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

Abstract

We consider $$\varepsilon $$-perturbed nonlinear Schrodinger equations of the form $$\begin{aligned} - \varepsilon ^2\Delta u + V(x)u = Q(x)f(u) \quad \text {in } \mathbb {R}^2, \end{aligned}$$where V and Q behave like $$(1+|x|)^{-\alpha }$$ with $$\alpha \in (0,2)$$ and $$(1+|x|)^{-\beta }$$ with $$\beta \in (\alpha , + \infty )$$, 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 $$\varepsilon >0$$, and in the semi-classical limit, these solutions concentrate at a global minimum point of $$\mathcal A=V/Q$$. Our existence result holds also when f has critical growth, for any $$\varepsilon >0$$.