Stationary nonlinear Schr"odinger equations in $BbbR^2$ with potentials vanishing at infinity

We deal with a class of 2-D stationary nonlinear Schrödinger equations (NLS) involving potentials V and weights Q decaying to zero at infinity as (1 + | x| α) - 1, α∈ (0 , 2) , and (1 + | x| β) - 1, β∈ (2 , + ∞) , respectively, and nonlinearities with exponential growth of the form exp γ0s2 for some γ0> 0. Working in weighted Sobolev spaces, we prove the existence of a bound state solution, i.e. a solution belonging to H1(R2). Our approach is based on a weighted Trudinger–Moser-type inequality and the classical mountain pass theorem.