Ground state solutions of the non-autonomous Schrödinger–Bopp–Podolsky system

Abstract In this paper, we consider the following non-autonomous Schrödinger–Bopp–Podolsky system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + V(x) q^2\phi = f(u)\\ \phi a^2 \Delta ^2 4\pi u^2 \end{array}\right. } \hbox { in }{\mathbb {R}}^3. \end{aligned}$$ - ? u + V ( x ) q 2 ? = f a 4 ? in R 3 . By using some original analytic techniques and new estimates of ground state energy, prove that admits a solution under mild assumptions on V f . final part give min-max characterization energy.