English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper, we are devoted to studying the positive solutions of thefollowing 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 theorigin, 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 begiven. For $\frac{n+\alpha}{n-2m}<p<\frac{n+2m}{n-2m}$ with $-2m<\alpha<2m$, weshow the super polyharmonic properties of solutions near the singularity, whichare essential tools in the study of polyharmonic equation. Using theseproperties, a classification of isolated singularities of positive solutions isestablished 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 behaviorof solutions near the singularity, i.e., either $x=0$ is a removablesingularity 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.

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