Energy bounds of sign-changing solutions to Yamabe equations on manifolds with boundary

TL;DR

This paper establishes lower energy bounds for sign-changing solutions to the Yamabe equation on manifolds with boundary, using the method of moving planes, which impacts the analysis of variational problems.

Contribution

It provides new lower bounds on the energy of sign-changing solutions to the Yamabe equation on manifolds with boundary, including a sharper bound via the moving planes method.

Findings

Sign-changing solutions have at least twice the energy of a standard bubble.

A sharper energy lower bound is established for sign-changing solutions.

The bounds extend the energy range for non-trivial weak limits of Palais-Smale sequences.

Abstract

We study the Yamabe equation in the Euclidean half-space. We prove that any sign-changing solution has at least twice the energy of a standard bubble. Moreover, a sharper energy lower bound of the sign-changing solution set is also established via the method of moving planes. This bound increases the energy range for which Palais-Smale sequences of related variational problem has a non-trivial weak limit.