Stable and finite Morse index solutions on $\mathbf {R}^n$ or on bounded domains with small diffusion
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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 $\epsilon$.References
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Additional Information
- E. N. Dancer
- Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
- Received by editor(s): July 26, 2002
- Received by editor(s) in revised form: October 21, 2003
- Published electronically: September 2, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 1225-1243
- MSC (2000): Primary 35B35
- DOI: https://doi.org/10.1090/S0002-9947-04-03543-3
- MathSciNet review: 2110438