Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials

We study the following fractional Schrödinger equation

$${\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}u+V(x)u=f(u),x\in {\mathbb{R}}^N,$$

where $s\in (0,1)$. Under some conditions on $f(u)$, we show that the problem has a family of solutions concentrating at any finite given local minima of $V$ provided that $V\in C({\mathbb{R}}^N,[0,+\mathrm{\infty }))$. All decay rates of $V$ are admissible. Especially, $V$ can be compactly supported. Different from the local case $s=1$ or the case of single-peak solutions, the nonlocal effect of the operator $(-\mathrm{\Delta }{)}^s$ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.