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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the classification of solutions of $ -\Delta u= e^u$ on $ \mathbb{R}^N$: Stability outside a compact set and applications


Authors: E. N. Dancer and Alberto Farina
Journal: Proc. Amer. Math. Soc. 137 (2009), 1333-1338
MSC (2000): Primary 35J60, 35B05, 35J25, 35B32
Published electronically: December 4, 2008
MathSciNet review: 2465656
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Abstract: In this short paper we prove that, for $ 3 \le N \le 9$, the problem $ -\Delta u = e^u$ on the entire Euclidean space $ \mathbb{R}^N$ does not admit any solution stable outside a compact set of $ \mathbb{R}^N$. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.


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Additional Information

E. N. Dancer
Affiliation: School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia
Email: normd@maths.usyd.edu.au

Alberto Farina
Affiliation: LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté de Mathématiques et d’Informatique, 33, rue Saint-Leu, 80039 Amiens, France
Email: alberto.farina@u-picardie.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09772-4
PII: S 0002-9939(08)09772-4
Received by editor(s): November 8, 2007
Published electronically: December 4, 2008
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2008 American Mathematical Society