Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics

Abstract

We present the sharp characterization of the behavior at the isolated singularities of positive solutions of some equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in $B\backslash \left\{0\right\},$ where $B\subset {\mathbb{R}}^N$ is the unit ball. The weights can be singular at x = 0. We present the sharp asymptotic behavior of positive solutions of the weighted elliptic equations at x = 0 and establish expansions of these solutions up to arbitrary orders. Asymptotic behavior at the isolated singularitie of positive solutions of elliptic equations without weights has been studied by many authors. We will obtain new results on the asymptotic behavior at the isolated singularities even for positive solutions of equations without weights in the subcritical case.

Keywords:

• Semilinear elliptic equation
• asymptotic behavior
• conical metric
• isolated singularity

2000 MR (MATHEMATICAL REVIEWS) SUBJECT CLASSIFICATION:

• 35J15
• 35J60
• 58J05

Additional information

Funding

The research of the first author is supported by NSFC (11171092, 11571093), the others are supported by NSFC (11526212, 11131007 and 11721101).