Multiplicity of solutions for Schrödinger-Poisson system with critical exponent in $\mathbb{R}^{3}$

Xueqin Peng; Gao Jia; Chen Huang; Xueqin Peng; Gao Jia; Chen Huang

doi:10.3934/math.2021126

AIMS Mathematics

2021, Volume 6, Issue 3: 2059-2077. doi: 10.3934/math.2021126

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Research article

Multiplicity of solutions for Schrödinger-Poisson system with critical exponent in $\mathbb{R}^{3}$

1.
College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China
2.
College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350117, China

Received: 19 October 2020 Accepted: 02 December 2020 Published: 09 December 2020
MSC : 35J62, 35B33

In this paper, we study the following Schrödinger-Poisson system with critical exponent

$ \begin{equation*} \begin{cases} -\Delta u-k(x)\phi u=\lambda h(x)|u|^{p-2}u+s(x)|u|^{4}u, \ ~~x\in\mathbb{R}^{3},\\ -\triangle\phi=k(x)u^{2}, \ \ \ \ \ \ \ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x\in\mathbb{R}^{3}, \\ \end{cases} \end{equation*} $

where $1 < p < 2$ and $\lambda > 0.$ Under suitable conditions on $k$, $h$ and $s$, we show that there exists $\lambda^{\ast}>0$ such that the above problem possesses infinitely many solutions with negative energy for each $\lambda\in(0, \lambda^{\ast})$. Moreover, we prove the existence of infinitely many solutions with positive energy. The main tools are the concentration compactness principle, $Z_{2}$ index theory and Fountain Theorem. These results extend some existing results in the literature.
- Schrödinger-Poisson system,
- critical exponent,
- variational method,
- genus
Citation: Xueqin Peng, Gao Jia, Chen Huang. Multiplicity of solutions for Schrödinger-Poisson system with critical exponent in $\mathbb{R}^{3}$[J]. AIMS Mathematics, 2021, 6(3): 2059-2077. doi: 10.3934/math.2021126

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Abstract

In this paper, we study the following Schrödinger-Poisson system with critical exponent

$ \begin{equation*} \begin{cases} -\Delta u-k(x)\phi u=\lambda h(x)|u|^{p-2}u+s(x)|u|^{4}u, \ ~~x\in\mathbb{R}^{3},\\ -\triangle\phi=k(x)u^{2}, \ \ \ \ \ \ \ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x\in\mathbb{R}^{3}, \\ \end{cases} \end{equation*} $

where $1 < p < 2$ and $\lambda > 0.$ Under suitable conditions on $k$, $h$ and $s$, we show that there exists $\lambda^{\ast}>0$ such that the above problem possesses infinitely many solutions with negative energy for each $\lambda\in(0, \lambda^{\ast})$. Moreover, we prove the existence of infinitely many solutions with positive energy. The main tools are the concentration compactness principle, $Z_{2}$ index theory and Fountain Theorem. These results extend some existing results in the literature.

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