# On divergent fractional Laplace equations

## TL;DR

This paper studies the divergent fractional Laplace operator, establishing local approximation, existence and multiplicity of solutions for Dirichlet problems, and new approximation results for nonlinear equations, extending classical fractional Laplacian theory.

## Contribution

It introduces new results on the divergent fractional Laplace operator, including local shadowing, solution existence, multiplicity, and nonlinear approximation, broadening understanding of fractional PDEs.

## Key findings

- Any function can be locally shadowed by a solution of the divergent fractional Laplace equation.
- Existence and characterization of solutions for the Dirichlet problem.
- New approximation results for nonlinear divergent fractional Laplace equations.

## Abstract

We consider the divergent fractional Laplace operator presented in [Dipierro-Savin-Valdinoci, Rev. Mat. Iberoam.] and we prove three types of results. Firstly, we show that any given function can be locally shadowed by a solution of a divergent fractional Laplace equation which is also prescribed in a neighborhood of infinity. Secondly, we take into account the Dirichlet problem for the divergent fractional Laplace equation, proving the existence of a solution and characterizing its multiplicity. Finally, we take into account the case of nonlinear equations, obtaining a new approximation results. These results maintain their interest also in the case of functions for which the fractional Laplacian can be defined in the usual sense.