Large solutions for subordinate spectral Laplacian

TL;DR

This paper constructs large solutions for a semilinear boundary value problem involving a subordinate spectral Laplacian, extending previous fractional Laplacian results and analyzing their explosion rates and regularity.

Contribution

It introduces a method to find large solutions for a broad class of non-local operators extending the spectral fractional Laplacian, including explosion rate bounds and interior regularity.

Findings

Established existence of large solutions for subordinate spectral Laplacian.

Derived upper bounds for explosion rates based on boundary distance and renewal functions.

Proved interior higher regularity results for solutions.

Abstract

We find a large solution to a semilinear Dirichlet problem in a bounded $C^{1,1}$ domain for a non-local operator $\phi(-\Delta\vert_{D})$, 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.