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

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