In this paper, we study the following Schrödinger–Born–Infeld system with concave and convex nonlinearities \(\begin{aligned} \left\{ \begin{aligned}&-\Delta u+u+\lambda \phi u=a(x)|u|^{p-2}u+b(x)|u|^{q-2}u, & \quad x \in {\mathbb {R}}^3,\\&-\text {div}\left( \frac{\nabla \phi }{\sqrt{1-|\nabla \phi |^2}}\right) = u^2, & \quad x \in {\mathbb {R}}^3,\\&u \left( x \right) \rightarrow 0, \quad \phi \left( x \right) \rightarrow 0, & \quad \text {as} \, \left| x \right| \rightarrow \infty , \end{aligned} \right. \end{aligned}\) where \(1<p<2<q<6,\) \(\lambda \ne 0\) and a(x), b(x) satisfy some suitable assumptions. Owing to monotonicity trick, Ekeland’s variational principle, and cut off technique, we obtain that the above system admits at least one positive energy solution and one negative energy solution in both the attractive (i.e., \(\lambda <0\) ) and repulsive (i.e., \(\lambda >0\) ) cases. In addition, replacing \(a(x)|u|^{p-2}u+b(x)|u|^{q-2}u\) by \(a(x)|u|^{p-2}u-b(x)|u|^{q-2}u\) with \(1<p<2<q<+\infty ,\) we obtain a sequence of solutions with negative energy levels for the above system in the repulsive case by using a variant of Clark’s theorem.
