由稳定Lévy算子驱动的半线性方程的孤立奇点与测度数据问题
Isolated Singularities and Measure Data Problems for Semilinear Equations Driven by Stable Lévy Operators
AI总结：
该论文研究由稳定Lévy算子驱动的半线性方程的孤立奇点与测度数据问题，证明了穿孔域内正解的性质、Dirichlet问题的临界参数及阈值处解的存在唯一性。
AI中文摘要：
我们研究由一致椭圆严格2s-稳定Lévy算子驱动的半线性方程的正解，其中s∈(0,1)。首先证明：在穿孔域D\{0}中，-Lu=u^p的每个正分布解都满足在D中 -Lu=u^p +kδ₀，其中k≥0；且当p≥d/(d-2s)时，必有k=0。接着研究对应Dirichlet问题，其中Dirac质量被有界正测度取代，建立了临界参数k_μ的存在性：低于该阈值时存在最小正解，高于时无解。在对称情形下，进一步证明阈值下方解的多重性，以及阈值处解的存在性与唯一性。
英文摘要：
We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2s$-stable Lévy operators, where $s\in (0,1)$. We first prove that every positive distributional solution of $-Lu=u^p$ in a punctured domain $D\setminus\{0\}$ satisfies $-Lu=u^p+kδ_0$ in $D$ for some $k\ge0$, and that necessarily $k=0$ whenever $p\ge d/(d-2s)$. We then study the corresponding Dirichlet problem in which the Dirac mass is replaced by a bounded positive measure, and establish the existence of a critical parameter $k_μ$: below this threshold minimal positive solutions exist, whereas above it the problem admits no solution. In the symmetric case, we further prove multiplicity below the threshold, as well as existence and uniqueness at the threshold itself.
