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# Stable and Finite Morse Index Solutions on Rn or on Bounded Domains with Small Diffusion

E. N. Dancer
Transactions of the American Mathematical Society
Vol. 357, No. 3 (Mar., 2005), pp. 1225-1243
Stable URL: http://www.jstor.org/stable/3845167
Page Count: 19
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## Abstract

In this paper, we study bounded solutions of $-\Delta u = f(u)$ on Rn (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 Ω with Dirichlet or Neumann boundary conditions for small ε.

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