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In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of RnOpen image in new window, n≥4Open image in new window. 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.

On a geometric equation involving the Sobolev trace critical exponent