In this paper, we prove the existence and concentration behavior of positive ground state solutions for the following generalized quasilinear Schrödinger equation −div(ε2g2(u)∇u)+ε2g(u)g′(u)|∇u|2+V(x)u=f(u),x∈R2, where g∈C1(R,(0,+∞)) and V is a continuous potential uniformly positive and unbounded from above. This study focuses on the key aspect of characterizing the nonlinear term f as having critical exponential growth with respect to the Trudinger–Moser inequality. By employing a change of variables, a version of the Trudinger–Moser inequality, a penalization method and the Mountain Pass arguments, we demonstrate the existence of ground states and establish their concentration behavior near a local minimum of V. Different from the Orlicz space used in do Ó and Severo J. Math. Phys. Calculus Var. Partial Differ. Equations 38, 275–315 (2010), we analyze the problem in the natural Hilbert space with some more general conditions. In particular, without imposing any monotonicity assumptions on f, we establish a mountain pass characterization of the least energy solution by carefully constructing a refined path. Notably, we remove the effective requirement on the limiting behavior lim inft→∞tf(t)eζ0t2α, which has been essential in prior works to compensate for the lack of compactness induced by the critical exponential nonlinearity. Our results extend some ones in Chen et al. Calculus Var. Partial Differ. Equations 63, 243 (2024).
Huang et al. (Sun,) studied this question.