Super polyharmonic property and asymptotic behavior of solutions to the higher order Hardy-Hénon equation near isolated singularities

Abstract

In this paper, we are devoted to studying the positive solutions of the following higher order Hardy-Hénon equation $$ (-Δ)^{m}u=|x|^αu^{p} \quad\mbox{in}~ B_{1}\setminus\{0\}\subset\mathbb{R}^{n} $$ with an isolated singularity at the origin, where $\alpha >-2m$, $m\ge 1$ 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 $p\ne \frac{n+4+2\alpha }{n-4}$ with $-4<\alpha \le 0$, we obtain the precise behavior of solutions near the singularity, i.e., either $x=0$ is a removable singularity or $$\underset{|x|\to 0}{\lim }|x{|}^{\frac{4+\alpha }{p-1}}u(x)=[A_0{]}^{\frac{1}{p-1}},$$ where $A_0>0$ is an exact constant.