American Institute of Mathematical Sciences

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

On coupled Dirac systems

 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.
Citation: Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 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. [2] B. Ammann, A variational Problem in Conformal Spin Geometry, Ph. D thesis, Habilitationsschift, Universität Hamburg 2003. [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. [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. [5] T. Bartsch and Y. Ding, Homoclinic solutions of an infinite-dimensional Hamiltonian system, Math. Z., 240 (2002), 289-310. doi: 10.1007/s002090100383. [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. [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. [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. [9] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883. [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. [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. [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. [13] P. Felmer, Periodic solutions of 'superquadratic' Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207. doi: 10.1006/jdeq.1993.1027. [14] T. Friedrich, Dirac Operators in Riemannian Geometry, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1997. doi: 10.1090/gsm/025. [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. [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. [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. [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. [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. [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. [21] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schröinger equation, Adv. Differential Equations, 3 (1998), 441-472. [22] H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton University Press, 1989. [23] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. [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. [25] M. Willem, Minimax Theorems, Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

show all references

References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Space, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003. [2] B. Ammann, A variational Problem in Conformal Spin Geometry, Ph. D thesis, Habilitationsschift, Universität Hamburg 2003. [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. [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. [5] T. Bartsch and Y. Ding, Homoclinic solutions of an infinite-dimensional Hamiltonian system, Math. Z., 240 (2002), 289-310. doi: 10.1007/s002090100383. [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. [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. [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. [9] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883. [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. [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. [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. [13] P. Felmer, Periodic solutions of 'superquadratic' Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207. doi: 10.1006/jdeq.1993.1027. [14] T. Friedrich, Dirac Operators in Riemannian Geometry, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1997. doi: 10.1090/gsm/025. [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. [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. [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. [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. [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. [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. [21] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schröinger equation, Adv. Differential Equations, 3 (1998), 441-472. [22] H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton University Press, 1989. [23] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. [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. [25] M. Willem, Minimax Theorems, Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.
 [1] Viktor L. Ginzburg and Basak Z. Gurel. The Generalized Weinstein--Moser Theorem. Electronic Research Announcements, 2007, 14: 20-29. doi: 10.3934/era.2007.14.20 [2] Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983 [3] Qiang Li. A kind of generalized transversality theorem for $C^r$ mapping with parameter. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1043-1050. doi: 10.3934/dcdss.2017055 [4] Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023 [5] Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893 [6] Denis Blackmore, Jyoti Champanerkar, Chengwen Wang. A generalized Poincaré-Birkhoff theorem with applications to coaxial vortex ring motion. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 15-33. doi: 10.3934/dcdsb.2005.5.15 [7] Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787 [8] Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 [9] Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248 [10] Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353 [11] Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555 [12] Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155 [13] Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska. Generalized Stokes system in Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125 [14] Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107 [15] M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703 [16] Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511 [17] Olaf Hansen. A global existence theorem for two coupled semilinear diffusion equations from climate modeling. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 541-564. doi: 10.3934/dcds.1997.3.541 [18] Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435 [19] Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549 [20] Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517

2016 Impact Factor: 1.099

Metrics

• PDF downloads (2)
• HTML views (4)
• Cited by (0)

[Back to Top]