In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional $p$-Laplacian: $$\left\{\begin{array}{cc}{(-\mathrm{\Delta })}_p^{\alpha }u=f(x,u,\nabla u),u>0\hfill & \text{in }\mathrm{\Omega },\hfill \\ u\equiv 0\hfill & \text{in }{ℝ}^n\setminus \mathrm{\Omega },\hfill \end{array}\right.$$ where $\mathrm{\Omega }$ is a bounded or an unbounded domain which is convex in $x_1$-direction, and ${(-\mathrm{\Delta })}_p^{\alpha }$ is the fractional $p$-Laplacian operator defined by $${(-\mathrm{\Delta })}_p^{\alpha }u(x)=C_{n,\alpha ,p}P.V.{\int }_{{ℝ}^n}\frac{|u(x)-u(y){|}^{p-2}[u(x)-u(y)]}{|x-y{|}^{n+\alpha p}}dy.$$ Under some mild assumptions on the nonlinearity $f(x,u,\nabla u)$, we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional $p$-Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].

在本文中，我们关注以下涉及小数的非线性方程的狄利克雷问题$p$-拉普拉斯算子：$$\left\{\begin{array}{cc}{(-\mathrm{\Delta })}_p^{\alpha }你=F(X,你,\nabla 你),你>0\hfill & \text{在 }\mathrm{\Omega },\hfill \\ 你\equiv 0\hfill & \text{在 }{ℝ}^n\setminus \mathrm{\Omega },\hfill \end{array}\right.$$在哪里$\mathrm{\Omega }$是一个有界或无界域，它是凸的$X_1$-方向，和${(-\mathrm{\Delta })}_p^{\alpha }$是分数$p$- 拉普拉斯算子定义为$${(-\mathrm{\Delta })}_p^{\alpha }你(X)=C_{n,\alpha ,p}磷.五.{\int }_{{ℝ}^n}\frac{|你(X)-你(\mathrm{是的}){|}^{p-2}[你(X)-你(\mathrm{是的})]}{|X-\mathrm{是的}{|}^{n+\alpha p}}d\mathrm{是的}.$$在非线性的一些温和假设下$F(X,你,\nabla 你)$，我们建立了涉及分数阶的非线性方程的正解的单调性和对称性$p$- 有界和无界域中的拉普拉斯算子。我们的结果是 Chen 和 Li [分数p-拉普拉斯算子的最大原理和解的对称性，Adv. 的扩展。数学。 335 (2018) 735–758] 和 Cheng等人。[分数拉普拉斯方程的最大原理及其应用，Commun. 当代。数学。 19（6）（2017）1750018]。