Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

Abstract

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup*inf type Harnack inequality of Schoen for integral equations.

55 pages. Added a remark explaining that if the solution satisfies either the lower bound or the upper bound then its Laplacian sign conditions hold near a non-removable singularity

References in corpus (2)

• On higher order extensions for the fractional Laplacian
• Existence and local uniqueness of bubbling solutions for poly-harmonic equations with critical growth

Cited by in corpus (5)

• On constant Q-curvature metrics with isolated singularities
• Isolated singularities of solutions to the Yamabe equation in dimension $6$
• Large singular solutions for conformal $Q$-curvature equations on ${\mathbb{S}}^n$
• Local behavior of positive solutions of higher order conformally invariant equations with a singular set
• The local behavior of positive solutions for higher order equation with isolated singularity