A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations

Yahui Niu

doi:10.3934/cpaa.2021027

Article Contents

2021, Volume 20, Issue 4: 1431-1445. Doi: 10.3934/cpaa.2021027

This issue Previous Article Expanding solutions of quasilinear parabolic equations Next Article Asymptotics for the higher-order derivative nonlinear Schrödinger equation

A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations

Yahui Niu^1,2,

1.
School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical, Sciences, Central China Normal University, Wuhan, 430079, China
2.
Department of Mathematical Sciences, Yeshiva University, New York, NY, USA

Received: August 31, 2020

Revised: November 30, 2020

Early access: March 2021

Published: April 2021

This work was supported by NSFC grant 11771166, Hubei Key Laboratory of Mathematical Sciences, Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46 and China Scholarship Council

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Abstract

In this paper, we proved a fractional Kirchhoff version of Hopf lemma for anti-symmetry functions and applied it to prove the symmetry and monotonicity of solutions for fractional Kirchhoff equations in the whole space by method of moving planes. We also obtain radially symmetry and monotonicity of solutions for fractional Kirchhoff equations in the unit ball. As far as we know, this is the first time to apply direct method of moving planes to fractional Kirchhoff problems.

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Mathematics Subject Classification: Primary: 35J60, 35B06.

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