We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983): $$-\Delta u=\lambda u+ |u|^{2^\ast-2}u, \quad u\in H_0^1(\Omega),$$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.

The Sandwich Pair theorems have presented very efficient ways to determine the existence of critical points or critical sequences for nonlinear differentiable functionals. In this paper, under rather weak hypotheses new relationships are established between sign-changing critical points and Sandwich Pairs or Linking Sandwich Pairs. The abstract results are demonstrated by applications on semi-linear elliptic equations. © 2013 Elsevier Ltd. All rights reserved.