Schrödinger-Poisson system without growth and the Ambrosetti-Rabinowitz conditions

Chen Huang; Gao Jia; Chen Huang; Gao Jia

doi:10.3934/math.2020090

AIMS Mathematics

2020, Volume 5, Issue 2: 1319-1332. doi: 10.3934/math.2020090

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Research article

Schrödinger-Poisson system without growth and the Ambrosetti-Rabinowitz conditions

Chen Huang ¹,
Gao Jia ^{2
,
,}

1.
Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China
2.
College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

Received: 02 November 2019 Accepted: 08 January 2020 Published: 20 January 2020
MSC : 35B45, 35J20, 35J50

We consider the following Schrödinger-Poisson system $ \left\{ \begin{array}{l} {\rm{ - }}\Delta u + V\left(x \right)u + \phi u = \lambda f\left(u \right)\; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}, \\ - \Delta \phi = {u^2}, \mathop {\lim }\limits_{|x| \to + \infty } \phi = 0, \; \; \; \; \; \; \; \; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}. \end{array} \right. $ Unlike most other papers on this problem, the Schrödinger-Poisson system without any growth and Ambrosetti-Rabinowitz condition is considered in this paper. Firstly, by Jeanjean's monotonicity trick and the mountain pass theorem, we prove that the problem possesses a positive solution for large value of $\lambda$. Secondly, we establish the multiplicity of solutions via the symmetric mountain pass theorem.
- Schrödinger-Poisson system,
- supercritical growth,
- variational methods,
- L^∞-estimate
Citation: Chen Huang, Gao Jia. Schrödinger-Poisson system without growth and the Ambrosetti-Rabinowitz conditions[J]. AIMS Mathematics, 2020, 5(2): 1319-1332. doi: 10.3934/math.2020090

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Abstract

We consider the following Schrödinger-Poisson system $ \left\{ \begin{array}{l} {\rm{ - }}\Delta u + V\left(x \right)u + \phi u = \lambda f\left(u \right)\; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}, \\ - \Delta \phi = {u^2}, \mathop {\lim }\limits_{|x| \to + \infty } \phi = 0, \; \; \; \; \; \; \; \; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}. \end{array} \right. $ Unlike most other papers on this problem, the Schrödinger-Poisson system without any growth and Ambrosetti-Rabinowitz condition is considered in this paper. Firstly, by Jeanjean's monotonicity trick and the mountain pass theorem, we prove that the problem possesses a positive solution for large value of $\lambda$. Secondly, we establish the multiplicity of solutions via the symmetric mountain pass theorem.

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