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Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

2018; De Gruyter; Volume: 18; Issue: 4 Linguagem: Inglês

10.1515/ans-2018-0012

2169-0375

Sławomir Rybicki, Naoki Shioji, Piotr Stefaniak,

Advanced Differential Equations and Dynamical Systems

Abstract The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in S n {S^{n}} . In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO ⁡ ( n ) {\operatorname{SO}(n)} -symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for SO ⁡ ( n ) {\operatorname{SO}(n)} -invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G -invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].