Super polyharmonic property and asymptotic behavior of solutions to the higher order Hardy-H\'enon equation near isolated singularities

TL;DR

This paper investigates positive solutions to higher order Hardy-Hénon equations near isolated singularities, establishing super polyharmonic properties, classifying singularities, and describing precise asymptotic behaviors.

Contribution

It introduces super polyharmonic properties for solutions and classifies isolated singularities for the fourth order case, advancing understanding of higher order Hardy-Hénon equations.

Findings

Super polyharmonic properties near singularities.

Classification of isolated singularities for the fourth order case.

Precise asymptotic behavior of solutions near singularities.

Abstract

In this paper, we are devoted to studying the positive solutions of the following higher order Hardy-H\'enon equation $$ (-\Delta)^{m}u=|x|^{\alpha}u^{p} \quad\mbox{in}~ B_{1}\setminus\{0\}\subset\mathbb{R}^{n} $$ with an isolated singularity at the origin, where $\alpha>-2m$, $m\geq1$ is an integer and $n>2m$. For $1<p<\frac{n+2m}{n-2m}$, singularity and decay estimates of solutions will be given. For $\frac{n+\alpha}{n-2m}<p<\frac{n+2m}{n-2m}$ with $-2m<\alpha<2m$, we show the super polyharmonic properties of solutions near the singularity, which are essential tools in the study of polyharmonic equation. Using these properties, a classification of isolated singularities of positive solutions is established for the fourth order case, i.e., $m=2$. Moreover, when $m=2$, $\frac{n+\alpha}{n-4}<p<\frac{n+4+\alpha}{n-4}$ and $p\neq \frac{n+4+2\alpha}{n-4}$ with $-4<\alpha\leq0$, we obtain the precise behavior of solutions near the singularity, i.e., either $x=0$ is a removable singularity or $$\lim_{|x|\rightarrow0}|x|^{\frac{4+\alpha}{p-1}}u(x)=[A_{0}]^{\frac{1}{p-1}},$$ where $A_0>0$ is an exact constant.