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

Guo, Yuxia; Liu, Jiaquan

J. Math. Anal. Appl. 359, No. 1, 28-38 (2009).

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.

Reviewer: Nils Ackermann (México)

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

Keywords:

strongly indefinite functional; Morse theory; critical group; topological degree; Maslov index; bifurcation

Citations:

Zbl 1139.58008; Zbl 1013.37023

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