Semi-classical Solutions For Fractional Schrodinger Equations With Potential Vanishing At Infinity

TL;DR

This paper investigates semi-classical solutions to fractional Schrödinger equations with potentials that vanish at infinity, demonstrating solution concentration at potential minima using variational and penalized methods.

Contribution

It introduces a novel approach to find solutions concentrating at potential minima for fractional Schrödinger equations with vanishing potentials.

Findings

Solutions concentrate at local minima of the potential.

Existence of solutions depends on potential behavior at infinity.

Variational and penalized techniques effectively find solutions.

Abstract

We study the following fractional Schr\"{o}dinger equation \begin{equation}\label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + Vu = |u|^{p - 2}u,\ \ x\in\,\,\mathbb{R}^N. \end{equation} We show that if the external potential $V\in C(\mathbb{R}^N;[0,\infty))$ has a local minimum and $p\in (2 + 2s/(N - 2s), 2^*_s)$, where $2^*_s=2N/(N-2s),\,N\ge 2s$, the problem has a family of solutions concentrating at the local minimum of $V$ provided that $\liminf_{|x|\to \infty}V(x)|x|^{2s} > 0$. The proof is based on variational methods and penalized technique. {\textbf {Key words}: } fractional Schr\"{o}dinger; vanishing potential; penalized technique; variational methods.