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In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of R n , n ≥ 4. Under the assumption that the prescribed function is flat near its critical point, we give precise estimates on the losses of the compactness, and we provide a new existence result of Bahri-Coron type. Moreover, we establish, under generic boundary condition, a Morse inequality at infinity, which gives a lower bound on the number of solutions to the above problem.

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