Liouville Type Theorem for Some Nonlocal Elliptic 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.