Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian

Zaizheng Li; Qidi Zhang

doi:10.3934/cpaa.2020293

Article Contents

2021, Volume 20, Issue 2: 835-865. Doi: 10.3934/cpaa.2020293

This issue Previous Article Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity Next Article Multiple positive solutions for coupled Schrödinger equations with perturbations

Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian

Zaizheng Li^1,2, , and
Qidi Zhang^3,

1.
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China
2.
Department of Mathematical Sciences, Yeshiva University, New York, NY 10033, USA
3.
Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

^* Corresponding author
^* Corresponding author

Received: June 30, 2020

Revised: September 30, 2020

Early access: December 2020

Published: February 2021

The first author was supported by CHINA SCHOLARSHIP COUNCIL

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Abstract

We prove a Hopf's lemma in the point-wise sense for fractional $ p $-Laplacian. The essential technique is to prove $ (-\Delta)^s_p u(x) $ is uniformly bounded in the unit ball $ B_1\subset\mathbb{R}^n $, where $ u(x) = (1-|x|^2)^s_{+} $. Also we study the global Hölder continuity of bounded positive solutions for $ (-\Delta)^s_p u(x) = f(x,u). $

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Mathematics Subject Classification: Primary: 35R11, 35B65; Secondary: 35J91.

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