| Summary: | In this paper we study the existence of ground states solutions for non-autonomous Schrödinger–Bopp–Podolsky system
\begin{equation*}
\begin{cases}
-\Delta u + u +\lambda K(x)\phi u = b(x)|u|^{p-2}u & \text{in} \ \mathbb{R}^{3}, \\
-\Delta \phi + a^2\Delta^2\phi = 4\pi K(x) u^{2} & \text{in}\ \mathbb{R}^{3},
\end{cases}
\end{equation*}
where $\lambda>0, 2<p\leq 4$ and both $K(x)$ and $b(x)$ are nonnegative functions in $\mathbb{R}^{3}$. Assuming that ${\lim_{\vert{x}\vert \to +\infty}}K(x)=K_\infty >0$ and ${\lim_{\vert{x}\vert \to +\infty}}b(x)=b_\infty >0$ and satisfying suitable assumptions, but not requiring any symmetry property on them. We show that the existence of a positive solution depends on the parameters $\lambda$ and $p$. We also establish the existence of ground state solutions for the case $3.18\approx\frac{1+\sqrt{73}}{3}<p\le{4}$.
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