Stability and instability results for sign-changing solutions to second-order critical elliptic equations

On a smooth, closed Riemannian manifold (M,g) of dimension n≥3, we consider the stationary Schrödinger equation Δgu+h0u=|u|2⁎−2u, where Δg:=−divg∇, h0∈C1(M) and 2⁎:=2nn−2. We prove that, up to perturbations of the potential function h0 in C1(M), the sets of sign-changing solutions that are bounded in H1(M) are precompact in the C2 topology. We obtain this result under the assumptions that (M,g) is locally conformally flat, n≥7 and h0≠n−24(n−1)Scalg at all points in M, where Scalg is the scalar curvature of the manifold. We then provide counterexamples in every dimension n≥3 showing the optimality of these assumptions.