Improved results for Klein-Gordon-Maxwell systems with general nonlinearity

Sitong Chen; Xianhua Tang

doi:10.3934/dcds.2018096

Article Contents

2018, Volume 38, Issue 5: 2333-2348. Doi: 10.3934/dcds.2018096

This issue Previous Article KdV-like solitary waves in two-dimensional FPU-lattices Next Article Conormal derivative problems for stationary Stokes system in Sobolev spaces

Improved results for Klein-Gordon-Maxwell systems with general nonlinearity

Sitong Chen and
Xianhua Tang^,

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

Author Bio: mathsitongchen@163.com (Sitong Chen); tangxh@mail.csu.edu.cn (Xianhua Tang)
^* Corresponding author
^* Corresponding author

Received: August 31, 2017

Revised: November 30, 2017

Published: June 2018

This work is partially supported by the Hunan Provincial Innovation Foundation for Postgraduate (No: CX2017B041) and the National Natural Science Foundation of China (No: 11571370)

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Abstract

This paper is concerned with the following Klein-Gordon-Maxwell system

$\left\{ \begin{align} &-\vartriangle u+\left[ m_{0}^{2}-{{(\omega +\phi )}^{2}} \right]u = f(u),\ \ \ \ \text{in}\ \ {{\mathbb{R}}^{3}}, \\ &\vartriangle \phi = (\omega +\phi ){{u}^{2}},\ \ \ \ \text{in}\ \ {{\mathbb{R}}^{3}}, \\ \end{align} \right.$

where $0 < ω≤ m_0$ and $f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$. By introducing some new tricks, we prove that the above system has 1) a ground state solution in the case when $0 < ω < m_0$ and $f$ is superlinear at infinity; 2) a nontrivial solution in the zero mass case, i.e. $ω = m_0$ and $f$ is super-quadratic at infinity. These results improve the related ones in the literature.

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Mathematics Subject Classification: 35J10, 35J20.

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