Global Bifurcation of Non-Radial Solutions for Symmetric Sub-linear Elliptic Systems on the Planar Unit Disc

In this paper, we prove a global bifurcation result for the existence of non-radial branches of solutions to the paramterized family of $\mathrm{\Gamma }$-symmetric problems $-\mathrm{\Delta }u=f(\alpha ,z,u)$, $u{\mathrm{\mid }}_{\mathrm{\partial }D}=0$ on the unit disc $D:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ with $u(z)\in {\mathbb{R}}^k$, where ${\mathbb{R}}^k$ is an orthogonal $\mathrm{\Gamma }$-representation, $f:\mathbb{R}\times \bar{D}\times {\mathbb{R}}^k\to {\mathbb{R}}^k$ is a sub-linear $\mathrm{\Gamma }$-equivariant continuous function, differentiable with respect to $u$ at zero and satisfying the conditions $f(\alpha ,e^{i\theta }z,u)=f(\alpha ,z,u)$ for all $\theta \in \mathbb{R}$ and $f(z,-u)=-f(z,u)$.