Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems

TL;DR

This paper investigates the asymptotic behavior of extremal functions for fractional Sobolev inequalities as the exponent p approaches infinity, revealing their relation to a minimization problem involving the s-Hölder seminorm.

Contribution

It characterizes the limit behavior of extremals and best constants in fractional Sobolev inequalities for large p, connecting them to a specific minimization problem.

Findings

Limit pairs relate to a minimization problem involving the s-Hölder seminorm.

As p approaches infinity, extremals converge to solutions of a related minimization problem.

The asymptotic analysis provides insight into the structure of extremals for fractional Sobolev inequalities.

Abstract

Let $\Omega$ be a smooth, bounded domain of $\mathbb{R}^{N}$, $\omega$ be a positive, $L^{1}$-normalized function, and $0<s<1<p.$ We study the asymptotic behavior, as $p\rightarrow\infty,$ of the pair $\left( \sqrt[p]{\Lambda_{p}% },u_{p}\right) ,$ where $\Lambda_{p}$ is the best constant $C$ in the Sobolev type inequality \[ C\exp\left( \int_{\Omega}(\log\left\vert u\right\vert ^{p})\omega \mathrm{d}x\right) \leq\left[ u\right] _{s,p}^{p}\quad\forall\,u\in W_{0}^{s,p}(\Omega) \] and $u_{p}$ is the positive, suitably normalized extremal function corresponding to $\Lambda_{p}$. We show that the limit pairs are closely related to the problem of minimizing the quotient $\left\vert u\right\vert _{s}/\exp\left( \int_{\Omega}(\log\left\vert u\right\vert )\omega \mathrm{d}x\right) ,$ where $\left\vert u\right\vert _{s}$ denotes the $s$-H\"{o}lder seminorm of a function $u\in C_{0}^{0,s}(\overline{\Omega}).$