2025/04/09 by Ziad Ghanem, Ghanem, Ziad, Crane, Casey
Mathematics · #Advanced Differential Equations and Dynamical Systems #Nonlinear Partial Differential Equations #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2504.06519
In this paper, we leverage the O(2) × \mathbb Z-equivariant Leray-Schauder degree and a novel characterization of the Burnside Ring A(O(2) × \mathbb Z2) presented by Ghanem in \citeGhanem1 to obtain (\rm i) an existence result for non-radial solutions to the problem -Δu = f(z,u) + Au, u|∂ D = 0 and (\rm ii) local and global bifurcation results for multiple branches of non-radial solutions to the one-parameter family of equations -Δu = f(z,u) + A(α)u, u|∂ D = 0, where D is the planar unit disc, u(z) ∈ \mathbb RN, A : \mathbb RN → \mathbb RN is an N × N matrix, A: \mathbb R → L(\mathbb RN) is a continuous family of N × N matrices and f: D × \mathbb RN → \mathbb RN is a sublinear, O(2) × \mathbb Z2-equivariant function of order o(|u|) as u approaches the origin in \mathbb RN.