On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth

We consider the equation -Deltau + V(x)u = f (x, u) for x epsilon R-2 where V : R-2 --> R is a positive potential bounded away from zero, and the nonlinearity f : R-2 x R --> R behaves like exp( alpha\u\(2)) as \u\ --> infinity. We also assume that the potential V(x) and the nonlinearity f (x, u) are asymptotically periodic at infinity. We prove the existence of at least one weak positive solution u epsilon H-1(R-2) by combining the mountain-pass theorem with Trudinger-Moser inequality and a version of a result due to Lions for critical growth in R-2. (C) 2003 Elsevier Ltd. All rights reserved.