Monotonicity in Half-spaces for p-Laplace Problems with a Sublinear Nonlinearity

We prove the monotonicity of positive solutions to the equation \(-\Delta _p u = f(u)\) $-{\mathrm{\Delta }}_{p}u=f(u)$ in \(\mathbb {R}^N_+\) ${\mathbb{R}}_{+}^N$ with zero Dirichlet boundary condition, where \(1<p<2\) $1<p<2$ and \(f:[0,+\infty )\rightarrow \mathbb {R}\) $f:[0,+\infty )\to \mathbb{R}$ is a continuous function which is positive and locally Lipschitz continuous in \((0,+\infty )\) $(0,+\infty )$ and \(\liminf _{t\rightarrow 0^+}\frac{f(t)}{t^{p-1}}>0\) ${lim\; inf}_{t\to 0^{+}}\frac{f(t)}{t^{p-1}}>0$ . Furthermore, we allow f to be sign-changing in the case \(\frac{2N+2}{N+2}<p<2\) $\frac{2N+2}{N+2}<p<2$ . The celebrated moving plane method will be used in the proofs of our results.