Liouville type results for systems of equations involving fractional Laplacian in exterior domains

Abstract

In this article we present a simple and unified probabilistic approach to prove nonexistence of positive super-solutions for systems of equations involving potential terms and the fractional Laplacian in an exterior domain. Such problems arise in the analysis of a priori estimates of solutions. The class of problems we consider in this article is quite general compared to the literature. The main ingredient for our proofs is the hitting time estimates for the symmetric α-stable process and probabilistic representation of the super-solutions.

Faber-Krahn type inequalities and uniqueness of positive solutions on metric measure spaces

“…The proof would be fairly routine and can be deduced from the proof of Theorem 4.3. For more details, we refer to [6], [7,Theorem 5.4].…”

Liouville type theorems for fractional and higher-order fractional Hénon-Lane-Emden systems

“…Not much work has been done on this question. As far as we know, the best results were obtained in [15] via the method of moving planes and [1] with probabilistic method.…”

“…A. Biswas [1] proved a more general Liouville type theorem for super-solutions for k = l = 0 through probabilistic approach. We give a proof of PDE method and generalize to higher-order fractional systems.…”

Liouville type theorems for fractional and higher-order fractional systems

“…A. Biswas [1] proved a similar Liouville type theorem for super-solutions of fractional systems with general nonlinearities through a probabilistic approach. We give a direct proof of PDE method for both fractional and higher-order fractional systems.…”

“…Not much work has been done on this question. As far as we know, the best results were obtained in [28] via the method of moving planes and [1] with probabilistic method.…”