On a geometric equation involving the Sobolev trace critical exponent

Abstract

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.

ON THE PRESCRIBED MEAN CURVATURE PROBLEM ON THE STANDARD n-DIMENSIONAL BALL

“…, y p ) ∈ C ∞ . Similar Morse lemma at infinity has been established for the problem (P ) on the sphere S n , n ≥ 3, under the hypothesis that the order of flatness at critical points of H is β ∈ [n − 2, n − 1[, see [3].…”

“…ρ plays a fundamental role in the existence of solutions to problem like (P ). Please see [3], such kind of phenomenon appears under (f ) β condition when β = n − 2. We assume the following A 0 ρ(y l1 , .…”

“…Similar phenomena occur for the scalar curvature problem, see [10], so the proof of Proposition 3.2 is similar to the corresponding statement in [10]. Before giving the proof of Theorem 3.1, we state the following notations extracted from [3].…”

“…We consider the case of u = p i=1 α iδi ∈ V (p, ε) such that there exist a i satisfying a i / ∈ ∪ y∈K B(y, ρ). In this region, the construction of the pseudo-gradient W proceeds exactly as the proof of (Theorem 3.2, of subset 2) of [3].…”