Standing waves for nonlinear Schrödinger equations involving critical growth of Trudinger-Moser type (Q903222)

From the abstract: In this paper, we deal with the following singularly perturbed elliptic problem \[ -\varepsilon^2 \Delta u+V(x) u=f(u), \quad u \in H^1(\mathbb{R}^2), \] where \(f(s)\) has critical growth of Trudinger-Moser type. In this paper, we construct a localized bound-state solution concentrating at an isolated component of the positive local minimum points of \(V\) as \(\varepsilon \rightarrow 0\) under certain conditions on \(f(s)\). Our results complete the analysis made in [\textit{J. Byeon} et al., Commun. Partial Differ. Equations 33, No. 6, 1113--1136 (2008; Zbl 1155.35344)] for the two-dimensional case, in the sense that, in that paper only the subcritical growth was considered.