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

This is the first of two articles dealing with the equation (-Delta)(s) upsilon = f (upsilon) in R-n, with s is an element of (0, 1), where (-Delta)(s) stands for the fractional Laplacian - the infinitesimal generator of a Levy process. This equation can be realized as a local linear degenerate elliptic equation in R-+(n+1) together with a nonlinear Neumann boundary condition on partial derivative R-+(n+1) =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 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 up arrow 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. (C) 2013 Elsevier Masson SAS. All rights reserved.