Standing waves for the Chern-Simons-Schrodinger equation with critical exponential growth

Abstract

In this paper, by combing the variational methods and Trudinger-Moser inequality, we study the existence and multiplicity of the positive standing wave for the following Chern-Simons-Schrödinger equation \begin{equation} -Δu+u +λ\left(\int_{0}^{\infty}\frac{h(s)}{s}u^{2}(s)ds+\frac{h^{2}(\vert x\vert)}{\vert x\vert^{2}}\right)u=f(x,u)+εk(x)\quad\quad \text{in}\,\,\mathbb{R}^2, \\ \end{equation} where $h(s)={\int }_0^s\frac{l}{2}{u}^2(l)dl$, $\lambda >0$ and the nonlinearity $f:{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$ behaves like $\text{exp}(\alpha |u{|}^2)$ as $|u|\to \infty$. For the case $\epsilon =0$, we can get a mountain-pass type solution.