On the Brézis–Nirenberg Problem

We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983): -Δ u=λ u+ |u|2⁻2u, u∈ H₀¹(Ω), where Ω is a bounded smooth domain of R N (N ≧ 7) and 2* is the critical Sobolev exponent. We show that, for each fixed λ > 0, this problem has infinitely many sign-changing solutions. In particular, if λ ≧ λ1, the Brézis–Nirenberg problem has and only has infinitely many sign-changing solutions except zero. The main tool is the estimates of Morse indices of nodal solutions.