  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 $\mathbb{R}^n$.
