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 $Γ$-symmetric problems $-Δu=f(α,z,u)$, $u|_{\partial D}=0$ 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 $Γ$-representation, $f: \mathbb{R} \times \overline{D} \times \mathbb{R}^k\to \mathbb{R}^k$ is a sub-linear $Γ$-equivariant continuous function, differentiable with respect to $u$ at zero and satisfying the conditions $f(α, e^{iθ}z,u)=f(α, z,u)$ for all $θ\in \mathbb{R}$ and $f(z,-u)=-f(z,u)$.