A note on bifurcations from eigenvalues of the Dirichlet-Laplacian with arbitrary multiplicity

TL;DR

This paper investigates bifurcations from eigenvalues of the Dirichlet-Laplacian with arbitrary multiplicity for a complex elliptic problem, analyzing solution existence, bifurcation structure, and stability in the context of complex Ginzburg-Landau equations.

Contribution

It characterizes bifurcation branches from eigenvalues of arbitrary multiplicity and analyzes their stability, extending understanding beyond simple eigenvalues.

Findings

Bifurcation branches are characterized starting from eigenvalues of arbitrary multiplicity.

The nature of bifurcations is discussed in specific cases.

Stability of bifurcating solutions under the complex Ginzburg-Landau flow is analyzed.

Abstract

In this short note, we consider the elliptic problem $$ \lambda \phi + \Delta \phi = \eta|\phi|^\sigma \phi,\quad \phi\big|_{\partial \Omega}=0,\quad \lambda, \eta \in \mathbb{C}, $$ on a smooth domain $\Omega\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature of such bifurcations in some specific cases. We conclude with the stability analysis of these branches under the complex Ginzburg-Landau flow.