A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems

We consider model semilinear elliptic equations of the type $$\begin{cases} - \mathrm{div} (A(x) \nabla u) = f u^{- \lambda}, \quad u > 0 \quad \text{in} \ \Omega, \\ u \in H_{0}^{1}(\Omega), \end{cases}$$ where $\Omega$ is a bounded domain in $\mathbf{R}^{N}$, $N \ge 1$, $A \in L^{\infty}(\Omega)^{N \times N}$ is a coercive matrix, $0 < \lambda \le 1$ and $f$ is a nonnegative function in $L^{1}_{loc}(\Omega)$, or more generally, nonnegative Radon measure on $\Omega$. We discuss $H^{1}$-stability of $u$ under a minimal assumption on $f$. Additionally, we apply the result to homogenization problems.