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Nontrivial solution for Schrödinger–Poisson equations involving a fractional nonlocal operator via perturbation methods

  • Xiaojing Feng
Article

Abstract

This paper focuses on the following Schrödinger–Poisson equations involving a fractional nonlocal operator \({\left\{\begin{array}{ll}-\Delta u+u+\phi u=f(x,u),&{\rm in}\ \mathbb{R}^3,\\(-\Delta)^{\alpha/2}\phi=u^2,\\lim_{|x|\to \infty}\phi(x)=0,&{\rm in}\ \mathbb{R}^3,\end{array}\right.}\) where \({\alpha \in (1,2]}\). Under certain assumptions, we obtain the existence of nontrivial solution of the above problem without compactness by using the methods of perturbation and the mountain pass theorem.

Keywords

Schrödinger–Poisson Perturbation methods Nontrivial solution 

Mathematics Subject Classification

35J20 35J60 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanxi UniversityTaiyuanPeople’s Republic of China

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