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