Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u=0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem \begin{equation*} \begin{cases} u^\eps \geq 0 & \mbox{in} \; \oeps,\\ \displaystyle - div \,A(x) D u^\eps = F(x,u^\eps) & \mbox{in} \; \oeps,\\ u^\eps = 0 & \mbox{on} \; \partial \oeps.\\ \end{cases} \end{equation*} In this problem $F(x,s)$ is a Carath\'eodory function such that\break $0 \leq F(x,s) \leq h(x)/\Gamma(s)$ 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 $\oeps$ 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 $u^\eps$ to the problem belongs to $H^1_0 (\oeps)$ 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 $u^\eps$ to the problem only belongs to $H_{\mbox{\tiny{loc}}}^1(\oeps)$ but in general does not belongs to $H^1_0 (\oeps)$ any more, even if $u^\eps$ vanishes on $\partial\oeps$ 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. In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the ``strange term" $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.