Compactness for a class of Yamabe-type problems on manifolds with boundary

In this paper we are interested in studying the set of positive solutions to the following Yamabe–type problem (⁎) { − Δ g u + a ( x ) u = 0 , in M , ∂ u ∂ η g + b ( x ) u = ( n − 2 ) u n n − 2 , on ∂ M , where ( M n , g ) is a compact Riemannian n−manifold, n ≥ 3 , with boundary, Δ g is the Laplace–Beltrami operator for g, η g is the pointing unit outward normal to ∂M, a ∈ C ∞ ( M ) and b ∈ C ∞ ( ∂ M ) . Assuming suitable hypotheses on the functions Ψ ( ξ ) = a ( ξ ) − n − 2 4 ( n − 1 ) R g ( ξ ) and Φ ( ξ ¯ ) = b ( ξ ¯ ) − n − 2 2 h g ( ξ ¯ ) , where R g denotes the scalar curvature of the metric g and h g is the mean curvature of ∂M, we prove some results of compactness for the set of positive solutions to the problem (⁎) .