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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stable and finite Morse index solutions on $\mathbf{R}^n$ or on bounded domains with small diffusion


Author: E. N. Dancer
Journal: Trans. Amer. Math. Soc. 357 (2005), 1225-1243
MSC (2000): Primary 35B35
Published electronically: September 2, 2004
MathSciNet review: 2110438
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Abstract: In this paper, we study bounded solutions of $- \Delta u = f (u)$ on $\mathbf{R}^n$ (where $n = 2$ and sometimes $n = 3$) and show that, for most $f$'s, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of $- \epsilon^2 \Delta u = f (u)$ on $\Omega$ with Dirichlet or Neumann boundary conditions for small $\epsilon$.


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Additional Information

E. N. Dancer
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03543-3
PII: S 0002-9947(04)03543-3
Received by editor(s): July 26, 2002
Received by editor(s) in revised form: October 21, 2003
Published electronically: September 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society