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

© 2015, Springer Basel. In this paper we consider singular quasilinear elliptic equations with quadratic gradient and a singular term with a variable exponent (Formula presented.) Here ¿ is an open bounded set of RR, ¿(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 H0 1(¿) to this problem when ¿(x) is strictly less than 2 in a strip around the boundary; while there is no solution in the energy space when there exists(Formula presented.) with (Formula presented.) such that ¿(x)>2 on ¿. Moreover, since we work by approximation we can analyze the behavior of the approximated solutions un in the case in which there is no solution in H0 1(¿).