Some results of nontrivial solutions for Klein–Gordon–Maxwell systems with local super-quadratic conditions

Abstract

The existence of nontrivial solutions for the following kind of Klein–Gordon–Maxwell system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u-(2\omega +\phi )\phi u=f(x,u),&{}\quad x\in {{\mathbb {R}}}^{3},\\ \Delta \phi =(\omega +\phi )u^{2},&{}\quad x\in {{\mathbb {R}}}^{3}, \end{array}\right. \end{aligned}$$ is investigated, where $$\omega >0$$ is a constant, $$V\in C({{\mathbb {R}}}^{3},{{\mathbb {R}}})$$ is either periodic or coercive and is allowed to be sign-changing, $$f\in C({{\mathbb {R}}}^{3}\times {{\mathbb {R}}},{{\mathbb {R}}})$$ and f is subcritical and local super-linear. Using local super-quadratic conditions and other suitable assumptions on the nonlinearity f(x, u) and the potential V(x), the existence of nontrivial solutions for the above system is established. The obtained results in this paper improve the related ones in the literature.