Nonlinear divergence equation with potentials vanishing in some direction at infinity: the exponential critical case

The purpose of this article is to state some weighted Sobolev embedding involving functions that vanishing only in a direction. In this setting we prove a weighted Trudinger–Moser type inequality and as an application, we addressed the existence of solutions to a class of elliptic equation of the form − div ( a ( x ) ∇ u ) + V ( x ) u = K ( x ) f ( u ) in R 2 , where the nonlinearity f has exponential critical growth in sense of Trudinger–Moser.