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

摘要

In this short note, we consider the elliptic problem lambda phi + Delta phi = etaphi(sigma)phi, phi(partial derivative Omega)= 0,lambda, eta is an element of C, on a smooth domain Omega subset of 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.