Stable and finite Morse index solutions on R-n or on bounded domains with small diffusion

摘要

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