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 $0 \leq F(x,s) \leq h(x)/Γ(s)$ a.e. $x\inΩ$ for every $s > 0$, with $h$ in some $L^r(Ω)$ and $Γ$ a $C^1([0, +\infty[)$ function such that $Γ(0) = 0$ and $Γ'(s) > 0$ for every $s > 0$. On the other hand the open sets $Ω^\varepsilon$ are obtained by removing many small holes from a fixed open set $Ω$ in such a way that a "strange term" $μu^0$ 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 $H^1_0 (Ω^\varepsilon)$ 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 $H_{\tiny loc}^1(Ω^\varepsilon)$ but in general does not belongs to $H^1_0 (Ω^\varepsilon)$ any more, even if $u^\varepsilon$ vanishes on $\partialΩ^\varepsilon$ 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.