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

and similarly we apply it to the solution v with now f = u µ and l = δ > q − 1 : since A q isW-q-C, we obtainII q−1u µ ≤ Cρ −q v δ δ. (3.5)ϕWe can assume that H ϕ uµ > 0. Indeed if H ϕ uµ = 0, then u = 0 in B(x 0 , ρ 0 ). Then ∇u = 0,thus v δ = 0 and then the estimates are trivially verified. Replacing (3.5) in (3.4) we deduceand similarly for u, henceIϕI (q−1)(p−1)v δ p−1−p−q≤ Cρ µ v δ µδ,ϕI 1v δ δ≤ Cρ −ξ ,ϕϕI 1u µ µ≤ Cρ −γ . (3.6)ϕMoreover, since A q is S-q-C, then from the usual weak Harnack inequality, since v ∈ L ∞ loc (Ω),and ϕ(x) = 1 in B(x 0 , ρ), with values in [0, 1] ,Similarlybecause A p is S-p-C.sup v ≤ CB(x 0 , ρ 2 )IB(x 0 ,ρ)v δ ! 1δsup u ≤ Cρ −γ ,B(x 0 , ρ 2 )I 1≤ v δ δ≤ Cρ −ξ .ϕ(ii) Case µ > p − 1, and δ ≤ q − 1. Here we still apply Corollary 2.2 with ρ ≤ ρ 02, ε ∈ (0, 1/4] ,and a function ϕ satisfying (2.8). Since µ > p − 1, we still obtain (3.4); and for any k > q − 1,and λ large enough,II (q−1)/ku µ ≤ C(ερ) −q v k , (3.7)and from Lemma 2.5,ϕI 1/kv k ≤ sup v ≤ Cε − Nq2δ 2ϕB(x 0 ,ρ(1+ε))ϕIB(x 0 ,ρ(1+2ε))v δ ! 1δ.Then with new constants C, setting m = q + δ −2 Nq 2 (q − 1), and h = (p − 1)µ −1 m,Iϕu µ ≤ Cε −m ρ −qIB(x 0 ,ρ(1+2ε))! (q−1)δv δ, (3.8)hence from (3.4) and (3.8),IB(x 0 ,ρ)Iv δ ≤ Cϕv δ ≤ Cρ −p Iϕ p−1u µ µ≤ Cε −h ρ − pµ+q(p−1)µIB(x 0 ,ρ(1+2ε))! (p−1)(q−1)δµv δ,14