Asymptotics for Sobolev extremals: the hyperdiffusive case

Abstract Let $$\Omega $$ be a bounded domain of $$\mathbb {R}^{N},$$ $$N\ge 2.$$ For $$p>N$$ and $$1\le q(p)<\infty $$ set $$ \lambda _{p,q(p)}:=\inf \left\{ \int _{\Omega }\left| \nabla u\right| ^{p}\textrm{d}x:u\in W_{0}^{1,p}(\Omega )\text { \ and \ }\int _{\Omega }\left| u\right| ^{q(p)}\textrm{d}x=1\right\} $$ and denote by $$u_{p,q(p)}$$ a positive extremal function corresponding to $$\lambda _{p,q(p)}$$ . We show that if $$\lim \limits _{p\rightarrow \infty }q(p)=\infty $$ , then $$\lim \limits _{p\rightarrow \infty }\lambda _{p,q(p)} ^{1/p}=\left\| d_{\Omega }\right\| _{\infty }^{-1}$$ , where $$d_{\Omega }$$ denotes the distance function to the boundary of $$\Omega .$$ Moreover, in the hyperdiffusive case: $$\lim \limits _{p\rightarrow \infty }\frac{q(p)}{p}=\infty ,$$ we prove that each sequence $$u_{p_{n},q(p_{n})},$$ with $$p_{n}\rightarrow \infty ,$$ admits a subsequence converging uniformly in $$\overline{\Omega }$$ to a viscosity solution to the problem $$ \left\{ \begin{array}{lll} -\Delta _{\infty }u=0 & \text {in} & \Omega \setminus M\\ u=0 & \text {on} & \partial \Omega \\ u=1 & \text {in} & M, \end{array} \right. $$ where M is a closed subset of the set of all maximum points of $$d_{\Omega }.$$