Prescribed Mean Curvature on the ball via a sign change in the derivative

Abstract

We consider the problem of finding a conformal metric on the $n$-dimensional Euclidean ball $B^n$, $n\geq 3$, with zero scalar curvature in the interior and prescribed mean curvature $H$ on the boundary $\partial B^n = S^{n-1}$. For rotationally symmetric $H=H(r)$ satisfying a flatness condition of order $\alpha\in(n-2,\,n-1)$ at each of its finitely many critical points in the region where $H>0$, we prove that the sign-change of $H'(r)$ in that region is sufficient for existence. Combined with the necessary condition of Liu and Wang, this yields a characterization: in this class, the problem is solvable if and only if $H$ is not monotone where it is positive. Along the way we prove that the single blow-up point of the subcritical approximation must be a local maximum of $H$, the boundary-trace analogue of a lemma used by Chen and Li on the sphere, and, under a separation condition on the values of the local maxima, we obtain multiple solutions. Our approach adapts the variational scheme of Chen and Li for the sphere to the boundary-trace setting, using subcritical approximation and the blow-up analysis of Escobar and Garc\'ia.