On the definition of the solution to a semilinear elliptic problem with a strong singularity at \(u = 0\)
type: Item
instance of: scholarly article
title: On the definition of the solution to a semilinear elliptic problem with a strong singularity at \(u = 0\)
zbMATH Open document ID: 1408.35026
author: D. Giachetti
author: Pedro J. Martínez-Aparicio
author: François Murat
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2018-11-22T00:00:00Z
review text: Let \(\Omega\) be a bounded open set in \(\mathbb{R}^N\), and let \(A:\Omega\rightarrow \mathbb{R}^{N}\times \mathbb{R}^{N}\) be a matrix with entries in \(L^\infty(\Omega)\). The authors consider the semilinear elliptic problem \[ \begin{cases} -\text{div}A(x)D u=F(x,u) & \text{in }\Omega,\\ u\geq 0 & \text{in } \Omega,\\ u=0 & \text{on } \partial \Omega, \end{cases}\tag{\(P\)} \] where the nonlinearity \(F:\Omega \times [0,+\infty[\rightarrow [0,+\infty]\) is a real extended Carathéodory function satisfying the growth condition \[ F(x,s)\leq \frac{h(x)}{\Gamma(s)}, \;\;\;\text{for almost all} \;\;x\in \Omega, \;\;\text{and for all} \;\;s\in ]0,+\infty[,\tag{1} \] for some nonnegative function \(\Gamma \in C^1([0,+\infty[)\), with \(\Gamma(0)=0\) and \(\Gamma'(s)>0\) for all \(s>0\), and some nonnegative function \(h\in L^r(\Omega)\), with \(r=\frac{2N}{N+2}\) if \(N\geq 3\), \(r>1\) if \(N=2\), and \(r=1\) if \(N=1\). The growth condition \((1)\) allows the nonlinearity \(F\) to be strongly singular at \(u=0\), and when this occurs, an appropriate definition of solution for problem \((P)\) is to be considered. In a recent paper, after introducing a new test function space \(\mathcal{V}(\Omega)\), the authors gave a definition of solution for problem \((P)\) under condition \((1)\), which is suggested by the notion of ``solution defined by transposition'' introduced by J. L. Lions and E. Magenes and by G. Stampacchia. This definition involves the truncation operators \(T_k(s)=\max\{-k,\min\{k,s\}\}\) and \(G_k(s)=s-T_k(s)\) and states that a nonnegative \(u\in L^2(\Omega)\cap H^1_{loc}(\Omega)\) is a solution to problem \((P)\) if, for each \(k>0\), the functions \(T_k(u)\) and \(G_k(u)\) satisfy, in a weak sense, a corresponding equation in divergence form. In another paper, the authors also introduced the definition of solution to problem \((P)\) for the case of mild singularities, that is when \(\Gamma(s)=s^\gamma\), with \(\gamma \in ]0,1]\). In this paper, the…
reviewed by: Giovanni Anello
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J75
zbMATH DE Number: 6983311
zbMATH Keywords: semilinear elliptic problems
zbMATH Keywords: singularity at \(u = 0\)
zbMATH Keywords: strong maximum principle
MaRDI profile type: Publication
full work available at URL: j.na.2018.04.023
OpenAlex ID: W2809375726
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DOI: 10.1016/J.NA.2018.04.023
description: scientific article; zbMATH DE number 6983311