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 Γ-symmetric problems -Δ u=f(α,z,u), u|∂ D=0 on the unit disc D:=∈ C : |z|<1\ with u(z)∈ Rᵏ, where Rᵏ is an orthogonal Γ-representation, f: R × D̄ × Rᵏ→ Rᵏ is a sub-linear Γ-equivariant continuous function, differentiable with respect to u at zero and satisfying the conditions f(α, eiθz,u)=f(α, z,u) for all θ∈ R and $f(z,-u)=-f(z,u)$.