Nodal solutions of elliptic equations with critical Sobolev exponents
type: Item
instance of: scholarly article
title: Nodal solutions of elliptic equations with critical Sobolev exponents
zbMATH Open document ID: 0702.35099
DOI: 10.1016/0022-0396(90)90093-5
author: F. V. Atkinson
author: Haïm Brezis
author: Lambertus A. Peletier
published in: Journal of Differential Equations
publication date: 1990-01-01T00:00:00Z
review text: The eigenvalue problem \(\Delta U=-\lambda U+| U|^{p-1}U\) in \(B=\{| x| <1\}\), \(U\neq 0\), \(U=0\) on \(\partial B\) is considered, where \(p=(N+1)/(N-2)\) and \(x=(x_ 1,...,x_ N)\) and radial solutions of variable sign are investigated. It is shown that for \(4\leq N\leq 6\) and small \(\lambda\), no radial solutions exist. Then the behaviour of the ``eigenvalues'' \(\lambda_ n\) corresponding to a solutions with \(n-1\) zeros are determined as \(| u_ n|_{\infty}\to \infty.\) A surprising phenomenon occurs, the \(\lambda_ n\) tend to well-defined numbers depending on the dimension N. The discussion is based on a subtle analysis of the ordinary differential equation \(y''+t^{-k}f(y)=0\).
reviewed by: Catherine Bandle
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 4150836
zbMATH Keywords: nodal solutions
zbMATH Keywords: critical Sobolev exponents
zbMATH Keywords: nonexistence
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: radial solutions
MaRDI profile type: Publication
full work available at URL: 0022-0396(90)90093-5
OpenAlex ID: W2057200865
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description: scientific article; zbMATH DE number 4150836