  We perform the homogenization of the semilinear elliptic problem
  \begin{equation*} \begin{cases} u^\varepsilon \geq 0 & \mbox{in} \;
\Omega^\varepsilon,\\ \displaystyle - div \,A(x) D u^\varepsilon =
F(x,u^\varepsilon) & \mbox{in} \; \Omega^\varepsilon,\\ u^\varepsilon = 0 &
\mbox{on} \; \partial \Omega^\varepsilon.\\ \end{cases} \end{equation*} In this
problem $F(x,s)$ is a Carath\'eodory function such that $0 \leq F(x,s) \leq
h(x)/\Gamma(s)$ a.e. $x\in\Omega$ for every $s > 0$, with $h$ in some
$L^r(\Omega)$ and $\Gamma$ a $C^1([0, +\infty[)$ function such that $\Gamma(0)
= 0$ and $\Gamma'(s) > 0$ for every $s > 0$. On the other hand the open sets
$\Omega^\varepsilon$ are obtained by removing many small holes from a fixed
open set $\Omega$ in such a way that a "strange term" $\mu u^0$ appears in the
limit equation in the case where the function $F(x,s)$ depends only on $x$.
  We already treated this problem in the case of a "mild singularity", namely
in the case where the function $F(x,s)$ satisfies $0 \leq F(x,s) \leq h(x)
(\frac 1s + 1)$. In this case the solution $u^\varepsilon$ to the problem
belongs to $H^1_0 (\Omega^\varepsilon)$ and its definition is a "natural" and
rather usual one.
  In the general case where $F(x,s)$ exhibits a "strong singularity" at $u =
0$, which is the purpose of the present paper, the solution $u^\varepsilon$ to
the problem only belongs to $H_{\tiny loc}^1(\Omega^\varepsilon)$ but in
general does not belongs to $H^1_0 (\Omega^\varepsilon)$ any more, even if
$u^\varepsilon$ vanishes on $\partial\Omega^\varepsilon$ in some sense.
Therefore we introduced a new notion of solution (in the spirit of the
solutions defined by transposition) for problems with a strong singularity.
This definition allowed us to obtain existence, stability and uniqueness
results.
