On the planar Schrodinger equation with indefinite linear part and critical growth nonlinearity

In the present paper, we develop a direct approach to find nontrivial solutions and ground state solutions for the following planar Schrodinger equation: {- Delta u + V(x)u = f ( x, u), x is an element of R-2, u is an element of H-1(R-2), where V(x) is an 1-periodic function with respect to x(1) and x(2), 0 lies in a gap of the spectrum of - Delta + V, and f ( x, t) behaves like +/- e(proportional to t2) as t -> +/- infinity uniformly on x is an element of R-2. Our theorems extend and improve the results of de Figueiredo-Miyagaki-Ruf (Calc Var Partial Differ Equ, 3(2):139-153, 1995), of de Figueiredo-do O-Ruf (Indiana Univ Math J, 53(4):1037-1054, 2004), of Alves-Souto-Montenegro (Calc Var Partial Differ Equ 43: 537-554, 2012), of Alves-Germano (J Differ Equ 265: 444-477, 2018) and of do O-Ruf (NoDEA 13: 167-192, 2006).