On the Schr\"odinger--Bopp--Podolsky system with indefinite potential: ground states, multiplicity and exponential decay

In this paper, we study the Schrödinger--Bopp--Podolsky system
\begin{equation*}
\begin{cases}
-\Delta u + V(x)u + \phi u = f(x,u), & \text{in } \mathbb{R}^3,
-\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2, & \text{in } \mathbb{R}^3.
\end{cases}
\end{equation*}
We consider the case where the potential \(V\) is indefinite so that the Schrödinger operator \(-\Delta + V\) has a finite-dimensional negative space. Under suitable assumptions on the potential \(V\) and nonlinearity $f(x,u)$, we prove the existence of nontrivial solutions via a local linking argument and Morse theory. Moreover, these solutions are shown to decay exponentially at infinity. Additionally, a ground state solution is obtained by minimization techniques. Finally, if \(f(x,u)\) is odd with respect to \(u\), we obtain an unbounded sequence of solutions using the symmetric mountain pass theorem.