AN ELLIPTIC EQUATION WITH A MILD SINGULARITY AT u = 0: EXISTENCE AND HOMOGENIZATION
Daniela Giachetti, Pedro J. Martínez−Aparicio, François Murat · arXiv (Cornell University) · 2015
In this paper we consider semilinear elliptic equations with singularities, whose prototype is the following \begin{equation*} \begin{cases} \displaystyle - div \,A(x) D u = f(x)g(u)+l(x)& \mbox{in} \; \Omega, u = 0 & \mbox{on} \; \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is an open bounded set of $\mathbb{R}^N,\, N\geq 1$, $A\in L^\infty(\Omega)^{N\times N}$ is a coercive matrix, $g:[0,+\infty)\rightarrow [0,+\infty]$ is continuous, and $0\leq g(s)\leq {{1}\over{s^\gamma}}+1$ $\forall s>0$, with $0 1$ if $N=2$, $r=1$ if $N=1$, $f(x), l(x)\geq 0$ a.e. $x \in \Omega$. We prove the existence of at least one nonnegative solution and a stability result; moreover uniqueness is also proved if $g(s)$ is nonincreasing or almost nonincreasing. Finally, we study the homogenization of these equations posed in a sequence of domains $\Omega^\epsilon$ obtained by removing many small holes from a fixed domain $\Omega$.