In this paper, we consider the following problem$\{\begin{array}{cc}{(-\mathrm{\Delta })}_p^{s}u(x)=f(u),u(x)>0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u(x)=0\hfill & \text{on}{\mathbb{R}}^n\setminus \mathrm{\Omega },\hfill \end{array}$ where ${(-\mathrm{\Delta })}_p^s$ is the fractional p-Laplacian with $p\ge 2$ and$\mathrm{\Omega }:=\{x=(x^{\prime },x_n)|x_n>\phi (x^{\prime })\}$ is an unbounded domain. In the case $\phi \equiv 0$, it reduces to the upper half space.

Without assuming any asymptotic behavior of u near infinity, we first develop narrow region principles in unbounded domains, then using the method of moving planes, we establish the monotonicity of the positive solutions.

In most previous literature, to apply the method of moving planes on unbounded domains, one usually needed to assume that the solution tends to zero in certain rate near infinity; or make a Kelvin transform, or divide the solution by a function, so that the new solution possesses such an asymptotic decay. Here we introduce a new idea, estimating the singular integral defining ${(-\mathrm{\Delta })}_p^s$ along a sequence of auxiliary functions at their maximum points. This way, we only require the solutions be bounded. We believe that this new method will become a useful tool in investigating qualitative properties of solutions for equations involving nonlinear nonlocal operators.

在本文中，我们考虑以下问题$\{\begin{array}{cc}{(-\mathrm{\Delta })}_{磷}^{秒}你(X)=F(你),你(X)>0\hfill & \text{在}\mathrm{\Omega },\hfill \\ 你(X)=0\hfill & \text{在}{\mathbb{电阻}}^n\setminus \mathrm{\Omega },\hfill \end{array}$ 在哪里 ${(-\mathrm{\Delta })}_{磷}^{秒}$是分数p -Laplacian 与$磷\ge 2$ 和$\mathrm{\Omega }：=\{X=(X^{\prime },X_n)|X_n>\phi (X^{\prime })\}$是一个无界域。在这种情况下$\phi \equiv 0$，它减少到上半部分空间。

在不假设u接近无穷大的任何渐近行为的情况下，我们首先在无界域中开发窄域原理，然后使用移动平面的方法，我们建立了正解的单调性。

在以前的大多数文献中，要在无界域上应用移动平面的方法，通常需要假设解在接近无穷大的某个速率下趋于零；或进行开尔文变换，或将解除以函数，使新解具有这种渐近衰减。这里我们引入一个新的想法，估计奇异积分定义${(-\mathrm{\Delta })}_{磷}^{秒}$沿着一系列辅助函数在它们的最大值点。这样，我们只要求解是有界的。我们相信，这种新方法将成为研究涉及非线性非局部算子的方程解的定性性质的有用工具。