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 \(\Gamma \) $\mathrm{\Gamma }$ -symmetric equations \(-\Delta u=f(\alpha ,z,u)\) $-\mathrm{\Delta }u=f(\alpha ,z,u)$ , \(u|_{\partial D}=0\) ${u|}_{\partial D}=0$ on the unit disc \(D:=\{z\in {\mathbb {C}} : |z|<1\}\) $D:=\{z\in \mathbb{C}:|z|<1\}$ with \(u(z)\in {\mathbb {R}}^k\) $u(z)\in {\mathbb{R}}^k$ , where \({\mathbb {R}}^k\) ${\mathbb{R}}^k$ is an orthogonal \(\Gamma \) $\mathrm{\Gamma }$ -representation, \(f: {\mathbb {R}} \times {{\overline{D}}}\times {\mathbb {R}}^k\rightarrow {\mathbb {R}}^k\) $f:\mathbb{R}\times \overline{D}\times {\mathbb{R}}^k\to {\mathbb{R}}^k$ is a sub-linear \(\Gamma \) $\mathrm{\Gamma }$ -equivariant continuous function, differentiable with respect to u at zero and satisfying the conditions \(f(\alpha , e^{i\theta }z,u)=f(\alpha , z,u)\) $f(\alpha ,e^{i\theta }z,u)=f(\alpha ,z,u)$ for all \(\theta \in {\mathbb {R}}\) $\theta \in \mathbb{R}$ and \(f(z,-u)=-f(z,u)\) $f(z,-u)=-f(z,u)$ .