Revisiting an idea of Brézis and Nirenberg

Let n 3 and Ω be a C1 bounded domain in Rn with 0 ∈ ∂Ω. Suppose ∂Ω is C2 at 0 and the mean curvature of ∂Ω at 0 is negative, we prove the existence of positive solutions for the equation: ⎧⎨⎩ u +λu n+2 n−2 + u2∗(s)−1 |x|s =0 inΩ, u =0 on∂Ω, (0.1) where λ > 0, 0 < s <2, 2∗(s) = 2(n−s) n−2 and n 4. For n = 3, the existence result holds for 0 < s <1. Under the same assumption of the domain Ω, for p 2∗(s) − 1, we also prove the existence of a positive solution for the following equation: ⎧⎨⎩ u −λup + u2∗(s)−1 |x|s =0 inΩ, u =0 on∂Ω, (0.2) where λ>0 and 1 p < n n− 2 . © 2010 Elsevier Inc. All rights reserved