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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 ofcomplex coefficients, motivated by the study of complex Ginzburg-Landauequations, breaks down the variational structure of the equation. We study theexistence of nontrivial solutions as bifurcations from the trivial solution.More precisely, we characterize the bifurcation branches starting fromeigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allowsus to discuss the nature of such bifurcations in some specific cases. Weconclude with the stability analysis of these branches under the complexGinzburg-Landau flow.

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