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

Abstract

This is the first of two articles dealing with the equation (−Δ)sv=f(v) in Rn, with s∈(0,1), where (−Δ)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 R+n+1 together with a nonlinear Neumann boundary condition on ∂R+n+1=Rn.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↑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.