A constructive method for obtaining the best Sobolev constants of the unit ball and their extremal functions Grey ERCOLE a, * , J´ ulio C´ esar do ESP ´ IRITO SANTO b , Eder Marinho MARTINS b † a Departamento de Matem´ atica - ICEx, Universidade Federal de Minas Gerais, Av. Antˆ onio Carlos 6627, Caixa Postal 702, 31270-901, Belo Horizonte, MG, Brazil. b Departamento de Matem´ atica - ICEB, Universidade Federal de Ouro Preto, Campus Universit´ ario Morro do Cruzeiro, 35400-000, Ouro Preto, MG, Brazil. May 31, 2015 Abstract Let B1 be the unit ball of R N ,N ≥ 2, and let p ⋆ = Np/(N − p) if 1 <p<N and p ⋆ = ∞ if p ≥ N. It is well known that, for each q ∈ [1,p ⋆ ), there exists a unique positive function wq ∈ W 1,p 0 (B1) such that ‖wq ‖ L q (B 1 ) = 1 and λq (B1) := min  ‖∇u‖ p L p (B 1 ) ‖u‖ p L q (B 1 ) :0 ≡ u ∈ W 1,p 0 (B1)  = ‖∇wq ‖ p L p (B 1 ) . Moreover, the extremal funcion wq is radial. In this paper we develop a constructive method for obtaining the pair (λq (B1),wq ) starting from ws for any 1 ≤ s < q. Since the extremal function w1 is explicitly known, the method is computationally practical, as our numerical tests show. 2010 Mathematics Subject Classification. 34L16; 35J25; 65N25 Keywords: Best Sobolev constant; extremal functions; inverse iteration method; p-Laplacian. 1 Introduction Let Ω be a bounded domain of R N ,N ≥ 2. It is well known that the Sobolev immersion W 1,p 0 (Ω)  → L q (Ω) is continuous if 1 ≤ q ≤ p ⋆ , where p ⋆ = Np/(N − p) if 1 <p<N and p ⋆ = ∞ if p ≥ N. As a consequence of this fact one can define λ q (Ω) := inf   Ω |∇u| p dx ( Ω |u| q dx ) p q :0 ≡ u ∈ W 1,p 0 (Ω)  . (1) In the case 1 ≤ q<p ⋆ the Sobolev immersion W 1,p 0 (Ω)  → L q (Ω) is also compact. Thus, in this case, one can show that λ q (Ω) is reached by at least one extremal function w q ∈ W 1,p 0 (Ω) and, moreover, this extremal function can be chosen positive and L q -normalized, that is, λ q (Ω) = ‖∇w q ‖ p p ,w q > 0 in Ω, and ‖w q ‖ q =1, (2) where ‖·‖ r = ( Ω |·| r dx ) 1 r stands for the standard norm of L r (Ω). In the remaining of this paper, w q denotes any function satisfying (2). * Corresponding author † E-mail addresses: grey@mat.ufmg.br (G. Ercole), cesares@iceb.ufop.br (J. C. E. Santo), eder@iceb.ufop.br (E. Martins). 1