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

$$\begin{aligned} -\Delta _p u + a(u)|\nabla u|^q = f(u) \text { in } \mathbb {R}^N_+, \quad u=0 \text { on } \partial \mathbb {R}^N_+, \end{aligned}$$

where \(p>2\), \(q\ge p-1\) and a, f are locally Lipschitz continuous functions such that f is positive on \((0,+\infty )\) 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.