scholarly journals Strongly indefinite functionals and multiple solutions of elliptic systems

2003 ◽  
Vol 355 (7) ◽  
pp. 2973-2989 ◽  
Author(s):  
D. G. De Figueiredo ◽  
Y. H. Ding
Keyword(s):  
Multiple Solutions ◽  
Elliptic Systems ◽  
Strongly Indefinite Functionals

scholarly journals Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

2018 ◽  
Vol 18 (4) ◽  
pp. 845-862 ◽  
Author(s):  
Sławomir Rybicki ◽  
Naoki Shioji ◽  
Piotr Stefaniak
Keyword(s):  
Degree Theory ◽  
Elliptic Systems ◽  
Geodesic Ball ◽  
Global Symmetry ◽  
Nontrivial Solutions ◽  
Strongly Indefinite Functionals ◽  
Nonlinear Anal ◽  
Connected Sets ◽  
Equivariant Bifurcation Theory ◽  
Closed Connected Sets

AbstractThe 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 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 {\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 {\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].

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