Non-existence of positive solutions for a higher order fractional equation

Xuewei Cui; Mei Yu

doi:10.3934/dcds.2019059

Article Contents

2019, Volume 39, Issue 3: 1379-1387. Doi: 10.3934/dcds.2019059

This issue Previous Article A Liouville theorem for the subcritical Lane-Emden system Next Article Regularity and classification of solutions to static Hartree equations involving fractional Laplacians

Non-existence of positive solutions for a higher order fractional equation

Xuewei Cui and
Mei Yu^,

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China

* Corresponding author: Mei Yu
* Corresponding author: Mei Yu

Received: November 30, 2017

Revised: March 31, 2018

Published: December 2018

This work was supported by the National Nature Science Foundation of China (Grant No.11271299 and No.11801446) and the Natural Science Basic Research Plan in Shaanxi Province of China (Program No.2016JM1023 and No.2018JQ1037.)

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Abstract

In this paper, we consider a nonlinear equation involving fractional Laplacian of higher order on the whole space. We establish the equivalence between the pseudo-differential equation and an integral equation by applying the maximum principle and the Liouville theorem. For positive solutions to the equation, we obtained non-existence by applying the method of moving planes.

Keywords:

Fractional Laplacian of higher order,
non-existence of solutions,
maximum principle,
Liouville theorem,
the method of moving planes in integral form.

Mathematics Subject Classification: Primary: 35A01, 35B09; Secondary: 35C15.

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