Non-degeneracy of the multi-bump solutions to the Brezis-Nirenberg problem

We revisit the well-known Brezis-Nirenberg problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u= u^{\frac{N+2}{N-2}}+\varepsilon u, &{}{{\text {in}}~\Omega },\\ u>0, &{}{{\text {in}}~\Omega },\\ u=0, &{}{\text {on}~\partial \Omega }, \end{array}\right. } \end{aligned}$$

where \(\varepsilon >0\) and \(\Omega \subset \mathbb {R}^N\) are a smooth bounded domain with \(N\ge 3\). The existence of multi-bump solutions to above problem for small parameter \(\varepsilon >0\) was obtained by Musso and Pistoia (Indiana Univ Math J 51:541–579, 2002). However, to our knowledge, whether the multi-bump solutions are non-degenerate that is open. Here, we give some straightforward answer on this question under some suitable assumptions for the Green’s function of \(-\Delta \) in \(\Omega \), which enriches the qualitative analysis on the solutions of Brezis-Nirenberg problem and can be viewed as a generalization of Grossi (Nonlinear Differ Equ Appl 12:227–241, 2005) where the non-degeneracy of a single-bump solution has been proved. And the main idea is the blow-up analysis based on the local Pohozaev identities.