Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part

We consider the equation ${\mathrm{\Delta }}_{g}u+hu=|u{|}^{2^{?}?2}u$ in a closed Riemannian manifold $(M,g)$, where $h\in C^{0,\theta }(M)$, $\theta \in (0,1)$ and $2^{?}=\frac{2n}{n?2}$, $n:=\dim ?(M)\ge 3$. 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 $n\ge 7$ and $h<\frac{n?2}{4(n?1)}{\text{Scal}}_g$ in M, where ${\text{Scal}}_g$ 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.