Existence of infinitely many small solutions for fractional Schrödinger-Poisson systems with sign-changing potential and local nonlinearity

Zonghu Xiu; Shengjun Li; Zhigang Wang; Zonghu Xiu; Shengjun Li; Zhigang Wang

doi:10.3934/math.2020442

AIMS Mathematics

2020, Volume 5, Issue 6: 6902-6912. doi: 10.3934/math.2020442

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Existence of infinitely many small solutions for fractional Schrödinger-Poisson systems with sign-changing potential and local nonlinearity

1.
Science and Information College, Qingdao Agricultural University, Qingdao, 266109, P.R. China
2.
School of Science, Hainan University, Haikou, 570228, P.R. China

Received: 07 July 2020 Accepted: 26 August 2020 Published: 07 September 2020
MSC : 35J60, 35J20

In this paper, we prove the existence of infinitely many small solutions for the following fractional Schr?dinger-Poisson system $ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^s u+V(x)u+\phi u = f(x,u),&x\in\mathbb{R}^3,\\ (-\Delta)^t\phi = u^2,& x\in\mathbb{R}^3, \end{array} \right. \end{equation*} $ where $(-\Delta)^\alpha$ denotes the fractional Laplacian of order $\alpha\in(0, 1)$ and $V$ is allowed to be sign-changing. We obtain infinitely many small solutions via a dual method. Our main tool is a critical point theorem which was established by Kajikiya.
- fractional Schrodinger-Poisson system,
- infinitely many small solutions,
- dual methods
Citation: Zonghu Xiu, Shengjun Li, Zhigang Wang. Existence of infinitely many small solutions for fractional Schrödinger-Poisson systems with sign-changing potential and local nonlinearity[J]. AIMS Mathematics, 2020, 5(6): 6902-6912. doi: 10.3934/math.2020442

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Abstract

In this paper, we prove the existence of infinitely many small solutions for the following fractional Schr?dinger-Poisson system $ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^s u+V(x)u+\phi u = f(x,u),&x\in\mathbb{R}^3,\\ (-\Delta)^t\phi = u^2,& x\in\mathbb{R}^3, \end{array} \right. \end{equation*} $ where $(-\Delta)^\alpha$ denotes the fractional Laplacian of order $\alpha\in(0, 1)$ and $V$ is allowed to be sign-changing. We obtain infinitely many small solutions via a dual method. Our main tool is a critical point theorem which was established by Kajikiya.

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