Multi-peak semiclassical bound states for Fractional Schr\"{o}dinger Equations with fast decaying potentials

Abstract

We study the following fractional Schr\"{o}dinger equation \begin{equation*}\label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} 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 $V\in C(\R^N,[0,+\infty))$. 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 $(-\Delta)^s$ 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.