Liouville type theorem of integral equation with anisotropic struture

2021/07/08 by Yating Niu, Niu, Yating
Computer Science · Mathematics · #35J61 45E10 45G15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2107.03902

openalex publication_date 2021/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we classify all positive solutions for the following integral equation: u(x)=∫_ℝⁿ+K_b(x,y)y_n^b f(u(y))dy, where b > 1 is a constant. Here K_b(x,y) is the Green function of the following homogeneous Neumann boundary problem \ \beginaligned -div(x^b_n ∇ u)amp;= f in ℝⁿ₊
(∂ u)/(∂ x_n)amp;= 0 on ∂ ℝⁿ₊ . \endaligned . By using the method of moving planes in integral form, we derive the symmetry of positive solutions. We also establish the equivalence between the integral equation and its corresponding partial differential equation. Similarly, the results can be generalized to the integral system.

Citations

• Classification of solutions of some nonlinear elliptic equations
• A symmetry problem in potential theory
• Classification of solutions for an integral equation

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