Stable and finite Morse index solutions on \(\mathbb R^n\) or on bounded domains with small diffusion.

*(English)*Zbl 1145.35369The 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.

Reviewer: Messoud A. Efendiev (München)

##### 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 |

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\textit{E. N. Dancer}, Trans. Am. Math. Soc. 357, No. 3, 1225--1243 (2005; Zbl 1145.35369)

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##### References:

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