Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations

Leyun Wu; Pengcheng Niu

doi:10.3934/dcds.2019069

Article Contents

2019, Volume 39, Issue 3: 1573-1583. Doi: 10.3934/dcds.2019069

This issue Previous Article Symmetry properties in systems of fractional Laplacian equations Next Article Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus

Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations

Leyun Wu and
Pengcheng Niu^,

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

* Corresponding author: Pengcheng Niu
* Corresponding author: Pengcheng Niu

Received: December 31, 2017

Revised: March 31, 2018

Published: December 2018

The authors are supported by the National Natural Science Foundation of China (No.11771354) and the first author also supported by Excellent Doctorate Cultivating Foundation of Northwestern Polytechnical University

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Abstract

In this paper, we consider the fractional p-Laplacian equation

$( - \Delta )_p^su(x) = f(u(x)), $

where the fractional p-Laplacian is of the form

$( - \Delta )_p^su(x) = {C_{n, s, p}}PV\int_{{\mathbb{R}^n}} {\frac{{{{\left| {u(x) - u(y)} \right|}^{p - 2}}(u(x) - u(y))}}{{{{\left| {x - y} \right|}^{n + sp}}}}} dy.$

By proving a narrow region principle to the equation above and extending the direct method of moving planes used in fractional Laplacian equations, we establish the radial symmetry in the unit ball and nonexistence on the half space for the solutions, respectively.

Keywords:

Mathematics Subject Classification: Primary: 35A01, 35B53; Secondary: 35B09.

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