Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls

TL;DR

This paper establishes the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls, utilizing a probabilistic tree-based approach that accommodates a broad class of polynomial nonlinearities.

Contribution

It introduces a novel probabilistic method based on (2s)-stable branching processes for solving fractional elliptic PDEs, allowing for unbounded polynomial nonlinearities.

Findings

Existence of solutions proven for small exterior conditions and nonlinearities.

Method applicable to a wide range of polynomial nonlinearities.

Numerical illustrations demonstrate effectiveness in high dimensions.

Abstract

We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension $d\geq 1$. Our approach relies on a tree-based probabilistic representation based on a (2s)-stable branching processes for all $s\in (0,1)$, and our existence results hold for sufficiently small exterior conditions and nonlinearity coefficients. In comparison with existing approaches, we consider a wide class of polynomial nonlinearities without imposing upper bounds on their maximal degree or number of terms. Numerical illustrations are provided in large dimensions.