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Stable and finite Morse index solutions on $$\mathbb R^n$$ or on bounded domains with small diffusion. (English) Zbl 1145.35369
The goal of this paper is to study the equation $-\Delta u= f(u)$ on $$\mathbb R^d$$ (or a half space). The author is interested in solutions which are weakly stable or have finite Morse index. He shows that for $$d= 2$$ and sometimes for $$d=3$$ these solutions are very simple and easy to understand. As an application of these ideas, the author presents a number of results on the solutions of $-\varepsilon^2\Delta u= f(u)\quad\text{in }\Omega,$ where $$\Omega$$ is a bounded open set in $$\mathbb R^2$$ (sometimes in $$\mathbb R^3$$) with Neumann or Dirichlet boundary conditions.

##### MSC:
 35J60 Nonlinear elliptic equations 35B25 Singular perturbations in context of PDEs 35B35 Stability in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
##### Keywords:
weakly stable solution; Morse index; small diffusion
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##### References:
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