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This paper is concerned with the following system: $$ \textstyle\begin{cases} {-\Delta u+\lambda A(x) u-K(x)(2 \omega +\phi ) \phi u=f(x, u)+h(x), \quad x \in \mathbb{R}^{3}}, \\ {\Delta \phi =K(x)(\omega +\phi ) u^{2}, \quad x \in \mathbb{R}^{3}}, \end{cases} $$ where $\lambda \geq 1$ is a parameter, $\omega >0$ is a constant and the potential A is sign-changing. Under the classic Ambrosetti–Rabinowitz condition and other suitable conditions, nontrivial solutions are obtained via the linking theorem and Ekeland’s variational principle. Especially speaking, we use a super-quadratic condition to replace the 4-superlinear condition which is usually used to show the existence of nontrivial solutions in many references. Our results improve the previous results in the literature.

Improved results of nontrivial solutions for a nonlinear nonhomogeneous Klein–Gordon–Maxwell system involving sign-changing potential