Singular semilinear elliptic equations in half-spaces

We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some situations, we can derive a precise estimate on the blow-up rate of $\frac{\partial u}{\partial\eta}$ as $x_N \to 0^+$, where $(\eta,e_N)>0$, and obtain a classification result. The main tools we use are the method of moving planes and the sliding method.