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} \; \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 \(0<\gamma\leq 1\) and \(f,l \in L^r(\Omega)\), \(r={{2N}\over{N+2}}\) if \(N\geq 3\), \(r>1\) 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 \(\Omega\).