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

We consider the equation −Δu+V(x)u=f(x,u) for View the MathML source where View the MathML source is a positive potential bounded away from zero, and the nonlinearity View the MathML source behaves like exp(α|u|2) as |u|→∞. 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 View the MathML source by combining the mountain-pass theorem with Trudinger–Moser inequality and a version of a result due to Lions for critical growth in View the MathML source.