1. Introduction
In this paper, we investigate the monotonicity properties of positive solutions to the following Dirichlet problem:
\begin{equation}\begin{cases}
(-\Delta)_p^\alpha u(x)=f(x,u,\nabla u) \qquad \text{in}\ \ E,\\ \quad u \gt 0 \qquad \text{in}\ \ E,\\
\quad u\equiv0 \qquad \text{in}\ \ \mathbb{R}^n\setminus E,
\end{cases}\end{equation} where 
$\alpha\in(0,1)$, 
$p\geq2$, 
$n\geq2$ and 
$E \subseteq \mathbb{R}^{n}$ is a coercive epigraph, that is, there exists a continuous function 
$\phi: \, \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ satisfying
\begin{equation*}
\lim_{|x'|\rightarrow+\infty}\phi(x')=+\infty,
\end{equation*} such that 
$E:=\{x=(x',x_{n}) \in \mathbb{R}^{n}|\,x_{n} \gt \phi(x')\}$, where 
$x':= (x_{1},...,x_{n-1}) \in \mathbb{R}^{n-1}$. The non-local operator 
$(-\Delta)_{p}^{\alpha}$ is the fractional 
$p$-Laplacian defined by
\begin{equation}
\begin{split}
(-\Delta)_p^\alpha u(x)&=C_{n,\alpha,p}P.V.\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-y|^{n+\alpha p}}dy\\
&=C_{n,\alpha,p}\lim_{\varepsilon\rightarrow0}\int_{\mathbb{R}^n\setminus {B_\varepsilon(x)}}\frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-y|^{n+\alpha p}}dy,
\end{split}\end{equation} where P.V. stands for the Cauchy principal value. In order to let the integral in the definition (1.2) of 
$(-\Delta)^{\alpha}_{p}$ makes sense in 
$E$, we require that
\begin{equation*}u\in C_{loc}^{1,1}(E)\cap \mathcal{L}_{\alpha p}(\mathbb{R}^{n})\end{equation*}with
\begin{equation*}\mathcal{L}_{\alpha p}(\mathbb{R}^{n}):=\left\{u:\mathbb{R}^{n}\rightarrow\mathbb{R} \,\Big| \int_{\mathbb{R}^n}\frac{|u(x)|^{p-1}}{1+|x|^{n+\alpha p}}dx \lt +\infty\right\}.\end{equation*} In the special case, 
$p=2$, 
$(-\Delta)^\alpha_p$ becomes the well-known fractional Laplacian 
$(-\Delta)^\alpha$. One can also show that, as 
$\alpha \rightarrow 1$, the fractional 
$p$-Laplacian converges to the regular 
$p$-Laplacian:
\begin{equation*} (-\Delta)^\alpha_p u(x) \rightarrow -\Delta_p u(x):=-\nabla\cdot\left(|\nabla u(x)|^{p-2}\nabla u(x)\right).\end{equation*} In recent years, fractional order operators have attracted more and more attentions. Besides various applications in fluid mechanics, molecular dynamics, relativistic quantum mechanics of stars (see e.g. [Reference Caffarelli and Vasseur9, Reference Constantin25]) and conformal geometry (see e.g. [Reference Chang and Gonzàlez11]), it also has many applications in probability and finance (see [Reference Bertoin3, Reference Cabré and Tan5]). The fractional Laplacians 
$(-\Delta)^{\alpha}$ (i.e., 
$(-\Delta)^{\alpha}_{p}$ with 
$p=2$) can be understood as the infinitesimal generators of stable Lévy diffusion processes (see [Reference Bertoin3]).
The non-local nature of the fractional 
$p$-Laplacians 
$(-\Delta)^{\alpha}_{p}$, the fractional Laplacians 
$(-\Delta)^{\alpha}$ and general fractional order operators makes them difficult to be investigated. To circumvent this, we basically have two approaches. One way is to define the non-local fractional Laplacian via Caffarelli and Silvestre’s extension method (see [Reference Caffarelli and Silvestre7]), which turns the non-local problem into a local one in higher dimensions. Another approach is to derive the integral representation formulae of solutions (see [Reference Chen, Li and Ou20, Reference Chen, Li and Ma21]). After establishing the equivalence between the fractional order equation and its corresponding integral equation, one can apply the method of moving planes (spheres) in integral forms, the method of scaling spheres or regularity lifting methods to study various properties of solutions to the fractional order equations involving fractional Laplacians. These two methods have been applied successfully to study equations involving fractional Laplacians, and a series of fruitful results have been derived (see [Reference Brandle, Colorado, de Pablo and Sanchez4, Reference Cabré and Tan5, Reference Caffarelli and Silvestre7, Reference Cao, Dai and Qin10, Reference Chen, Li and Ou20, Reference Dai, Liu and Lu27, Reference Dai, Liu and Qin28, Reference Dai, Peng and Qin30, Reference Dai and Qin32, Reference Dai and Qin34, Reference Frank, Lenzmann and Silvestre39, Reference Silvestre47] and the references therein).
However, as far as we know, except the fractional Laplacian, there has not been any extension methods that work for other non-local operators, such as the uniformly elliptic non-local operators and fully nonlinear non-local operators (see [Reference Caffarelli and Silvestre8] for the introductions of these operators) including the fractional 
$p$-Laplacian 
$(-\Delta)^{\alpha}_{p}$. Moreover, when using the above two approaches, we often need to impose some additional conditions on the solutions, which would not be necessary if we consider the pseudo-differential equation directly. Therefore, it is desirable to develop direct methods without going through extension methods or integral representation formulae.
In [Reference Chen, Li and Li19], Chen, Li and Li introduced the direct method of moving planes for the fractional Laplacian which has been applied to obtain symmetry, monotonicity, and Liouville type theorems of solutions for various semi-linear equations involving the fractional Laplacian. In [Reference Chen, Li and Li17], Chen, Li and Li refined the direct method of moving planes, so that it can be applied to fully nonlinear non-local problem in the case that the operator is non-degenerate in certain sense. In order to investigate the degenerate fractional 
$p$-Laplacian 
$(-\Delta)_{p}^{\alpha}$, Chen and Li [Reference Chen and Li16] introduced some new ideas, among which a significant one is a variant of the Hopf Lemma, the key boundary estimate, which plays the role of the narrow region principle in the second step of the direct method of moving planes. For more applications about direct blowing-up and rescaling argument, direct method of moving spheres, direct method of scaling spheres and sliding method for various non-local problems, please see [Reference Chen, Li and Li18, Reference Chen, Li and Zhang22, Reference Dai and Qin31–Reference Dai, Qin and Wu36, Reference Liu45] and the references therein.
As far as we know, the authors in previous literature mentioned above mainly deal with the nonlinear terms 
$f(u)$ or 
$f(x,u)$. Cheng [Reference Cheng23] and Cheng, Huang and Li [Reference Cheng, Huang and Li24] considered the nonlinear term 
$f(x, u,\nabla u)$ for the fractional Laplacian 
$(-\Delta)^{\alpha}$, they obtained the monotonicity and symmetry of positive solutions in bounded domains and unbounded domains which are convex in 
$x_{1}$ direction. Recently, Dai, Liu and Wang [Reference Dai, Liu and Wang29] proved the monotonicity and symmetry of positive solutions to the nonlinear equations (1.1) involving the fractional 
$p$-Laplacians 
$(-\Delta)^{\alpha}_{p}$ with nonlinearity 
$f(x,u,\nabla u)$ in both bounded and unbounded domains which are convex in 
$x_{1}$ direction. Their results extended those in Chen and Li [Reference Chen and Li16] for nonlinearity 
$f(u)$ and unit ball or 
$\mathbb{R}^{n}$ to nonlinear term 
$f(x,u,\nabla u)$ and general bounded or unbounded domains which are convex in 
$x_{1}$ direction, and also extend the results in Cheng, Huang and Li [Reference Cheng, Huang and Li24] for fractional Laplaican 
$(-\Delta)^{\alpha}$ to general fractional 
$p$-Laplacian 
$(-\Delta)^{\alpha}_{p}$ with 
$p\geq2$. For monotonicity and symmetry results of solutions to equations involving regular Laplacian with nonlinear term 
$f(x, u,\nabla u)$ or generalizations to fully nonlinear equations, please refer to [Reference Berestycki and Nirenberg1, Reference Berestycki and Nirenberg2, Reference Li41, Reference Li42, Reference Liu45, Reference Wang and Niu48].
In this paper, by applying the direct method of moving planes for 
$(-\Delta)^{\alpha}_{p}$, we will investigate the strict monotonicity properties of positive solutions to the Dirichlet problems (1.1) involving 
$(-\Delta)^{\alpha}_{p}$ in a coercive epigraph 
$E$. For more articles concerning the method of moving planes for fractional or higher order PDEs and integral equations, please see [Reference Caffarelli, Gidas and Spruck6, Reference Chang and Yang12–Reference Chen and Li15, Reference Dai, Fang and Qin26–Reference Dai, Liu and Wang29, Reference Dai and Qin31, Reference Dai and Qin33, Reference Dai, Qin and Wu36–Reference Felmer and Wang38, Reference Gidas, Ni and Nirenberg40, Reference Li and Zhu43, Reference Lin44, Reference Lu and Zhu46, Reference Wei and Xu49, Reference Zhuo, Chen, Cui and Yuan50] and the references therein.
Let 
$\mathcal{F}$ be the collection of functions 
$f(x, u, \mathbf{p})$: 
$\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ such that for any 
$M \gt 0$, 
$\forall u_1,u_2\in[-M, M]$ and 
$\forall x, \mathbf{p}\in\mathbb{R}^n$,
\begin{equation*}|f(x, u_1, \mathbf{p})-f(x, u_2, \mathbf{p})|\leq C_M |u_1-u_2| \qquad \text{for some}\ \ C_M \gt 0.\end{equation*} Our strict monotonicity results in a coercive epigraph 
$E$ is the following theorem.
Theorem 1.1. Let 
$E$ be a coercive epigraph, and let 
$u \in {\mathcal{L}}_{\alpha p}(\mathbb{R}^{n})\cap C^{1,1}_{loc}(E)\cap C(\bar{E})$ be a solution to (1.1) with 
$\alpha \in (0,1)$, 
$p\geqslant 2$ and 
$f(x,u,\textbf{p})\in \mathcal{F}$ satisfies
\begin{equation}
\left\{
\begin{aligned}
&f(x',x_{n},u,p_{1},...,p_{n})\leqslant
f(x',\bar{x}_{n},u,p_{1},...,-p_{n}),\\
&\forall \, x_{n} \geqslant \min_{\mathbb{R}^{n-1}}{\phi},\
p_{n} \geqslant 0,\
\bar{x}_{n}\geqslant x_{n}.\\
\end{aligned}
\right.
\end{equation} Then 
$u$ is strictly monotone increasing in 
$x_{n}$.
Remark 1.2. Typical forms of 
$f(x,u,\mathbf{p})$ which satisfy all the assumptions in Theorem 1.1 include: 
$f(x,u,\mathbf{p})=u^{q}(1+|\mathbf{p}|^{2})^{\frac{\sigma}{2}}$ with 
$q\geq1$ and 
$\sigma\leq0$, 
$f(x,u,\mathbf{p})=e^{\kappa u}(1+|\mathbf{p}|^{2})^{\frac{\sigma}{2}}$ with 
$\kappa\in\mathbb{R}$ and 
$\sigma\leq0$, and 
$f(x,u,\mathbf{p})=K(x)(1+|\mathbf{p}|^{2})^{\frac{\sigma}{2}}$ with 
$\sigma\in\mathbb{R}$, 
$K(x)=K(x',|x_{n}|)$ and 
$K(r,x')$ is non-increasing w.r.t. 
$r\in[0,+\infty)$.
Remark 1.3. Comparing with the monotonicity and symmetry results for 
$(-\Delta)^{\alpha}_{p}$ in general unbounded domains 
$\Omega$ derived by Dai, Liu and Wang in [Reference Dai, Liu and Wang29], we do not need to assume the following growth condition on 
$f(x, u, \mathbf{p})$: for some 
$s \gt p-2$,
\begin{equation}
\frac{|f(x, u_1, \mathbf{p})-f(x, u_2, \mathbf{p})|}{|u_1-u_2|}
\leq C(|u_1|^s+|u_2|^s) \qquad \text{as}\ \ u_1, u_2\rightarrow0.
\end{equation} In the following, we will use 
$C$ to denote a general positive constant that may depend on 
$\alpha$, 
$p$ and 
$n$, and whose value may differ from line to line.
2. Proof of Theorem 1.1
In this section, by using the direct method of moving planes for 
$(-\Delta)^{\alpha}_{p}$, we will carry out our proof of Theorem 1.1 and derive the strict monotonicity of positive solutions to (1.1) in a coercive epigraph 
$E$.
To this end, we need some standard notations. Without loss of generality, we assume
\begin{equation}
\inf _{x \in \Omega} x_{n}=\min _{\mathbb{R}^{n-1}} \phi=0.
\end{equation} For arbitrary 
$\lambda \gt 0$, let
\begin{equation}
T_{\lambda}:=\{x\in \mathbb{R}^{n}|x_{n}=\lambda\},
\end{equation}be the moving planes,
\begin{equation}
\Sigma_{\lambda}:=\{x\in \mathbb{R}^{n}|x_{n} \lt \lambda\},
\end{equation}be the region below the plane, and
\begin{equation}
x^{\lambda}:=(x_{1},x_{2},...,2\lambda-x_{n}),
\end{equation} be the refection of 
$x$ with respect to the plane 
$T_{\lambda}$.
To compare the values of 
$u(x)$ with 
$u_{\lambda}(x):=u(x^{\lambda})$, we define
\begin{equation}
w_{\lambda}(x):=u_{\lambda}(x)-u(x).
\end{equation}In order to carry out the method of moving planes, we need the following lemma on the key boundary estimate from [Reference Chen and Li16], which is a variant of the Hopf Lemma and plays the role of the narrow region principle in the second step of the method of moving planes.
Lemma 2.1 ([Reference Chen and Li16], Theorem 3)
Assume that 
$w_{\lambda_0}\geqslant 0$ and 
$w_{\lambda_0}\not\equiv0$ in 
$\Sigma_{\lambda_0}$. Suppose 
$\lambda_k\searrow \lambda_0$ and 
$x^k\in \Sigma_{\lambda_k}$, such that
\begin{equation*}w_{\lambda_k}(x^k)=\min_{\Sigma_{\lambda_k}}w_{\lambda_k}\leq0 \quad \text{and} \quad x^k\rightarrow \hat{x}\in \partial \Sigma_{\lambda_0}.\end{equation*} Let 
$\delta_{k}:=\delta_{x^{k},\lambda_{k}}=dist(x^{k},T_{\lambda_{k}})=|\lambda_{k}-(x^{k})_{n}|$. Then
\begin{equation*}\overline{\lim_{\delta_k\rightarrow0+}}\frac{1}{\delta_k}\left\{(-\Delta)_p^\alpha u_{\lambda_k}(x^{k})-(-\Delta)_p^\alpha u(x^k)\right\} \lt 0.\end{equation*} We also need the strong maximum principle on 
$(-\Delta)^{\alpha}_{p}$ to derive the strict monotonicity.
Lemma 2.2 (Strong maximum principle)
Assume 
$u \in \mathcal{L}_{\alpha p}(\mathbb{R}^{n})$ and 
$u_{\lambda}(x)-u(x)\geqslant 0$ in 
$\Sigma_{\lambda}$. If there exists some 
$\ \hat{x} \in \Sigma_{\lambda}$ such that 
$u_{\lambda}(\hat{x})-u(\hat{x})= 0$, 
$u$ is 
$C^{1,1}_{loc}$ near 
$\hat{x}$ and 
${\hat{x}}^{\lambda}$ and 
${(-\Delta)}^{\alpha}_{p}u_{\lambda}(\hat{x})-{(-\Delta)}^{\alpha}_{p}u(\hat{x})\geqslant 0$, then 
$u_{\lambda}(x)=u(x)$ a.e. in 
$\mathbb{R}^{n}$.
Proof. Through direct calculations, we have
\begin{align*}
0&\leqslant (-\Delta)_p^\alpha u_{\lambda}(\hat{x})-(-\Delta)_p^\alpha u(\hat{x})\\
&=C_{n,\alpha,p}P.V.\int_{\mathbb{R}^n}\frac{G\big(u_{\lambda}(\hat{x})-u_{\lambda}(y)\big)-G\big(u(\hat{x})-u(y)\big)}{|\hat{x}-y|^{n+\alpha p}}dy\\
&=C_{n,\alpha,p}P.V.\int_{\Sigma_{\lambda}}\frac{G\big(u_{\lambda}(\hat{x})-u_{\lambda}(y)\big)-G\big(u(\hat{x})-u(y)\big)}{|\hat{x}-y|^{n+\alpha p}}dy\\
&\ \ +C_{n,\alpha,p}P.V.\int_{\Sigma_{\lambda}}\frac{G\big(u_{\lambda}(\hat{x})-u(y)\big)-G\big(u(\hat{x})-u_{\lambda}(y)\big)}{|\hat{x}-y^{\lambda}|^{n+\alpha p}}dy\\
&= C_{n,\alpha,p}P.V.\int_{\Sigma_{\lambda}}\left(\frac{1}{|\hat{x}-y|^{n+\alpha p}}-\frac{1}{|\hat{x}-y^{\lambda}|^{n+\alpha p}}\right)\times\\
&\quad \Big(G\big(u_{\lambda}(\hat{x})-u_{\lambda}(y)\big)-G\big(u(\hat{x})-u(y)\big)\Big)dy\\
&\leqslant 0,
\end{align*} where 
$G(t):=|t|^{p-2}t$ and we have used 
$u_{\lambda}(\hat{x})=u(\hat{x})$. Thus we can deduce that
\begin{equation}
u_{\lambda}(y)=u(y)\ \ \ \ \text{a.e. in}\,\, \Sigma.
\end{equation}Then it follows that
\begin{equation}
u_{\lambda}(y)=u(y)\ \ \ \ \text{a.e. in}\,\, \mathbb{R}^{n}.
\end{equation}This finishes our proof of Lemma 2.2.
Now, we assume that 
$u \gt 0$ is a solution to Dirichlet problem (1.1). Since 
$E$ is a coercive epigraph, 
$\Sigma_{\lambda}\cap E$ is always bounded for every 
$\lambda \gt 0$. One can easily obtain that, for any 
$\lambda \gt 0$,
\begin{equation}
w_{\lambda}(x)\geqslant 0,\ \ \ \ w_{\lambda}(x)\not\equiv 0\ \ \ \ in\ \Sigma_{\lambda}\backslash E.
\end{equation} We aim at proving that 
$w_{\lambda} \gt 0$ in 
$\Sigma_{\lambda}\cap E$ for every 
$\lambda \gt 0$, which gives the desired strict monotonicity.
We will carry out the method of moving planes in two steps.
$Step\ 1$. We will first show that, there exists an 
$\eta \gt 0$ sufficiently close to 0 such that
\begin{equation}
w_{\lambda}(x)\geqslant 0\ \ \ \ in\ \Sigma_{\lambda}\cap E,\ \ \forall\ 0 \lt \lambda\leqslant \eta.
\end{equation} Suppose (2.9) does not hold, then there exists a sequence 
$\{\lambda_{k}\}$ satisfying 
$\lambda_{k} \gt 0$ and 
$\lambda_{k} \rightarrow 0$ as 
$k \rightarrow +\infty$ such that
\begin{equation}
\inf_{\Sigma_{\lambda_{k}}\cap E}w_{\lambda_{k}} = \inf_{\lambda \in (0,\lambda_{k}]}\inf_{x \in \Sigma_{\lambda}}w_{\lambda}(x) \lt 0.
\end{equation} Consequently, there exists 
$x^{k} \in \Sigma_{\lambda_{k}}\cap E$ such that
\begin{equation}
w_{\lambda_{k}}(x_{k}) = \inf_{\Sigma_{\lambda_{k}}\cap E}w_{\lambda_{k}} = \inf_{\Sigma_{\lambda_{k}}}w_{\lambda_{k}} \lt 0.
\end{equation} It follows directly from (2.10) and (2.11) that 
$\frac{\partial w_{\lambda}}{\partial \lambda}|_{\lambda = \lambda_{k}}(x^{k})\leqslant 0,$ and hence, 
$(\partial_{x_{n}}u)[(x^{k})^{\lambda_{k}}]\leqslant 0.$ Note that 
$x^{k}$ is the interior minimum of 
$w_{\lambda_{k}}(x)$, then one has 
$\nabla_{x}w_{\lambda_{k}}(x^{k})=0$, i.e.,
\begin{equation}
\nabla_{x}u_{\lambda_{k}}(x^{k})=\nabla_{x}u(x^{k}).
\end{equation}By the assumption in Theorem 1.1 and the equation (1.1), we have
\begin{align}
&\ \ \ \ {(-\Delta)}^{\alpha}_{p}u_{\lambda_{k}}(x^{k})-{(-\Delta)}^{\alpha}_{p}u(x^{k})\nonumber\\
&=f((x^{k})^{\lambda_{k}}, u_{\lambda_{k}}(x^{k}), (\nabla_{x}u)\left[(x^{k})^{\lambda_{k}}\right])-f(x^{k}, u(x^{k}), (\nabla_{x}u)(x^{k}))\nonumber\\
&\geqslant f(x^{k}, u_{\lambda_{k}}(x^{k}), (\nabla_{x}u)(x^{k})) - f(x^{k}, u(x^{k}), (\nabla_{x}u)(x^{k}))\nonumber\\
&=:c_{\lambda_{k}}(x^{k})w_{\lambda_{k}}(x^{k}),\end{align}where
\begin{equation}
c_{\lambda}(x):=\frac{f(x, u_{\lambda}(x), (\nabla_{x}u)(x))-f(x, u(x), (\nabla_{x}u)(x))}{u_{\lambda}(x)-u(x)}.
\end{equation} Since 
$f(x,u,\mathbf{p})\in\mathcal{F}$ and 
$u \in C(\mathbb{R}^n)$, we can deduce that 
$c_{\lambda_{k}}(x^{k})$ is uniformly bounded and the bound is independent of 
$k$.
On the other hand, by Lemma 2.1, we derive that there exists some positive constant 
$c_0 \gt 0$ such that
\begin{equation}
(-\Delta)_p^\alpha u_{\lambda_k}(x^k)-(-\Delta)_p^\alpha u(x^k)\leqslant -c_0 \delta_k, \quad\,\, \text{as } \ \delta_k\rightarrow0.
\end{equation} Let 
$\overline{x^k}:=((x^k)',\lambda_k)\in T_{\lambda_k}$, then
\begin{equation*}w_{\lambda_k}(\overline{x^k})=w_{\lambda_k}(x^k)+\nabla_x w_{\lambda_k}(x^k)\cdot(\overline{x^k}-x^k)+o(|\overline{x^k}-x^k|).\end{equation*}It follows from (2.12) that
\begin{equation}
w_{\lambda_k}(x^k)=o_{k}(1)\delta_k.
\end{equation} From (2.13), (2.15) and (2.16), one can derive immediately a contradiction. Hence there exists an 
$\eta \gt 0$ such that (2.9) holds for any 
$0 \lt \lambda\leqslant \eta$.
Furthermore, it follows from the Lemma 2.2 (Strong maximum principle) that
\begin{equation}
w_{\lambda}(x) \gt 0\ \ \ \ \rm{in}\ \Sigma_{\lambda}\cap E,\ \ \forall\ 0 \lt \lambda\leqslant \eta.
\end{equation} Step 2. Step 1 provides a starting point to move the plane 
$T_\lambda$ upward along the 
$x_{n}$-axis. Now we move the plane upward further as long as (2.17) holds to its limiting position. To this end, let us define
\begin{equation}
\lambda_{0}:=\sup\{\lambda \gt 0\ |\ w_{\mu} \gt 0\ \text{in} \,\, \Sigma_{\mu} \cap \Omega,\ \forall\ 0 \lt \mu \lt \lambda\}.
\end{equation} We aim to prove that 
$\lambda_0=+\infty$ via contradiction arguments. Suppose on the contrary that 
$\lambda_0 \lt +\infty$, we first show that
\begin{equation}
w_{\lambda_0} \gt 0 \quad \text{in} \,\, E\cap\Sigma_{\lambda_0}.
\end{equation} Note that 
$w_{\lambda_0}\geqslant0$ in 
$E\cap\Sigma_{\lambda_0}$, suppose (2.19) does not hold, then there exists 
$\bar{x}\in E\cap\Sigma_{\lambda_0}$ such that 
$w_{\lambda_0}(\bar{x})=0$. One can calculate directly that
\begin{align}
&\quad (-\Delta)_p^\alpha u_{\lambda_0}(\bar{x})-(-\Delta)_p^\alpha u(\bar{x})\nonumber\\
&=C_{n,\alpha,p}P.V.\int_{\mathbb{R}^n}\frac{G\big(u_{\lambda_0}(\bar{x})-u_{\lambda_0}(y)\big)-G\big(u(\bar{x})-u(y)\big)}{|\bar{x}-y|^{n+\alpha p}}dy\nonumber\\
&=C_{n,\alpha,p}P.V.\int_{\Sigma_{\lambda_0}}\frac{G\big(u_{\lambda_0}(\bar{x})-u_{\lambda_0}(y)\big)-G\big(u(\bar{x})-u(y)\big)}{|\bar{x}-y|^{n+\alpha p}}dy\nonumber\\
&\ \ +C_{n,\alpha,p}P.V.\int_{\Sigma_{\lambda_0}}\frac{G\big(u_{\lambda_0}(\bar{x})-u(y)\big)-G\big(u(\bar{x})-u_{\lambda_0}(y)\big)}{|\bar{x}-y^{\lambda_0}|^{n+\alpha p}}dy\nonumber\\
&=C_{n,\alpha,p}P.V.\int_{\Sigma_{\lambda_0}}\left(\frac{1}{|\bar{x}-y|^{n+\alpha p}}-\frac{1}{|\bar{x}-y^{\lambda_0}|^{n+\alpha p}}\right)\times\nonumber\\
&\quad \Big(G\big(u_{\lambda_0}(\bar{x})-u_{\lambda_0}(y)\big)-G\big(u(\bar{x})-u(y)\big)\Big)dy\nonumber\\
&\quad +C_{n,\alpha,p}P.V.\bigg[\int_{\Sigma_{\lambda_0}}\frac{G\big(u_{\lambda_0}(\bar{x})-u_{\lambda_0}(y)\big)-G\big(u(\bar{x})-u_{\lambda_0}(y)\big)}{|\bar{x}-y^{\lambda_0}|^{n+\alpha p}}dy\nonumber\\
&\quad +\int_{\Sigma_{\lambda_0}}\frac{
G\big(u_{\lambda_0}(\bar{x})-u(y)\big)-G\big(u(\bar{x})-u(y)\big)}{|\bar{x}-y^{\lambda_0}|^{n+\alpha p}}dy\bigg]\nonumber\\
&=:I_1+I_2.\end{align} For 
$I_1$, we can infer from the mean value theorem that
\begin{align}
I_1&=C_{n,\alpha,p}(n+\alpha p)\int_{\Sigma_{\lambda_0}}\frac{G'\big(\theta(y)\big)}{\eta(y)^{n+\alpha p+1}}\left(w_{\lambda_0}(\bar{x})-w_{\lambda_0}(y)\right)\left(|\bar{x}-y^{\lambda_0}|-|\bar{x}-y|\right)dy\nonumber\\
&\leqslant 2C_{n,\alpha,p}(n+\alpha p)\int_{\Sigma_{\lambda_0}}G'\big(\theta(y)\big)\left(w_{\lambda_0}(\bar{x})-w_{\lambda_0}(y)\right)
\frac{\left({\lambda_0}-(\bar{x})_{n}\right)\left({\lambda_0}-y_{n}\right)}{|\bar{x}-y^{\lambda_0}|^{n+\alpha p+2}}dy\nonumber\\
&\leqslant \bar{C}_{n,\alpha,p}\left[\lambda_0-(\bar{x})_{n}\right]\int_{\Sigma_{\lambda_0}}G'\big(\theta(y)\big)\left({\lambda_0}-y_{n}\right)
\frac{w_{\lambda_0}(\bar{x})-w_{\lambda_0}(y)}{|\bar{x}-y^{\lambda_0}|^{n+\alpha p+2}}dy,
\end{align} where 
$|\bar{x}-y| \lt \eta(y) \lt |\bar{x}-y^{\lambda_0}|$, 
$u_{\lambda_0}(\bar{x})-u_{\lambda_0}(y)\leqslant \theta(y)\leqslant u(\bar{x})-u(y)$ and 
$G'\big(\theta(y)\big)\geqslant 0$ but 
$G'\big(\theta(y)\big)\not\equiv0$ in 
$\Sigma_{\lambda_{0}}$.
It is obvious that 
$I_2=0$ due to 
$w_{\lambda_0}(\bar{x})=0$. Consequently, since 
$w_{\lambda_0}\not\equiv0$ in 
$\Sigma_{\lambda_0}$, similar to (2.13), we deduce from (2.21) that
\begin{align}
0&\leqslant (-\Delta)_p^\alpha u_{\lambda_0}(\bar{x})-(-\Delta)_p^\alpha u(\bar{x})-c_{\lambda_{0}}(\bar{x})w_{\lambda_0}(\bar{x})=I_{1}+I_{2} \nonumber\\
&\leqslant \bar{C}_{n,\alpha,p}\left[\lambda_0-(\bar{x})_{n}\right]\int_{\Sigma_{\lambda_0}}G'\big(\theta(y)\big)\left(\lambda_0-y_{n}\right)
\frac{-w_{\lambda_0}(y)}{|\bar{x}-y^{\lambda_0}|^{n+\alpha p+2}}dy\nonumber\\
& \lt 0,
\end{align} where we have used the fact that 
$G'\big(\theta(y)\big) \gt 0$ and 
$w_{\lambda_0} \gt 0$ in 
$E^{\lambda_{0}}\setminus E\subset\Sigma_{\lambda_{0}}$ (
$S^{\lambda}$ is the reflection of the set 
$S$ with respect to the plane 
$T_{\lambda}$), which yields a contradiction. Hence we arrive at (2.19), i.e., 
$w_{\lambda_0} \gt 0$ in 
$E\cap\Sigma_{\lambda_0}$.
Suppose that 
$\lambda_{0} \lt +\infty$, by the definition of 
$\lambda_0$, there exists a sequence 
$+\infty \gt \lambda_k\searrow\lambda_0$ and 
$x^k\in E\cap\Sigma_{\lambda_k}$ such that
\begin{equation*}
w_{\lambda_k}(x^k)=\min_{\Sigma_{\lambda_k}}w_{\lambda_k} \lt 0 \ \ \text{and}\ \ \nabla_{x} w_{\lambda_k}(x^k)=0.
\end{equation*} There exists a subsequence of 
$\{x^k\}$ (still be denoted by 
$\{x^k\}$) that converges to some point 
$\tilde{x}\in\overline{E\cap\Sigma_{\lambda_{0}}}$. By the continuity of 
$w_{\lambda}(x)$ and its derivatives with respect to both 
$x$ and 
$\lambda$, we have
\begin{equation}
w_{\lambda_0}(\tilde{x})=0\ \ \text{and hence}\ \ \tilde{x}\in\partial\left(E\cap\Sigma_{\lambda_0}\right),\ \ \text{and}\ \ \nabla_{x} w_{\lambda_0}(\tilde{x})=0.
\end{equation} Suppose that 
$\tilde{x}\not\in T_{\lambda_0}$, i.e., 
$\lambda_{0}-(\tilde{x})_{n}=d_{0} \gt 0$. Then, similar to the inequalities (2.20), (2.21) and (2.22) (replacing 
$\lambda_{0}$, 
$\tilde{x}$ by 
$\lambda_{k}$, 
$x^{k}$ therein), we can deduce from the Lebesgue’s dominated convergence theorem that
\begin{align}
&\quad 0\leqslant \lim_{k\rightarrow+\infty}\left\{(-\Delta)_p^\alpha u_{\lambda_{k}}(x^{k})-(-\Delta)_p^\alpha u(x^{k})-c_{\lambda_{k}}(x^{k})w_{\lambda_{k}}(x^{k})\right\}\nonumber\\
&=\lim_{k\rightarrow+\infty}\bar{C}_{n,\alpha,p}\left[\lambda_{k}-(x^{k})_{n}\right]\int_{\Sigma_{\lambda_{k}}}\left(\lambda_{k}-y_n\right)
\frac{G\big(u_{\lambda_{k}}(x^{k})-u_{\lambda_{k}}(y)\big)-G\big(u(x^{k})-u(y)\big)}{|x^{k}-y^{\lambda_{k}}|^{n+\alpha p+2}}dy\nonumber\\
&\quad+ \lim_{k\rightarrow+\infty}C_{n,\alpha,p}\bigg[\int_{\Sigma_{\lambda_{k}}}\frac{G\big(u_{\lambda_{k}}(x^{k})-u_{\lambda_{k}}(y)\big)
-G\big(u(x^{k})-u_{\lambda_{k}}(y)\big)}{|x^{k}-y^{\lambda_{k}}|^{n+\alpha p}}dy \nonumber \\
& \qquad\qquad\qquad\qquad +\int_{\Sigma_{\lambda_{k}}}\frac{
G\big(u_{\lambda_{k}}(x^{k})-u(y)\big)-G\big(u(x^{k})-u(y)\big)}{|x^{k}-y^{\lambda_{k}}|^{n+\alpha p}}dy\bigg] \nonumber\\
&=\bar{C}_{n,\alpha,p}\left[\lambda_{0}-(\tilde{x})_{n}\right]\int_{\Sigma_{\lambda_{0}}}\left(\lambda_{0}-y_n\right)
\frac{G\big(u_{\lambda_{0}}(\tilde{x})-u_{\lambda_{0}}(y)\big)-G\big(u(\tilde{x})-u(y)\big)}{|\tilde{x}-y^{\lambda_{0}}|^{n+\alpha p+2}}dy \nonumber\\
&=-\bar{C}_{n,\alpha,p}\left[\lambda_{0}-(\tilde{x})_{n}\right]\int_{\Sigma_{\lambda_{0}}}\left(\lambda_{0}-y_n\right)G'\big(\theta(y)\big)
\frac{w_{\lambda_0}(y)}{|\tilde{x}-y^{\lambda_{0}}|^{n+\alpha p+2}}dy,\end{align} where 
$u_{\lambda_0}(\tilde{x})-u_{\lambda_0}(y)\leq\theta(y)\leq u(\tilde{x})-u(y)$, 
$G'\big(\theta(y)\big)\geqslant 0$ in 
$\Sigma_{\lambda_{0}}$ and 
$G'\big(\theta(y)\big) \gt 0$ in 
$E^{\lambda_{0}}\cap\Sigma_{\lambda_{0}}$. Inequality (2.24) implies immediately that
\begin{equation*}\int_{\Sigma_{\lambda_{0}}}\left(\lambda_{0}-y_n\right)G'\big(\theta(y)\big)
\frac{w_{\lambda_0}(y)}{|\tilde{x}-y^{\lambda_{0}}|^{n+\alpha p+2}}dy\leqslant 0.\end{equation*}Since
\begin{equation*}\int_{\Sigma_{\lambda_{0}}\backslash E}\left(\lambda_{0}-y_n\right)G'\big(\theta(y)\big)
\frac{w_{\lambda_0}(y)}{|\tilde{x}-y^{\lambda_{0}}|^{n+\alpha p+2}}dy\geqslant0,\end{equation*}it follows immediately that
\begin{equation*}\int_{\Sigma_{\lambda_{0}}\cap E}\left(\lambda_{0}-y_n\right)G'\big(\theta(y)\big)
\frac{w_{\lambda_0}(y)}{|\tilde{x}-y^{\lambda_{0}}|^{n+\alpha p+2}}dy\leqslant0.\end{equation*} So we obtain 
$w_{\lambda_{0}}\leqslant 0$ in 
$E\cap\Sigma_{\lambda_{0}}$, which contradicts (2.19).
Therefore, it follows that 
$\tilde{x}\in T_{\lambda_0}$ and hence
\begin{equation*}\delta_{k}:=dist(x^{k},T_{\lambda_{k}})=\lambda_{k}-(x^{k})_{n}\rightarrow0, \quad \text{as} \,\, k\rightarrow+\infty.\end{equation*}Then, similar to (2.16), we can get
\begin{equation}
\frac{w_{\lambda_k}(x^k)}{\delta_k}\rightarrow0, \quad \text{as} \,\, k\rightarrow+\infty.
\end{equation} By Lemma 2.1 and equation (1.1), there exists a positive constant 
$c_0 \gt 0$ such that, for 
$\delta_k$ sufficiently small,
\begin{equation}
c_{\lambda_{k}}(x^k)\frac{w_{\lambda_k}(x^k)}{\delta_k}\leqslant \frac{1}{\delta_{k}}\left\{(-\Delta)_p^\alpha u_{\lambda_k}(x^{k})-(-\Delta)_p^\alpha u(x^k)\right\}\leqslant -c_0 \lt 0,
\end{equation}where
\begin{equation*}c_{\lambda_{k}}(x^k)=\frac{f\big(x^k,u_{\lambda_k}(x^k),(\nabla_xu)(x^k)\big)
-f\left(x^k,u(x^k),(\nabla_xu)(x^k)\right)}{u_{\lambda_k}(x^k)-u(x^k)},\end{equation*} is uniformly bounded and the bound is independent of 
$k$, since 
$f(x, u, \mathbf{p})\in\mathcal{F}$ and 
$u \in C(\mathbb{R}^n)$. It follows immediately that (2.25) and (2.26) will lead to a contradiction. Thus we must have 
$\lambda_{0}=+\infty$.
As a consequence, we have arrived at
\begin{equation*}
w_\lambda \gt 0 \quad \text{in} \,\, \Sigma_\lambda\cap E, \quad \forall \,\, 0 \lt \lambda \lt +\infty.
\end{equation*} For any 
$(x',x_n)$, 
$(x',\bar{x}_n)\in E$ with 
$0 \lt x_n \lt \bar{x}_n$, one can take 
$\lambda=\frac{x_n+\bar{x}_n}{2}$. Then we have
\begin{equation*}u(x',x_n) \lt u(x',\bar{x}_n),\end{equation*} and hence 
$u(x',x_n)$ is strictly increasing in 
$x_n$-direction in 
$E$. This concludes our proof of Theorem 1.1.
Funding
Wei Dai is supported by the National Science and Technology Major Project (2022ZD0116401), the NNSF of China (No. 12222102 and No. 12426659) and the Fundamental Research Funds for the Central Universities. Jingze Fu is supported by the National Science and Technology Major Project (2022ZD0116401) and the Fundamental Research Funds for the Central Universities.
Acknowledgements
The authors are deeply grateful to the anonymous referees for their careful reading and very valuable comments and suggestions that improved the presentation of the paper.
