Variational theory for the resonant T-curvature equation

In this paper, we study the resonant prescribed T-curvature problem on a compact 4-dimensional Riemannian manifold with boundary. We derive sharp energy and gradient estimates of the associated Euler-Lagrange functional to characterize the critical points at infinity of the associated variational problem under a non-degeneracy on a naturally associated Hamiltonian function. Using this, we derive a Morse type lemma at infinity around the critical points at infinity. Using the Morse lemma at infinity, we prove new existence results of Morse theoretical type. Combining the Morse lemma at infinity and the Liouville version of the Barycenter technique of Bahri–Coron (Commun Pure Appl Math 41–3:253–294, 1988) developed in Ndiaye (Adv Math 277(277):56–99, 2015), we prove new existence results under a topological hypothesis on the boundary of the underlying manifold, the selection map at infinity, and the entry and exit sets at infinity.