Monotonicity of solutions to degenerate p-Laplace problems with a gradient term in half-spaces

We establish the monotonicity of positive solutions to the problem \(\begin{aligned} -\Delta _p u + a(u)|\nabla u|^q = f(u) \text { in } \mathbb {R}^N_+, \quad u=0 \text { on } \partial \mathbb {R}^N_+, \end{aligned}\) $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{p}u+a(u){|\mathrm{\nabla }u|}^q=f(u)\text{in}{\mathbb{R}}_{+}^N,u=0\text{on}\partial {\mathbb{R}}_{+}^N,\end{array}$$ where \(p>2\) $p>2$ , \(q\ge p-1\) $q\ge p-1$ and a, f are locally Lipschitz continuous functions such that f is positive on \((0,+\infty )\) $(0,+\infty )$ and it is either sublinear or superlinear near 0. The main tool we use is the refined method of moving planes for quasilinear elliptic problems in half-spaces.