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) no. 1, pp. 23-53.

This is the first of two articles dealing with the equation ${(-\Delta )}^{s}v=f\left(v\right)$ in ${ℝ}^n$, with $s\in (0,1)$, where ${(-\Delta )}^s$ 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 ${ℝ}_{+}^{n+1}$ together with a nonlinear Neumann boundary condition on $\partial {ℝ}_{+}^{n+1}={ℝ}^n$.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 $s\uparrow 1$, 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.

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