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

Abstract

In this paper we consider singular quasilinear elliptic equations with quadratic gradient and a singular term with a variable exponent    −∆u + |∇u| 2 uγ(x) = f in Ω, u = 0 on ∂Ω. Here Ω is an open bounded set of R N , γ(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 H1 0 (Ω) to this problem when γ(x) is strictly less than 2 in a strip around the boundary; while there is no solution in the energy space when there exists Γ ⊂ ∂Ω with |Γ|N−1 > 0 such that γ(x) > 2 on Γ. 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 H1 0 (Ω).