Blowing-up solutions for second-order critical elliptic equations: the impact of the scalar curvature

2019/12/19 by Frédéric Robert, Jérôme Vétois, Robert, Frédéric +1 · 1 citation
Mathematics · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.1912.09376

Abstract

Given a closed manifold (Mⁿ,g), n≥ 3, Olivier Druet proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation Δ_gu+h₀u=u^(n+2)/(n-2), ugt;0\hbox in M is that h₀∈ C¹(M) touches the Scalar curvature somewhere when n≥ 4 (the condition is different for n=6). In this paper, we prove that Druet's condition is also sufficient provided we add its natural differentiable version. For n≥ 6, our arguments are local. For the low dimensions n∈\4,5\, our proof requires the introduction of a suitable mass that is defined only where Druet's condition holds. This mass carries global information both on h₀ and (M,g).

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