Struwe's Decomposition for a Polyharmonic Operator on a compact Riemannian Manifold with or without Boundary

TL;DR

This paper extends Struwe's decomposition to high-order elliptic operators on compact Riemannian manifolds, accounting for boundary effects and providing a detailed analysis of Palais-Smale sequences as sums of bubbles.

Contribution

It generalizes Struwe's 1984 result to polyharmonic operators on manifolds, including boundary cases and boundary-adjacent bubbles.

Findings

Decomposition of Palais-Smale sequences into bubbles.

Inclusion of boundary effects in bubble formation.

Application to smooth bounded domains in Euclidean space.

Abstract

Given a high-order elliptic operator on a compact manifold with or without boundary, we perform the decomposition of Palais-Smale sequences for a nonlinear problem as a sum of bubbles. This is a generalization of the celebrated 1984 result of Struwe. Unlike the case of second-order operators, bubbles close to the boundary might appear. Our result includes the case of a smooth bounded domain of $\mathbb{R}^n$.