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

Abstract

We deal with the existence of positive solutions for the following fractional Schrödinger equation $$ \varepsilon ^{2s} (-Δ)^{s} u + V(x) u = f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\epsilon >0$ is a parameter, $s\in (0,1)$, $N>2s$, $(-\Delta {)}^s$ is the fractional Laplacian operator, and $V:{\mathbb{R}}^N\to \mathbb{R}$ 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.

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