Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

Abstract

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass $$\begin{array}{c}\left\{\begin{array}{cc}-\mathrm{\Delta }\mathit{u}+\mathit{\varphi }\mathit{u}=\mathrm{(}{\mathit{I}}_{\mathit{\alpha }}\ast \mathit{F}\mathrm{(}\mathit{u}\mathrm{)}\mathrm{)}\mathit{f}\mathrm{(}\mathit{u}\mathrm{)}\mathrm{,}\hfill & \mathit{x}\in {\mathbb{R}}^2\mathrm{,}\hfill \\ \mathrm{\Delta }\mathit{\varphi }={\mathit{u}}^2\mathrm{,}\hfill & \mathit{x}\in {\mathbb{R}}^2\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $\mathit{\alpha }\in \mathrm{(}0\mathrm{,}2\mathrm{)}$ , ${\mathit{I}}_{\mathit{\alpha }}:{\mathbb{R}}^2\to \mathbb{R}$ is the Riesz potential, $\mathit{f}\in \mathcal{C}\mathrm{(}\mathbb{R}\mathrm{,}\mathbb{R}\mathrm{)}$ is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.