Global bifurcation of solutions to elliptic systems with system and domain symmetries

2025/10/23 by Stefaniak, Piotr
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.20462

Abstract

We study parameterized elliptic systems on symmetric domains with additional system symmetries. We prove the existence of continua of nontrivial solutions bifurcating from the constant branch determined by a critical point of the potential, without assuming nondegeneracy, via the degree for equivariant gradient maps. Our assumptions are formulated in terms of the right-hand side. When the domain is a compact symmetric space, the bifurcating solutions break symmetry at every nonzero level. Under additional assumptions on the right-hand side, the continua are unbounded.

Citations

• Periodic solutions of autonomous S¹-symmetric Newtonian systems
• Unbounded sets of solutions of non-cooperative elliptic systems on symmetric spaces
• The Equivariant Spectral Flow and Bifurcation for Functionals with Symmetries -- Part I
• Bifurcations from degenerate orbits of solutions of nonlinear elliptic systems
• Bifurcations from the orbit of solutions of the Neumann problem

Related

• Bifurcations from degenerate orbits of solutions of nonlinear elliptic systems
• Symmetry breaking of solutions of non-cooperative elliptic systems
• Unbounded sets of solutions of non-cooperative elliptic systems on symmetric spaces
• Global Bifurcation of Non-Radial Solutions for Symmetric Sub-linear Elliptic Systems on the Planar Unit Disc
• Connected sets of solutions of symmetric elliptic systems