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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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021279
References:
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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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