doi: 10.3934/dcdsb.2021279
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Nonstationary homoclinic orbit for an infinite-dimensional fractional reaction-diffusion system

1. 

Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, Hubei 443002, China

2. 

College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

3. 

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

* Corresponding author: Linfeng Mei

Received  August 2021 Early access November 2021

Fund Project: The first author is supported by the Natural Science Foundation of Hubei Province of China: Grant No. 2021CFB473, the second author is supported by the Natural Science Foundation of China: Grant No. 11771125

This paper study nonstationary homoclinic-type solutions for a fractional reaction-diffusion system with asymptotically linear and local super linear nonlinearity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is not super quadratic at infinity globally. These enable us to develop a direct approach and new tricks to overcome the difficulties. We establish the existence of homoclinic orbit under some weak assumptions on nonlinearity.

Citation: Peng Chen, Linfeng Mei, Xianhua Tang. Nonstationary homoclinic orbit for an infinite-dimensional fractional reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021279
References:
[1]

A. AlaediB. AhmadM. Kirane and R. Lassoued, Global existence and large time behavior of solutions of a time behavior of solutions of a time fractional reaction diffusion system, Frac. Calc. Appl. Anal., 23 (2020), 390-407.  doi: 10.1515/fca-2020-0019.

[2]

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

[3]

H. Brézis and L. Nirenberg, Characterization of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 225-326.

[4]

P. ChenZ. CaoS. Chen and X. Tang, Ground state for a fractional reaction-diffusion system, J. Appl. Anal. Comput., 11 (2021), 556-567.  doi: 10.11948/20200349.

[5]

S. ChenA. FiscellaP. Pucci and X. Tang, Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differ. Equ., 268 (2020), 2672-2716.  doi: 10.1016/j.jde.2019.09.041.

[6]

P. ClémentP. Felmer and E. Mitidieri, Homoclinic orbits for a class of infinite dimensional Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 367-393. 

[7]

D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functions and multiple solutions of elliptic systems, Trans. Amer. Math. Soc., 355 (2003), 2973-2989.  doi: 10.1090/S0002-9947-03-03257-4.

[8]

D. G. de Figueiredo and P. L. Felmer, On superquadiatic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 97-116.  doi: 10.1090/S0002-9947-1994-1214781-2.

[9]

Y. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709639.

[10]

Y. Ding and Q. Guo, Homoclinic solutions for an anomalous diffusion system, J. Math. Anal. Appl., 466 (2018), 860-879.  doi: 10.1016/j.jmaa.2018.06.028.

[11]

Y. DingS. Luan and M. Willem, Solutions of a system of diffusion equations, J. Fixed Point Theory Appl., 2 (2007), 117-139.  doi: 10.1007/s11784-007-0023-8.

[12]

Y. Ding and T. Xu, Effect of external potentials in a coupled system of multi-component incongruent diffusion, Topol. Method. Nonl. Anal., 54 (2019), 715-750.  doi: 10.12775/tmna.2019.066.

[13]

Y. Ding and T. Xu, Concentrating patterns of reaction-diffusion systems: A variational approach, Trans. Amer. Math. Soc., 369 (2017), 97-138.  doi: 10.1090/tran/6626.

[14]

W. Kryszewski and A. Szulkin, An infinite dimensional Morse theorem with applications, Trans. Amer. Math. Soc., 349 (1997), 3181-3234.  doi: 10.1090/S0002-9947-97-01963-6.

[15]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.

[16]

J.-L. Lions, Contrôe Optimal de Systèmes Gouvernés par des Équations aux Dérivées Particlles, (French) Dunod and Gauthier-Villars, Paris, 1968.

[17] G. Molica BisciV. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 2016.  doi: 10.1017/CBO9781316282397.
[18]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.

[19]

K. M. Saad and J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Physica A., 509 (2018), 703-716.  doi: 10.1016/j.physa.2018.05.137.

[20]

P. SantoroJ. de PaulaE. Lenzi and L. Evangelista, Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell, J. Chem. Phys., 135 (2011), 114704. 

[21]

A. Szulkin and T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.

[22]

X. TangS. ChenX. Lin and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differ. Equ., 268 (2020), 4663-4690.  doi: 10.1016/j.jde.2019.10.041.

[23]

X. H. Tang, Non-nehari manifold method for asymptotically linear schrodinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.  doi: 10.1017/S144678871400041X.

[24]

X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwanese J. Math., 18 (2014), 1957-1979.  doi: 10.11650/tjm.18.2014.3541.

[25]

X. H. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.

[26]

J. L. Vázquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion, Lecture Notes in Mathematics, 2186 (2017), 205-278. 

[27]

J. WangJ. Xu and F. Zhang, Infinitely many solutions for diffusion equations without symmetry, Nonlinear Anal., 74 (2011), 1290-1303.  doi: 10.1016/j.na.2010.10.002.

[28]

Y. Wei and M. Yang, Existence of solutions for a system of diffusion equations with spectrum point zero, Z. Angew. Math. Phys., 65 (2014), 325-337.  doi: 10.1007/s00033-013-0334-0.

[29]

M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627.  doi: 10.1016/j.na.2009.11.009.

[30]

M. Yang, Nonstationary homoclinic orbits for an infinite-dimensional Hamiltonian system, J. Math. Phys., 51 (2010), 102701, 11 pp. doi: 10.1063/1.3488967.

[31]

M. YangZ. Shen and Y. Ding, On a class of infinite-dimensional Hamiltonian systems with asymptotically periodic nonlinearities, Chin. Ann. Math. Ser. B, 32 (2011), 45-58.  doi: 10.1007/s11401-010-0625-0.

[32]

J. ZhangX. Tang and W. Zhang, Ground state solutions for superquadratic Hamiltonian elliptic systems with gradient terms, Nonlinear Anal., 95 (2014), 1-10.  doi: 10.1016/j.na.2013.07.027.

[33]

F. Zhao and Y. Ding, On a diffusion system with bounded potential, Discrete Contin. Dyn. Syst., 23 (2009), 1073-1086.  doi: 10.3934/dcds.2009.23.1073.

[34]

L. Zhao and F. Zhao, On ground state solutions for superlinear Hamiltonian elliptic systems, Z. Angew. Math. Phys., 64 (2013), 403-418.  doi: 10.1007/s00033-012-0258-0.

show all references

References:
[1]

A. AlaediB. AhmadM. Kirane and R. Lassoued, Global existence and large time behavior of solutions of a time behavior of solutions of a time fractional reaction diffusion system, Frac. Calc. Appl. Anal., 23 (2020), 390-407.  doi: 10.1515/fca-2020-0019.

[2]

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

[3]

H. Brézis and L. Nirenberg, Characterization of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 225-326.

[4]

P. ChenZ. CaoS. Chen and X. Tang, Ground state for a fractional reaction-diffusion system, J. Appl. Anal. Comput., 11 (2021), 556-567.  doi: 10.11948/20200349.

[5]

S. ChenA. FiscellaP. Pucci and X. Tang, Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differ. Equ., 268 (2020), 2672-2716.  doi: 10.1016/j.jde.2019.09.041.

[6]

P. ClémentP. Felmer and E. Mitidieri, Homoclinic orbits for a class of infinite dimensional Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 367-393. 

[7]

D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functions and multiple solutions of elliptic systems, Trans. Amer. Math. Soc., 355 (2003), 2973-2989.  doi: 10.1090/S0002-9947-03-03257-4.

[8]

D. G. de Figueiredo and P. L. Felmer, On superquadiatic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 97-116.  doi: 10.1090/S0002-9947-1994-1214781-2.

[9]

Y. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709639.

[10]

Y. Ding and Q. Guo, Homoclinic solutions for an anomalous diffusion system, J. Math. Anal. Appl., 466 (2018), 860-879.  doi: 10.1016/j.jmaa.2018.06.028.

[11]

Y. DingS. Luan and M. Willem, Solutions of a system of diffusion equations, J. Fixed Point Theory Appl., 2 (2007), 117-139.  doi: 10.1007/s11784-007-0023-8.

[12]

Y. Ding and T. Xu, Effect of external potentials in a coupled system of multi-component incongruent diffusion, Topol. Method. Nonl. Anal., 54 (2019), 715-750.  doi: 10.12775/tmna.2019.066.

[13]

Y. Ding and T. Xu, Concentrating patterns of reaction-diffusion systems: A variational approach, Trans. Amer. Math. Soc., 369 (2017), 97-138.  doi: 10.1090/tran/6626.

[14]

W. Kryszewski and A. Szulkin, An infinite dimensional Morse theorem with applications, Trans. Amer. Math. Soc., 349 (1997), 3181-3234.  doi: 10.1090/S0002-9947-97-01963-6.

[15]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.

[16]

J.-L. Lions, Contrôe Optimal de Systèmes Gouvernés par des Équations aux Dérivées Particlles, (French) Dunod and Gauthier-Villars, Paris, 1968.

[17] G. Molica BisciV. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 2016.  doi: 10.1017/CBO9781316282397.
[18]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.

[19]

K. M. Saad and J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Physica A., 509 (2018), 703-716.  doi: 10.1016/j.physa.2018.05.137.

[20]

P. SantoroJ. de PaulaE. Lenzi and L. Evangelista, Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell, J. Chem. Phys., 135 (2011), 114704. 

[21]

A. Szulkin and T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.

[22]

X. TangS. ChenX. Lin and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differ. Equ., 268 (2020), 4663-4690.  doi: 10.1016/j.jde.2019.10.041.

[23]

X. H. Tang, Non-nehari manifold method for asymptotically linear schrodinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.  doi: 10.1017/S144678871400041X.

[24]

X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwanese J. Math., 18 (2014), 1957-1979.  doi: 10.11650/tjm.18.2014.3541.

[25]

X. H. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.

[26]

J. L. Vázquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion, Lecture Notes in Mathematics, 2186 (2017), 205-278. 

[27]

J. WangJ. Xu and F. Zhang, Infinitely many solutions for diffusion equations without symmetry, Nonlinear Anal., 74 (2011), 1290-1303.  doi: 10.1016/j.na.2010.10.002.

[28]

Y. Wei and M. Yang, Existence of solutions for a system of diffusion equations with spectrum point zero, Z. Angew. Math. Phys., 65 (2014), 325-337.  doi: 10.1007/s00033-013-0334-0.

[29]

M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627.  doi: 10.1016/j.na.2009.11.009.

[30]

M. Yang, Nonstationary homoclinic orbits for an infinite-dimensional Hamiltonian system, J. Math. Phys., 51 (2010), 102701, 11 pp. doi: 10.1063/1.3488967.

[31]

M. YangZ. Shen and Y. Ding, On a class of infinite-dimensional Hamiltonian systems with asymptotically periodic nonlinearities, Chin. Ann. Math. Ser. B, 32 (2011), 45-58.  doi: 10.1007/s11401-010-0625-0.

[32]

J. ZhangX. Tang and W. Zhang, Ground state solutions for superquadratic Hamiltonian elliptic systems with gradient terms, Nonlinear Anal., 95 (2014), 1-10.  doi: 10.1016/j.na.2013.07.027.

[33]

F. Zhao and Y. Ding, On a diffusion system with bounded potential, Discrete Contin. Dyn. Syst., 23 (2009), 1073-1086.  doi: 10.3934/dcds.2009.23.1073.

[34]

L. Zhao and F. Zhao, On ground state solutions for superlinear Hamiltonian elliptic systems, Z. Angew. Math. Phys., 64 (2013), 403-418.  doi: 10.1007/s00033-012-0258-0.

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