Nonexistence results for quasilinear elliptic problems with singular lower order terms

We establish a general Hardy-type inequality and apply it to investigate the nonexistence of positive solutions to the nonlinear elliptic problem$ -\Delta u\ge \lambda \rho(x) u^p +\mu(x)\frac{|\nabla u|^s}{u^{\theta}}\quad \text{in}\ \Omega , $in bounded domains in $ \mathbb{R}^N $, $ N \geq 2 $, and unbounded exterior domains in $ \mathbb{R}^N $, $ N \geq 3 $. We also discuss the corresponding results for the related systems of partial differential equations.