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In this paper, we study bounded solutions of −∆u = 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 − 2 ∆u = f (u) on Ω with Dirichlet or Neumann boundary conditions for small .doi:10.1090/s0002-9947-04-03543-3 fatcat:ylswsqxklnbsnnxvgqqm7s7r7a