A Liouville theorem for the higher-order fractional Laplacian

Abstract

We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the differential equations. We first derive nonexistence of positive solutions, often known as the Liouville type theorem, for the integral and differential equations. Then through an delicate iteration, we show symmetry for positive solutions.

A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions

“…The technique is the standard one introduced in [29], with the adaptations to the nonlocal setting provided by [5]. It relies in the Liouville theorems obtained in [37] and [17], the monotonicity of solutions in half-spaces proved in [4] and our new Theorem 1. For every k, take a point x k ∈ Ω where u k achieves its maximum.…”

A priori bounds and existence of solutions for some nonlocal elliptic problems

“…Then by bootstrapping using again Proposition 2.8 in [52] we would actually have v ∈ C ∞ (R N ). In particular we deduce that v is a strong solution of (−∆) s v = v p in R N in the sense of [60]. However, since p < N +2s N −2s , this contradicts for instance Theorem 4 in [60] (see also [25]).…”

“…It is worthy of mention at this point that the corresponding Liouville theorems are already available (cf. [60,25,43,32]).…”

A Hopf lemma and regularity for fractional $ p $-Laplacians

“…Another useful method to study the fractional Laplacian is the integral equations method, which turns a given fractional Laplacian equation into its equivalent integral equation, and then various properties of the original equation can be obtained by investigating the integral equation, see [7,14,29] and references therein.…”