Singular semilinear elliptic equations in half-spaces

We prove the monotonicity of positive solutions to the problem $-\mathrm{\Delta }u=f(u)$ in ${\mathbb{R}}_{+}^N:=\{(x^{\mathrm{\prime }},x_N)\in {\mathbb{R}}^N\mid x_N>0\}$ 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 $\frac{\mathrm{\partial }u}{\mathrm{\partial }\eta }$ 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.