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

In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form -Delta u + V(x)u = f (x, u) + lambda a(x)vertical bar u vertical bar(q-2) u, x is an element of R-2, (0.1) where lambda > 0, q is an element of (1, 2), a is an element of L2/(2-q) (R-2), V(x), and f (x, t) are 1-periodic with respect to x, and f (x, t) has critical exponential growth at t = infinity. By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e. lambda > 0) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.