Multiple semiclassical states for fractional Schrodinger equations with asymptotically linear nonlinearities

Abstract

In this paper, we consider the singularly perturbed fractional Schr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alpha u+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is a small parameter, $\alpha\in(0,1)$, $(-\Delta)^\alpha$ is the fractional Laplacian operator of order $\alpha$, $V$ possesses global minimum points, and $f$ is asymptotically linear at infinity. We investigate the relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its global minimum. We also construct multiple concentrating solutions if $V$ has several strict global minimum points. In particular, some new tricks and the method of Nehari manifold dependent on a suitable restricted set are introduced to overcome the difficulty resulting from the appearance of asymptotically linear nonlinearity.