Stable and finite Morse index solutions on 𝐑ⁿ or on bounded domains with small diffusion

Dancer, E.

doi:10.1090/S0002-9947-04-03543-3

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

Trans. Amer. Math. Soc. 357 (2005), 1225-1243

DOI: https://doi.org/10.1090/S0002-9947-04-03543-3

Published electronically: September 2, 2004

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