Multiple Solutions for Schrödinger–Bopp–Podolsky Systems with Steep Potential Well and Concave-convex Nonlinearities

In this paper, we study the following Schrödinger–Bopp–Podolsky system \[ \begin{cases} -\Delta u + \lambda V(x)u + l(x) \phi u = f(x) |u|^{p-2}u + g(x) |u|^{q-2}u &\textrm{in $\mathbb{R}^{3}$}, \\ -\Delta \phi + a^{2} \Delta^{2} \phi = l(x) u^{2} &\textrm{in $\mathbb{R}^{3}$}, \end{cases} \] where $1 \lt q \lt 2 \lt p \lt 4$, $\lambda,a \gt 0$ and $V \in C(\mathbb{R}^{3},\mathbb{R})$ is a steep potential well. Such a problem has been rarely studied via variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold, because Palais–Smale sequences may not be bounded. By introducing an innovative constraint method of the Nehari manifold and combining the Ekeland variational principle, we prove the existence and multiplicity of nontrivial solutions for the above problem when $\lambda$ is sufficiently large.