论文标题
超级多谐属性和渐近行为对高阶Hardy-Hénon方程的渐近性行为靠近孤立的奇异性
Super polyharmonic property and asymptotic behavior of solutions to the higher order Hardy-Hénon equation near isolated singularities
论文作者
论文摘要
在本文中，我们致力于研究以下更高级的hardy-hénon方程$$（-Δ）^{m} u = | x | x | x |^|^au^{p} \ quad \ quad \ mbox {in} 〜b_ {1}奇异性在原点，其中$α> -2M $，$ m \ geq1 $是整数，$ n> 200万美元。对于$ 1 <p <\ frac {n+2m} {n-2m} $，将给出解决方案的奇异性和衰减估计。对于$ \ frac {n+α} {n-2m} <p <\ frac {n+2m} {n-2m} {n-2m} $，$ -2M <α<2m $，我们显示了附近解决方案的超级多谐属性，这是奇异性工具的重要工具。使用这些属性，为第四阶情况（即$ m = 2 $）建立了孤立奇点的分类。此外，当$ m = 2 $，$ \ frac {n+α} {n-4} <p <p <\ frac {n+4+α} {n-4} {n-4} $和$ p \ neq \ neq \ frac {n+4+4+2α} {n-44} {n-4} {n-4} {n-4} {n-4} $ with $ -4 <a partive y solution conse in solution condies in I.E. $ x = 0 $是可移动的奇异性或$$ \ lim_ {| x | \ rightarrow0} | x |^{\ frac {4+α} {p-1}}}} = [x）= [a_ {0}]
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 $α>-2m$, $m\geq1$ is an integer and $n>2m$. For $1<p<\frac{n+2m}{n-2m}$, singularity and decay estimates of solutions will be given. For $\frac{n+α}{n-2m}<p<\frac{n+2m}{n-2m}$ with $-2m<α<2m$, 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 $m=2$, $\frac{n+α}{n-4}<p<\frac{n+4+α}{n-4}$ and $p\neq \frac{n+4+2α}{n-4}$ with $-4<α\leq0$, we obtain the precise behavior of solutions near the singularity, i.e., either $x=0$ is a removable singularity or $$\lim_{|x|\rightarrow0}|x|^{\frac{4+α}{p-1}}u(x)=[A_{0}]^{\frac{1}{p-1}},$$ where $A_0>0$ is an exact constant.
