Positive solutions for singular elliptic problems depending on the gradient

We investigate the existence of asymptotically vanishing positive solutions of the following class of singular elliptic problems $ \Delta u(x)+f(x,u(x))-b(x)(u(x))^{-1}||\nabla u(x)||^{2}+g(x)x\cdot\nabla u(x) = 0\text{ in }\Omega_{R} = \left\{ x\in\mathbb{R}^{n},||x||>R>2\right\} $, $ n>2 $. Our approach is based on the sub-supersolution method for bounded sub-domains, as well as the unbounded domain approximation method combined with some classical convergence procedure. The rate of decay of the solutions is also described. In the first part of the paper we formulate some auxiliary results concerning solutions for the case when $ b\equiv0 $. These results play the crucial role in the proof of the existence of solution for our singular problem.