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

Abstract

In this paper, we prove a global bifurcation result for the existence of non-radial branches of solutions to the paramterized family of $\Gamma$-symmetric problems $-\Delta u=f(\alpha ,z,u)$, $u{|}_{\partial D}=0$ 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 $\Gamma$-representation, $f:\mathbb{R}\times \overline{D}\times {\mathbb{R}}^k\to {\mathbb{R}}^k$ is a sub-linear $\Gamma$-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 $f(z,-u)=-f(z,u)$.

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