On conformally invariant equation \((-\Delta)^p u-K(x)u^{\frac{N+2p}{N-2p}}=0\) and its generalizations
type: Item
instance of: scholarly article
title: On conformally invariant equation \((-\Delta)^p u-K(x)u^{\frac{N+2p}{N-2p}}=0\) and its generalizations
zbMATH Open document ID: 1030.35062
DOI: 10.1007/BF02505961
author: Guozhen Lu
author: Xingwang Xu
author: Juncheng Wei
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2003-10-19T00:00:00Z
review text: The authors prove several existence and nonexistence results for positive entire solutions of conformally invariant equations of the form \[ (-\Delta)^p u - K(x) u^q = 0 \quad\text{in}\quad \mathbb{R}^N, \] where \(N\geq 2p+1\), \(q>1\), and \(K\) is a Hölder continuous function. They prove the existence of infinitely many positive solutions if \(K\) decays sufficiently fast at infinity, and the nonexistence of positive solutions if \(K\) grows sufficiently fast at infinity. They also establish a Kazdan-Warner type condition for nonexistence of solutions decaying at infinity. The results are similar in spirit to those obtained by \textit{W.-M. Ni} [Indiana Univ. Math. J. 31, 493-529 (1982; Zbl 0496.35036)] and many subsequent authors for the second-order case.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35G99
zbMATH DE Number: 1993717
zbMATH Keywords: polyharmonic operator
zbMATH Keywords: conformally invariant equation
reviewed by: John Urbas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1993717