Asymptotics for the best Sobolev constants and their extremal functions

Abstract

Let Ω be a bounded and smooth domain of R N , N ≥ 2, and consider

Minimization of quotients with variable exponents

“…Thus, in order to overcome the difficulties imposed by the fact that the exponents depend on x, we adapt arguments developed by Franzina and Lindqvist in [18], where p(x) = q(x). Actually, our results in the present paper generalize those of [8] for variable exponents and complement the approach of [18].…”

“…The distinction seems to be due to the Dirac delta that appears in the right-hand term of the Euler-Lagrange equation (2) when q(x) is replaced by jq(x) and j is taken to infinity. The same distinction appears when p and q are constant, as one can check from [8] and [20].…”

“…(Ω) into the Lebesgue space L q(x) (Ω), where the variable exponents satisfy In this paper we study the behavior of the least Rayleigh quotients when the functions p(x) and q(x) become arbitrarily large. Our script is based on the paper [8], where these functions are constants. Thus, in order to overcome the difficulties imposed by the fact that the exponents depend on x, we adapt arguments developed by Franzina and Lindqvist in [18], where p(x) = q(x).…”

“…In [8], Ercole and Pereira first studied the behavior, when q → ∞, of the positive minimizers w q corresponding to λ q := min ∇u L p (Ω) : u ∈ W 1,p 0 (Ω) in u L q (Ω) = 1 , for a fixed p > N. An L ∞ -normalized function u p ∈ W 1,p 0 (Ω) is obtained as the uniform limit in Ω of a sequence w qn , with q n → ∞. Such a function is positive in Ω, assumes its maximum only at a point x p and satisfies −∆ p u = Λ p δ xp in Ω u = 0 on ∂Ω, where Λ p := min ∇u L p (Ω) : u ∈ W 1,p 0 (Ω) in u L ∞ (Ω) = 1 and δ xp denotes the Dirac delta distribution concentrated at x p .…”

Extremal functions for Morrey’s inequality in convex domains

“…which was derived by Ercole and Pereira in [9]. Here ∆ p ψ := div(|Dψ| p−2 Dψ) is the p-Laplacian, and x 0 is the unique point for which |u| is maximized in Ω.…”