On the 3-D Brezis–Nirenberg problem with small parameter

In 2023, H. Brezis [Some of my favorite open problems, Rend. Lincei 34(2) (2023) 307–335] published a list of his “favorite open problems”, which he described as challenges he had “raised throughout his career and has resisted so far”. In this paper, we shall provide a partial answer to this question by presenting the existence of sign-changing solutions to the equation whenever the parameter is small enough. Our construction is based on the building blocks of Del Pino–Musso–Pacard–Pistoia sign-changing solutions to Yamabe problem.