Factorization into kk-bubbles for Palais–Smale maps to potential type energy functionals

We prove a decomposition into generalized bubbles for Palais–Smale sequences associated with potential energy functionals for vector-valued function spaces. The study is motivated by the compactness question for solutions of critical potential systems, for which the existence problem was recently addressed. We also present some examples of the existence of radial generalized bubbles.