The Hopf bifurcation theorem in Hilbert spaces for abstract semilinear equations

TL;DR

This paper extends the Hopf bifurcation theorem to Hilbert spaces for abstract semilinear equations, removing compactness constraints and enabling applications to unbounded domains.

Contribution

It provides a more general Hopf bifurcation theorem in Hilbert spaces without compactness assumptions, broadening its applicability.

Findings

The theorem applies to semilinear equations in unbounded domains.

It removes the need for compactness conditions in bifurcation analysis.

The result generalizes classical theorems to wider functional settings.

Abstract

We prove a Hopf bifurcation theorem in Hilbert spaces for abstract semilinear equations, which improves a classical result by Crandall and Rabinowitz in the case where basic spaces are Hilbert spaces. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular, our theorem can be applied to semilinear equations in unbounded domains of $\mathbb{R}^n$.