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In this paper, we study bounded solutions of -Δu = f(u) on 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 -e 2 Δu - f(u) on Ω with Dirichlet or Neumann boundary conditions for small e.

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