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

$$\begin{aligned} -\text {div}\left \left( a(x)+|u|^\theta \right) \nabla u\right] +\frac{\theta }{2}|u|^{\theta -2}u|\nabla u|^2=|u|^{p-2}u \quad \text {in}\ \Omega \end{aligned}$$

with zero Dirichlet boundary condition is proved. Here \(\theta >0\) and a(x) is a measurable function satisfying \(0<\alpha \le a(x)\le \beta \). The equation involves singularity when \(0<\theta \le 1\). As a main novelty with respect to corresponding results in the literature, we only assume \(\theta +2<p<\frac{2^*}{2}(\theta +2)\). The proof relies on a perturbation method and a critical point theory for E-differentiable functionals.