A Morse theoretical approach for the boundary mean curvature problem on B4

In this paper we investigate existence as well as multiplicity of scalar flat metric of prescribed boundary mean curvature on the standard 4-dimensional ball. Due to the existence of critical point at infinity, the standard variational methods cannot be applied. To overcome this difficulty, we prove that in a neighborhood of critical points at infinity, a Morse lemmas at infinity reduction holds, then develop a whole Morse theory of this noncompact variational problem. In particular we establish, under generic boundary condition Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of the critical point at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As further application of this Morse theoretical approach, we prove more existence results and extend a topological invariant introduced by A. Bahri.