August  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 and 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.

[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. ChenJ. JostJ. Li and G. Wang, Dirac-harmonic maps, Math. Z., 254 (2006), 409-432.  doi: 10.1007/s00209-006-0961-7.

[11]

Q. ChenJ. 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. ChenJ. JostJ. Li and G. Wang, Dirac-harmonic maps, Math. Z., 254 (2006), 409-432.  doi: 10.1007/s00209-006-0961-7.

[11]

Q. ChenJ. 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.

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