Large solutions for subordinate spectral Laplacian
Ivan Biočić, Vanja Wagner
TL;DR
This work develops a general semilinear theory for non-local operators $\phi(-\Delta|_{D})$, extending the spectral fractional Laplacian, by leveraging the subordinate killed Brownian motion framework. It proves the existence of large solutions with boundary blow-up under a Keller-Osserman-type condition linking $f$ and $\phi$, and establishes interior Hölder regularity for distributional solutions, including a strong regularity result for $\phi(-\Delta)$. The analysis relies on sharp Green and Poisson kernel estimates, a renewal function that governs boundary behavior, and a monotone approximation scheme complemented by a well-constructed supersolution. Together these results advance understanding of nonlinear boundary blow-up phenomena for a broad class of non-local operators in dimensions $d\ge3$, with implications for related Lévy-type diffusion models.
Abstract
We find a large solution to a semilinear Dirichlet problem in a bounded $C^{1,1}$ domain for a non-local operator $φ(-Δ\vert_{D})$, an extension of the infinitesimal generator of a subordinate killed Brownian motion. The setting covers and extends the case of the spectral fractional Laplacian. The upper bound for the explosion rate of the large solution is obtained, and is given in terms of the renewal function, distance to the boundary, and the Keller-Osserman-type transformation of the nonlinearity. Additionally, we prove interior higher regularity results for this operator.