# Liouville Type Theorem for Some Nonlocal Elliptic Equations

## TL;DR

This paper establishes Liouville theorems for certain nonlocal elliptic equations involving nonlinearities and boundary conditions, showing under specific assumptions that no nontrivial positive solutions exist, extending classical results to nonlocal cases.

## Contribution

It extends Liouville theorems from local elliptic equations to nonlocal equations with nonlinear boundary conditions using the moving plane method.

## Key findings

- No nontrivial positive solutions under certain conditions
- Extension of classical Liouville theorems to nonlocal problems
- Application of the moving plane method to nonlocal equations

## Abstract

In this paper, we prove some Liouville theorem for the following elliptic equations involving nonlocal nonlinearity and nonlocal boundary value condition $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u(y)=\intpr \frac{ F(u(x',0))}{|(x',0)-y|^{N-\alpha}}dx'g(u(y)), &y\in\R, \\ \\ \displaystyle \frac{\partial u}{\partial \nu}(x',0)=\intr \frac{G(u(y))}{|(x',0)-y|^{N-\alpha}}\,dy f(u(x',0)), &(x',0)\in\partial \mathbb R_+^N, \end{array} \right. $$ where $\mathbb R_+^N=\{x\in \mathbb R^N:x_N>0\}$, $f,g,F,G$ are some nonlinear functions. Under some assumptions on the nonlinear functions $f,g,F,G$, we will show that this equation doesn't possess nontrivial positive solution. We extend the Liouville theorems from local problems to nonlocal problem. We use the moving plane method to prove our result.