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

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 $p\to \infty ,$ of the pair $\left( \sqrt[p]{Λ_{p}% },u_{p}\right) ,$ where ${\Lambda }_p$ 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 ${\Lambda }_p$. We show that the limit pairs are closely related to the problem of minimizing the quotient ${\left|u\right|}_s/\exp \left({\int }_{\Omega }(\log \left|u\right|)\omega \mathrm{d}x\right),$ where ${\left|u\right|}_s$ denotes the $s$-Hölder seminorm of a function $u\in C_0^{0,s}(\overline{\Omega }).$

24 pages, version accepted for publication in Annali di Matematica Pura ed Applicata