Sign-changing blow-up for scalar curvature type equations

Overview & Synopsis

Given (M, g) a compact Riemannian manifold of dimension n ≥ 3, we are interested in the existence of blowing-up sign-changing families (uε) ε>0 ∈ C 2,θ (M), θ ∈ (0, 1), of solutions to ∆guε + huε = |uε| 4 n−2 −ε uε in M , where ∆g := −divg(∇) and h ∈ C 0,θ (M) is a potential. We prove that such families exist in two main cases: in small dimension n ∈ {3, 4, 5, 6} for any potential h or in dimension 3 ≤ n ≤ 9 when h ≡ n−2 4(n−1) Scalg. These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet [11] and Khuri–Marques–Schoen [19].