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

TL;DR

This paper establishes a global bifurcation theory for non-radial solutions of symmetric sub-linear elliptic systems on the unit disc, revealing new branches of solutions with specific symmetry properties.

Contribution

It proves the existence of non-radial solution branches for symmetric elliptic systems on the disc, extending bifurcation theory to sub-linear, equivariant contexts.

Findings

Existence of non-radial solution branches with symmetry

Global bifurcation structure established

Solutions are sub-linear and $ ext{Gamma}$-equivariant

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 $f(z,-u)=-f(z,u)$.