Bound state solutions for fractional Schrödinger-Poisson systems

Xinsheng Du; Qi Li; Zengqin Zhao; Gen Li

doi:10.3934/mfc.2021023

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2022, Volume 5, Issue 1: 57-66. Doi: 10.3934/mfc.2021023

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Bound state solutions for fractional Schrödinger-Poisson systems

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

^* Corresponding author: Qi Li
^* Corresponding author: Qi Li

Received: June 2021

Revised: September 2021

Early access: October 2021

Published: February 2022

The first author is supported by NSF of Shandong Province (ZR2020MA005)

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Abstract

In this work, we investigate a class of fractional Schrödinger - Poisson systems

$ \begin{equation*} \left\{\begin{array}{ll}(-\triangle)^s u +V(x)u+\lambda\phi u = \mu u+|u|^{p-1}u, & x\in\ \mathbb{R}^3, \\(-\triangle)^s \phi = u^2, & x\in\ \mathbb{R}^3, \end{array}\right. \end{equation*} $

where $ s\in(\frac{3}{4}, 1) $, $ p\in(3, 5) $, $ \lambda $ is a positive parameter. By the variational method, we show that there exists $ \delta(\lambda)>0 $ such that for all $ \mu\in[\mu_1, \mu_1+\delta(\lambda)) $, the above fractional Schrödinger -Poisson systems possess a nonnegative bound state solutions with positive energy. Here $ \mu_1 $ is the first eigenvalue of $ (-\triangle)^s +V(x) $.

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Mathematics Subject Classification: 35K70, 58E05.

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