A singular elliptic equation with natural growth in the gradient and a variable exponent

In this paper we consider singular quasilinear elliptic equations with quadratic gradient and a singular term with a variable exponent Here \({\Omega}\) is an open bounded set of \({\mathbb{R}^N}\), \({\gamma(x)}\) is a positive continuous function and f is positive function that belongs to a certain Lebesgue space. We show, among other results, that there exists a solution in the natural energy space \({H^1_0 (\Omega)}\) to this problem when \({\gamma (x)}\) is strictly less than 2 in a strip around the boundary; while there is no solution in the energy space when there exists \({\Gamma \subset \partial \Omega}\) with \({|\Gamma|_{N-1} > 0}\) such that \({\gamma(x) > 2}\) on \({\Gamma}\). Moreover, since we work by approximation we can analyze the behavior of the approximated solutions \({u_n}\) in the case in which there is no solution in \({H_0^1(\Omega)}\).