Nonlinear elliptic equations on \(S_n\) in the supercritical case
type: Item
instance of: scholarly article
title: Nonlinear elliptic equations on \(S_n\) in the supercritical case
zbMATH Open document ID: 0946.35030
DOI: 10.1016/S0007-4497(99)80012-8
author: Thierry Aubin
author: Athanase Cotsiolis
published in: Bulletin des Sciences Mathématiques
publication date: 2000-10-10T00:00:00Z
review text: The authors study some elliptic nonlinear differential equations on the sphere \(S_n\) \((n\geq 3)\) with data invariant under the group \(G=O(m)\times O(k)\subset O(n+1)\) with \(m+k=n+1\) and \(k\geq m\geq 2.\) In the first part of the paper they establish the value of the best Sobolev constant of the embedding of \(H= \{ u\in H_1(S_n):u\) invariant under \(G \}\) into \(L_q\), \(q=2k/(k-2)\). Then they apply the result to prove the existence of a smooth positive solution for the equation \[ \Delta u+a(x)u=f(x)|u|^{p-1}u\text{ on } S_n, \] where \(a\) and \(f\) are smooth and \(G\)-invariant, \(\Delta+a\) is coercive and \(f\) is positive somewhere, \(n\geq 4\), \(p=q\). This extends previous results by \textit{A. Cotsiolis} and \textit{D. Iliopoulos} [C. R. Acad. Sci. Paris, Sér. I 313, No.~9, 607-609 (1991; Zbl 0747.58044); Bull. Sci. Math. 119, No.~5, 419-431 (1995; Zbl 0836.35044)]. The authors provide a similar existence result for the equation on \(S_3\) \[ \Delta+a(x)=f(x)e^u. \]
reviewed by: Gabriella Caristi
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 1268640
zbMATH Keywords: elliptic nonlinear differential equations on manifolds
zbMATH Keywords: group invariance
zbMATH Keywords: Sobolev embedding
MaRDI profile type: Publication
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full work available at URL: s0007-4497(99)80012-8
OpenAlex ID: W1987964125
description: scientific article; zbMATH DE number 1268640