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

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.

Keywords

• Nonhomogeneous Klein-Gordon-Maxwell equations
• multiple solutions
• Pohozaev identity
• variational method