Liouville theorem for elliptic equations with mixed boundary value conditions and finite Morse indices
- Xueqiao Wang^{1} and
- Xiongjun Zheng^{1}Email author
https://doi.org/10.1186/s13660-015-0867-1
© Wang and Zheng 2015
Received: 18 July 2015
Accepted: 15 October 2015
Published: 5 November 2015
Abstract
In this paper, we establish Liouville type theorem for boundedness solutions with finite Morse index of the following mixed boundary value problems: \(-\Delta u=|u|^{p-1}u\) in \(\mathbb{R}^{N}_{+}\), \(\frac{\partial u}{\partial\nu}=|u|^{q-1}u\) on \(\Gamma_{1}\), \(\frac{\partial u}{\partial\nu}=0\) on \(\Gamma_{0}\), and \(-\Delta u=|u|^{p-1}u\) in \(\mathbb{R}^{N}_{+}\), \(\frac{\partial u}{\partial\nu}=|u|^{q-1}u\) on \(\Gamma_{1}\), \(u=0\) on \(\Gamma_{0}\), where \(\mathbb{R}^{N}_{+} =\{x\in\mathbb{R}^{N}:x_{N}>0\}\), \(\Gamma_{1}=\{x\in \mathbb{R}^{N}:x_{N}=0,x_{1}<0\}\) and \(\Gamma_{0}=\{x\in\mathbb{R}^{N}:x_{N}=0,x_{1}>0\}\). The exponents p, q satisfy the conditions in Theorem 1.1.
Keywords
1 Introduction
In this paper, we establish a Liouville type theorem in the upper half space \(\mathbb{R}^{N}_{+}\) with mixed boundary conditions.
On the other hand, it is puzzling if problems (1.1) and (1.2) admit sign-changing solutions. A partial answer came from [3] by assuming additionally that solutions have finite Morse indices. It was proved in [3] that problems (1.1) and (1.2) do not possess nontrivial bounded solution with finite Morse index provided \(1< p<\frac{N+2}{N-2}\). In applications, fortunately, one may find critical points with finite Morse indices by the mountain pass theorem and saddle point theorem and so on, it allows one to establish for instance as in [4], the existence result for indefinite nonlinearities.
Recently, it is investigated by many authors various type of Liouville theorems for solutions with finite Morse indices, such as problems with Neumann boundary condition, Dirichlet-Neumann mixed boundary and nonlinear boundary conditions etc.; see [5–10] and references therein.
Our main results are as follows.
Theorem 1.1
If \(1< p\leq\frac{N+2}{N-2}\), \(1< q\leq\frac{N}{N-2}\), and \((p,q)\neq (\frac{N+2}{N-2},\frac{N}{N-2})\), then problems (1.3) and (1.4) do not possess nontrivial bounded solution with finite Morse index.
Theorem 1.1 will be proved in the next section. We first prove that a finite Morse index implies certain integrable conditions on u. Then by the Pohozaev identity, we show the nonexistence result.
2 Proof of Theorem 1.1
In this section, we establish the Liouville type theorem for bounded solutions of problems (1.3) and (1.4) with finite Morse indices, that is, we show that such solutions must be trivial. We assume in this section that p, q in (1.3) and (1.4) satisfying \(1< p\leq\frac{N+2}{N-2}\), \(1< q\leq\frac{N}{N-2}\), and \((p,q)\neq (\frac{N+2}{N-2},\frac{N}{N-2})\).
Lemma 2.1
Proof
Next, we show that a finite Morse index implies u satisfying a certain integrable condition. More precisely, we have the following lemma.
Lemma 2.2
Proof
We only prove the results for problem (1.3). For problem (1.4), the proof can proceed similarly.
The next lemma is the well-known local Pohozaev identity for elliptic problems with nonlinear boundary value condition.
Lemma 2.3
Proof
The proof of this lemma is standard, we give it here for completeness. We deal only with problem (1.3). The proof for problem (1.4) is almost the same except that different boundary value conditions were used. We omit the details.
Proof of Theorem 1.1
Declarations
Acknowledgements
The work is supported by National Natural Science Foundation of China (No:11271170) and GAN PO 555 program of Government of Jiangxi Province.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Gidas, B, Spruck, J: A priori bounds of positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equ. 6, 883-901 (1981) MATHMathSciNetView ArticleGoogle Scholar
- Gidas, B, Spruck, J: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525-598 (1981) MATHMathSciNetView ArticleGoogle Scholar
- Bahri, A, Lions, PL: Solutions of superlinear elliptic equations and their Morse indices. Commun. Pure Appl. Math. 45, 1205-1215 (1992) MATHMathSciNetView ArticleGoogle Scholar
- Ramos, M, Terracini, S, Troestler, C: Superlinear indefinite elliptic problems and Pohozaev type identities. J. Funct. Anal. 159, 596-628 (1998) MATHMathSciNetView ArticleGoogle Scholar
- Harrabi, A, Ahmedou, M, Rebhi, S, Selmi, A: A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case. Manuscr. Math. 137, 525-544 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Harrabi, A, Rebhi, S, Selmi, S: Solutions of superlinear equations and their Morse indices I. Duke Math. J. 94, 141-157 (1998) MATHMathSciNetView ArticleGoogle Scholar
- Harrabi, A, Rebhi, S, Selmi, S: Solutions of superlinear equations and their Morse indices II. Duke Math. J. 94, 159-179 (1998) MathSciNetView ArticleGoogle Scholar
- Yu, X: Solutions of mixed boundary problems and their Morse indices. Nonlinear Anal. TMA 96, 146-153 (2014) MATHView ArticleGoogle Scholar
- Yu, X: Liouville theorem for elliptic equations with nonlinear boundary value conditions and finite Morse indices. J. Math. Anal. Appl. 421, 436-443 (2015) MATHMathSciNetView ArticleGoogle Scholar
- Yu, X: Solutions of fractional Laplacian equations and their Morse indices. J. Differ. Equ. 260(1), 860-871 (2016) View ArticleGoogle Scholar
- Yu, X: Liouville Type Theorems for Two Mixed Boundary Value Problems with General Nonlinearities. arXiv:1410.5157