Ambrosetti-Prodi type results for Dirichlet problems of fractional Laplacian-like operators

TL;DR

This paper extends Ambrosetti-Prodi type results to nonlinear Dirichlet problems involving fractional Laplacians, introducing a new functional integration approach for non-local operators.

Contribution

It provides the first robust technique using functional integration for Ambrosetti-Prodi results in non-local fractional Laplacian problems.

Findings

Established Ambrosetti-Prodi results for fractional Laplacian problems.

Developed a new functional integration-based method for non-local operators.

Addressed both semi-linear and super-linear nonlinearities.

Abstract

We establish Ambrosetti--Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for the fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.