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

In 2023, H. Brezis [2] 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 signchanging 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.