Single peak solutions for critical Schrödinger equation in dimension two

We are concerned with the concentration behavior of semiclassical states of the following Schrödinger equation$ \begin{cases} -\varepsilon^{2}\Delta v+V(x)v = f(v) \ \ \ \mbox {in} \ \ \mathbb{R}^{2},\\ v>0,\ v(x)\to0 \ \text{as}\ |x|\to+\infty.\;\;& \end{cases} $Here $ V $ possesses a local maximum point, and $ f $ has critical growth at infinity in the sense of Trudinger–Moser inequality. We seek solutions for the critical equation near the ground state level $ c_0 $ around a local maximum of $ V $ as $ \varepsilon\to0 $. Due to the lack of information about the uniqueness or nondegeneracy of the limiting equation, variational method is applied here, and a sensitive lower gradient estimate should be made when the local centroids of the functions are away from the local maximum of $ V $. We obtain this estimate by using the method of Zhang and Zhang [Calc. Var. Partial Differential Equations 63: 220, 2024], which is different from the ideas of Del Pino and Felmer [Math. Ann. 324:1-32, 2002], where a special gradient flow with high regularity is used and in Byeon and Tanaka [J. Eur. Math. Soc. 315:1859-1899, 2013; Mem. Amer Math. Soc. 229:viii+89, 2014], where an additional translation flow is introduced. Finally, we show the existence of semiclassical state solutions, moreover, we prove that these solutions have exponential decay property and concentrate at a local maximum of the potential $ V $.