论文标题
关于Dirichlet-Laplacian的特征值分叉的注释
A note on bifurcations from eigenvalues of the Dirichlet-Laplacian with arbitrary multiplicity
论文作者
论文摘要
在简短的说明中，我们考虑椭圆问题 $$ λϕ+ δϕ =η| ϕ |^σϕ，\ quad ϕ \ big | _ {\ partialω} = 0，\quadλ，η\ in \ mathbb {c}， $$ 在平滑域上$ω\ subset \ mathbb {r}^n $，$ n \ ge 1 $。复杂系数的存在是由研究复杂的金堡 - 兰道方程的研究，破坏了方程的变分结构。我们将非平凡溶液的存在作为琐碎解决方案的分叉。更准确地说，我们表征了分支分支，从任意多重性的Dirichlet-Laplacian的特征值开始。这使我们能够在某些特定情况下讨论此类分叉的性质。我们以对复杂的金兹堡 - 兰道流下这些分支的稳定性分析结束。
In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. 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.
