Monotonicity in half-spaces for singular quasilinear elliptic problems involving the gradient

Abstract

We study positive solutions to the problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $\gamma>0$, $0<q\le p$, $\vartheta\ge0$ and $f:[0,+\infty)\to\mathbb{R}$ is a locally Lipschitz continuous function. We describe the behavior of solutions and their derivatives near the boundary. Then we exploit that information and the moving plane method to prove the monotonicity of solutions in the $x_N$-direction. This result holds for $W^{1,p}_{\rm loc}(\mathbb{R}^2_+)\cap L^\infty_{\rm loc}(\overline{\mathbb{R}^2_+})$ solutions in dimension two and for $W^{1,p}_{\rm loc}(\mathbb{R}^N_+)$ solutions which are bounded in strips in higher dimensions. Most of our results are new even in the case $\vartheta=0$ or $p=2$.