Journal of Function Spaces
Volume 2016 (2016), Article ID 3495170, 6 pages
http://dx.doi.org/10.1155/2016/3495170

Research Article

Liouville Theorem for Some Elliptic Equations with Weights and Finite Morse Indices

Qiongli Wu,1,2 Liangcai Gan,¹ and Qingfeng Fan³

¹Electronic Information School, Wuhan University, Hubei 430071, China
²Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Hubei 430071, China
³Université de Versailles Saint-Quentin, Laboratoire Données et Algorithmes pour une Ville Intelligente et Durable (DAVID), 78035 Versailles, France

Received 11 December 2015; Revised 20 April 2016; Accepted 21 April 2016

Academic Editor: Simone Secchi

Copyright © 2016 Qiongli Wu et al. 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.

Abstract.

We establish the nonexistence of solution for the following nonlinear elliptic problem with weights: in , where is a positive parameter. Suppose that , for or , for ; we will show that this equation does not possess nontrivial bounded solution with finite Morse index.

1. Introduction

Liouville theorems are very important in proving a priori bound of solutions for elliptic equations. As far as we know, the most powerful tool for proving the a priori bound is the blow-up method. During the blow-up process, we first suppose that the solutions are unbounded; then we can scale the sequence of solutions. Finally, we get a nontrivial solution for the limit equation. On the other hand, if we can prove that the limit equation does not possess a nontrivial solution, then we get a contradiction. So the solutions must be bounded. From the above statements, it is easy to see that the most important ingredient in proving the a priori bound is the nonexistence result for the limit equation. These kinds of nonexistence results are usually called Liouville type theorems.

For elliptic equations, the first Liouville theorem was proved by Gidas and Spruck in [1], in which the authors proved that the following equation

does not possess positive solutions provided . Moreover, it was also proved that the exponent is optimal in the sense that problem (1) indeed possesses a positive solution for . So the exponent is usually called the critical exponent for problem (1). However, this Liouville theorem is not sufficient for proving the a priori bound since the solutions may blow up on the boundary of the domain. In order to overcome this difficulty, the authors studied the limit equation in the half space

in [2]. They proved the above equation also does not possess positive solutions provided . The above two Liouville theorems are what we need to prove a priori bound for positive solutions of nonlinear elliptic equations in bounded domain. Later, Chen and Li obtained similar nonexistence results for the above two equations in [3] by using the moving plane method.

At the same time, elliptic equations with weights

were also widely studied and there are many existence and nonexistence results for problem (3). If , we say this problem is a Hardy type equation, while for , we say this problem is a Henon type equation. For the Hardy type problem, it can be proved that this problem does not possess positive solution provided by using the moving plane method as in [3]. However, for the Henon type equation, this proof of the nonexistence result is completely open up to now. The main difference between the two cases lies in that, for , the weight is decreasing in , so the moving plane method works. However, for , the weight is increasing in , so the moving plane method does not work.

On the other hand, we note that the above-mentioned results only claim that the above equations do not possess positive solution. A natural and more difficult question is whether the above equations possess sign-changing solution. However, this question is also completely open up to now. A partial answer was given in [4] in which the authors assume the solution has finite Morse index; then they proved the nonexistence result for this kind of solution. To prove this result, the author first deduced some integrable conditions on the solution based on finite Morse index; then they use the Pohozaev identity to prove the nonexistence result. After this work, there are many extensions on similar problems. For example, Harrabi et al. extended these results to more general nonlinear problems in [5, 6]. The corresponding Neumann boundary value problems were studied in [7]; Yu studied the mixed boundary problems, the nonlinear boundary value problem, and the fractional Laplacian equation in [8], [9], and [10], respectively.

In this paper, inspired by the above works, we study another problem, that is, the following elliptic equation with weight:

where is a positive parameter. We are mainly concerned with the nonexistence of solution with finite Morse index. Because of the interaction of and , in order to prove the nonexistence result, we need to add a new bound for the exponent of the weight. More precisely, we have the following result.

Theorem 1. Suppose that , for or , for ; let be a bounded solution for problem (4) with ; then , where is the Morse index of .

The rest of this paper is devoted to the proof of the above theorem. We first deduce some inequality based on finite Morse index; then we derive some integral conditions on this solution. Finally, we use the Pohozaev inequality to prove the above theorem. In the following, we denote by a positive constant, which may vary from line to line.

2. Proof of Theorem 1

In this section, we always assume the conditions in Theorem 1 hold. We establish the nonexistence of finite Morse index solution for problem (4). For this purpose, we first recall the definition of Morse index. Let be a solution of problem (4); we define

as the Morse index for , where

and hence

Also, for further use, we need to define some cut-off function. Let ; we define as follows:

Moreover, we assume that for and for . In the same spirit of [8–10], we have the following result.

Lemma 2. Let be a solution of (4) with finite Morse index; then there exists such that

for any .

Proof. The proof is the same as [8–10]; we omit it.

The next lemma is the key ingredient in the proof of Theorem 1.

Lemma 3. Let be a bounded solution of problem (4) with finite Morse index and let , satisfy the assumptions in Theorem 1; then one has

Proof. We will use the information of finite Morse index of to prove our result. We first prove that . By Lemma 2, there exists a positive constant , such that

for any . Then by the definition of , we conclude that

A direct calculation shows that the right hand side of (12) equals

On the other hand, if we multiply (4) by and integrate by parts, then we have

Inserting (14) into (12), then we get

In particular, we have

If , we already have and since the right hand sides of (16) and (17) are bounded by a positive constant independent of . This proves the result for . So in the following, we always assume that . We deduce from (16) and the Holder inequality that

Suppose on the contrary that is infinite; then we have

for some and large enough. Denote , , and ; if we iterate the above inequality times, then we get

with . Since is bounded, a direct calculation shows that the right hand side of (20) is of order with

as . In particular, we can choose large enough, such that . Then it follows from (20) that

as , which is impossible. So we get .
Next, we show that . By the same spirit as the above, we deduce from (17) that

Suppose on the contrary that is infinite; then we have

for some and large enough. Denote and ; similarly, if we iterate the above inequality times, then we get

with . By the boundedness of , a direct calculation shows that the right hand side of (25) is of order with

as . So we can still choose large enough, such that . Then it follows from (25) that

as , which is a contradiction. So we get that .
Finally, we show that . For this purpose, we first choose a cut-off function such that for and for . For any , we multiply (4) by and integrate by parts; then we get

Since

we infer from the above two equations that

By the assumption for and the fact that , we get

by letting in the above equation. This finishes the proof of this lemma.

In order to complete the proof of Theorem 1, we need the following Pohozaev identity for problem (4).

Lemma 4. Suppose that is a solution of (4); then the following identity holds:

Proof. The proof of this lemma is standard; we give the details to keep this paper self-contained.
Multiplying (4) by and integrating in , then we get

The left hand side of (33) equals

while the right hand side of (33) equals

Combining the above two equations together, then we get the above local Pohozaev identity for problem (4).

With the above preparations, we can prove Theorem 1 now.

Proof of Theorem 1. First, since , , and by Lemma 3, then there exists a sequence such that

as . Let in the local Pohozaev identity and let ; then we get

Next, multiplying (4) by and integrating by parts, then we get

We infer from the above two identities that

By the assumptions on the exponents and in Theorem 1, we have

For (39) to hold, the only possibility is

which finally implies .

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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