caa a22 4500 605496498 CHVBK 20210128100536.0 cr unu---uuuuu 210128e20151001xx s 000 0 eng 10.1007/s10231-014-0425-3 doi (NATIONALLICENCE)springer-10.1007/s10231-014-0425-3 Ercole Grey Departamento de Matemática - ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Caixa Postal 702, 30161-970, Belo Horizonte, MG, Brazil aut Regularity results for the best-Sobolev-constant function [Elektronische Daten] [Grey Ercole] We derive some new regularity results for the best-Sobolev-constant function defined by $$\begin{aligned} q\in [1,p^{\star }]\mapsto \lambda _{q}:=\inf \left\{ \left\| \nabla u\right\| _{p}^{p}\,/\,\left\| u\right\| _{q}^{p}:u\in W_{0}^{1,p} (\Omega )\setminus \{0\}\right\} , \end{aligned}$$ q ∈ [ 1 , p ⋆ ] ↦ λ q : = inf ∇ u p p / u q p : u ∈ W 0 1 , p ( Ω ) \ { 0 } , where $$\Omega $$ Ω is a bounded and smooth domain of $$\mathbb {R}^{N},\,1<p<N$$ R N , 1 < p < N and $$p^{\star }:=\frac{Np}{N-p}$$ p ⋆ : = N p N - p . In a previous work, we proved that this function is absolutely continuous and thus its derivative $$\lambda _{q}^{\prime }$$ λ q ′ exists at almost all $$q\in [1,p^{\star }]$$ q ∈ [ 1 , p ⋆ ] . In this paper, we prove that $$\lambda _{q}^{\prime }$$ λ q ′ exists if, and only if, the functional $$\begin{aligned} I_{q}(u):=\int \limits _{\Omega }\left| u\right| ^{q}\log \left| u\right| \hbox {d}x \end{aligned}$$ I q ( u ) : = ∫ Ω u q log u d x is constant on the set $$E_{q}$$ E q of the $$L^{q}$$ L q -normalized extremal functions corresponding to $$\lambda _{q}$$ λ q . Moreover, we prove that the existence of $$\lambda _{q}^{\prime }$$ λ q ′ is also equivalent to the continuity at $$q$$ q of the function $$s\in [1,p^{\star })\mapsto I_{s}(u_{s})$$ s ∈ [ 1 , p ⋆ ) ↦ I s ( u s ) , where $$u_{s}$$ u s is any function in $$E_{s}$$ E s . It follows from these results that $$\lambda _{q}^{\prime }$$ λ q ′ exists and is continuous if $$q\in [1,p]$$ q ∈ [ 1 , p ] and $$\Omega $$ Ω is a general bounded domain and also if $$q\in (p,p^{\star })$$ q ∈ ( p , p ⋆ ) and $$\Omega $$ Ω is a ball. After deriving some estimates for $$I_{q}(u_{q})$$ I q ( u q ) , we also prove, under an expected asymptotic behavior of $$u_{q}$$ u q , as $$q\rightarrow p^{\star }$$ q → p ⋆ , that $$\lambda _{q}$$ λ q is $$\alpha $$ α -Hölder continuous in $$[1,p^{\star }]$$ [ 1 , p ⋆ ] for any $$0<\alpha <1$$ 0 < α < 1 . As a consequence, this Hölder regularity holds for a general bounded domain $$\Omega $$ Ω when $$p=2$$ p = 2 and also for a ball when $$p>1$$ p > 1 . Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 Best-Sobolev-constant function nationallicence $$p$$ p -Laplacian nationallicence Sobolev immersion nationallicence Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/5(2015-10-01), 1381-1392 0373-3114 194:5<1381 2015 194 10231 https://doi.org/10.1007/s10231-014-0425-3 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-014-0425-3 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Ercole Grey Departamento de Matemática - ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Caixa Postal 702, 30161-970, Belo Horizonte, MG, Brazil aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/5(2015-10-01), 1381-1392 0373-3114 194:5<1381 2015 194 10231