Abstract
We first study the linear eigenvalue problem for a planar Dirac system in the open half-line and describe the nodal properties of its solutions by means of the rotation number. We then give a global bifurcation result for a planar nonlinear Dirac system in the open half-line. As an application, we provide a global continuum of solutions of the nonlinear Dirac equation which have a special form.
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Balabane M., Cazenave T., Douady A., Merle F.: Existence of excited states for a nonlinear Dirac field. Commun. Math. Phys. 119, 153–176 (1988)
Balabane M., Cazenave T., Vazquez L.: Existence of standing waves for Dirac fields with singular nonlinearities. Commun. Math. Phys. 133, 53–74 (1990)
Cacciafesta F.: Global small solutions to the critical radial Dirac equation with potential. Nonlinear Anal. 74, 6060–6073 (2011)
Capietto A., Dambrosio W.: Planar Dirac-type systems: the eigenvalue problem and a global bifurcation result. J. Lond. Math. Soc. 81, 477–498 (2010)
Capietto A., Dambrosio W., Papini D.: A global bifurcation result for a second order singular equation. Rend. Istit. Mat. Univ. Trieste. Special Issue in honour of Prof. F. Zanolin. 44, 173–185 (2012)
Coppel, W.A.: Dichotomies in stability theory. Lectures Notes in Mathematics, vol. 629 (1978)
Dancer N.: Boundary-value problems for ordinary differential equations on infinite intervals. Proc. Lond. Math. Soc. 30, 76–94 (1975)
Dancer N.: Boundary-value problems for ordinary differential equations of infinite intervals. II. Q. J. Math. Oxford Ser. 28, 101–115 (1977)
Ding J., Xu J., Zhang F.: Solutions of super-linear Dirac equations with general potentials. Differ. Equ. Dyn. Syst. 17, 235–256 (2009)
Ding Y., Ruf B.: Solutions of a nonlinear Dirac equation with external fields. Arch. Rational Mech. Anal. 190, 57–82 (2008)
Dong Y., Xie J.: Admissible solutions for Dirac equations with singular and non-monotone nonlinearity. Proc. Edinb. Math. Soc. 54, 363–371 (2011)
Dunford, N., Schwartz, J.: Linear Operators—Part II: spectral theory. Interscience Publishers, New York (1963)
Eastham, M.S.P.: The asymptotic solution of linear differential systems. London Math. Society Monographs New Series (1989)
Esteban M.J.: An overview on linear and nonlinear Dirac equations. Discrete Contin. Dyn. Syst. 8, 381–397 (2002)
Kalf H., Schmidt K.M.: Spectral stability of the Coulomb-Dirac Hamiltonian with anomalous magnetic moment. J. Differ. Equ. 205, 408–423 (2004)
Rabier P.J., Stuart C.: Global bifurcation for quasilinear elliptic equations on \({\mathbb{R}^{N}}\) . Math. Z. 237, 85–124 (2001)
Rañada, A.F.: On nonlinear classical Dirac fields and quantum physics. In: Old and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology, pp. 363–376. Plenum, New York (1983)
Schmid H., Tretter C.: Singular Dirac systems ans Sturm-Liouville problems nonlinear in the spectral parameter. J. Differ. Equ. 181, 511–542 (2002)
Schmid H., Tretter C.: Eigenvalue accumulation for Dirac operators with spherically symmetric potential. J. Phys. A. 37, 8657–8674 (2004)
Secchi S., Stuart C.: Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete Contin. Dyn. Syst. 9, 1493–1518 (2003)
Soler M.: Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy. Phys. Rev. D. 1, 2766–2769 (1970)
Stuart C.: Global properties of components of solutions of non-linear second order differential equations on the half-line. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(2), 265–286 (1975)
Thaller, B.: The Dirac Equation. Text and Monographs in Physics (1992)
Weidmann, J.: Spectral theory of ordinary differential equations. Lectures Notes in Mathematics, vol. 1258 (1987)
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Under the auspices of GNAMPA-I.N.d.A.M., Italy. The work of the first two authors has been performed in the frame of the M.I.U.R. Project ’Topological and Variational Methods in the Study of Nonlinear Phenomena’; the work of the third author has been performed in the frame of the M.I.U.R. Project ’Nonlinear Control: Geometrical Methods and Applications’ and of the GNAMPA-I.N.d.A.M. project “Equazione di evoluzione degeneri e singolari: controllo e applicazioni”.
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Capietto, A., Dambrosio, W. & Papini, D. Linear and nonlinear eigenvalue problems for Dirac systems in unbounded domains. Nonlinear Differ. Equ. Appl. 22, 263–299 (2015). https://doi.org/10.1007/s00030-014-0282-1
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DOI: https://doi.org/10.1007/s00030-014-0282-1