An existence result for sign-changing solutions of the Brézis–Nirenberg problem. (October 2018)

• Record Type: Journal Article
• Authors: He, Tieshan; Zhang, Chaolong; Wu, Dongqing; Liang, Kaihao
• Is Part Of: Applied mathematics letters. Volume 84(2018)
• Page Start: 90
• Page End: 95
• Publication Date: 2018-10
• Subjects: Critical exponent -- Nodal solutions -- Variational approach -- Gradient flow; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2018.04.020
• Languages: English
• ISSNs: 0893-9659

Abstract

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.