On the Brézis-Nirenberg problem

Abstract

We study the following Brézis-Nirenberg problem (Comm Pure Appl Math 36:437-477, 1983): where Ω is a bounded smooth domain of RN (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. © 2010 The Author(s).