Sign‐changing solutions to the Yamabe problem on manifolds with boundary

Abstract Let be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to having constant scalar curvature on and constant mean curvature on its boundary is equivalent, in analytic terms, to finding a positive solution to a nonlinear boundary‐value problem with critical growth. While the existence of positive solutions to this problem is by now well understood, the existence of sign‐changing (nodal) solutions remains largely open. In this work, we establish the existence of least‐energy sign‐changing solutions when the manifold is positive and the mean curvature of the boundary is a non‐negative constant. More precisely, we prove that if and has a nonumbilic boundary point, then the problem admits least‐energy nodal solutions. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates.