Existence results for a singular quasilinear elliptic equation

Abstract

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.