论文标题
高阶分数拉普拉斯的超级多谐属性
On super polyharmonic property of high-order fractional Laplacian
论文作者
论文摘要
令$ 0 <α<2 $，$ p \ geq 1 $，$ m \ in \ mathbb {n} _+$。考虑$ u $是PDE \ BEGIN {equation} \ label {Abstract PDE}的积极解决方案 （-Δ）^{\fracα{2}+m} u（x）= u^p（x）\ quad \ text {in} \ mathbb {r}^n。 \ end {equation} cao，dai and Qin（美国数学学会的交易，2021年）表明，在\ Mathcal {l}_α$中的条件下，PDE拥有超级Polyharmonic属性$（δ）^{k+\ \ \ \ \ \fracα{k+\fracα{k+\fracα{2} u {在本文中，我们显示了另一种超级多谐属性$（-Δ）^k u> 0 $ for $ k = 1，... m $在不同条件下$（ - δ）^mu \ in \ Mathcal {l}_α$ and $（queq）0 $。两种超级多谐属性都可以导致PDE与积分方程之间的等效性$ u（x）= \ int _ {\ Mathbb {r}^n} \ frac {u^p（y）} {| x-y |^y |^{n-2m-α}}} dy $。
Let $0<α<2$, $p\geq 1$, $m\in\mathbb{N}_+$. Consider $u$ to be the positive solution of the PDE \begin{equation}\label{abstract PDE} (-Δ)^{\fracα{2}+m} u(x)=u^p(x) \quad\text{in }\mathbb{R}^n. \end{equation} Cao, Dai and Qin( Transactions of the American mathematical society, 2021) showed that, under the condition $u\in\mathcal{L}_α$, the PDE possesses super polyharmonic property $(-Δ)^{k+\fracα{2}}u\geq 0$ for $k=0,1,...,m-1$. In this paper, we show another kind of super polyharmonic property $(-Δ)^k u> 0$ for $k=1,...m$ under different conditions $(-Δ)^mu\in\mathcal{L}_α$ and $(-Δ)^m u\geq 0$. Both kinds of super polyharmonic properties can lead to the equivalence between the PDE and the integral equation $u(x)=\int_{\mathbb{R}^n}\frac{u^p(y)}{|x-y|^{n-2m-α}}dy$.
