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We provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the standard n dimensional ball, n ≥ 3, with respect to some scalar flat metric. Because of the presence of some critical nonlinearity, blow up phenomena occur and existence results are highly nontrivial since one has to overcome topological obstructions. Our approach consists of, on one hand, developing a Morse theoretical approach to this problem through a Morse-type reduction of the associated Euler-Lagrange functional in a neighborhood of its critical points at Infinity and, on the other hand, extending to this problem some topological invariants introduced by A. Bahri in his study of Yamabe type problems on closed manifolds.Keywords Boundary mean curvature · Intersection number · Morse theory · Morse lemma at infinity · Critical points at infinity · Morse inequalities · Boundary operator · Topological methods Mathematics Subject Classification (2000) 58E05 · 35J65 · 53C21 · 35B40