An existence result for sign-changing solutions of the Brézis–Nirenberg problem

Abstract We consider the Brezis–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.