Global Bifurcation of Non-Radial Solutions for Symmetric Sub-linear Elliptic Systems on the Planar Unit Disc
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929478146457600 |
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| author | Ghanem, Ziad Crane, Casey Liu, Jingzhou |
| author_facet | Ghanem, Ziad Crane, Casey Liu, Jingzhou |
| contents | In this paper, we prove a global bifurcation result for the existence of non-radial branches of solutions to the paramterized family of $Γ$-symmetric problems $-Δu=f(α,z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in \mathbb{C} : |z|<1\}$ with $u(z)\in \mathbb{R}^k$, where $\mathbb{R}^k$ is an orthogonal $Γ$-representation, $f: \mathbb{R} \times \overline{D} \times \mathbb{R}^k\to \mathbb{R}^k$ is a sub-linear $Γ$-equivariant continuous function, differentiable with respect to $u$ at zero and satisfying the conditions $f(α, e^{iθ}z,u)=f(α, z,u)$ for all $θ\in \mathbb{R}$ and $f(z,-u)=-f(z,u)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Bifurcation of Non-Radial Solutions for Symmetric Sub-linear Elliptic Systems on the Planar Unit Disc Ghanem, Ziad Crane, Casey Liu, Jingzhou Analysis of PDEs In this paper, we prove a global bifurcation result for the existence of non-radial branches of solutions to the paramterized family of $Γ$-symmetric problems $-Δu=f(α,z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in \mathbb{C} : |z|<1\}$ with $u(z)\in \mathbb{R}^k$, where $\mathbb{R}^k$ is an orthogonal $Γ$-representation, $f: \mathbb{R} \times \overline{D} \times \mathbb{R}^k\to \mathbb{R}^k$ is a sub-linear $Γ$-equivariant continuous function, differentiable with respect to $u$ at zero and satisfying the conditions $f(α, e^{iθ}z,u)=f(α, z,u)$ for all $θ\in \mathbb{R}$ and $f(z,-u)=-f(z,u)$. |
| title | Global Bifurcation of Non-Radial Solutions for Symmetric Sub-linear Elliptic Systems on the Planar Unit Disc |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.13495 |