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

• Published in: Nonlinear differential equations and applications Vol. 30; no. 3
• Main Authors: Correia, Simão, Figueira, Mário
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-05-2023; Springer Nature B.V
• Subjects: Analysis; Bifurcations; Dirichlet problem; Eigenvalues; Flow stability; Landau-Ginzburg equations; Mathematics; Mathematics and Statistics; Stability analysis; 35B35; 35Q56; Bifurcation; 35B10; 35B32; Bound-states; Complex Ginzburg-Landau
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-023-00846-y

In this short note, we consider the elliptic problem
λ
ϕ
+
Δ
ϕ
=
η
|
ϕ
|
σ
ϕ
,
ϕ
|
∂
Ω
=
0
,
λ
,
η
∈
C
,
on a smooth domain
Ω
⊂
R
N
,
N
⩾
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.