In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere (S n , h). We derive from this analysis some a priori estimates in dimension 5 and 6. On S 5 these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On S 6 we prove the existence of at least one solution when an index formula associated to this conformal invariant is different from zero.
Estimates for isolated simple blow up pointsIn this section we study the properties of isolated simple blow up points for equation (3). We first prove some Harnack type inequalities. In the following, given r > 0, B r will denote the open ball of radius r centred at 0 in R n , and B r its closure.
Abstract. This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the C 2 -norm and the total Leray-Schauder degree of all solutions is equal to −1 . Then we deduce from this compactness result the existence of at least one solution to our problem.MSC classification: 35J60, 53C21, 58G30.
abstract. -We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.
In this paper, we perform a fine blow-up analysis for a boundary value elliptic equation involving the critical trace Sobolev exponent related to the conformal deformation of the metrics on the standard ball, namely the problem of prescribing the boundary mean curvature. From this analysis some a priori estimates in low dimension are obtained. With these estimates, we prove the existence of at least one solution when an index-counting formula associated to the prescribed mean curvature is different from zero. r
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