Ground state solutions for the Chern-Simons-Schrödinger equation with zero mass potential

Abstract

In this paper, we consider the nonlinear Chern-Simons-Schrödinger equation \begin{equation*} -\Delta u+ \Big ( \frac{h_u^2(|x|)}{|x|^2} +\int^{\infty}_{|x|}\frac{h_u(s)}{s}u^2(s)ds \Big ) u =-a|u|^{p-2}u+f(u),\,\, x\in\mathbb R^2, \end{equation*} where $a > 0$, $p\in (2,3)$ and \begin{equation*} h_u(s)=\int^s_0\frac{\tau}{2}u^2(\tau)d\tau =\frac{1}{4\pi}\int_{B_s}u^2(x)dx \end{equation*} is the so-called Chern-Simons term, $f$ has sub-critical exponential growth $\lim_{|t|\to\infty}\frac{|f(t)|}{e^{\alpha t^2}}=0$ for every $\alpha > 0$. Under some wild assumptions on $f$, we prove the existence of a ground state solution, a mountain-pass type solution and infinitely many geometrically distinct of solutions for the above equation.