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Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials
The index bundle and multiparameter bifurcation for discrete dynamical systems
1. | Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Torun, Poland |
2. | School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, United Kingdom |
We develop a K-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author.
References:
[1] |
A. Abbondandolo and P. Majer,
On the global stable manifold, Studia Math., 177 (2006), 113-131.
doi: 10.4064/sm177-2-2. |
[2] |
D. Arlt,
Zusammenziehbarkeit der allgemeinen linearen Gruppe des Raumes $c_0$ der Nullfolgen, Invent. Math., 1 (1966), 36-44.
doi: 10.1007/BF01389697. |
[3] |
M. F. Atiyah,
Thom complexes, Proc. London Math. Soc.(3), 11 (1961), 291-310.
doi: 10.1112/plms/s3-11.1.291. |
[4] |
M. F. Atiyah and I. M. Singer,
The index of elliptic operators. Ⅳ, Ann. of Math.(2), 93 (1971), 119-138.
doi: 10.2307/1970756. |
[5] | |
[6] |
T. Bartsch,
The global structure of the zero set of a family of semilinear Fredholm maps, Nonlinear Anal, 71 (1991), 313-331.
doi: 10.1016/0362-546X(91)90074-B. |
[7] |
G. E. Bredon,
Topology and Geometry, Graduate Texts in Mathematics, 139 Springer, 1993.
doi: 10.1007/978-1-4757-6848-0. |
[8] |
W. A. Coppel,
Dichotomies in Stability Theory, Lecture Notes in Math., vol. 629, Springer-Verlag, New York, 1978. |
[9] |
M. Crabb and I. James,
Fibrewise Homotopy Theory, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1998.
doi: 10.1007/978-1-4471-1265-5. |
[10] |
A. Dold,
ÜUber fasernweise Homotopieäquivalenz von Faserräumen, Math. Z., 62 (1955), 111-136.
doi: 10.1007/BF01180627. |
[11] |
P. M. Fitzpatrick and J. Pejsachowicz,
Nonorientability of the index bundle and several-parameter bifurcation, J. Funct. Anal., 98 (1991), 42-58.
doi: 10.1016/0022-1236(91)90090-R. |
[12] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981. |
[13] |
W. Hurewicz and H. Wallmann,
Dimension Theory, Princeton Mathematical Series, 4 Princeton University Press, 1941. |
[14] |
T. Hüls,
Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31.
doi: 10.1080/10236190902932742. |
[15] |
K. Jänich,
Vektorraumbündel und der Raum der Fredholmoperatoren, Math. Ann., 161 (1965), 129-142.
doi: 10.1007/BF01360851. |
[16] |
S. Lang,
Differential and Riemannian Manifolds, Third edition, Graduate Texts in Mathematics, 160 Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4182-9. |
[17] |
H. B. Lawson and M. -L. Michelsohn,
Spin Geometry, Princeton Mathematical Series, 38 Princeton University Press, Princeton, NJ, 1989. |
[18] |
J. P. May,
A Concise Course in Algebraic Topology, Chicago University Press, 2nd edition, 1999. |
[19] |
J. W. Milnor and J. D. Stasheff,
Characteristic Classes, Princeton University Press, 1974. |
[20] |
K. J. Palmer,
Exponential dichotomies and transversal homoclinic points, Journal of Differential Equations, 55 (1984), 225-256.
doi: 10.1016/0022-0396(84)90082-2. |
[21] |
K. J. Palmer,
Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported, 1 (1988), 265-306.
|
[22] |
E. Park,
Complex Topological K-theory, Cambridge Studies in Advanced Mathematics, 111 Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511611476. |
[23] |
J. Pejsachowicz,
K-theoretic methods in bifurcation theory, Fixed point theory and its applications (Berkeley, CA, 1986), Contemp. Math, 72 (1988), 193-206.
doi: 10.1090/conm/072/956492. |
[24] |
J. Pejsachowicz,
Index bundle, Leray-Schauder reduction and bifurcation of solutions of nonlinear elliptic boundary value problems, Topol. Methods Nonlinear Anal., 18 (2001), 243-267.
doi: 10.12775/TMNA.2001.033. |
[25] |
J. Pejsachowicz,
Bifurcation of homoclinics, Proc. Amer. Math. Soc., 136 (2008), 111-118.
doi: 10.1090/S0002-9939-07-09088-0. |
[26] |
J. Pejsachowicz, Bifurcation of homoclinics of Hamiltonian systems,
Proc. Amer. Math. Soc., 136 (2008). |
[27] |
J. Pejsachowicz,
Bifurcation of Fredholm maps Ⅰ. The index bundle and bifurcation, Topol.
Methods Nonlinear Anal, 38 (2011), 115-168.
|
[28] |
J. Pejsachowicz,
Bifurcation of Fredholm maps Ⅱ. The dimension of the set of bifurcation points, Topol. Methods Nonlinear Anal., 38 (2011), 291-305.
|
[29] |
J. Pejsachowicz and R. Skiba,
Global bifurcation of homoclinic trajectories of discrete dynamical systems, Central European Journal of Mathematics, 10 (2012), 2088-2109.
doi: 10.2478/s11533-012-0121-8. |
[30] |
J. Pejsachowicz and R. Skiba,
Topology and homoclinic trajectories of discrete dynamical systems, Discrete and Continuous Dynamical Systems, Series S, 6 (2013), 1077-1094.
doi: 10.3934/dcdss.2013.6.1077. |
[31] |
J. Pejsachowicz,
The index bundle and bifurcation from infinity of solutions of nonlinear elliptic boundary value problems, J. Fixed Point Theory Appl., 17 (2015), 43-64.
doi: 10.1007/s11784-015-0237-0. |
[32] |
O. Perron,
Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
[33] |
C. Pötzsche,
Nonautonomous bifurcation of bounded solutions Ⅰ: A Lyapunov-Schmidt approach, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 739-776.
doi: 10.3934/dcdsb.2010.14.739. |
[34] |
C. Pötzsche,
Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961.
doi: 10.3934/cpaa.2011.10.937. |
[35] |
C. Pötzsche,
Bifurcations in nonautonomous dynamical systems: Results and tools in discrete time, Proceedings of the International Workshop Future Directions in Difference Equations, 69 (2011), 163-212.
|
[36] |
S. Secchi and C. A. Stuart,
Global Bifurcation of homoclinic solutions of Hamiltonian systems, Discrete Contin. Dyn. Syst., 9 (2003), 1493-1518.
doi: 10.3934/dcds.2003.9.1493. |
[37] |
M. Starostka and N. Waterstraat,
A remark on singular sets of vector bundle morphisms, Eur. J. Math., 1 (2015), 154-159.
doi: 10.1007/s40879-014-0010-8. |
[38] |
N. Waterstraat,
The index bundle for Fredholm morphisms, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 299-315.
|
[39] |
N. Waterstraat, A remark on bifurcation of Fredholm maps accepted for publication in Adv. Nonlinear Anal., arXiv: 1602.02320 [math. FA]
doi: 10.1515/anona-2016-0067. |
[40] |
M. G. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov,
Banach bundles and linear operators, Russian Math. Surveys, 30 (1975), 101-157.
|
show all references
References:
[1] |
A. Abbondandolo and P. Majer,
On the global stable manifold, Studia Math., 177 (2006), 113-131.
doi: 10.4064/sm177-2-2. |
[2] |
D. Arlt,
Zusammenziehbarkeit der allgemeinen linearen Gruppe des Raumes $c_0$ der Nullfolgen, Invent. Math., 1 (1966), 36-44.
doi: 10.1007/BF01389697. |
[3] |
M. F. Atiyah,
Thom complexes, Proc. London Math. Soc.(3), 11 (1961), 291-310.
doi: 10.1112/plms/s3-11.1.291. |
[4] |
M. F. Atiyah and I. M. Singer,
The index of elliptic operators. Ⅳ, Ann. of Math.(2), 93 (1971), 119-138.
doi: 10.2307/1970756. |
[5] | |
[6] |
T. Bartsch,
The global structure of the zero set of a family of semilinear Fredholm maps, Nonlinear Anal, 71 (1991), 313-331.
doi: 10.1016/0362-546X(91)90074-B. |
[7] |
G. E. Bredon,
Topology and Geometry, Graduate Texts in Mathematics, 139 Springer, 1993.
doi: 10.1007/978-1-4757-6848-0. |
[8] |
W. A. Coppel,
Dichotomies in Stability Theory, Lecture Notes in Math., vol. 629, Springer-Verlag, New York, 1978. |
[9] |
M. Crabb and I. James,
Fibrewise Homotopy Theory, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1998.
doi: 10.1007/978-1-4471-1265-5. |
[10] |
A. Dold,
ÜUber fasernweise Homotopieäquivalenz von Faserräumen, Math. Z., 62 (1955), 111-136.
doi: 10.1007/BF01180627. |
[11] |
P. M. Fitzpatrick and J. Pejsachowicz,
Nonorientability of the index bundle and several-parameter bifurcation, J. Funct. Anal., 98 (1991), 42-58.
doi: 10.1016/0022-1236(91)90090-R. |
[12] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981. |
[13] |
W. Hurewicz and H. Wallmann,
Dimension Theory, Princeton Mathematical Series, 4 Princeton University Press, 1941. |
[14] |
T. Hüls,
Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31.
doi: 10.1080/10236190902932742. |
[15] |
K. Jänich,
Vektorraumbündel und der Raum der Fredholmoperatoren, Math. Ann., 161 (1965), 129-142.
doi: 10.1007/BF01360851. |
[16] |
S. Lang,
Differential and Riemannian Manifolds, Third edition, Graduate Texts in Mathematics, 160 Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4182-9. |
[17] |
H. B. Lawson and M. -L. Michelsohn,
Spin Geometry, Princeton Mathematical Series, 38 Princeton University Press, Princeton, NJ, 1989. |
[18] |
J. P. May,
A Concise Course in Algebraic Topology, Chicago University Press, 2nd edition, 1999. |
[19] |
J. W. Milnor and J. D. Stasheff,
Characteristic Classes, Princeton University Press, 1974. |
[20] |
K. J. Palmer,
Exponential dichotomies and transversal homoclinic points, Journal of Differential Equations, 55 (1984), 225-256.
doi: 10.1016/0022-0396(84)90082-2. |
[21] |
K. J. Palmer,
Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported, 1 (1988), 265-306.
|
[22] |
E. Park,
Complex Topological K-theory, Cambridge Studies in Advanced Mathematics, 111 Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511611476. |
[23] |
J. Pejsachowicz,
K-theoretic methods in bifurcation theory, Fixed point theory and its applications (Berkeley, CA, 1986), Contemp. Math, 72 (1988), 193-206.
doi: 10.1090/conm/072/956492. |
[24] |
J. Pejsachowicz,
Index bundle, Leray-Schauder reduction and bifurcation of solutions of nonlinear elliptic boundary value problems, Topol. Methods Nonlinear Anal., 18 (2001), 243-267.
doi: 10.12775/TMNA.2001.033. |
[25] |
J. Pejsachowicz,
Bifurcation of homoclinics, Proc. Amer. Math. Soc., 136 (2008), 111-118.
doi: 10.1090/S0002-9939-07-09088-0. |
[26] |
J. Pejsachowicz, Bifurcation of homoclinics of Hamiltonian systems,
Proc. Amer. Math. Soc., 136 (2008). |
[27] |
J. Pejsachowicz,
Bifurcation of Fredholm maps Ⅰ. The index bundle and bifurcation, Topol.
Methods Nonlinear Anal, 38 (2011), 115-168.
|
[28] |
J. Pejsachowicz,
Bifurcation of Fredholm maps Ⅱ. The dimension of the set of bifurcation points, Topol. Methods Nonlinear Anal., 38 (2011), 291-305.
|
[29] |
J. Pejsachowicz and R. Skiba,
Global bifurcation of homoclinic trajectories of discrete dynamical systems, Central European Journal of Mathematics, 10 (2012), 2088-2109.
doi: 10.2478/s11533-012-0121-8. |
[30] |
J. Pejsachowicz and R. Skiba,
Topology and homoclinic trajectories of discrete dynamical systems, Discrete and Continuous Dynamical Systems, Series S, 6 (2013), 1077-1094.
doi: 10.3934/dcdss.2013.6.1077. |
[31] |
J. Pejsachowicz,
The index bundle and bifurcation from infinity of solutions of nonlinear elliptic boundary value problems, J. Fixed Point Theory Appl., 17 (2015), 43-64.
doi: 10.1007/s11784-015-0237-0. |
[32] |
O. Perron,
Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
[33] |
C. Pötzsche,
Nonautonomous bifurcation of bounded solutions Ⅰ: A Lyapunov-Schmidt approach, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 739-776.
doi: 10.3934/dcdsb.2010.14.739. |
[34] |
C. Pötzsche,
Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961.
doi: 10.3934/cpaa.2011.10.937. |
[35] |
C. Pötzsche,
Bifurcations in nonautonomous dynamical systems: Results and tools in discrete time, Proceedings of the International Workshop Future Directions in Difference Equations, 69 (2011), 163-212.
|
[36] |
S. Secchi and C. A. Stuart,
Global Bifurcation of homoclinic solutions of Hamiltonian systems, Discrete Contin. Dyn. Syst., 9 (2003), 1493-1518.
doi: 10.3934/dcds.2003.9.1493. |
[37] |
M. Starostka and N. Waterstraat,
A remark on singular sets of vector bundle morphisms, Eur. J. Math., 1 (2015), 154-159.
doi: 10.1007/s40879-014-0010-8. |
[38] |
N. Waterstraat,
The index bundle for Fredholm morphisms, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 299-315.
|
[39] |
N. Waterstraat, A remark on bifurcation of Fredholm maps accepted for publication in Adv. Nonlinear Anal., arXiv: 1602.02320 [math. FA]
doi: 10.1515/anona-2016-0067. |
[40] |
M. G. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov,
Banach bundles and linear operators, Russian Math. Surveys, 30 (1975), 101-157.
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