This is the first of two articles dealing with the equation ${(-\Delta )}^{s}v=f\left(v\right)$ in ${\mathbb{R}}^{n}$, 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 ${\mathbb{R}}_{+}^{n+1}$ 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.

@article{AIHPC_2014__31_1_23_0, author = {Cabr\'e, Xavier and Sire, Yannick}, title = {Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, publisher = {Elsevier}, volume = {31}, number = {1}, year = {2014}, pages = {23-53}, doi = {10.1016/j.anihpc.2013.02.001}, zbl = {1286.35248}, mrnumber = {3165278}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2014__31_1_23_0} }

Cabré, Xavier; Sire, Yannick. Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, pp. 23-53. doi : 10.1016/j.anihpc.2013.02.001. http://www.numdam.org/item/AIHPC_2014__31_1_23_0/

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