Multiple solutions for Schrödinger-Poisson systems with critical nonlocal term

Abstract

This paper is concerned with the existence of positive bound state solutions for Schrödinger-Poisson systems with critical nonlocal term: \begin{equation} \begin{cases} -\Delta u=\phi|u|^{3}u+\lambda Q(x)|u|^{q-2}u &\text{in } \mathbb{R}^3, \\ -\Delta \phi=|u|^5 & \text{in } \mathbb{R}^3. \end{cases} \tag{$\mathcal{P}$} \end{equation} Under certain assumptions on $Q$ and $\lambda$, we prove that $(\mathcal{P})$ has multiple positive bound state solutions by decomposition the Nehari manifold and fine estimates.