Existence and concentration of positive solutions to generalized Chern-Simons-Schrödinger system with critical exponential growth

Abstract

We are concerned with a class of generalized Chern-Simons-Schrödinger systems$\{\begin{array}{c}-\Delta u+\lambda V\left(x\right)u+A_{0}u+\sum _{j=1}^2{A}_j^{2}u=f\left(u\right),\\ {\partial }_1{A}_2-{\partial }_2{A}_1=-\frac{1}{2}|u{|}^2,{\partial }_1{A}_1+{\partial }_2{A}_2=0,\\ {\partial }_1{A}_0=A_2|u{|}^2,{\partial }_2{A}_0=-A_1|u{|}^2,\end{array}$ where $\lambda >0$ denotes a sufficiently large parameter, $V:R^2\to R$ admits a potential well $\Omega \triangleq \text{int}{V}^{-1}\left(0\right)$ and the nonlinearity f fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. Under some suitable assumptions on V and f, based on variational method together with some new technical analysis, we are able to get the existence of positive solutions for some large $\lambda >0$, and the asymptotic behavior of the obtained solutions is also investigated when $\lambda \to +\infty$.