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

Abstract

In this paper, we study bounded solutions of − Δ u = f ( u ) - \Delta u = f (u) on R n \mathbf {R}^n (where n = 2 n = 2 and sometimes n = 3 n = 3 ) and show that, for most f 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 − ϵ 2 Δ u = f ( u ) - \epsilon ^2 \Delta u = f (u) on Ω \Omega with Dirichlet or Neumann boundary conditions for small ϵ \epsilon .