We study the existence of semi-classical bound states of the nonlinear Schrödinger equation \begin{linenomath} -\varepsilon2\Delta u+V(x)u=f(u),\quad x\in {\bf R}^N,\end{linenomath} where N ≥ 3;, Ï is a positive parameter; V:R N → [0, ∞) satisfies some suitable assumptions. We study two cases: if f is asymptotically linear, i.e., if lim|t| → ∞ f(t)/t=constant, then we get positive solutions. If f is superlinear and f(u)=|u| p-2 u+|u| q-2 u, 2* > p > q > 2, we obtain the existence of multiple sign-changing semi-classical bound states with information on the estimates of the energies, the Morse indices and the number of nodal domains. For this purpose, we establish a mountain cliff theorem without the compactness condition and apply a new sign-changing critical point theorem. © 2013 Cambridge Philosophical Society.

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.