In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem. 1. New Results on Scalar Curvature Problem In this paper, we revisit a problem having a geometric origin. Namely, let be the standard -dimensional half-sphere endowed with its standard Riemannian metric . Given a function , we consider the problem of finding a metric in the conformal class of such that and , where is the scalar curvature of and is the mean curvature of with respect to . Let be such a metric “conformal” to ; then the above problem amounts to find a smooth positive solution to the following PDE: where is the outward normal vector with respect to the metric . Problems related to (2) were widely studied by various authors [1–12]. Note that, to solve the problem (2), the function has to be positive somewhere. Moreover, there exist topological obstructions, as Kazdan-Warner obstructions for the scalar curvature problem on (see [13]). Therefore we are led to seek sufficient conditions to set on , so that the problem (2) has solutions. In addition to the existence problem, we address, in this paper, the question of the number of such metrics in the conformal class of , with prescribed scalar curvature and zero boundary mean curvature. In [3, 4, 6, 11], the authors have studied the problem (2). The methods of [6, 11] involve a fine blow-up analysis of some subcritical approximations and the use of topological degree tools. However, the methods of [3, 4] make use of algebraic topological and dynamical tools, coming from the theory of critical points at infinity (see [14]); we also have addressed this problem in [12], using similar tools. The main contribution of the present work is to generalize certain previous existence results of？？ [3, 12] and to give new existence results to which we add multiplicity results, using tools coming from the theory of critical points at infinity. In the first part of this paper, we provide existence and multiplicity results under perturbative hypothesis. In order to state our results, we introduce the following notations and assumptions. Through the whole of this paper, we assume that has a finite set of nondegenerate critical points, ordered such that We define the following sets: Let be a pseudogradient of of Morse-Smale
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