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A Note on an Equation with Critical Exponent

A Note on an Equation with Critical Exponent E. N. DANCER In this paper, we consider the existence of non-trivial positive solutions of im+2)l{m 2) -A w = u - in Q, u = 0 on dQ, (1) where m > 2, and Q is a smooth bounded domain in R . If Q is star-shaped it follows easily from the Pokojaev identity (see [6]) that there is no non-trivial positive solution of (1). On the other hand if Q has non-trivial homology in some positive dimension, then Bahri and Coron [1] prove that (1) has a non-trivial positive solution. If m = 3, they show that their assumption is equivalent to Q being not contractible. Thus it becomes of interest to study (1) when Q is contractible but not star-shaped. We make a small contribution to this by constructing, for each m > 2 a domain Q homeomorphic to a ball for which (1) has a non-trivial positive solution. This solution is non-degenerate and hence continues to exist for all nearby domains. Thus the existence of solutions depends on more than the topology of the domain. Our result is obtained by a simple combination of ideas and results in Bahri and Coron [1], Saut and Teman http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Note on an Equation with Critical Exponent

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/20.6.600
Publisher site
See Article on Publisher Site

Abstract

E. N. DANCER In this paper, we consider the existence of non-trivial positive solutions of im+2)l{m 2) -A w = u - in Q, u = 0 on dQ, (1) where m > 2, and Q is a smooth bounded domain in R . If Q is star-shaped it follows easily from the Pokojaev identity (see [6]) that there is no non-trivial positive solution of (1). On the other hand if Q has non-trivial homology in some positive dimension, then Bahri and Coron [1] prove that (1) has a non-trivial positive solution. If m = 3, they show that their assumption is equivalent to Q being not contractible. Thus it becomes of interest to study (1) when Q is contractible but not star-shaped. We make a small contribution to this by constructing, for each m > 2 a domain Q homeomorphic to a ball for which (1) has a non-trivial positive solution. This solution is non-degenerate and hence continues to exist for all nearby domains. Thus the existence of solutions depends on more than the topology of the domain. Our result is obtained by a simple combination of ideas and results in Bahri and Coron [1], Saut and Teman

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1988

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