A semilinear elliptic equation with a mild singularity at $u=0$: existence and homogenization

Abstract

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} \; Ω,\\ u = 0 & \mbox{on} \; \partial Ω,\\ \end{cases} \end{equation*} where $\Omega$ is an open bounded set of ${\mathbb{R}}^N,N\ge 1$, $A\in L^{\infty }(\Omega {)}^{N\times N}$ is a coercive matrix, $g:[0,+\infty )\to [0,+\infty ]$ is continuous, and $0\le g(s)\le \frac{1}{s^{\gamma }}+1$ $\forall s>0$, with $0<\gamma \le 1$ and $f,l\in L^r(\Omega )$, $r=\frac{2N}{N+2}$ if $N\ge 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, $f(x),l(x)\ge 0$ 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 $\Omega$.

References in corpus (1)

• Inverse Problem for a Curved Quantum Guide

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