Radial solutions of a semilinear elliptic equation at a critical exponent
type: Item
instance of: scholarly article
title: Radial solutions of a semilinear elliptic equation at a critical exponent
zbMATH Open document ID: 0679.35033
DOI: 10.1007/BF00281356
author: Christopher K. R. T. Jones
published in: Archive for Rational Mechanics and Analysis
publication date: 1988-01-01T00:00:00Z
review text: I consider the problem of finding radial solutions of a semilinear elliptic equation which are not necessarily positive. The case of interest occurs when the nonlinear term is a power with a critical exponent plus some lower order term. It is known from the work of Brezis and Nirenberg that the lower order terms can force either the existence or non-existence of positive solutions depending on its power. In this paper it is shown that a similar effect occurs for non-positive solutions but the power cut-off is different. This leads to the interesting case in which a positive solution exists but there are no non-positive ones. My techniques are to construct a compactified phase portrait using appropriate scalings. The existence is then determined by tracking the contortions of an appropriate invariant manifold.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B99
zbMATH DE Number: 4113130
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: critical exponent
zbMATH Keywords: non-positive solutions
MaRDI profile type: Publication
Wikidata QID: Q57919742
cites work: Emden-Fowler equations involving critical exponents
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Q4180600
cites work: Symmetry and related properties via the maximum principle
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: On the Infinitely Many Solutions of a Semilinear Elliptic Equation
cites work: Variational and topological methods in nonlinear problems
full work available at URL: https://doi.org/10.1007/bf00281356
OpenAlex ID: W1979886387
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description: scientific article; zbMATH DE number 4113130