Existence and concentration of nontrivial solutions for a class critical quasilinear Schrödinger equations with steep potential well

This paper considers the existence and concentration of nontrivial solutions for the following quasilinear Schrödinger equation with steep potential well$\begin{equation} -\Delta u + \lambda V(x)u + \dfrac{\kappa}{2}[\Delta (u^2)]u = h(u) + |u|^{2^{*}-2} u, \quad x \in \mathbb{R}^{N},\quad\quad\quad \text{(QS)} \end{equation}$where $ \kappa > 0 $ is a parameter, $ N \geq 3 $, and $ \lambda > 0 $. Here, $ h $ is a continuous function satisfying certain growth conditions. By using a change of variables technique and $ L^\infty $-estimate, we establish the existence of nontrivial solutions for sufficiently small $ \kappa > 0 $ and sufficiently large $ \lambda > 0 $. Furthermore, we investigate the asymptotic behavior of the nontrivial solutions as $ \lambda \to +\infty $.