Moving planes for nonlinear fractional Laplacian equation with negative powers

Miaomiao Cai,Li Ma

doi:10.3934/dcds.2018201

Abstract

In this paper, we study symmetry properties of positive solutions to the fractional Laplace equation with negative powers on the whole space. We can use the direct method of moving planes introduced by Jarohs-Weth-Chen-Li-Li to prove one particular result below. If \begin{document} $u∈ C^{1, 1}_{loc}(\mathbb{R}^{n})\cap L_{α}$ \end{document} satisfies \begin{document}$(-Δ)^{α/2}u(x)+u^{-β}(x) = 0, x∈ \mathbb{R}^n, $ \end{document} with the growth/decay property \begin{document}$u(x) = a|x|^{m}+o(1), as |x| \to ∞, $ \end{document} where \begin{document} $\frac{α}{β+1} , \begin{document} $a>0$ \end{document} is a constant, then the positive solution \begin{document} $u(x)$ \end{document} must be radially symmetric about some point in \begin{document} $\mathbb{R}^{n}$ \end{document} . Similar result is also true for Henon type nonlinear fractional Laplace equation with negative powers.

Highlights

Analytical problems with negative powers arise naturally in the thin film equations and electrostatic micro-electromechanical system (MEMS) device [1] [2] [8] [16] [17] [13] [9] [11] [21] [26], singular minimal surface equations [25], Lichnerowicz equations in general relativity [20] [23] [21] [22] [28], and prescribed curvature equations in conformal geometry [29]
Thanks to the works [5] and [15] ( [12], [27], and [10]), we can use the direct method of moving planes to study symmetry properties of non-negative solutions to the equation (1)
In the second part of the paper, we study the symmetry properties of non-negative solutions to Henon type fractional Laplace equation

Summary

Introduction

Analytical problems with negative powers arise naturally in the thin film equations and electrostatic micro-electromechanical system (MEMS) device [1] [2] [8] [16] [17] [13] [9] [11] [21] [26], singular minimal surface equations [25], Lichnerowicz equations in general relativity [20] [23] [21] [22] [28], and prescribed curvature equations in conformal geometry [29]. Thanks to the works [5] and [15] ( [12], [27], and [10]), we can use the direct method of moving planes to study symmetry properties of non-negative solutions to the equation (1).

Results

Conclusion