Spike solutions for nonlinear Schrödinger equations in 2D with vanishing potentials

We consider ε-perturbed nonlinear Schrödinger equations of the form -ε2Δu+V(x)u=Q(x)f(u)inR2,where V and Q behave like (1 + | x|) -α with α∈ (0 , 2) and (1 + | x|) -β with β∈ (α, + ∞) , respectively. When f has subcritical exponential growth—by means of a weighted Trudinger–Moser-type inequality and the mountain pass theorem in weighted Sobolev spaces—we prove the existence of nontrivial mountain pass solutions, for any ε> 0 , and in the semi-classical limit, these solutions concentrate at a global minimum point of A= V/ Q. Our existence result holds also when f has critical growth, for any ε> 0.