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

TL;DR

This paper investigates multi-peak solutions to fractional Schrödinger equations with fast decaying potentials, demonstrating the existence of solutions concentrating at local minima, using penalized and variational methods.

Contribution

It introduces new results on multi-peak solutions for fractional Schrödinger equations with general decaying potentials, including compact support, highlighting the nonlocal effects.

Findings

Solutions concentrate at local minima of V

All decay rates of V are admissible

Multiple bumps influence each other due to nonlocal effects

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.