Unbounded Branches of Non-Radial Solutions to Semilinear Elliptic Systems on a Disc and their Patterns

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 $(\mathrm{i})$ an existence result for non-radial solutions to the problem $-\mathrm{\Delta }u=f(z,u)+Au$, $u{\mathrm{\mid }}_{\mathrm{\partial }D}=0$ and $(\mathrm{i}\mathrm{i})$ local and global bifurcation results for multiple branches of non-radial solutions to the one-parameter family of equations $-\mathrm{\Delta }u=f(z,u)+\mathbf{\text{A}}(\alpha )u$, $u{\mathrm{\mid }}_{\mathrm{\partial }D}=0$, where $D$ is the planar unit disc, $u(z)\in {\mathbb{R}}^N$, $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is an $N\times N$ matrix, $\mathbf{\text{A}}:\mathbb{R}\to L({\mathbb{R}}^N)$ is a continuous family of $N\times N$ matrices and $f:\bar{D}\times {\mathbb{R}}^N\to {\mathbb{R}}^N$ is a sublinear, $O(2)\times {\mathbb{Z}}_2$-equivariant function of order $o(\mathrm{\mid }u\mathrm{\mid })$ as $u$ approaches the origin in ${\mathbb{R}}^N$.