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

By Daniela Giachetti, Pedro J. Martínez‐Aparicio, F. Murat. 2015

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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 $\Omega$.

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Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness (2016, background): “only assumed in a weaker sense than the usual one of traces, i.e. uγ+1 2 ∈ H1 0(Ω). Note that, if γ>1, then solutions with inﬁnite energy do exist, even for smooth data ([34]). Let us also mention [25, 26] where, in order to deal with homogenization issues, existence and uniqueness of ﬁnite energ”