Asymptotics for Sobolev extremals: the hyperdiffusive case

Let $\mathrm{\Omega }$ be a bounded, smooth domain of ${\mathbb{R}}^N,$ $N\ge 2.$ For $p>N$ and $1\le q(p)<\mathrm{\infty }$ set $${\lambda }_{p,q(p)}:=\inf \{{\int }_{\mathrm{\Omega }}{\mid \mathrm{\nabla }u\mid }^p\mathrm{d}x:u\in W_0^{1,p}(\mathrm{\Omega })\text{ and }{\int }_{\mathrm{\Omega }}{\mid u\mid }^{q(p)}\mathrm{d}x=1\}$$ and let $u_{p,q(p)}$ denote a corresponding positive extremal function. We show that if $\underset{p\to \mathrm{\infty }}{\lim }q(p)=\mathrm{\infty }$, then $\underset{p\to \mathrm{\infty }}{\lim }{\lambda }_{p,q(p)}^{1\mathrm{/}p}={\parallel d_{\mathrm{\Omega }}\parallel }_{\mathrm{\infty }}^{-1}$, where $d_{\mathrm{\Omega }}$ denotes the distance function to the boundary of $\mathrm{\Omega }\mathrm{.}$ Moreover, in the hyperdiffusive case: $\underset{p\to \mathrm{\infty }}{\lim }\frac{q(p)}{p}=\mathrm{\infty },$ we prove that each sequence $u_{p_n,q(p_n)},$ with $p_n\to \mathrm{\infty },$ admits a subsequence converging uniformly in $\bar{\mathrm{\Omega }}$ to a viscosity solution to the problem \left\{ \begin{array} [c]{lll} -Δ_{\infty}u=0 & \text{in} & Ω\setminus M\\ u=0 & \text{on} & \partialΩ\\ u=1 & \text{in} & M, \end{array} \right. where $M$ is a closed subset of the set of all maximum points of $d_{\mathrm{\Omega }}\mathrm{.}$