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-Schr\"odinger equation \begin{equation} -\Delta u+u +\lambda\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)+\epsilon k(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 $f:\mathbb{R}^2\times \mathbb{R}\rightarrow \mathbb{R}$ behaves like $\text{exp}(\alpha\vert u\vert^{2})$ as $\vert u\vert\rightarrow \infty$. For the case $\epsilon=0$, we can get a mountain-pass type solution.