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This is the first of two articles dealing with the equation $(-\Delta)^{s} v=f(v)$ in $\mathbb{R}^{n}$, with $s\in (0,1)$, where $(-\Delta)^{s}$ stands forthe fractional Laplacian ---the infinitesimal generator of a L\'evy 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 boundedincreasing solutions in all of $\mathbb{R}$. These necessary conditions (whichwill be proven in a follow-up paper to be also sufficient for the existence ofa bounded increasing solution) are derived from an equality and an estimateinvolving a Hamiltonian ---in the spirit of a result of Modica for theLaplacian. In addition, we study regularity issues, as well as maximum andHarnack principles associated to the equation.

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