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Multi‐Peak Solutions for a Wide Class of Singular Perturbation Problems
Author(s) -
Wei Juncheng,
Winter Matthias
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079900719x
Subject(s) - degenerate energy levels , mathematics , bounded function , singular perturbation , perturbation (astronomy) , mathematical analysis , domain (mathematical analysis) , boundary (topology) , boundary value problem , pure mathematics , physics , quantum mechanics
This paper concerns a wide class of singular perturbation problems arising from such diverse fields as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. The corresponding elliptic equations in a bounded domain without any symmetry assumptions are studied. It is assumed that the mean curvature of the boundary hasM ¯isolated, non‐degenerate critical points. Then it is shown that for any positive integer M ⩽ M ¯there exists a stationary solution with M local peaks which are attained on the boundary and which lie close to these critical points. The method is based on Lyapunov–Schmidt reduction.