Unbounded Branches of Non-Radial Solutions to Semilinear Elliptic Systems on a Unit Ball in $\mathbb R^3$ and Their Patterns

Abstract

We investigate symmetry-breaking phenomena in semilinear elliptic systems on the unit ball in $\mathbb{R}^3$, focusing on the emergence of non-radial solution branches with prescribed spatial and internal symmetries. Extending previous scalar results, we develop a framework for systems equivariant under $G := O(3) \times \Gamma \times \mathbb{Z}_2$, where $\Gamma$ is a finite group encoding coupling symmetries. Using the $G$-equivariant Leray--Schauder degree and Burnside ring techniques, we derive computable criteria for the existence of unbounded branches of non-radial solutions and classify their isotropy types. Our approach accommodates non-simple eigenvalue multiplicities and provides explicit bifurcation conditions in terms of spectral resonance between coupling eigenvalues and spherical Laplacian modes. Applications to coupled spherical oscillators illustrate how Platonic symmetries and internal permutations interact to produce complex patterns. These results establish a general method for detecting and characterizing symmetry-breaking in high-dimensional elliptic systems.