Liouville type theorem of integral equation with anisotropic struture

TL;DR

This paper classifies all positive solutions of a specific anisotropic integral equation with a Neumann boundary condition, establishing symmetry and equivalence to a PDE, and generalizing to systems.

Contribution

It provides a Liouville type theorem for anisotropic integral equations, using the method of moving planes, and links solutions to their PDE counterparts.

Findings

All positive solutions are classified.

Symmetry of solutions is established.

Equivalence between integral and PDE forms is shown.

Abstract

In this paper, we classify all positive solutions for the following integral equation: \begin{equation} u(x)=\int_{\mathbb{R}^n_+}K_b(x,y)y_n^b f(u(y))dy, \end{equation} where $ b > 1$ is a constant. Here $ K_b(x,y)$ is the Green function of the following homogeneous Neumann boundary problem \begin{equation} \left\{ \begin{aligned} -\text{div}(x^{b}_n \nabla u)&= f \quad in \mathbb{R}^n_+ \\ \frac{\partial u}{\partial x_n}&= 0 \quad on \ \partial \mathbb{R}^n_+ . \end{aligned} \right. \end{equation} 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.