论文标题
某些分数拉普拉斯方程的奇异性和超谐感的可移动性
Removability of singularities and superharmonicity for some fractional Laplacian equations
论文作者
论文摘要
我们研究了对$$（-Δ）^γu= fu^p \ quad \ text {in} \ mathbb r^n \setMinusς$$的一些定性属性（包括可移动的奇异性和超谐感）。这里（0，\ frac {n} {2}）$。除其他外，我们首先证明，如果$σ$是$ \ mathbb r^n $的紧凑型设置，带有Assouad dimension $ \ bf d $（不一定是整数），$ {\ bf d} <n-2γ$，和$ u \ inl_γ（\ Mathbb r^n）\ cap l^p_ p _} r^n \setMinusς}）$是某些$$ p> \ frac {n- \ bf d} {n- {\ bf d}-2γ}，$ u \ in l^p_ {loc}（loc}}（\ nocbb r^n）$和$ us $ n y n n y n n n n n in l^p _ p_p_}（l^p_}（loc p_}），然后，我们证明了所有$σ\ in（0，γ）$，如果$σ= ϕ $。
We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-Δ)^γu=fu^p\quad\text{in }\mathbb R^n\setminusΣ$$ which are singular at $Σ$. Here $γ\in (0, \frac{n}{2})$. Among other things, we first prove that if $Σ$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d}<n-2γ$, and $ u\in L_γ(\mathbb R^n)\cap L^p_{loc}({\mathbb R^n\setminusΣ})$ is a non-negative solution for some $$p>\frac{n-\bf d}{n-{\bf d}-2γ},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solution in $\mathbb R^n$. Then we prove that $ (-Δ)^σu >0$ for all $ σ\in (0, γ)$, if $Σ=ϕ$.
