Monotonicity of positive solutions to quasilinear elliptic equations in half-spaces with a changing-sign nonlinearity (Q2146312)

The authors study the monotonicity properties of the weak solutions of the elliptic problem \[ \begin{cases} -\Delta_p u=f(u), & \text{ in }\mathbb{R}_+^{N},\\ u(x',y)>0, & \text{ in }\mathbb{R}_+^{N},\\ u(x',0)=0, & \text{ on }\partial \mathbb{R}_+^{N},\tag{P} \end{cases} \] where \(N\geq 2\), \(p\in \left(\frac{2N+2}{N+2},2\right)\) and \(f:[0,\infty)\rightarrow \mathbb{R}\) is a \(C^1\) function such that \(f(0)\geq 0\) which may be sign-changing. The following result is proved: assume that \(f^{-1}(0)\) is a discrete set and let \(u\in C_{\mathrm{loc}}^{1,\alpha}(\overline{\mathbb{R}_+^{N}})\), \(\alpha \in (0,1)\), be a weak solution of problem (P) whose gradient is bounded on every strip \(\mathbb{R}^{N-1}\times (0,\lambda)\). Then \(u(x',\cdot)\) is increasing and one has \[ \frac{\partial u}{\partial y}\geq 0, \text{ in }\mathbb{R}^N_+ \] and \[ \frac{\partial u}{\partial y}>0,\text{ in }\{x\in \mathbb{R}_+^N: f(u(x))\neq 0\}. \] Ingredients of the proof are comparison and maximum principles and the moving plane method.