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 \(d_{\Omega }.\)
