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Analytic imperfect bifurcation theorem and the Liouville‐Gel’fand equation on a perturbed annular domain
Author(s) -
Kan Toru,
Miyamoto Yasuhito
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100213
Subject(s) - mathematics , imperfect , bifurcation , degenerate energy levels , bifurcation theory , mathematical analysis , domain (mathematical analysis) , saddle node bifurcation , inverse , transcritical bifurcation , nonlinear system , geometry , physics , philosophy , linguistics , quantum mechanics
An imperfect bifurcation is studied in a framework of functional analysis. We obtain a sufficient condition for an imperfect bifurcation and classify the types of imperfect bifurcations when an analyticity condition is imposed. It is almost impossible to ascertain the assumptions of existing imperfect bifurcation theorems when the degenerate solution is not constant. Our sufficient condition does not require the inverse of the linearized operator. As a nontrivial application of our imperfect bifurcation theorem, we are concerned with the Liouville‐Gel’fand equation on a two‐dimensional perturbed annular domain.