Analytic imperfect bifurcation theorem and the Liouville‐Gel’fand equation on a perturbed annular domain
Toru Kan, Yasuhito Miyamoto · Mathematische Nachrichten · 2013
Abstract 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.