Existence results for a singular quasilinear elliptic equation

Let (Omega subset mathbb R^N) be a bounded domain with smooth boundary. Existence of a positive solution to the quasilinear equation $$begin{aligned} -text {div}left[ left( a(x)+|u|^theta right) nabla uright] +frac{theta }{2}|u|^{theta -2}u|nabla u|^2=|u|^{p-2}u quad text {in} Omega end{aligned}$$with zero Dirichlet boundary condition is proved. Here (theta u003e0) and a(x) is a measurable function satisfying (0u003calpha le a(x)le beta ). The equation involves singularity when (0u003ctheta le 1). As a main novelty with respect to corresponding results in the literature, we only assume (theta +2u003cpu003cfrac{2^*}{2}(theta +2)). The proof relies on a perturbation method and a critical point theory for E-differentiable functionals.