Radial symmetry for a generalized nonlinear fractional p-Laplacian problem

Wenwen Hou,Guotao Wang,Ravi P Agarwal,Lihong Zhang

doi:10.15388/namc.2021.26.22358

Abstract

This paper first introduces a generalized fractional p-Laplacian operator (–Δ)sF;p. By using the direct method of moving planes, with the help of two lemmas, namely decay at infinity and narrow region principle involving the generalized fractional p-Laplacian, we study the monotonicity and radial symmetry of positive solutions of a generalized fractional p-Laplacian equation with negative power. In addition, a similar conclusion is also given for a generalized Hénon-type nonlinear fractional p-Laplacian equation.

Highlights

Fractional-order differential equations are very suitable for describing materials and processes with memory and heritability, and their description of complex systems has the advantages of simple modeling, clear physical meaning of parameters and accurate description
We are concerned with a generalized nonlinear fractional p-Laplacian equation with negative power
Before we prove Theorem 2, we first consider the singularity of equation (2) at the origin

Summary

Introduction

Fractional-order differential equations are very suitable for describing materials and processes with memory and heritability, and their description of complex systems has the advantages of simple modeling, clear physical meaning of parameters and accurate description. Singular elliptic equations modeling steady states of van der Waals force driven thin films have been mathematically rigorously studied with no flux Neumann boundary condition They gave a complete description of all continuous radially symmetric solutions. In [6], Ma and Cai studied the following nonlinear fractional Laplacian equation with negative powers by using the direct method of moving planes:. (−∆)α/2v(x) + v−β(x) = 0, x ∈ Rn. The above results of all encourage us to further study a generalized nonlinear fractional p-Laplacian equation with negative powers by the direct method of moving planes.

PV F

Due to

Then we have