Revisiting an idea of Brézis and Nirenberg

Abstract

Let n 3 and Ω be a C 1 bounded domain in R n with 0 ∈ ∂Ω. Suppose ∂Ω is C 2 at 0 and the mean curvature of ∂Ω at 0 is negative, we prove the existence of positive solutions for the equation: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:where λ > 0 and 1 p < n n − 2 .

Infinitely many solutions for quasilinear elliptic equations involving double critical terms and boundary geometry

“…Recently, some attention is paid to elliptic problems with double critical terms together with boundary geometry conditions on the domain. For instance, among other problems, Hsia, Lin and Wadade [23] considered the following equation where µ > 0. Note that equation (1.5) is the special case of equation (1.1) when p = 2 and a ≡ 0.…”

“…Note that equation (1.5) is the special case of equation (1.1) when p = 2 and a ≡ 0. Assuming that 0 ∈ ∂Ω and the mean curvature of ∂Ω at 0 is negative, Hsia, Lin and Wadade [23] proved the existence of positive solutions to equation (1.5) for all 0 < s < 2 when N ≥ 4, and for 0 < s < 1 when N = 3. For more results in this respect, we refer to e.g.…”

Multiple positive solutions for a nonlinear elliptic equation involving Hardy-Sobolev-Maz’ya term

“…With similar argument, Deng and Peng in [15] obtained multiple positive solutions for (1.1) μ with Neumann boundary condition in the case λ = 0, t = 0. For more other results, we refer the readers to [18][19][20] and the reference therein.…”

Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

“…Lin et al [37,[45][46][47] and Ghoussoub and Robert [31] when γ = 0. It consists of considering domains where the singularity 0 is on the boundary.…”