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 ${\mathbb {R}}^N$ ($N\ge 3$), $\lambda \in {\mathbb {R}}$, $1<q< 2$, $0\le \mu \in L^\infty (\Omega )$ and the datum $f\in L^p(\Omega )$, for some $p>\frac{N}{2}$, 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 $\Omega$.