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In this paper, we consider the singularly perturbed fractionalSchr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alphau+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is asmall parameter, $\alpha\in(0,1)$, $(-\Delta)^\alpha$ is the fractionalLaplacian operator of order $\alpha$, $V$ possesses global minimum points, and$f$ is asymptotically linear at infinity. We investigate the relationshipbetween the number of positive solutions and the topology of the set where thepotential $V$ attains its global minimum. We also construct multipleconcentrating solutions if $V$ has several strict global minimum points. Inparticular, some new tricks and the method of Nehari manifold dependent on asuitable restricted set are introduced to overcome the difficulty resultingfrom the appearance of asymptotically linear nonlinearity.

Multiple semiclassical states for fractional Schrodinger equations with asymptotically linear nonlinearities