Rabinowitz alternative for non-cooperative elliptic systems on geodesic balls

Sławomir Rybicki; Naoki Shioji; Piotr Stefaniak

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arXiv · 1703.08417

Rabinowitz alternative for non-cooperative elliptic systems on geodesic balls

Sławomir Rybicki·Naoki Shioji·Piotr Stefaniak

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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 $S^n$. 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. Go{\l}\c{e}biewska, S. Rybicki, \emph{Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals}, Nonl. Anal. {74} (2011), 1823-1834..

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Sławomir Rybicki, Naoki Shioji, Piotr Stefaniak. 2017-03-24. Rabinowitz alternative for non-cooperative elliptic systems on geodesic balls. https://arxiv.org/abs/1703.08417

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