Asymptotics for Sobolev extremals: the hyperdiffusive case

Abstract

Let $\Omega$ be a bounded, smooth domain of ${\mathbb{R}}^N,$ $N\ge 2.$ For $p>N$ and $1\le q(p)<\infty$ set \[ λ_{p,q(p)}:=\inf\left\{ \int_Ω\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(Ω)\text{ \ and \ }\int_{Ω}\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} \] and let $u_{p,q(p)}$ denote a corresponding positive extremal function. We show that if $\underset{p\to \infty }{\lim }q(p)=\infty$, then $\underset{p\to \infty }{\lim }{\lambda }_{p,q(p)}^{1/p}={\Vert d_{\Omega }\Vert }_{\infty }^{-1}$, where $d_{\Omega }$ denotes the distance function to the boundary of $\Omega .$ Moreover, in the hyperdiffusive case: $\underset{p\to \infty }{\lim }\frac{q(p)}{p}=\infty ,$ we prove that each sequence $u_{p_n,q(p_n)},$ with $p_n\to \infty ,$ admits a subsequence converging uniformly in $\overline{\Omega }$ 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 $d_{\Omega }.$

16 pages