Elliptic and parabolic semilinear problems without conditions at infinity
type: Item
instance of: scholarly article
title: Elliptic and parabolic semilinear problems without conditions at infinity
zbMATH Open document ID: 0678.35034
DOI: 10.1007/BF00281214
published in: Archive for Rational Mechanics and Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let \(m\geq 1\) and \(N\geq 1\) be integers and let \(q=(p+1)/p\) be the conjugate Hölder exponent of a number \(p+1>2\). For \(f\in L^ q_{loc}({\mathbb{R}}^ N)\), one principal result is the existence of a distribution \(u\in W^{m,2}_{loc}({\mathbb{R}}^ N)\cap L^{p+1}_{loc}({\mathbb{R}}^ N),\) satisfying \((-\Delta)^ mu+| u|^{p-1}u=f\) in \({\mathbb{R}}^ N\). This solution u is unique if, in addition, \(p\leq (N+2m)/(N-2m).\) Similar theorems are proved for the parabolic equation \[ u_ t+(-\Delta)^ mu+| u|^{p-1}u=f. \] Such problems require neither conditions at infinity nor limitations on the growth of f at infinity. The proofs are based on local estimates obtained from a new weighted Gårding-type inequality (related to a Caccioppoli-type inequality) not requiring boundary conditions. Additional theorems concern elliptic equations with strong nonlinearities of order zero and equations with unbounded measures f. Second order results of the present type were obtained by \textit{H. Brezis} [Appl. Math. Optimization 12, 271-282 (1984; Zbl 0562.35035)].
reviewed by: Charles A. Swanson
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 4111167
zbMATH Keywords: semilinear
zbMATH Keywords: even order
zbMATH Keywords: Hölder exponent
zbMATH Keywords: existence
zbMATH Keywords: distribution
zbMATH Keywords: unique
zbMATH Keywords: conditions at infinity
zbMATH Keywords: limitations on the growth
zbMATH Keywords: local estimates
zbMATH Keywords: strong nonlinearities of order zero
zbMATH Keywords: equations with unbounded measures
author: Francisco Bernis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4111167