On the Brézis–Nirenberg Problem.

We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983): 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. [ABSTRACT FROM AUTHOR]