On Schrödinger-Poisson equations with a critical nonlocal term

Abstract

In this paper, we study the following non-autonomous Schrödinger-Poisson equation with a critical nonlocal term and a critical nonlinearity: $ \begin{equation*} \left\{\begin{aligned} & -\Delta u +V(x) u + \lambda \phi |u|^3 u = f(u) + (u^+)^5,\ \ {\rm in } \ \ \ \ \mathbb{R}^3,\\ & -\Delta \phi = |u|^5, \ \ {\rm in } \ \ \ \ \mathbb{R}^3. \end{aligned}\right. \end{equation*} $ First, we consider the case that the nonlinearity satisfies the Berestycki-Lions type condition with critical growth. Second, we consider the case that $ \mathrm{int}V^{-1}(0) $ is contained in a spherical shell. By using variational methods, we obtain the existence and asymptotic behavior of positive solutions.

Keywords

• schrödinger-poisson equation
• critical nonlocal term
• critical nonlinearity
• variational method