𝜆−2𝑥,𝑥−𝑦𝑢𝑦^𝑞=𝜆𝑑𝑦𝑑𝑥2_ℝ^𝑁𝜂|𝑥|𝑅𝑣^𝑝(𝑥)⋅_ℝ𝑁𝑥−𝑦^𝜆−2⟨𝑥−𝑦,𝑥−𝑦⟩𝑢𝑦^𝑞+𝜆𝑑𝑦𝑑𝑥2_ℝ^𝑁𝜂|𝑥|𝑅𝑣^𝑝(𝑥)⋅_ℝ𝑁𝑥−𝑦^𝜆−2⟨𝑥+𝑦,𝑥−𝑦⟩𝑢𝑦^𝑞=𝜆𝑑𝑦𝑑𝑥2_ℝ^𝑁𝜂|𝑥|𝑅𝑣^1+𝑝(𝑥)+𝜆𝑑𝑥2_ℝ^𝑁𝜂|𝑥|𝑅𝑣^𝑝(𝑥)⋅_ℝ𝑁𝑥−𝑦^𝜆−2⟨𝑥+𝑦,𝑥−𝑦⟩𝑢𝑦^𝑞𝑑𝑦𝑑𝑥, (19) it can be checked as in [19] that _{ℝ𝑁ℝ𝑁}𝑥−𝑦^𝜆−2⟨𝑥+𝑦,𝑥−𝑦⟩𝑢𝑦^𝑞𝑣(𝑥)^𝑝𝑑𝑦𝑑𝑥<∞. (20) Hence, by letting 𝑅→∞ in (19) we get lim_𝑅→∞𝜆_ℝ𝑁𝜂|𝑥|𝑅𝑣^𝑝(𝑥)⋅_ℝ𝑁𝑥−𝑦^𝜆−2𝑥,𝑥−𝑦𝑢𝑦^𝑞=𝜆𝑑𝑦𝑑𝑥2_ℝ^𝑁𝑣^1+𝑝(𝑥)𝜆𝑑𝑥+2⋅_{ℝ𝑁ℝ𝑁}𝑥−𝑦^𝜆−2⋅𝑥+𝑦,𝑥−𝑦𝑢^𝑞𝑦𝑣^𝑝(𝑥)𝑑𝑦𝑑𝑥. (21) We infer from (13), (18), and (21) that −𝑁1+𝑝_ℝ^𝑁𝑣^1+𝑝(𝑥)=𝜆𝑑𝑥2_ℝ^𝑁𝑣^1+𝑝(𝑥)+𝜆𝑑𝑥2_ℝ^𝑁_ℝ^𝑁𝑥−𝑦^𝜆−2⋅𝑥+𝑦,𝑥−𝑦𝑢^𝑞𝑦𝑣^𝑝(𝑥)𝑑𝑦𝑑𝑥. (22) Similarly, we can prove that −𝑁1+𝑝_ℝ^𝑁𝑢^1+𝑞(𝑥)=𝜆𝑑𝑥2_ℝ^𝑁𝑢^1+𝑞(𝑥)+𝜆𝑑𝑥2_ℝ^𝑁_ℝ^𝑁𝑥−𝑦^𝜆−2⋅𝑥+𝑦,𝑥−𝑦𝑢^𝑞(𝑥)𝑣^𝑝𝑦=𝜆𝑑𝑦𝑑𝑥2_ℝ^𝑁𝑢^1+𝑞(𝑥)−𝜆𝑑𝑥2_ℝ^𝑁_ℝ^𝑁𝑥−𝑦^𝜆−2⋅𝑥+𝑦,𝑥−𝑦𝑢^𝑞𝑦𝑣^𝑝(𝑥)𝑑𝑥𝑑𝑦. (23) The above two equations imply that −𝑁1+𝑝_ℝ^𝑁𝑢^1+𝑞(𝑥)𝑁𝑑𝑥−1+𝑝_ℝ^𝑁𝑣^1+𝑝(𝑥)=𝜆𝑑𝑥2_ℝ^𝑁𝑢^1+𝑞(𝑥)𝑑𝑥+_ℝ𝑁𝑣^1+𝑝(𝑥)𝑑𝑥. (24) On the other hand, since 𝑣(𝑥)^1+𝑝=𝑣(𝑥)^𝑝_ℝ𝑁𝑥−𝑦^𝜆𝑢^𝑞𝑦𝑑𝑦, (25) we have _ℝ𝑁𝑣^1+𝑝(𝑥)=𝑑𝑥_{ℝ𝑁ℝ𝑁}𝑥−𝑦^𝜆𝑢^𝑞𝑦𝑣(𝑥)^𝑝𝑑𝑦𝑑𝑥 (26) by taking into account that 𝑣(𝑥)∈𝐿^1+𝑝(ℝ^𝑁). Similarly, we have _ℝ𝑁𝑢^1+𝑞(𝑥)=𝑑𝑥_{ℝ𝑁ℝ𝑁}𝑥−𝑦^𝜆𝑢^𝑞𝑦𝑣(𝑥)^𝑝𝑑𝑦𝑑𝑥. (27) Then it follows from (26) and (27) that _ℝ𝑁𝑢^1+𝑞(𝑥)𝑑𝑥=_ℝ𝑁𝑣^1+𝑝(𝑥)𝑑𝑥. (28) Finally, we infer from (24) and (28) that 1+11+𝑝𝜆1+𝑞=−𝑁. (29) This completes the proof of Theorem 1.

3. Proof of Theorem 2

We assume that 𝑞<0 without loss of generality. First, we note that by Lemma  3.11.3 in [20], we have, for all 𝑟>0, 1𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑢(𝑥)𝑑𝑥=_ℝ𝑁1𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑥−𝑦^𝜆𝑑𝑥𝑣^𝑝𝑦𝑑𝑦≤𝐶_ℝ𝑁𝑦^𝜆𝑣^𝑝𝑦𝑑𝑦=𝐶𝑢(0). (30) Similarly, we have 1𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑣(𝑥)𝑑𝑥≤𝐶𝑣(0) (31) for any 𝑟>0.

If we choose 𝛼=𝑞/(𝑞−1), 𝛽=1−𝑞, and  𝛿=(𝑞−1)/𝑞, then we can infer from the Holder inequality that 11=𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑢(𝑥)^−𝛼𝑢(𝑥)^𝛼≤1𝑑𝑥𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑢(𝑥)^−𝛼𝛽𝑑𝑥^1/𝛽⋅1𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑢(𝑥)^𝛼𝛿𝑑𝑥^1/𝛿=1𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑢(𝑥)^𝑞𝑑𝑥^1/(1−𝑞)⋅(𝐶𝑢(0))^{𝑞/(𝑞−1)}. (32) That is, (𝐶𝑢(0))^𝑞≤1𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑢^𝑞(𝑥)𝑑𝑥. (33) Since 𝜆<0, so if |𝑥|<𝑟, then we have 𝑟^𝜆<|𝑥|^𝜆. Multiplying both sides of (33) by 𝜔_𝑁𝑟^𝑁+𝜆, we get 𝐶^𝑞𝜔_𝑁𝑟^𝑁+𝜆𝑢(0)^𝑞≤𝜔_𝑁𝑟^𝑁+𝜆1𝜔_𝑁𝑟^𝑁_𝐵𝑟(0)𝑢^𝑞(𝑥)𝑑𝑥=𝑟^𝜆_𝐵𝑟(0)𝑢^𝑞(𝑥)≤𝑑𝑥_𝐵𝑟(0)|𝑥|^𝜆𝑢^𝑞(𝑥)𝑑𝑥=𝑣(0). (34) Since −𝑁<𝜆<0, we have 𝑁+𝜆>0. Hence the left-hand side of (34) goes to infinity as 𝑟→∞, which is a contradiction. This completes the proof.

Acknowledgments

The first author is supported by SZPP (no. KQCX20120802140634893), Guangdong S&T Major Project (no. 2012A080104014), and SRF for ROCS, SEM. Xiaohui Yu is supported by NSFC no. 11101291. The authors would like to thank the anonymous referees for their valuable suggestions.