In this paper, we show the existence of solutions for the coupled Dirac system
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: |
[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.![]() ![]() ![]() |