Electron. J. Diff. Equ., Vol. 2015 (2015), No. 102, pp. 1-12.

Existence and multiplicity of solutions for nonhomogeneous Klein-Gordon-Maxwell equations

Liping Xu, Haibo Chen

Abstract:
This article concerns the nonhomogeneous Klein-Gordon-Maxwell equation
$$\displaylines{ -\Delta u+u-(2\omega +\phi)\phi u=|u|^{p-1}u +h(x), \quad\text{in }\mathbb{R}^3,\cr \Delta \phi=(\omega +\phi)u^2,\quad\text{in }\mathbb{R}^3, }$$
where $\omega>0$ is constant, $p\in(1,5)$. Under appropriate assumptions on h(x), the existence of at least two solutions is obtained by applying the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.

Submitted January 21, 2015. Published April 16, 2015.
Math Subject Classifications: 35J20, 35J65, 35J60.
Key Words: Nonhomogeneous Klein-Gordon-Maxwell equations; multiple solutions; Pohozaev identity; variational method.

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Liping Xu
School of Mathematics and Statistics
Central South University
Changsha 410075, China
email: x.liping@126.com
Haibo Chen
School of Mathematics and Statistics
Central South University
Changsha 410075, China
email: math_chb@csu.edu.cn

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