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

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.