Some remarks on elliptic equations with singular potentials and mixed boundary conditions

Abstract

We study the problems {-div(|x|-2γ∇u) = λ|x|-2(γ+1)u + f(x, u), u ≥ 0 in Ω, (0.1) B(u) = 0 on ∂Ω where -∞ < γ 0, Ω ⊂ IRN is a smooth bounded domain with 0 ∈ Ω, and B(u) = uχ∑1 + |x| -2γ∂u/∂vχ∑2, mixed boundary condition. Mainly we will be interested in the behavior of the solutions to (0.1) close to the critical constant in the Hardy-Sobolev inequality, ΛN,γ(Ω, ∑1).