Singular quasilinear elliptic problems with changing sign datum: existence and homogenization

Abstract

We study singular quasilinear elliptic equations whose model is $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\,\Delta u = \lambda u + \mu (x) \frac{|\nabla u|^q}{|u|^{q-1}}+f(x) &{} \quad \hbox {in}\,\, \Omega , \\ u=0 &{}\quad \hbox {on} \,\, \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a bounded smooth domain of \({\mathbb {R}}^N\) (\(N\ge 3\)), \(\lambda \in {\mathbb {R}}\), \(1 \frac{N}{2}\), may change sign. We prove existence of solution and we deal with the homogenization problem posed in a sequence of domains \(\Omega ^\varepsilon \) obtained by removing many small holes from a fixed domain \(\Omega \).