Singular semilinear elliptic equations in half-spaces

We prove the monotonicity of positive solutions to the problem -Δ u = f(u) in RN₊ := \(x',xN)N xN>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 ∂ u/∂η as xN → 0⁺, where (η,eN)>0, and obtain a classification result. The main tools we use are the method of moving planes and the sliding method.