On the Yamabe problem and the scalar curvature problems under boundary conditions

In this paper we prove some existence results concerning a problem arising in conformal differential geometry. Consider a smooth metric g onB = {x ∈ R : |x| < 1}, the unit ball onR, n ≥ 3, and let∆g, Rg, νg, hg denote, respectively, the Laplace-Beltrami operator, the scalar curvature of (B, g), the outward unit normal to∂B = Sn−1 with respect tog and the mean curvature of (Sn−1, g). Given two smooth functions R′ andh′, we will be concerned with the existence of positive solutionsu ∈ H 1(B) of −4(n− 1) (n− 2)∆gu+ Rgu = R ′u n+2 n−2 , in B;