An existence result for sign-changing solutions of the Brézis–Nirenberg problem. (October 2018) - He, Tieshan; Zhang, Chaolong; Wu, Dongqing; Liang, Kaihao
Abstract: We consider the Brézis–Nirenbergproblem: { − Δ u = | u | 2 ⋆ − 2 u + λ u in Ω, u = 0 on ∂ Ω, where Ω is a smooth bounded domain in R N, N ≥ 3, 2 ⋆ = 2 N N − 2 is the critical Sobolev exponent and λ > 0 . Our main result asserts that if N ≥ 4 then there exists a pair of sign-changing solutions of the problem for every λ ∈ ( 0, λ 1 ( Ω ) ), λ 1 ( Ω ) being the first eigenvalue of − Δ in Ω with Dirichlet boundary conditions, while if N = 3 then a pair of sign-changing solutions exists for λ slightly smaller than λ 1 ( Ω ) . Our approach uses variational methods together with flow invariance arguments.