A direct method of moving planes for the Logarithmic Laplacian☆
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 -Laplacian Schrödinger equation involving logarithmic nonlinearity. For the details of this method on radial symmetry and monotonicity of solutions of fractional Laplacian or fractional -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 as . The representation of the Logarithmic Laplacian is as follows.
Proposition 1.1 [17] If is Dini continuous at some point , is an open subset and , then we have the following integral representation where
and , where is the Gamma function, is the Digamma function, is the Euler Mascheroni constant and is a sphere with 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 be a hyperplane in , where . Let be the reflection of about the plane . Define
Theorem 2.1 Assume that is a bounded Lipschitz domain and on . Moreover, let be a continuous function on , a Dini continuous function in and if then [17]
Theorem 2.2 Assume that is a bounded Lipschitz domain in and
The direct method of moving planes and its application
In this section, we consider the nonlinear Schrödinger equation
Theorem 3.1 Let be a positive solution of (3.1). If then must be radially symmetric and monotone decreasing about some point in .
Proof Let be a point in , select any direction as the direction. For , let
When is negative at some points, it is easy to verify that
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Radial symmetry and monotonicity of the positive solutions for k-Hessian equations
2023, Applied Mathematics LettersRadial solution of the Logarithmic Laplacian system
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2024, Rendiconti del Circolo Matematico di PalermoAN ANISOTROPIC TEMPERED FRACTIONAL p-LAPLACIAN MODEL INVOLVING LOGARITHMIC NONLINEARITY
2024, Evolution Equations and Control TheorySymmetry of Ancient Solution for Fractional Parabolic Equation Involving Logarithmic Laplacian
2023, Fractal and FractionalSymmetry of Positive Solutions for Lane-Emden Systems Involving the Logarithmic Laplacian
2023, Acta Applicandae Mathematicae
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Supported by National Natural Science Foundation of China (No.12001344) and the Graduate Innovation Program of Shanxi, China (No.2020SY337).
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All authors contributed equally to this work.