A nonhomogeneous elliptic problem involving critical growth in dimension two ✩

In this paper we study a class of nonhomogeneous Schrödinger equations − u + V (x)u = f (u)+h(x) in the whole two-dimension space where V (x) is a continuous positive potential bounded away from zero and which can be “large” at the infinity. The main difficulty in this paper is the lack of compactness due to the unboundedness of the domain besides the fact that the nonlinear term f (s) is allowed to enjoy the critical exponential growth by means of the Trudinger–Moser inequality. By combining variational arguments and a version of the Trudinger–Moser inequality, we establish the existence of two distinct solutions when h is suitably small.