Asymptotic behavior of positive solutions of some nonlinear elliptic equations on cylinders

TL;DR

This paper analyzes the long-term behavior of positive solutions to certain nonlinear elliptic equations on cylinders, unifying the understanding of singularities in equations like Yamabe and Hardy-Hénon.

Contribution

It provides a comprehensive description of the asymptotic behavior of solutions near the boundary of cylinders, linking various classical equations under a unified framework.

Findings

Quantitative asymptotic descriptions of solutions near the boundary.

Unification of singularity analysis for multiple elliptic equations.

Insights into the structure of solutions on cylindrical domains.

Abstract

We establish quantitative asymptotic behavior of positive solutions of a family of nonlinear elliptic equations on the half cylinder near the end. This unifies the study of isolated singularities of some semilinear elliptic equations, such as the Yamabe equation, Hardy-H\'enon equation etc.