On a periodic Schrödinger equation involving periodic and nonperiodic nonlinearities in R2

We study the existence of solutions for the nonlinear Schrödinger equation−Δu+V(x)u=f(x,u)inR2, where the potential V is 1-periodic, 0 lies in a spectral gap from the spectrum of the Schrödinger operator S=−Δ+V and the nonlinearity f(x,t) has exponential growth in the sense of Trudinger-Moser. The main feature here is that f(x,t) is allowed to be both periodic and nonperiodic in the x variable. Our proofs rely on a linking theorem and the Lions concentration compactness principle.