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

Abstract

This is the first of two articles dealing with the equation $(-\mathrm{\Delta }{)}^{s}v=f(v)$ in ${\mathbb{R}}^n$, with $s\in (0,1)$, where $(-\mathrm{\Delta }{)}^s$ stands for the fractional Laplacian — the infinitesimal generator of a Lévy 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. Our proofs are uniform as $s\uparrow 1$, establishing in the limit the corresponding known results for the Laplacian.

In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.