On a class of nonlocal Schrödinger equations with exponential growth

Abstract

We consider the following singularly perturbed Schrödinger equation involving the $ \frac Ns $-fractional Laplacian operator,$$ \varepsilon^{N}(-\Delta)_{ \frac Ns } ^{s}u+V(x)|u|^{\frac{N}{s}-2}u=f(u) \quad \text{in } \mathbb R^{N},$$where $\varepsilon$ is a positive parameter, $s\in (0,1)$, the potential $V$ is positive and away from zero, and $f$ is a Trudinger-Moser type nonlinearity. By using penalization methods andLusternik-Schnirelmann's theory, we examine existence, multiplicity and concentration of non-trivial non-negative solutions for small values of $\varepsilon$.