Unbounded Branches of Non-Radial Solutions to Semilinear Elliptic Systems on a Disc and their Patterns
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| Format: | Preprint |
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2025
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| author | Ghanem, Ziad Crane, Casey |
| author_facet | Ghanem, Ziad Crane, Casey |
| contents | In this paper, we leverage the $O(2) \times \mathbb Z$-equivariant Leray-Schauder degree and a novel characterization of the Burnside Ring $A(O(2) \times \mathbb Z_2)$ presented by Ghanem in \cite{Ghanem1} to obtain $(\rm i)$ an existence result for non-radial solutions to the problem $-Δu = f(z,u) + Au$, $u|_{\partial D} = 0$ and $(\rm ii)$ local and global bifurcation results for multiple branches of non-radial solutions to the one-parameter family of equations $-Δu = f(z,u) + \textbf{A}(α)u$, $u|_{\partial D} = 0$, where $D$ is the planar unit disc, $u(z) \in \mathbb R^N$, $A : \mathbb R^N \rightarrow \mathbb R^N$ is an $N \times N$ matrix, $\textbf{A}: \mathbb R \rightarrow L(\mathbb R^N)$ is a continuous family of $N \times N$ matrices and $f: \overline D \times \mathbb R^N \rightarrow \mathbb R^N$ is a sublinear, $O(2) \times \mathbb Z_2$-equivariant function of order $o(|u|)$ as $u$ approaches the origin in $\mathbb R^N$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06519 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unbounded Branches of Non-Radial Solutions to Semilinear Elliptic Systems on a Disc and their Patterns Ghanem, Ziad Crane, Casey Analysis of PDEs In this paper, we leverage the $O(2) \times \mathbb Z$-equivariant Leray-Schauder degree and a novel characterization of the Burnside Ring $A(O(2) \times \mathbb Z_2)$ presented by Ghanem in \cite{Ghanem1} to obtain $(\rm i)$ an existence result for non-radial solutions to the problem $-Δu = f(z,u) + Au$, $u|_{\partial D} = 0$ and $(\rm ii)$ local and global bifurcation results for multiple branches of non-radial solutions to the one-parameter family of equations $-Δu = f(z,u) + \textbf{A}(α)u$, $u|_{\partial D} = 0$, where $D$ is the planar unit disc, $u(z) \in \mathbb R^N$, $A : \mathbb R^N \rightarrow \mathbb R^N$ is an $N \times N$ matrix, $\textbf{A}: \mathbb R \rightarrow L(\mathbb R^N)$ is a continuous family of $N \times N$ matrices and $f: \overline D \times \mathbb R^N \rightarrow \mathbb R^N$ is a sublinear, $O(2) \times \mathbb Z_2$-equivariant function of order $o(|u|)$ as $u$ approaches the origin in $\mathbb R^N$. |
| title | Unbounded Branches of Non-Radial Solutions to Semilinear Elliptic Systems on a Disc and their Patterns |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.06519 |