Regularity results for positive solutions of a semilinear elliptic system
type: Item
instance of: scholarly article
title: Regularity results for positive solutions of a semilinear elliptic system
zbMATH Open document ID: 1030.35042
DOI: 10.1007/BF02505951
author: A. Boccuto
author: Enzo Mitidieri
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2003-10-19T00:00:00Z
review text: The authors study regularity of positive weak solutions \((u,v)\) of the semilinear elliptic system \[ -\Delta u = v^p,\quad -\Delta v = u^q \quad \text{\;in\;} \Omega. \tag{1} \] Here, \(\Omega\subset \mathbb{R}^n\) is an open set, \(n \geq 3\), \(p\geq 1\), \(q\geq 1\), \(pq >1\). In order that \((u,v)\) is a positive weak solution, it is only required that \(v^p, u^q \in L^1 (\Omega)\), that \(u,v\geq 0\) and that (1) is satisfied in the distributional sense. Whether these solutions are in general regular or not depends on whether the exponents \(p,q\) are below or above a certain critical line. The authors prove that any positive weak solution \((u,v)\) to (1) is regular, provided that the exponents satisfy \[ \max \{ (p+1), (q+1) \} > \frac{n-2}{2}(pq-1). \] In the reverse case \(\max \{ (p+1), (q+1) \} \leq \frac{n-2}{2}(pq-1)\), they can still prove that \(u\in W^{1,r}_{\text{loc}} (\Omega)\), \(v\in W^{1,s}_{\text{loc}} (\Omega)\) for \(1<r< 2p(q+1)/(pq+2p+1)\), \(1<s< 2q(p+1)/(pq+2q+1)\). This improves the information, which one directly gets from the fact that \(\Delta u, \Delta v \in L^1 (\Omega)\). Moreover, applications to biharmonic equations are given.
reviewed by: Hans-Christoph Grunau
Mathematics Subject Classification ID: 35J45
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 1993711
zbMATH Keywords: semilinear elliptic system
zbMATH Keywords: weak solution
zbMATH Keywords: biharmonic equation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1993711