Competition reaction–absorption in some elliptic and parabolic problems related to the Caffarelli–Kohn–Nirenberg inequalities

We consider the following elliptic problem: −div(|x|−2γ ∇u) + u |x|2(γ+1) = f (x,u), u 0 in Ω, u|∂Ω = 0 (1) and the corresponding parabolic version    ut − div(|x|−2γ ∇u) + u |x|2(γ+1) = f (x,u), u 0 in Ω ×(0,T ), u(x, t) =0 for(x, t) ∈ ∂Ω ×(0,T ), u(x, 0) = ϕ(x) if x ∈ Ω, (2) Ω ⊂ RN is a smooth bounded domain with 0 ∈ Ω and −∞<γ < N−2 2 . We will consider in particular the critical case where f (x,u) = λ uα |x|2(γ+1) and α 1. Conditions on γ , λ, α are given to obtain existence and nonexistence result.  2005 Elsevier Inc. All rights reserved