Nonlinear elliptic problems with critical exponent in shrinking annuli
type: Item
instance of: scholarly article
title: Nonlinear elliptic problems with critical exponent in shrinking annuli
zbMATH Open document ID: 0619.35044
DOI: 10.1007/BF01474177
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164347
review text: Consider the nonlinear boundary value problems \[ \Delta u+u^{(N+2)/(N- 2)}=0\quad in\quad \epsilon <| x| <R,\quad u=0\quad on\quad | x| =\epsilon \quad and\quad on\quad | x| =R \] \[ \Delta v+v^{(N+2)/(N-2)}=0\quad in\quad \epsilon <| x| <R,\quad \partial v/\partial n=0\quad on\quad | x| =\epsilon,\quad v=0\quad on\quad | x| =R. \] Here \(\partial /\partial n\) stands for the outer normal derivative and \(x=(x_ 1,x_ 2,...,x_ N)\), is an arbitrary point in \(R^ N\). Recently it has been shown that both problems have a unique positive radially symmetric solution. In this paper the precise behaviour as \(\epsilon\) \(\to 0\) of \(u_{\max}(\epsilon)\), \(v_{\max}(\epsilon)\) and of the value r(\(\epsilon)\) where u achieves its maximum is computed. Moreover it is shown that v and u tend after suitable normalization to a specific multiple of the Green's function of the Laplace operator in the sphere. The arguments are based on a phase plane analysis. The results are then used to discuss nonlinear Dirichlet problems in more general doubly connected domains.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 4004736
zbMATH Keywords: critical exponent
zbMATH Keywords: shrinking annuli
zbMATH Keywords: nonlinear boundary value problems
zbMATH Keywords: positive radially symmetric solution
zbMATH Keywords: Green's function
zbMATH Keywords: Laplace operator
zbMATH Keywords: phase plane analysis
zbMATH Keywords: doubly connected domains
author: Catherine Bandle
author: Lambertus A. Peletier
MaRDI profile type: Publication
cites work: Emden-Fowler equations involving critical exponents
cites work: Nonlinear elliptic problems in annular domains
cites work: Q3748635
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Oscillation and Nonoscillation of Solutions of Generalized Emden-Fowler Equations
cites work: Q3687093
cites work: FURTHER STUDIES OF EMDEN'S AND SIMILAR DIFFERENTIAL EQUATIONS
cites work: Symmetry and related properties via the maximum principle
cites work: Remarks on some quasilinear elliptic equations
cites work: On a Class of Nonlinear Second-Order Differential Equations
cites work: Uniqueness and nonuniqueness for positive radial solutions of Δu + f(u, r) = 0
cites work: On the Generalized Emden–Fowler Equation
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description: scientific article; zbMATH DE number 4004736