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

We are concerned with a class of generalized Chern-Simons-Schrödinger systems{−Δu+λV(x)u+A0u+∑j=12Aj2u=f(u),∂1A2−∂2A1=−12|u|2,∂1A1+∂2A2=0,∂1A0=A2|u|2,∂2A0=−A1|u|2, where λ>0 denotes a sufficiently large parameter, V:R2→R admits a potential well Ω≜intV−1(0) 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 λ>0, and the asymptotic behavior of the obtained solutions is also investigated when λ→+∞.