Multiple Solutions for $4$-superlinear Klein-Gordon-Maxwell System Without Odd Nonlinearity

Abstract

In this paper, we study the following Klein-Gordon-Maxwell system\[\begin{cases} -\Delta u + V(x)u - (2\omega + \phi)\phi u = f(x,u), &x \in \mathbb{R}^3, \\ \Delta \phi = (\omega + \phi) u^2, &x \in \mathbb{R}^3,\end{cases}\]where the nonlinearity $f$ and the potential $V$ are allowed to be sign-changing. Under some appropriate assumptions on $V$ and $f$, we obtain the existence of two different solutions of the system via the Ekeland variational principle and the Mountain Pass Theorem.