Concentrating solutions for a class of nonlinear fractional Schr\"odinger equations in $\mathbb{R}^{N}$

Abstract

We deal with the existence of positive solutions for the following fractional Schr\"odinger equation $$ \varepsilon ^{2s} (-\Delta)^{s} u + V(x) u = f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian operator, and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous positive function. Under the assumptions that the nonlinearity $f$ is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of $V$ as $\varepsilon$ tends to zero.