American Institute of Mathematical Sciences

Advanced
August  2017, 37(8): 4329-4346. doi: 10.3934/dcds.2017185

On coupled Dirac systems

Wenmin Gong 1,, and  Guangcun Lu 2,

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

* Corresponding author: Wenmin Gong

The first author is supported by the NSF (grant no. 11571194) of China.
The second author is Partially supported by the NNSF (grant no. 10971014 and 11271044) of China.

Received  October 2016 Revised  February 2017 Published  April 2017

In this paper, we show the existence of solutions for the coupled Dirac system
$\left\{ \begin{aligned}Du=\frac{\partial H}{\partial v}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\\Dv=\frac{\partial H}{\partial u}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\end{aligned} \right.$
where $M$ is an $n$-dimensional compact Riemannian spin manifold, $D$ is the Dirac operator on $M$, and $H:\Sigma M\oplus \Sigma M\to \mathbb{R}$ is a real valued superquadratic function of class $C^1$ in the fiber direction with subcritical growth rates. Our proof relies on a generalized linking theorem applied to a strongly indefinite functional on a product space of suitable fractional Sobolev spaces. Furthermore, we consider the $\mathbb{Z}_2$-invariant $H$ that includes a nonlinearity of the form
$H(x,u,v)=f(x)\frac{|u|^{p+1}}{p+1}+g(x)\frac{|v|^{q+1}}{q+1},$
where $f(x)$ and $g(x)$ are strictly positive continuous functions on $M$ and $p, q>1$ satisfy
$\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-1}{n}.$
In this case we obtain infinitely many solutions of the coupled Dirac system by using a generalized fountain theorem.
Keywords: Coupled Dirac system, generalized fountain theorem, generalized linking theorem, strongly indefinite functionals.
Mathematics Subject Classification: Primary: 53C27, 57R58, 58E05, 58J05.
Citation: Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Space, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

B. Ammann, A variational Problem in Conformal Spin Geometry, Ph. D thesis, Habilitationsschift, Universität Hamburg 2003. Google Scholar

[3]

B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions, Commun. Anal. Geom., 17 (2009), 429-479.  doi: 10.4310/CAG.2009.v17.n3.a2.  Google Scholar

[4]

S. Angenent and R. van der Vorst, A superquadratic indefinite elliptic system and its MorseConley-Floer homology, Math. Z., 231 (1999), 203-248.  doi: 10.1007/PL00004731.  Google Scholar

[5]

T. Bartsch and Y. Ding, Homoclinic solutions of an infinite-dimensional Hamiltonian system, Math. Z., 240 (2002), 289-310.  doi: 10.1007/s002090100383.  Google Scholar

[6]

T. Bartsch and Y. Ding, Periodic solutions of superlinear beam and membrane equations with perturbations from symmetry, Nonlinear Analysis, 44 (2001), 727-748.  doi: 10.1016/S0362-546X(99)00302-8.  Google Scholar

[7]

T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal., 20 (1993), 1205-1216.  doi: 10.1016/0362-546X(93)90151-H.  Google Scholar

[8]

C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 405 (2013), 438-452.  doi: 10.1016/j.jmaa.2013.04.018.  Google Scholar

[9]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.  doi: 10.1007/BF01389883.  Google Scholar

[10]

Q. Chen, J. Jost, J. Li and G. Wang, Dirac-harmonic maps, Math. Z., 254 (2006), 409-432.  doi: 10.1007/s00209-006-0961-7.  Google Scholar

[11]

Q. Chen, J. Jost and G. Wang, Nonlinear Dirac equations on Riemann surfaces, Ann. Global Anal. Geom., 33 (2008), 253-270.  doi: 10.1007/s10455-007-9084-6.  Google Scholar

[12]

P. Felmer and D. G. deFigueiredo, On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 99-116.  doi: 10.1090/S0002-9947-1994-1214781-2.  Google Scholar

[13]

P. Felmer, Periodic solutions of 'superquadratic' Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207.  doi: 10.1006/jdeq.1993.1027.  Google Scholar

[14]

T. Friedrich, Dirac Operators in Riemannian Geometry, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1997. doi: 10.1090/gsm/025.  Google Scholar

[15]

T. Friedrich, On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phy., 28 (1998), 143-157.  doi: 10.1016/S0393-0440(98)00018-7.  Google Scholar

[16]

N. Ginoux, The Dirac Spectrum, Lecture Notes in Math. , vol. 1976, Springer, Dordrechtheidelberg-London-New York, 2009. doi: 10.1007/978-3-642-01570-0.  Google Scholar

[17]

W. Gong and G. Lu, On Dirac equation with a potential and critical Sobolev exponent, Commun. Pure Appl. Anal., 14 (2015), 2231-2263.  doi: 10.3934/cpaa.2015.14.2231.  Google Scholar

[18]

J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58.  doi: 10.1006/jfan.1993.1062.  Google Scholar

[19]

T. Isobe, Existence results for solutions to nonlinear Dirac equations on compact spin manifolds, Manuscripta math, 135 (2011), 329-360.  doi: 10.1007/s00229-010-0417-6.  Google Scholar

[20]

T. Isobe, Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds, J.Funct. Anal., 260 (2011), 253-307.  doi: 10.1016/j.jfa.2010.09.008.  Google Scholar

[21]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schröinger equation, Adv. Differential Equations, 3 (1998), 441-472.   Google Scholar

[22]

H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton University Press, 1989.  Google Scholar

[23]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun. Pure Appl. Math., 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.  Google Scholar

[24]

S. Raulot, A Sobolev-like inequality for the Dirac operator, J. Funct. Anal., 256 (2009), 1588-1617.  doi: 10.1016/j.jfa.2008.11.007.  Google Scholar

[25]

M. Willem, Minimax Theorems, Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Space, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

B. Ammann, A variational Problem in Conformal Spin Geometry, Ph. D thesis, Habilitationsschift, Universität Hamburg 2003. Google Scholar

[3]

B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions, Commun. Anal. Geom., 17 (2009), 429-479.  doi: 10.4310/CAG.2009.v17.n3.a2.  Google Scholar

[4]

S. Angenent and R. van der Vorst, A superquadratic indefinite elliptic system and its MorseConley-Floer homology, Math. Z., 231 (1999), 203-248.  doi: 10.1007/PL00004731.  Google Scholar

[5]

T. Bartsch and Y. Ding, Homoclinic solutions of an infinite-dimensional Hamiltonian system, Math. Z., 240 (2002), 289-310.  doi: 10.1007/s002090100383.  Google Scholar

[6]

T. Bartsch and Y. Ding, Periodic solutions of superlinear beam and membrane equations with perturbations from symmetry, Nonlinear Analysis, 44 (2001), 727-748.  doi: 10.1016/S0362-546X(99)00302-8.  Google Scholar

[7]

T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal., 20 (1993), 1205-1216.  doi: 10.1016/0362-546X(93)90151-H.  Google Scholar

[8]

C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 405 (2013), 438-452.  doi: 10.1016/j.jmaa.2013.04.018.  Google Scholar

[9]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.  doi: 10.1007/BF01389883.  Google Scholar

[10]

Q. Chen, J. Jost, J. Li and G. Wang, Dirac-harmonic maps, Math. Z., 254 (2006), 409-432.  doi: 10.1007/s00209-006-0961-7.  Google Scholar

[11]

Q. Chen, J. Jost and G. Wang, Nonlinear Dirac equations on Riemann surfaces, Ann. Global Anal. Geom., 33 (2008), 253-270.  doi: 10.1007/s10455-007-9084-6.  Google Scholar

[12]

P. Felmer and D. G. deFigueiredo, On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 99-116.  doi: 10.1090/S0002-9947-1994-1214781-2.  Google Scholar

[13]

P. Felmer, Periodic solutions of 'superquadratic' Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207.  doi: 10.1006/jdeq.1993.1027.  Google Scholar

[14]

T. Friedrich, Dirac Operators in Riemannian Geometry, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1997. doi: 10.1090/gsm/025.  Google Scholar

[15]

T. Friedrich, On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phy., 28 (1998), 143-157.  doi: 10.1016/S0393-0440(98)00018-7.  Google Scholar

[16]

N. Ginoux, The Dirac Spectrum, Lecture Notes in Math. , vol. 1976, Springer, Dordrechtheidelberg-London-New York, 2009. doi: 10.1007/978-3-642-01570-0.  Google Scholar

[17]

W. Gong and G. Lu, On Dirac equation with a potential and critical Sobolev exponent, Commun. Pure Appl. Anal., 14 (2015), 2231-2263.  doi: 10.3934/cpaa.2015.14.2231.  Google Scholar

[18]

J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58.  doi: 10.1006/jfan.1993.1062.  Google Scholar

[19]

T. Isobe, Existence results for solutions to nonlinear Dirac equations on compact spin manifolds, Manuscripta math, 135 (2011), 329-360.  doi: 10.1007/s00229-010-0417-6.  Google Scholar

[20]

T. Isobe, Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds, J.Funct. Anal., 260 (2011), 253-307.  doi: 10.1016/j.jfa.2010.09.008.  Google Scholar

[21]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schröinger equation, Adv. Differential Equations, 3 (1998), 441-472.   Google Scholar

[22]

H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton University Press, 1989.  Google Scholar

[23]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun. Pure Appl. Math., 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.  Google Scholar

[24]

S. Raulot, A Sobolev-like inequality for the Dirac operator, J. Funct. Anal., 256 (2009), 1588-1617.  doi: 10.1016/j.jfa.2008.11.007.  Google Scholar

[25]

M. Willem, Minimax Theorems, Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[1]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374
[2]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016
[3]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395
[4]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[5]

Ethan Akin, Julia Saccamano. Generalized intransitive dice II: Partition constructions. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021005
[6]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
[7]

Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021098
[8]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056
[9]

Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3717-3735. doi: 10.3934/dcdsb.2020254
[10]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002
[11]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027
[12]

Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182
[13]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019
[14]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020
[15]

Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249
[16]

Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270
[17]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[18]

Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021056
[19]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[20]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (83)
  • HTML views (66)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences