On the planar Schrödinger 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 Schrödinger equation: - Δ u + V ( x ) u = f ( x , u ) , x ∈ R 2 , u ∈ 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 - Δ + V , and f ( x , t ) behaves like ± e α t 2 as t → ± ∞ uniformly on x ∈ 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 Ó-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 Ó-Ruf (NoDEA 13: 167–192, 2006).