Yamabe equations on half-spaces

Given smooth functions K(x) on Ω̄ and h(x) on ∂Ω, and a smooth domain in Rn, n≥3, the elliptic equation: Δu+K(x)uτ = 0 and u>0 in Ω, ∂u/∂v+ 1/2 (n-2)H(x)u = h(x)uσ on ∂Ω, where τ = (n+2)/(n-2) is the critical Sobolev exponent, σ = n/(n-2) is the critical Sobolev exponent on the boundary, v is the unit outer normal of ∂Ω and H(x) is the mean curvature, is analyzed. The Yamabe equation K(x) and h(x), K(x) on h(x) on half-space R+n = {(x′,xn)∈Rn:xn>0} Δu+K(x)uτ = 0 and u>0 R+n, ∂u/∂v = h(x)uσ on ∂R+n is considered. Here K(x) is a bounded smooth function on R+n and is bounded from below by a positive constant, and h(x) is a bounded smooth function on ∂R+n = {(x′,0): x′∈Rn-1}. It is recalled that the unit ball is conformal to R+n and the Yamabe equation is conformally invariant.