Existence of axially symmetric solutions for a kind of planar Schrödinger-Poisson system

Abstract

In this paper, we study the following kind of Schrödinger-Poisson system in $ { \mathbb{R}}^{2} $ $ \begin{equation*} \left\{\begin{array}{ll} -\Delta u+V(x)u+\phi u = K(x)f(u), \ \ \ x\in{ \mathbb{R}}^{2}, \\ \Delta \phi = u^{2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in{ \mathbb{R}}^{2}, \end{array}\right. \end{equation*} $ where $ f\in C({ \mathbb{R}}, { \mathbb{R}}) $, $ V(x) $ and $ K(x) $ are both axially symmetric functions. By constructing a new variational framework and using some new analytic techniques, we obtain an axially symmetric solution for the above planar system. Our result improves and extends the existing works.

Keywords

• existence
• axially symmetric
• ground state solution
• logarithmic convolution potential
• planar schrödinger-poisson system