Isolated Singularities and Measure Data Problems for Semilinear Equations Driven by Stable Lévy Operators

We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2s$-stable L\'evy 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 $-Lu=u^p+k\delta_0$ 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 $k_\mu$: 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.