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

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 $\mathbb{R}^n$.