Stationary nonlinear Schrödinger equations in R2 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 (Formula presented.), (Formula presented.), and (Formula presented.), (Formula presented.), respectively, and nonlinearities with exponential growth of the form (Formula presented.) for some (Formula presented.). Working in weighted Sobolev spaces, we prove the existence of a bound state solution, i.e. a solution belonging to (Formula presented.). Our approach is based on a weighted Trudinger–Moser-type inequality and the classical mountain pass theorem.