Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials

Xianhua Tang; Sitong Chen

doi:10.3934/dcds.2017214

Article Contents

2017, Volume 37, Issue 9: 4973-5002. Doi: 10.3934/dcds.2017214

This issue Previous Article Fiber bunching and cohomology for Banach cocycles over hyperbolic systems Next Article Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation

Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials

Xianhua Tang^, and
Sitong Chen

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

* Corresponding author
* Corresponding author

Received: January 31, 2017

Revised: March 31, 2017

Published: October 2017

This work is partially supported by the National Natural Science Foundation of China (No: 11571370)

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Abstract

This paper is dedicated to studying the following Schrödinger-Poisson problem

$\left\{ \begin{array}{ll} -\triangle u+V(x)u+φ u=f(u), \ \ \ \ x∈ \mathbb{R}^{3},\\ -\triangle φ=u^2, \ \ \ \ x∈ \mathbb{R}^{3}, \end{array} \right. $

where $V(x)$ is weakly differentiable and $f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$. By introducing some new tricks, we prove the above problem admits a ground state solution of Nehari-Pohozaev type and a least energy solution under mild assumptions on $V$ and $f$. Our results generalize and improve the ones in [D. Ruiz, J. Funct. Anal. 237 (2006) 655-674], [J.J. Sun, S.W. Ma, J. Differential Equations 260 (2016) 2119-2149] and some related literature.

Keywords:

Schrödinger-Poisson problem,
ground state solution of Nehari-Pohozaev type,
the least energy solutions.

Mathematics Subject Classification: 35J10, 35J20.

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