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

Abstract

We perform the homogenization of the semilinear elliptic problem \begin{equation*} \begin{cases} u^\varepsilon \geq 0 & \mbox{in} \; Ω^\varepsilon,\\ \displaystyle - div \,A(x) D u^\varepsilon = F(x,u^\varepsilon) & \mbox{in} \; Ω^\varepsilon,\\ u^\varepsilon = 0 & \mbox{on} \; \partial Ω^\varepsilon.\\ \end{cases} \end{equation*} In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\Gamma (s)$ 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 ${\Gamma }^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\Omega }^{\epsilon }$ 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 $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\Omega }^{\epsilon })$ 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 $u^{\epsilon }$ to the problem only belongs to $H_{loc}^1({\Omega }^{\epsilon })$ but in general does not belongs to $H_0^1({\Omega }^{\epsilon })$ any more, even if $u^{\epsilon }$ vanishes on $\partial {\Omega }^{\epsilon }$ 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.