Keller-Osserman estimates for some quasilinear elliptic ... - LMPT

Lemma 2.5 Let A p be S-p-C, and u ∈ W 1,ploc(Ω) be nonnegative, such that−A p u ≦ 0 in Ω;then for any s > 0, there exists a constant C = C(N, p, s, K 1,p , K 2,p ), such that for any ballB(x 0 , 2ρ) ⊂ Ω and any ε ∈ 0, 1 2,sup u ≤ Cε − Np2s 2B(x 0 ,ρ)IB(x 0 ,ρ(1+ε))u s ! 1s. (2.13)Proof. From a slight adaptation of the usual case where ε = 1 2, for any l > p − 1, there existsC = C(N, l) > 0 such that for any ε ∈ 0, 1 2,sup u ≦ Cε −NB(x 0 ,ρ)IB(x 0 ,ρ(1+ε))u l ! 1l. (2.14)Thus we can assume s ≤ p − 1. We fix for example l = p, and define a sequence (ρ n ) by ρ 0 = ρ,and ρ n = ρ(1 + ε 2 + ... + ( ε 2 )n ) for any n ≥ 1, and we set M n = sup B(x0 ,ρ n) u p . From (2.14) weobtain, with new constants C = C(N, p),M n ≦ C( ρ n+1− 1) −Np uρ nIB(x p ≤ C( ε I0 ,ρ n+1 ) 2 )−(n+1)Np u p .B(x 0 ,ρ n+1 )From the Young inequality, for any δ ∈ (0, 1), and any r < 1, we obtainM n ≦ C( ε I2 )−(n+1)Np Mn+11−r u prB(x 0 ,ρ n+1 )≦ δM n+1 + rδ 1−1/r (C( ε 2 )−(n+1)Np ) 1 rIB(x 0 ,ρ n+1 )u pr ! 1r.Defining κ = rδ 1−1/r C 1 rand b = ( ε 2 )−Np/r , we findM n ≦ δM n+1 + b n+1 κIB(x 0 ,ρ n+1 )u pr ! 1r.Taking δ = 1 2band iterating, we obtainM 0 =supB(x 0 ,ρ)u p ≦ δ n+1 M n+1 + bκ≦ δ n+1 M n+1 + 2bκInX(δb) ii=0B(x 0 ,ρ n+1 )u pr ! 1r.IB(x 0 ,ρ n+1 )u pr ! 1r10