The asymptotic behavior of Palais-Smale sequences on manifolds with boundary

TL;DR

This paper analyzes the asymptotic behavior of Palais-Smale sequences for Yamabe-type equations on manifolds with boundary, showing convergence to solutions plus bubbles formed by rescaling Euclidean solutions.

Contribution

It provides a detailed description of the asymptotic behavior of Palais-Smale sequences, including the formation of bubbles, on manifolds with boundary.

Findings

Sequences converge to solutions plus bubbles

Bubbles are rescaled fundamental solutions of Euclidean equations

Results extend understanding of Yamabe problem on manifolds with boundary

Abstract

We describe the asymptotic behavior of Palais-Smale sequences associated to certain Yamabe-type equations on manifolds with boundary. We prove that each of those sequences converges to a solution of the limit equation plus a finite number of "bubbles" which are obtained by rescaling fundamental solutions of the corresponding Euclidean equations.