Asymptotics for Sobolev extremals: the hyperdiffusive case

Let Ω be a bounded, smooth domain of RN, N≥2. For $p>N$ and 1≤ q(p)<∞ set \[ λp,q(p):=inf\{ ∫Ω ∇ u ᵖdx:u∈ W₀1,p(Ω) \ and \ ∫Ω u q(p)dx=1\} \] and let up,q(p) denote a corresponding positive extremal function. We show that if limp→∞q(p)=∞, then limp→∞λp,q(p)1/p= dΩ ∞⁻¹, where dΩ denotes the distance function to the boundary of Ω. Moreover, in the hyperdiffusive case: limp→∞q(p)/p=∞, we prove that each sequence u_pₙ,q(pₙ), with pₙ→∞, admits a subsequence converging uniformly in Ω̄ to a viscosity solution to the problem \[ \{ {array} [c]{lll} -Δ∞u=0 & in & Ω M\\ u=0 & on & ∂Ω\\ u=1 & in & M, {array} . \] where M is a closed subset of the set of all maximum points of dΩ.