A direct method of moving planes for the Logarithmic Laplacian

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Abstract

In this paper, a direct method of moving planes for the Logarithmic Laplacian is developed. Some key concepts of the method such as Maximum principle for anti-symmetric functions, Narrow region principle and Decay at infinity are discussed. As an application of the direct method of moving planes, the symmetry and monotonicity of the positive solutions of two nonlinear equations involving Logarithmic Laplacian are obtained.

Introduction

In recent years, fractional Laplacian is found to be of great interest as it can be used to model physical phenomena of diverse nature, for instance, see [1], [2], [3]. The method of moving planes is a powerful tool to study the radial solutions of fractional Laplacian equation. This method, introduced by Alexanderoff in the early 1950s, has been developed by many researchers later, for instance, see [4], [5], [6], [7], [8], [9], [10]. In 2017, Chen and Li [11] developed the direct method of moving planes for the fractional Laplacian. The method is directly applicable to nonlocal problems on bounded as well as unbounded domains. In [12], this method was applied to investigate the radial symmetry of standing waves for the nonlinear fractional p-Laplacian Schrödinger equation involving logarithmic nonlinearity. For the details of this method on radial symmetry and monotonicity of solutions of fractional Laplacian or fractional p-Laplacian equations and systems, we refer the reader to the works presented in [13], [14], [15], [16].

Recently, Chen and Weth [17] introduced the concept of a new fractional Laplacian operator called Logarithmic Laplacian. They first discussed the origin of the Logarithmic Laplacian and derived its related properties. Then they studied the functional analytic framework of Dirichlet problems involving the Logarithmic Laplacian on bounded domains and characterized the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of (Δ)s as s0. The representation of the Logarithmic Laplacian is as follows.

Proposition 1.1 [17]

If ϑ is Dini continuous at some point xRN, ΛRN is an open subset and xΛ, then we have the following integral representation (Δ)Lϑ(x)=CNΛϑ(x)ϑ(y)|xy|NdyCNRNΛϑ(y)|xy|Ndy+[hΛ(x)+μN]ϑ(x),where hΛ(x)=CNB1(x)Λ1|xy|NdyCNΛB1(x)1|xy|Ndy, CN=ππ2Γ(N2) and μN=2log2+κ(N2)β, where Γ is the Gamma function, κ=ΓΓ is the Digamma function, β=Γ(1) is the Euler Mascheroni constant and B1(x) is a sphere with x as its center and 1 as its radius.

In this paper, we study the Logarithmic Laplacian by the direct method of moving planes. At first, we prove a series of maximum principles for the Logarithmic Laplacian. Then two nonlinear Logarithmic Laplacian equations are investigated with the aid of the direct method of moving planes.

Section snippets

Several maximum principles

Let us begin this section with some notation and symbols.

Let T={x={x1,x}RNx1=ν,forsomeνR}be a hyperplane in RN, where x=(x2,,xN). Let xˆ=(2νx1,x2,,xN)be the reflection of x about the plane T. Define H={xRN|x1<ν}andHˆ={xxˆH}.

Theorem 2.1

[17]

Assume that ΛRN is a bounded Lipschitz domain and hΛ+μN0 on Λ. Moreover, let ϑL01(RN) be a continuous function on Λ¯, a Dini continuous function in Λ and if (Δ)Lϑ0inΛ,ϑ0onRNΛ,then ϑ>0inΛorϑ0a.e.inRN.

Theorem 2.2

Assume that Λ is a bounded Lipschitz domain in H and hH+μN

The direct method of moving planes and its application

In this section, we consider the nonlinear Schrödinger equation (Δ)Lϑ+ϑ=ϑp,xRN.

Theorem 3.1

Let ϑL01(RN) be a positive solution of (3.1). If lim|x|ϑ(x)<(1p)1p1,1<p<,then ϑ must be radially symmetric and monotone decreasing about some point in RN.

Proof

Let x0 be a point in RN, select any direction as the x1direction. For ν<x10, let Tν={xRNx1=ν},Σν={xRN|x1<ν},xν=(2νx1,x2,,xN), ϑν(x)=ϑ(xν),ϖν(x)=ϑν(x)ϑ(x),Σν={xΣνϖν(x)<0}.When ϖν is negative at some points, it is easy to verify that (Δ)Lϖν(x)+(1pϑp

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Supported by National Natural Science Foundation of China (No.12001344) and the Graduate Innovation Program of Shanxi, China (No.2020SY337).

1

All authors contributed equally to this work.

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