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

We deal with the existence of positive solutions for the following fractional Schr"odinger equation $${\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}u+V(x)u=f(u)in{\mathbb{R}}^N,$$ where $\epsilon >0$ is a parameter, $s\in (0, 1)$,$N>2s$,$(-\Delta)^{s}$isthefractionalLaplacianoperator,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 $\epsilon$ tends to zero.