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

Abstract

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