Rabinowitz alternative for non-cooperative elliptic systems on geodesic balls

2017/03/24 by Sławomir Rybicki, Rybicki, Sławomir, Naoki Shioji +3 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities #math.AP

paper · pdf · doi:10.48550/arxiv.1703.08417

arxiv created 2017/03/24 · openalex publication_date 2017/03/24 · arxiv updated 2017/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Sⁿ. In particular, we have shown that if the geodesic ball is a hemisphere, then 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)-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)-invariant strongly indefinite functionals defined in A. Gołȩbiewska, S. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonl. Anal. 74 (2011), 1823-1834..

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• Unbounded sets of solutions of non-cooperative elliptic systems on symmetric spaces

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