LAUSR

Abstract We consider the equation Δ g u + h u = | u | 2 ⁎ − 2 u in a closed Riemannian manifold ( M , g ) , where h ∈ C 0 , θ ( M ) , θ ∈ ( 0 , 1 ) and 2 ⁎ = 2 n n − 2 , n : = dim ⁡ ( M ) ≥ 3 . We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is a priori bounded. We obtain this result under the conditions that n ≥ 7 and h n − 2 4 ( n − 1 ) Scal g in M, where Scal g is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function h.