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

In this paper, by combing the variational methods and Trudinger-Moser inequality, we study the existence and multiplicity of the positive standing waves for the following Chern-Simons-Schrödinger equation(0.1)−Δu+u+λ(∫0∞h(s)su2(s)ds+h2(|x|)|x|2)u=f(x,u)+ϵk(x) inR2, where h(s)=∫0sl2u2(l)dl, λ>0 and the nonlinearity f:R2×R→R behaves like exp(α|u|2) as |u|→∞. For the case ϵ=0, we can get a mountain-pass type solution.