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

TL;DR

This paper investigates how solutions to symmetric elliptic systems change as parameters vary, revealing bifurcation phenomena and symmetry-breaking behaviors on symmetric domains using equivariant degree theory.

Contribution

It introduces a new approach to analyze bifurcations in elliptic systems with symmetries without requiring nondegeneracy, expanding understanding of solution structures.

Findings

Existence of solution continua bifurcating from constant solutions.

Symmetry breaking occurs at every nonzero bifurcation point.

Under certain conditions, bifurcating solutions form unbounded continua.

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.