Existence and Symmetry of Solutions for a Class of Fractional Schrödinger–Poisson Systems

Yongzhen Yun,Tianqing An,Guoju Ye

doi:10.3390/math9101149

Abstract

In this paper, we investigate a class of Schrödinger–Poisson systems with critical growth. By the principle of concentration compactness and variational methods, we prove that the system has radially symmetric solutions, which improve the related results on this topic.

Highlights

In recent years, fractional equations or systems have been studied extensively by researchers due to their various applications in various fields, such as obstacle problems, electrical circuits, quantum mechanics, and phase transitions; see [1,2,3,4,5,6,7] and their references
It is important to mention that Laskin in [7] established the following timedependent Schrödinger equation involving a fractional Laplacian when he expanded the Feynman path integral, from Brownian-like to Lévy-like quantum mechanical paths i g(x, t), (x, t) ∈ R3 × R
Gu et al in [19] only studied the existence of a positive solution by variational methods, and there are no relevant articles that consider the existence of radially symmetric solutions of the fractional Schrödinger– Poisson system with critical growth

Summary

Introduction

Fractional equations or systems have been studied extensively by researchers due to their various applications in various fields, such as obstacle problems, electrical circuits, quantum mechanics, and phase transitions; see [1,2,3,4,5,6,7] and their references. The variational method, they obtained the existence of a positive solution for (2). Azzollini and Pomponio in [15] considered system (3) by variational methods; they established the existence of a ground state solution when potential V(x) is a positive constant or non-constant. Fractional Schrödinger–Poisson systems have received lots of attention in recent years, and many of the works have studied the existence of solutions of it; see [18,19,20,21,22] and their references. Gu et al in [19] only studied the existence of a positive solution by variational methods, and there are no relevant articles that consider the existence of radially symmetric solutions of the fractional Schrödinger– Poisson system with critical growth.

Preliminaries

Conclusions

Methods