渐近线性Dirac方程的相对Morse指标及其多解性

摘要/Abstract

摘要：

该文主要研究Dirac方程周期解的存在性和多重性. 通过引入相对Morse指标对相应的线性Dirac方程进行分类, 并给出解存在的扭转性条件.

Abstract:

This paper is concerned with the existence and multiplicity of periodic solutions for Dirac equation. We will establish a relative Morse index theory to classify the associated linear Dirac equation. Under a general twist condition for the nonlinear part via the relative Morse index, existence of multiple solutions are obtained.

Key words: Relative Morse index, Periodic solution of Dirac equation, Twist condition

中图分类号:

O177.91

单远. 渐近线性Dirac方程的相对Morse指标及其多解性[J]. 数学物理学报, 2023, 43(1): 69-81.

Shan Yuan. Relative Morse Index and Multiple Solutions for Asymptotically Linear Dirac Equation[J]. Acta mathematica scientia,Series A, 2023, 43(1): 69-81.

参考文献 34

[1]	Abbondandolo A. Morse Theory for Hamiltonian Systems. London: Chapman Hall, 2001
[2]	Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14: 349-381 doi: 10.1016/0022-1236(73)90051-7
[3]	Bartsch T, Ding Y. Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math Nachr, 2006, 279: 1267-1288 doi: 10.1002/mana.200410420
[4]	Chen C, Hu X. Maslov index for homoclinic orbits of Hamiltonian systems. Ann Inst H Poincaré Anal Non Linaire, 2007, 24: 589-603
[5]	Conley C, Zehnder E. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm Pure Appl Math, 1984, 37: 207-253 doi: 10.1002/cpa.3160370204
[6]	Ding Y. Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations. Calc Var Partial Differential Equations, 2010, 38: 75-109 doi: 10.1007/s00526-009-0279-5
[7]	Ding Y. Infinitely many solutions for a class of nonlinear Dirac equations without symmetry. Nonlinear Anal, 2009, 70: 921-935 doi: 10.1016/j.na.2008.01.022
[8]	Ding Y. Semi-classical ground states concerntrating on the nonlinear potential for a Dirac equation. J Differential Equations, 2010, 249: 1015-1034 doi: 10.1016/j.jde.2010.03.022
[9]	Ding Y, Liu X. Semi-classical limits of ground states of a nonlinear Dirac equation. J Differential Equations, 2012, 252: 4962-4987 doi: 10.1016/j.jde.2012.01.023
[10]	Ding Y, Liu X. Periodic waves of nonlinear Dirac equations. Nonlinear Anal, 2014, 109: 252-267 doi: 10.1016/j.na.2014.06.015
[11]	Ding Y, Liu X. Periodic solutions of a Dirac equation with concave and convex nonlinearities. J Differential Equations, 2015, 258: 3567-3588 doi: 10.1016/j.jde.2015.01.013
[12]	Ding Y, Liu X. Periodic solutions of an asymptotically linear Dirac equation. Annali di Matematica, 2017, 196: 717-735 doi: 10.1007/s10231-016-0592-5
[13]	Ding Y, Ruf B. Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities. SIAM J Math Anal, 2012, 44: 3755-3785 doi: 10.1137/110850670
[14]	Ding Y, Ruf B. Solutions of a nonlinear Dirac equation with external fields. Arch Ration Mech Anal, 2008, 190: 1007-1032
[15]	Ding Y, Shan Y. Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations (II). arXiv:1104.1670v1
[16]	Ding Y, Wei J. Stationary states of nonlinear Dirac equations with general potentials. Rev Math Phys, 2008, 20: 1007-1032 doi: 10.1142/S0129055X0800350X
[17]	Ekeland I. Une theorie de Morse pour les systemes hamiltoniens convexes. Ann Inst H Poincaré Anal Non Linaire, 1984, 1: 19-78
[18]	Esteban M, Séré E. An overview on linear and nonlinear Dirac equations. Discrete Contin Dyn Syst, 2002, 8: 281-397
[19]	Esteban M, Séré E. Stationary states of the nonlinear Dirac equation: a variational approach. Commun Math Phys, 1995, 171: 323-350 doi: 10.1007/BF02099273
[20]	Li G B, Zhou H S. The existence of a positive solution to a asymptotically linear scalar field equations. Proc R Soc Edinburgh Ser A, 2000, 130: 81-105
[21]	Liu C. Maslov-type index theory for symplectic paths with Lagrangian boundary conditions. Adv Nonlinear Stud, 2007, 7: 131-161 doi: 10.1515/ans-2007-0107
[22]	Liu C. A note on the relations between the various index theories. J Fixed Point Theory Appl, 2017, 19: 617-648 doi: 10.1007/s11784-016-0368-y
[23]	Long Y. A Maslov-type index theory for symplectic paths. Topol Methods Nonlinear Anal, 1997, 10: 47-78
[24]	Long Y. Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Sci China, 1990, 33: 1409-1419
[25]	Long Y, Zehnder E. Morse theory for forced oscillations of asymptotically linear Hamiltonian systems// Alberverio S, et al. Stochastic Processes, Phiscs and Geometry. Teaneck: World Scientific, 1990: 528-563
[26]	Long Y, Zhu C. Maslov type index theory for symplectic paths and spectral flow (II). Chin Ann Math, 2000, 21B: 89-108
[27]	Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence, RI: American Mathematical Society, 1986
[28]	Ranada A F. Classical nonlinear dirac field models of extended particles//Barut A O. Quantum Theory, Groups, Fields and Particles. Reidel: Amsterdam, 1982
[29]	Shan Y. Morse index and multiple solutions for the asymptotically linear Schrödinger type equation. Nonlinear Anal, 2013, 89: 170-178 doi: 10.1016/j.na.2013.05.014
[30]	Shan Y. A twist condition and multiple solutions of unbounded self-adjoint operator equation with symmetries. J Math Anal Appl, 2014, 410: 597-606 doi: 10.1016/j.jmaa.2013.08.067
[31]	Struwe M. Variational Methods:Applications to Nonlinear Partifal Differential Equations and Hamiltonian Systems. New York: Springer, 1990
[32]	Thaller B. The Dirac Equation, Texts and Monographs in Physics. Berlin: Springer, 1992
[33]	Wang Q, Liu C. A new index theory for linear self-adjoint operator equations and its applications. J Differential Equations, 2016, 260: 3749-3784 doi: 10.1016/j.jde.2015.10.046
[34]	Zhu C, Long Y. Maslov type index theory for symplectic paths and spectral flow (I). Chin Ann Math, 1999, 20B: 413-424