Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates

TL;DR

This paper investigates fractional Laplacian equations, establishing necessary conditions for solutions, analyzing regularity, and deriving maximum principles, with a focus on bounded increasing solutions in the entire space.

Contribution

It introduces necessary conditions for solutions of fractional Laplacian equations using Hamiltonian estimates and studies their regularity and maximum principles.

Findings

Necessary conditions on the nonlinearity for solution existence

Regularity and maximum principles for fractional Laplacian equations

Hamiltonian estimates analogous to Modica's result for the Laplacian

Abstract

This is the first of two articles dealing with the equation $(-\Delta)^{s} v= f(v)$ in $\mathbb{R}^{n}$, with $s\in (0,1)$, where $(-\Delta)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a L\'evy process. This equation can be realized as a local linear degenerate elliptic equation in $\mathbb{R}^{n+1}_+$ together with a nonlinear Neumann boundary condition on $\partial \mathbb{R}^{n+1}_+=\mathbb{R}^{n}$. In this first article, we establish necessary conditions on the nonlinearity $f$ to admit certain type of solutions, with special interest in bounded increasing solutions in all of $\mathbb{R}$. These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian ---in the spirit of a result of Modica for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.