  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.
