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

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 us...

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Bibliographic Details
Main Authors An, Xiaoming, Duan, Lipeng, Peng, Yanfang
Format Journal Article
LanguageEnglish
Published 08.07.2019
Subjects
Mathematics - Analysis of PDEs
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Summary: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.
DOI:10.48550/arxiv.1907.03908