Standing waves for nonlinear Schrodinger equations involving critical growth of Trudinger-Moser type

In this paper, we deal with the following singularly perturbed elliptic problem -epsilon(2) Delta u + V(x)u = f(u), u is an element of H-1(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 epsilon -> 0 under certain conditions on f(s). Our results complete the analysis made in Byeon et al. (Commun Partial Differ Equ 33: 1113-1136, 2008) for the two-dimensional case, in the sense that, in that paper only the subcritical growth was considered.