Liouville type theorem of integral equation with anisotropic struture

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.