Minimizing sequences for conformally invariant integrals of higher order (Q353120)

For mappings from \(S^{2m}\) to a compact manifold \(N\subset\mathbb{R}^n\), the authors study the Paneitz energies -- those energies of order \(m\) which are invariant under conformal transformations of \(S^{2m}\) . The behavior of minimizing sequences for such Paneitz energies within a given free homotopy class \(\alpha\in\pi_{2m}(N)\) is investigated. It is proved that either a subsequence converges in \(W^{m,2}\) to a minimizer of the Paneitz energy, or one has only weak convergence, and Paneitz energy concentrates at finitely many points of \(S^{2m}\). As well, it is shown that conformal rescalings reveal bubbles -- i.e., Paneitz-polyharmonic spheres, that are responsible for both energy loss and topology changes in the limit of the minimizing sequence.