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

Cabré, Xavier ; Sire, Yannick

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 23-53 / Harvested from Numdam

Résumé

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.

Détail

Publié le : 2014-01-01

MR 3165278 | Zbl 1286.35248 | 1 citation dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2013.02.001

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