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

Abstract

Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ 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 $n\ge7$ and $M$ 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.