On the Schrödinger--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} -Δu + V(x)u + φu = f(x,u), & $\text{in }$ $\mathbb{R}$^3, -Δφ+ a^2 Δ^2 φ= 4πu^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 $-\mathrm{\Delta }+V$ 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.