On the planar Schrödinger-Poisson system with zero mass and critical exponential growth

Abstract

This paper is concerned with the following planar Schrödinger-Poisson system with zero mass \begin{equation*} \begin{cases} -\Delta u+\lambda \phi u=f(x,u), \;\; & x\in {\mathbb R}^{2},\\ \Delta \phi=2\pi u^2, \;\; & x\in {\mathbb R}^{2}, \end{cases} \end{equation*} where $\lambda > 0$ and $f\in \mathcal{C}(\mathbb R^2\times\mathbb R, \mathbb R)$ is of subcritical or critical exponential growth in the sense of Trudinger-Moser. By using some new analytical approaches, we prove that the above system has axially symmetric solutions under weak assumptions on $\lambda$ and $f$. This seems the first result on the planar Schrödinger-Poisson system with zero mass.