On divergent fractional Laplace equations

Abstract

We consider the divergent fractional Laplace operator presented in [5] and we prove three types of results.

Given u : Rn → R and s ∈ (0, 1), to define the fractional Laplacian of u,(−∆)su(x) := lim ρ0 u(x) − u(y) Rn\Bρ(x) |x − y|n+2s dy, (1.1)one typically needs two main requisites on the function u: u has to be sufficiently smooth in the vicinity of x, for instance u ∈ Cγ(Bδ(x)) for some δ > 0 and γ > 2s, u needs to have a controlled growth at infinity, for instance|u(x)| Rn 1 + |x|n+2s dx < +∞. (1.2)in [5] we have recently introduced a new notion of “divergent” fractional Laplacian, which can be used even when condition (1.2) is violated
In [5] we have recently introduced a new notion of “divergent” fractional Laplacian, which can be used even when condition (1.2) is violated. This notion takes into account the case of functions with polynomial growth, for which the classical definition in (1.1) makes no sense, and it recovers the classical definition for functions with controlled growth such as in (1.2)
The notion of divergent fractional Laplacian possesses several interesting features and technical advantages, including suitable Schauder estimates in which the full smooth Hölder norm of the solution is controlled by a suitable seminorm of the nonlinearity