Nonlinear degenerate elliptic equations in weighted Sobolev spaces

Abstract

We study the existence of solutions for the nonlinear degenerated elliptic problem -div α(x, u, ∇u) = ƒ in Ω u = 0 on ∂Ω, where Ω is a bounded open set in ℝN, N ≥ 2, α is a Carathéodory function having degenerate coercivity α(x, u, ∇u)∇u ≥ v(x)b(|u|)|∇u|p, 1 < p < N, v(∙) is the weight function, b is continuous and ƒ ∈ Lr(Ω).