Abstract:

This article concerns the following Klein-Gordon-Maxwell system $\begin{equation*} \begin{cases} -\Delta u+u-(2\omega+\phi)\phi u=\lambda Q(x)f(u), & x\in \mathbb{R}^{3},\\ \Delta \phi=(\omega+\phi)u^2, &x\in \mathbb{R}^{3}, \end{cases} \end{equation*}$ where $\omega> 0$ is a constant, $\lambda> 0$ is a parameter, $Q$ is a positive function. When the nonlinear term $f$ is sublinear at infinity, two nontrivial solutions for the system are established via variational methods and three critical points theorem. Furtermore, when $f$ is sublinear only in a neighbourhood of the origin, existence and multiplicity of non-trivial solutions are obtained via variational methods and critical point theorem. Our result completes some recent works concerning the multiplicity of solutions of this system.

Key words: Klein-Gordon-Maxwell system, Variational methods, Sublinearity, Critical point theorem, Multiplicity