Multi-bump solutions of Schrödinger–Bopp–Podolsky system with steep potential well - Li Wang; Jun Wang; Jixiu Wang (Electronic Journal of Qualitative Theory of Differential Equations)
In this paper, we study the existence of multi-bump solutions for the following Schrödinger–Bopp–Podolsky system with steep potential well: \begin{equation*} \begin{cases} -\Delta u+(\lambda V(x)+V_0(x))u+K(x)\phi u= |u|^{p-2}u, &x\in \mathbb{R}^3,\\ -\Delta \phi+a^2\Delta^2\phi=K(x) u^2, &x\in \mathbb{R}^3, \end{cases} \end{equation*} where $p \in(4,6), a>0$ and $\lambda$ is a parameter. We require that $V(x) \geq 0$ and has a bounded potential well $\Omega=V^{-1}(0)$. Combining this with other suitable assumptions on $\Omega, V_{0}$ and $K$, when $\lambda$ is large enough, we obtain the existence of multi-bump-type solutions $u_{\lambda}$ by using variational methods.