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

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

Problem Statement
The paper investigates positive solutions to semilinear equations driven by uniformly elliptic strictly $2s$-stable Lévy operators ($s \in (0, 1)$) in a bounded domain $D \subset \mathbb{R}^d$ ($d \ge 1$). The study focuses on two interconnected problems:

1. The Isolated Singularity Problem: Analyzing positive distributional solutions to $-Lu = u^p$ in the punctured domain $D \setminus \{0\}$, where $L$ is the generator of a strictly $2s$-stable Lévy process. The operator $L$ is defined via a Lévy measure $\nu$ that is not necessarily symmetric and satisfies a uniform non-degeneracy condition.
2. The Measure Data Dirichlet Problem: Studying the exterior Dirichlet problem $-Lu = u^p + k\mu$ in $D$ with $u=0$ in $D^c$, where $\mu$ is a bounded positive Borel measure and $k > 0$ is a parameter.

The primary goal is to establish the existence, qualitative properties, and removability of singularities, as well as the existence of a critical parameter $k_\mu$ governing solvability, under general structural assumptions that include anisotropic and non-symmetric stable generators.

Methodology and Novelty
The authors explicitly distinguish their approach from existing literature on the fractional Laplacian $(-\Delta)^s$. Standard techniques for the fractional Laplacian rely heavily on:

• The Caffarelli–Silvestre extension (localizing the nonlocal operator).
• Explicit formulas and sharp pointwise estimates for the Green function (e.g., $G_D(x,y) \asymp G(x,y)(\delta_D(x)/|x-y|)^s (\delta_D(y)/|x-y|)^s$).
• Smoothness of the jump measure density and specific Harnack inequalities.

The paper notes that these specific properties are unavailable for general anisotropic and non-symmetric stable Lévy operators. Consequently, the authors develop a methodology based on probabilistic potential theory and operator-theoretic techniques, avoiding the extension method and pointwise Green function estimates.

Key methodological features include:

• Tail Spaces ($T_\nu(V)$): To handle the nonlocal nature of $L$, the authors introduce a space of functions $T_\nu(V)$ ensuring that $Lu$ is well-defined as a distribution. This space accounts for the integrability of the solution outside the test function's support.
• Riesz Decomposition and Dynkin's Formula: The analysis of isolated singularities relies on Dynkin's formula and the Riesz decomposition theorem for excessive functions, rather than classical PDE regularity arguments.
• Compactness via Deny's Theorem: To construct solutions for the measure data problem without explicit supersolutions (which are unavailable due to the lack of pointwise Green bounds), the authors combine Deny's theorem with probabilistic arguments to establish the compactness of the nonlinear Green operator.
• Fixed Point and Variational Methods: Existence for small parameters is derived via Schauder's fixed-point theorem. Multiplicity and stability results in the symmetric case utilize variational methods, specifically the Mountain Pass theorem and energy functionals involving Bregman divergences.

Key Results

1. Removability and Singularity Structure:

   • Every positive distributional solution $u$ of $-Lu = u^p$ in $D \setminus \{0\}$ belongs to the tail space $T_\nu(D)$ and satisfies $-Lu = u^p + k\delta_0$ in $D$ for some $k \ge 0$.
   • Removability Threshold: If $p \ge d/(d-2s)$, then necessarily $k=0$. The singularity is removable.
   • Asymptotic Behavior: If $p < d/(d-2s)$ and $k > 0$, the solution behaves asymptotically like the Green function $G_D(x, 0)$ near the origin ($k G_D(x, 0) \le u(x) \le c k G_D(x, 0)$). The authors note that, unlike the isotropic fractional Laplacian case, the singular profile is not necessarily isotropic ($|x|^{2s-d}$) but depends on the spectral measure and direction of approach.

2. Critical Parameter and Solvability:

   • For the problem $-Lu = u^p + k\mu$, there exists a critical parameter $k_\mu \in (0, \infty)$.
   • Subcritical ($k < k_\mu$): A minimal positive solution $u_k$ exists.
   • Supercritical ($k > k_\mu$): No positive solution exists.
   • Critical ($k = k_\mu$): At most one positive solution exists.

3. Symmetric Case (Multiplicity and Stability):

   • Assuming $L$ is symmetric, the minimal solution $u_k$ for $k < k_\mu$ is stable.
   • Multiplicity: For every $k < k_\mu$, there exists a second positive solution $v_k > u_k$.
   • Existence at Threshold: A solution exists at $k = k_\mu$, implying uniqueness at the critical parameter.

Significance and Claims
The paper claims that its principal contribution lies not only in the results themselves but in the methods developed to establish them under broad structural assumptions. The authors assert that their analysis demonstrates that the principal phenomena of removability and critical thresholds persist even when the operator lacks the specific analytic properties (explicit Green functions, smoothness of densities) typically required in the fractional Laplacian literature.

By utilizing the killed resolvent and probabilistic potential theory, the authors provide a systematic framework for semilinear nonlocal equations driven by general stable Lévy operators. The work extends the classical theory of isolated singularities (developed for the Laplacian and fractional Laplacian by authors such as Lions, Brezis, Véron, and Chen-Quaas) to the anisotropic and non-symmetric setting, overcoming the lack of pointwise estimates through functional analytic and probabilistic compactness arguments.

The paper concludes that the "principal removability and threshold phenomena persist under these broad structural assumptions," validating the use of probabilistic tools as a robust alternative to extension methods and pointwise estimates in nonlocal analysis.