Local estimates for conformal Q-curvature equations

2021/07/09 by Tianling Jin, Hui Yang, Jin, Tianling +1
Computer Science · Mathematics · #35B40 #35J91 #45M05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2107.04437

openalex publication_date 2021/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive local estimates of positive solutions to the conformal Q-curvature equation (-Δ)^m u = K(x) u^{(n+2m)/(n-2m)} ~~~~~~ in ~ Ω\backslash Λ near their singular set Λ, where Ω⊂ ℝⁿ is an open set, K(x) is a positive continuous function on Ω, Λ is a closed subset of ℝⁿ, 2 ≤ m < n/2 and m is an integer. Under certain flatness conditions at critical points of K on Λ, we prove that u(x) ≤ C [dist(x, Λ)]^-(n-2m)/2 when the upper Minkowski dimension of Λ is less than (n-2m)/2.

Citations

• Uniqueness theorems through the method of moving spheres
• Conformally flat manifolds, Kleinian groups and scalar curvature
• Asymptotic behavior of solutions to the σ k -Yamabe equation near isolated singularities
• Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
• Classification of solutions for an integral equation
• The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation
• On constant Q-curvature metrics with isolated singularities
• Isolated singularities of solutions to the Yamabe equation in dimension 6

Related

• Large singular solutions for conformal Q-curvature equations on \mathbbSⁿ
• Local behavior of positive solutions of higher order conformally invariant equations with a singular set
• The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds
• Asymptotic Behavior of Positive Solutions of the Equation Δu + K u^{(n + 2)/(n - 2)} = 0 in Rⁿ and Positive Scalar Curvature
• On locally conformally flat manifolds with finite total Q-curvature