# Semi-classical analysis for Fractional Schr\"{o}dinger Equations with fast decaying potenials

## TL;DR

This paper analyzes fractional Schr"{o}dinger equations with rapidly decaying potentials, demonstrating the existence of solutions concentrating at local minima using penalized techniques, even when the potential decays arbitrarily or is compactly supported.

## Contribution

It introduces a novel approach to find solutions concentrating at local minima for fractional Schr"{o}dinger equations with highly decaying potentials, expanding the class of potentials considered.

## Key findings

- Solutions concentrate at local minima of V(x).
- Existence of solutions for a broad class of decaying potentials.
- Applicable to potentials with arbitrary decay or compact support.

## Abstract

We study the following fractional Schr\"{o}dinger equation \begin{equation*}\label{eq0.1} \epsilon^{2s}(-\Delta)^s u + V(x)u = |u|^{p - 2}u, \,\,x\in\,\,\mathbb{R}^N, \end{equation*} where $s\in (0,\,1)$, $N>2s$, $p>1$ is subcritical and $V(x)$ is a nonnegative continuous potential. We use penalized technique to show that the problem has a family of solutions concentrating at a positive local minimum of $V(x)$ provided that $\frac{2s}{N-2s}+2<p<\frac{2N}{N-2s}$. The novelty is that $V$ can decay arbitrarily or even be compactly supported.