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

Abstract

<abstract><p>We study the following fractional Schrödinger equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} $\end{document} </tex-math> </disp-formula></p> <p>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,+\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.</p></abstract>