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

We consider the Brézis-Nirenbergproblem: {−Δu=|u|2⋆−2u+λuinΩ,u=0on∂Ω,where Ω is a smooth bounded domain in RN,N≥3,2⋆=2NN−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,λ1(Ω)), λ1(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 λ1(Ω). Our approach uses variational methods together with flow invariance arguments.