Asymptotics for the best Sobolev constants and their extremal functions

Let $\mathrm{\Omega }$ be a bounded domain of ${\mathbf{R}}^N,$ $N\ge 2.$ Let, for $p>N,$ $${\mathrm{\Lambda }}_p(\mathrm{\Omega }):=\inf \{{\parallel \mathrm{\nabla }u\parallel }_p^p:u\in W_0^{1,p}(\mathrm{\Omega })and{\parallel u\parallel }_{\mathrm{\infty }}=1\}\mathrm{.}$$ We first prove that $$\underset{p\to \mathrm{\infty }}{\lim }{\mathrm{\Lambda }}_p(\mathrm{\Omega }{)}^{\frac{1}{p}}=\frac{1}{{\parallel \rho \parallel }_{\mathrm{\infty }}},$$ where $\rho$ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions of ${\mathrm{\Lambda }}_p(\mathrm{\Omega })$ converge (as $p\to \mathrm{\infty }$) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured $\mathrm{\Omega }\mathrm{.}$