## 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

### Citations:

Zbl 1139.58008; Zbl 1013.37023
Full Text:

### References:

 [1] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear anal., 7, 981-1012, (1983) · Zbl 0522.58012 [2] Chang, K.C., Infinite dimensional Morse theory and multiple solutions, (1993), Birkhäuser [3] Chang, K.C.; Ghoussoub, N., The Conley index and the critical groups via an extension of gromoll – merer theory, Topol. methods nonlinear anal., 7, 77-93, (1996) · Zbl 0898.58006 [4] Chang, K.C.; Liu, J.Q.; Liu, M.J., Nontrivial periodic solutions for strong resonance Hamiltonian systems, Ann. inst. H. Poincaré, 14, 103-117, (1997) · Zbl 0881.34061 [5] Chang, K.C.; Wang, Z.Q., Notes on the bifurcation theorem, J. fixed point theory appl., 1, 195-208, (2007) · Zbl 1139.58008 [6] Chow, S.-N.; Lauterbach, R., A bifurcation theorem for critical points of variational problems, Nonlinear anal., 12, 51-61, (1988) · Zbl 0659.58007 [7] Costa, D.G.; Magalhaes, C., A variational approach to subquadratic perturbations of elliptic system, J. differential equations, 111, 103-122, (1994) · Zbl 0803.35052 [8] Costa, D.G.; Magalhaes, C., A variational approach to noncooperative elliptic systems, Nonlinear anal., 25, 699-715, (1998) · Zbl 0852.35039 [9] Dancer, E.N., A note on bifurcation from infinity, Quart. J. math. Oxford ser., 25, 81-84, (1974) · Zbl 0282.47021 [10] Guo, Y.X.; Liu, J.Q., Morse theory for strongly indefinite functional, Nonlinear anal., 48, 831-851, (2002) · Zbl 1013.37023 [11] Guo, Y.X., Computations of critical groups at a degenerate critical point for strongly indefinite functionals, J. math. anal. appl., 256, 462-477, (2001) · Zbl 0982.58009 [12] Krasnoselski, M.A., Topological methods in the theory of nonlinear integral equations, (1964), Macmillan New York [13] Liu, J.Q., Bifurcation for potential operators, Nonlinear anal., 15, 345-353, (1990) · Zbl 0705.47052 [14] Rabinowitz, P.H., On bifurcations from infinity, J. differential equations, 14, 462-475, (1973) · Zbl 0272.35017 [15] Rabinowitz, P.H., A bifurcation theorem for potential operators, J. funct. anal., 25, 412-424, (1977) · Zbl 0369.47038 [16] Schmitt, K.; Wang, Z.Q., On bifurcation from infinity for potential operators, Differential integral equations, 4, 933-943, (1991) · Zbl 0736.58014 [17] Toland, J.F., Bifurcation and asymptotic bifurcation for noncompact nonsymmetric potential operators, Proc. roy. soc. Edinburgh, 73, 137-147, (1975) · Zbl 0341.47042
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