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

TL;DR

The paper studies sign-changing (nodal) solutions to the Escobar problem on manifolds with boundary by developing a refined variational framework based on conformal invariants. It proves the existence of least-energy nodal solutions in two key scalar-curvature configurations (scalar-flat with constant boundary mean curvature and constant scalar curvature with minimal boundary) for n ≥ 7 with a nonumbilic boundary point, and shows that the minimal-boundary problem has infinitely many nodal solutions on the unit ball when n ≥ 5. The approach hinges on sharp energy estimates for constructed test functions (bubbles) near the boundary and a careful minmax/constraint analysis that leverages Escobar-type inequalities and conformal invariants. Additionally, the ball case is handled via symmetry reductions of the Yamabe problem on ℝ^n_+, yielding an infinite family of sign-changing solutions. The results extend the understanding of nodal phenomena in conformal geometry with boundary and provide concrete criteria under which least-energy nodal solutions exist.

Abstract

Let $(M, g)$ be a $n-$dimensional compact Riemannian manifold with boundary. The Escobar 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 in two particular cases: the scalar-flat problem, where the scalar curvature on $M$ is zero and the mean curvature of its boundary is constant, and the minimal boundary problem, where the mean curvature of the boundary vanishes and the scalar curvature of $M$ is constant. More precisely, we prove that if $n\ge7$ and $M$ has a nonumbilic boundary point, then both problems admit least-energy nodal solutions. In addition, we show that when $n\ge5$, the minimal boundary problem possesses infinitely many sign-changing solutions on the unit ball. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates derived from Escobar's work.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof