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

Let $\mathrm{\Omega }$ be a smooth, bounded domain of ${\mathbb{R}}^N$, $\omega$ be a positive, $L^1$-normalized function, and $0<s<1<p\mathrm{.}$ We study the asymptotic behavior, as $p\to \mathrm{\infty },$ of the pair $\left( \sqrt[p]{\Lambda_{p}% },u_{p}\right) ,$where$\Lambda_{p}$isthebestconstant$C$ in the Sobolev type inequality $$C\exp ({\int }_{\mathrm{\Omega }}(\log {\mid u\mid }^p)\omega \mathrm{d}x)\le {\left[u\right]}_{s,p}^p\mathrm{\forall }u\in W_0^{s,p}(\mathrm{\Omega })$$ and $u_p$ is the positive, suitably normalized extremal function corresponding to ${\mathrm{\Lambda }}_p$. 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}$denotesthe$s$-H\ddot{o}lderseminormofafunction$u\in C{0}^{0,s}(\overline{\Omega}).$