Multi-bump solutions of Schrödinger–Poisson equations with steep potential well

Abstract

In this paper, we study a system of Schrodinger–Poisson equation $$\left\{\begin{array}{ll}-\Delta u+(\lambda a(x)+a_0(x))u+K(x)\phi u=|u|^{p-2}u, & \quad x \in\mathbb{R}^3, \\- \Delta \phi=K(x)u^2,& \quad x \in \mathbb{R}^3 \end{array} \right.$$ where $${p \in (4,6)}$$ and $${\lambda}$$ is a parameter. We require that $${a(x) \geq 0}$$ and has a bounded potential well $${\Omega = a^{-1}(0)}$$ . Combining this with other suitable assumptions on Ω, a 0 and K, we obtain the existence of multi-bump-type solution $${u_\lambda}$$ when $${\lambda}$$ is large via variational methods.