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

TL;DR

The paper develops a general framework for symmetry-breaking in semilinear elliptic systems on the unit ball in ${\mathbb R}^3$ by leveraging a $G$-equivariant Leray–Schauder degree with Burnside-ring coefficients. It reformulates the problem in a functional-analytic setting, performs a detailed $G$-isotypic decomposition, and derives explicit degree-based and parity criteria that guarantee unbounded branches of non-radial solutions with prescribed isotropy. Local bifurcation is tied to the nontriviality of the local bifurcation invariant $\omega_G(\alpha_0)$ and its orbit-type coefficients, while global behavior is described via the Rabinowitz alternative under a discrete critical set. The motivating two-spherical-oscillator example with $\Gamma=S_2$ yields concrete resonance and parity conditions linking coupling eigenvalues to spherical harmonics, producing actionable criteria for symmetry-breaking patterns and illustrating interactions between Platonic spatial symmetries and internal permutations.

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 $G := O(3) \times Γ\times \mathbb{Z}_2$, where $Γ$ 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.

Figures (3)

• Figure 1: Graph of the sigmoid function $\zeta(\alpha)$.
• Figure 2: Visualization of the normalized angular eigenfunctions $u_{3,k}(\phi,\theta):= P_3^k(\cos\phi)\cos k\theta$ and $v_{3,k} := P_3^k(\cos \phi)\sin k\theta$ for $k=0,1,2,3$.
• Figure 3: Representative solutions admitting maximal non-radial symmetries for the antiphase ($j=1$) isotypic component. These patterns are constructed as linear combinations of the normalized angular eigenfunctions $u_{m,k}(\theta,\phi) := P_m^k(\cos \phi)\cos k\theta$: $O(2)^-$ fixes $u_{1,0}$, $\mathbb O_1$ fixes $u_{4,0} + \sqrt{{10}/{7}} u_{4,4}$, $\mathbb O_1^-$ and $\mathbb O_1^+$ fix $u_{3,0} + \sqrt{{8}/{5}} u_{3,3}$, and $\mathbb I_1$ fixes $u_{6,0} + \sqrt{{28}/{11}} u_{6,5}$. Note that $\mathbb O_1^-$ and $\mathbb O_1^+$ fix the same subspace through different group actions (on odd and even Fourier modes, respectively). For the in-phase case ($j=0$), solutions with the same spatial symmetries exist where the patterns on both spheres are identical.

Theorems & Definitions (15)

• Lemma 1.1
• Lemma 1.2
• proof
• Theorem 1.1
• Theorem 1.2
• Remark 3.1
• Theorem 4.1
• proof
• Theorem 5.1
• Theorem 5.2