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Bifurcation for strongly indefinite functional and applications to Hamiltonian system and noncooperative elliptic system. (English) Zbl 1187.37024

This article covers the extension of bifurcation results given in [K.-C. Chang, Z.-Q. Wang, J. Fixed Point Theory Appl. 1, No. 2, 195–208 (2007; Zbl 1139.58008)] to the case of strongly indefinite functionals, i.e., where positive and negative eigenspaces of the second differential at the bifurcation point are infinite dimensional.
Consider functionals \[ f_\lambda:=\frac12((A+B)u,u)-\frac12\lambda(Ju,u)+g(u) \] defined on a real Hilbert space \(H\). Here \(\lambda\) is a real parameter, \((\cdot,\cdot)\) denotes the inner product, \(A\), \(B\) and \(J\) denote linear, bounded self-adjoint operators in \(H\), \(B\) is compact, and \(g: H\to\mathbb{R}\) is differentiable and such that \(G:=g'\) gives a Lipschitz continuous compact nonlinear operator that satisfies \(G(u)=o(\left\| u\right\|)\) as \(u\to0\).
In earlier work [Nonlinear Anal., Theory Methods Appl. 48, No. 6(A), 831–851 (2002; Zbl 1013.37023)] the authors define critical groups and a degree theory for dynamically isolated critical sets of the negative pseudogradient flow generated by \(f_\lambda\), via a Galerkin approximation scheme. Using these notions and employing the Maslov index of \(B\) with respect to \(A\) they formulate two abstract theorems stating the existence of bifurcation from \(0\) and from \(\infty\) under conditions on the critical groups and indices of critical sets.
These theorems are applied to a periodic Hamiltonian system and to a non-cooperative elliptic system.

MSC:

37B30 Index theory for dynamical systems, Morse-Conley indices
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47H11 Degree theory for nonlinear operators
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
35J50 Variational methods for elliptic systems
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