Notions of sublinearity and superlinearity for nonvariational elliptic systems
type: Item
instance of: scholarly article
title: Notions of sublinearity and superlinearity for nonvariational elliptic systems
zbMATH Open document ID: 1119.35017
author: Boyan Sirakov
published in: Discrete and Continuous Dynamical Systems
publication date: 2005-06-21T00:00:00Z
review text: This paper deals with \(n\times n\) elliptic systems of partial differential equations of the form \[ \left\{\begin{matrix} -\Delta \mathbf{u} =\text\textbf{f}(\text\textbf{u})\,\, \text{in } \Omega,\\ \text\textbf{u}=\text\textbf{0}\,\, \text{on } \partial \Omega.\end{matrix}\right. \] Here \(\Omega\subset {\mathbb R}^N\) is a bounded smooth domain, \(\mathbf{u }=(u_1,u_2,\cdots,u_n)\) is a vector function, and \(\mathbf{f}(\text\textbf{u})=(f_1(\text\textbf{u}),\cdots,f_n(\text\textbf{u}))\) is Hölder continuous on \({\mathbb R}^n_+=[0,\infty)^n\) such that \(f_i(\mathbf{0})=0\) for \(i=1,\cdots, n\). The author introduces the notions of sublinearity and superlinearity for such systems and provides conditions for the existence of positive solutions. These generalize earlier results on particular \(2\times 2\) systems by authors [\textit{Y. Liu}, On a class of semilinear elliptic systems, Proc. R. Soc. Edinb. 124 A, 609--620 (1994; Zbl 0806.35038)] and \textit{C. Alves} and \textit{D. G. Figueiredo} [''Nonvariational elliptic systems,'' Discrete Contin. Dyn. Syst. 8, No. 2, 289--302 (2002; Zbl 1162.35356)].
reviewed by: Yaping Liu
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 2177518
zbMATH Keywords: elliptic systems
zbMATH Keywords: existence
zbMATH Keywords: topological methods
zbMATH Keywords: fixed point
zbMATH Keywords: sublinear
zbMATH Keywords: superlinear
MaRDI profile type: Publication
full work available at URL: dcds.2005.13.163
OpenAlex ID: W2094848355
DOI: 10.3934/DCDS.2005.13.163
description: scientific article; zbMATH DE number 2177518