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

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 $\epsilon$.