A SHARP TRUDINGER-MOSER TYPE INEQUALITY IN R2

In this paper, we establish a sharp Trudinger-Moser type inequality for a class of Schrodinger operators in R-2. We obtain a result related to the compactness of the embedding of a subspace of W-1,W-2(R-2) into the Orlicz space L-phi(R-2) determined by phi(t) = E-beta t2-1. Our result is similar to one obtained by Adimurthi and Druet for smooth bounded domains in R-2, which is closely related to a compactness result proved by Lions. Furthermore, similarly to what has been done by Carleson and Chang, we prove the existence of an extremal function for this Trudinger-Moser inequality by performing a blow-up analysis. Trudinger-Moser type inequalities have a wide variety of applications to the study of nonlinear elliptic partial differential equations involving the limiting case of Sobolev inequalities and have received considerable attention in recent years.