Ground state solutions for nonlinear fractional Kirchhoff–Schrödinger–Poisson systems

Li Wang; Tao Han; Kun Cheng; Jixiu Wang

doi:10.1515/ijnsns-2019-0205

Published by De Gruyter September 25, 2020

Ground state solutions for nonlinear fractional Kirchhoff–Schrödinger–Poisson systems

Li Wang , Tao Han , Kun Cheng and Jixiu Wang

From the journal International Journal of Nonlinear Sciences and Numerical Simulation

https://doi.org/10.1515/ijnsns-2019-0205

Showing a limited preview of this publication:

Abstract

In this paper, we study the existence of ground state solutions for the following fractional Kirchhoff–Schrödinger–Poisson systems with general nonlinearities:

{ ( a + b [ u ] s 2 ) ( − Δ ) s u + u + ϕ ( x ) u = ( | x | − μ ∗ F ( u ) ) f ( u ) in ℝ 3 , ( − Δ ) t ϕ ( x ) = u 2 in ℝ 3 ,

where

[ u ] s 2 = ∫ ℝ 3 | ( − Δ ) s 2 u | 2 d x = ∬ ℝ 3 × ℝ 3 | u ( x ) − u ( y ) | 2 | x − y | 3 + 2 s d x d y ,

s , t ∈ ( 0 , 1 ) with 2 t + 4 s > 3 , 0 < μ < 3 − 2 t , f : ℝ 3 × ℝ → ℝ satisfies a Carathéodory condition and (−Δ)^s is the fractional Laplace operator. There are two novelties of the present paper. First, the nonlocal term in the equation sets an obstacle that the bounded Cerami sequences could not converge. Second, the nonlinear term f does not satisfy the Ambrosetti–Rabinowitz growth condition and monotony assumption. Thus, the Nehari manifold method does not work anymore in our setting. In order to overcome these difficulties, we use the Pohozǎev type manifold to obtain the existence of ground state solution of Pohozǎev type for the above system.

Keywords: critical points; fractional Laplace operator; Pohozǎev type manifold; variational methods

MSC 2010: 35R11; 35B38; 35A15; 35J10

Corresponding author: Jixiu Wang, School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang, 441053, PR China, E-mail: wangjixiu127@aliyun.com

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11701178

Funding source: Science and Technology Project of Jiangxi Provincial Education Department

Award Identifier / Grant number: GJJ190337

Award Identifier / Grant number: GJJ180737

Funding source: Natural Science Foundation program of Jiangxi Provincial

Award Identifier / Grant number: 20202BABL201011

Acknowledgments

The authors would like to thank the anonymous referee for valuable comments and suggestions on improving the presentation of manuscript. The work was supported by the National Natural Science Foundation of China (No. 11701178) and Science and Technology Project of Jiangxi Provincial Education Department (GJJ190337, GJJ180737).

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: The work was supported by the National Natural Science Foundation of China (No. 11701178), Natural Science Foundation program of Jiangxi Provincial(20202BABL201011), and Natural Science Foundation of Jiangxi Educational Committee (GJJ190337, GJJ180737).
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] E. Di Nezza, G. Palatucci, and E. Valdinoci, “Hitchhiker’s guide to the fractional Sobolev spaces,” Bull. Sci. Math., vol. 136, pp. 521–573, 2012, https://doi.org/10.1016/j.bulsci.2011.12.004.10.1016/j.bulsci.2011.12.004Search in Google Scholar

[2] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge, Cambridge University Press, 1996.Search in Google Scholar

[3] S. Kurihura, “Large-amplitude quasi-solitons in superfluid films,” J. Phys. Soc. Japan, vol. 50, pp. 3262–3267, 1981, https://doi.org/10.1143/JPSJ.50.3262.10.1143/JPSJ.50.3262Search in Google Scholar

[4] L. Caffarelli, J.-M. Roquejoffre, and Y. Sire, “Variational problems for free boundaries for the fractional Laplacian,” J. Eur. Math. Soc., vol. 12, pp. 1151–1179, 2010, https://doi.org/10.4171/jems/226.10.4171/JEMS/226Search in Google Scholar

[5] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Boca Raton, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, 2004.Search in Google Scholar

[6] L. Caffarelli, S. Salsa, and L. Silvestre, “Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian,” Invent Math., vol. 171, pp. 425–461, 2008, https://doi.org/10.1007/s00222-007-0086-6.10.1007/s00222-007-0086-6Search in Google Scholar

[7] L. Silvestre, “Regularity of the obstacle problem for a fractional power of the Laplace operator,” Commun. Pure Appl. Math., vol. 60, pp. 67–112, 2007, https://doi.org/10.1002/cpa.20153.10.1002/cpa.20153Search in Google Scholar

[8] L. Silvestre, “On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion,” Adv. Math., vol. 226, pp. 2020–2039, 2011, https://doi.org/10.1016/j.aim.2010.09.007.10.1016/j.aim.2010.09.007Search in Google Scholar

[9] G. Alberti, G. Bouchitté, and P. Seppecher, “Phase transition with the line-tension effect,” Arch. Ration. Mech. Anal., vol. 144, pp. 1–46, 1998, https://doi.org/10.1007/s002050050111.10.1007/s002050050111Search in Google Scholar

[10] G. Molica Bisci, V. Rădulescu, and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge, Cambridge University Press, 2016.10.1017/CBO9781316282397Search in Google Scholar

[11] A. Fiscella and E. Valdinoci, “A critical Kirchhoff type problem involving a nonlocal operator,” Nonlinear Anal., vol. 94, pp. 156–170, 2014, https://doi.org/10.1016/j.na.2013.08.011.10.1016/j.na.2013.08.011Search in Google Scholar

[12] V. Benci and D. Fortunato, “An eigenvalue problem for the Schrödinger–Maxwell equations,” Topol. Methods Nonlinear Anal., vol. 11, pp. 283–293, 1998, https://doi.org/10.12775/tmna.1998.019.10.12775/TMNA.1998.019Search in Google Scholar

[13] V. Benci and D. Fortunato, “Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equationNonlinear Anal.s,” Rev. Math. Phys., vol. 14, pp. 409–420, 2002, https://doi.org/10.1142/s0129055x02001168.10.1142/S0129055X02001168Search in Google Scholar

[14] O. Sanchez and J. Soler, “Long time dynamics of Schrödinger–Poisson–Slater systems,” J. Stat. Phys., vol. 114, pp. 179–204, 2004, https://doi.org/10.1023/b:joss.0000003109.97208.53.10.1023/B:JOSS.0000003109.97208.53Search in Google Scholar

[15] D. Lü, “A note on Kirchhoff-type equations with Hartree-type nonlinearities,” Nonlinear Anal., vol. 99, pp. 35–48, 2014, https://doi.org/10.1016/j.na.2013.12.022.10.1016/j.na.2013.12.022Search in Google Scholar

[16] W. Zou, “Variant fountain theorems and their applications,” Manuscripta Math., vol. 104, pp. 343–358, 2001, https://doi.org/10.1007/s002290170032.10.1007/s002290170032Search in Google Scholar

[17] S. Liang and V. Rădulescu, “Existence of infinitely many solutions for degenerate Kirchhoff-type Schrodinger-Choquard equations,” Electron. J. Differential Equations, vol. 2017, pp. 1–17, 2017.Search in Google Scholar

[18] S. J. Chen and C.-L. Tang, “High energy solutions for the superlinear Schrödinger-Maxwell equations,” Nonlinear Anal., vol. 71, pp. 4927–4934, 2009, https://doi.org/10.1016/j.na.2009.03.050.10.1016/j.na.2009.03.050Search in Google Scholar

[19] J. Sun, “Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations,” J. Math. Anal. Appl., vol. 390, pp. 514–522, 2012, https://doi.org/10.1016/j.jmaa.2012.01.057.10.1016/j.jmaa.2012.01.057Search in Google Scholar

[20] L. Wang, J. Wang, and X. Li, “Infinitely many solutions to quasilinear Schrödinger equations with critical exponent,” Electron. J. Qual. Theor. Differ. Equ., vol. 5, pp. 1–16, 2019, https://doi.org/10.14232/ejqtde.2019.1.5.10.14232/ejqtde.2019.1.5Search in Google Scholar

[21] G. Zhao, X. Zhu, and Y. Li, “Existence of infinitely many solutions to a class of Kirchhoff–Schrödinger–Poisson system,” Appl. Math. Comput., vol. 256, pp. 572–581, 2015, https://doi.org/10.1016/j.amc.2015.01.038.10.1016/j.amc.2015.01.038Search in Google Scholar

[22] R. Kajikiya, “A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations,” J. Funct. Anal., vol. 225, pp. 352–370, 2005, https://doi.org/10.1016/j.jfa.2005.04.005.10.1016/j.jfa.2005.04.005Search in Google Scholar

[23] O. Nikan, H. Jafari, and A. Golbabai, “Numerical analysis of the fractional evolution model for heat flow in materials with memory,” Alexandria Eng. J., vol. 59, pp. 2627–2637, 2020, https://doi.org/10.1016/j.aej.2020.04.026.10.1016/j.aej.2020.04.026Search in Google Scholar

[24] O. Nikan, J. A. Tenreiro Machado, A. Golbabai, and T. Nikazad, “Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media,” Int. Commun. Heat Mass Tran., vol. 111, 2020, Art no. 104443, https://doi.org/10.1016/j.icheatmasstransfer.2019.104443.10.1016/j.icheatmasstransfer.2019.104443Search in Google Scholar

[25] O. Nikan, J. A. Tenreiro Machado, Z. Avazzadeh, and H. Jafari, “Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics,” J. Adv. Res., 2020, https://doi.org/10.1016/j.jare.2020.06.018.10.1016/j.jare.2020.06.018Search in Google Scholar PubMed PubMed Central

[26] Y. Esmaeelzade Aghdam, H. Mesgrani, M. Javidi, and O. Nikan, “A computational approach for the space-time fractional advection-diffusion equation arising in contaminant transport through porous media,” Eng. Comput., 2020, https://doi.org/10.1007/s00366-020-01021-y.10.1007/s00366-020-01021-ySearch in Google Scholar

[27] H. Rezazadeh, M. S. Osman, M. Eslami, M. Mirzazadeh, Q. Zhou, S. A. Badri, and A. Korkmaz, “Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations,” Nonlinear Eng., vol. 8, no. 1, pp. 224–230, 2019, https://doi.org/10.1515/nleng-2018-0033.10.1515/nleng-2018-0033Search in Google Scholar

[28] M. S. Osman, “New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics,” Pramana - J. Phys., vol. 93, p. 26, 2019, https://doi.org/10.1007/s12043-019-1785-4.10.1007/s12043-019-1785-4Search in Google Scholar

[29] D. Lu, K. U. Tariq, M. S. Osman, D. Baleanu, M. Younis, and M. M. A. Khater, “New analytical wave structures for the (3+1)-dimensional Kadomtsev–Petviashvili and the generalized Boussinesq models and their applications,” Results Phys., vol. 14, September 2019, Art no. 102491, https://doi.org/10.1016/j.rinp.2019.102491.10.1016/j.rinp.2019.102491Search in Google Scholar

[30] M. S. Osman, D. Lu, and M. M. A. Khater, “A study of optical wave propagation in the nonautonomous Schrödinger-Hirota equation with power-law nonlinearity,” Results Phys., vol. 13, 2019, Art no. 102157, https://doi.org/10.1016/j.rinp.2019.102157.10.1016/j.rinp.2019.102157Search in Google Scholar

[31] M. S. Osman and A.-M. Wazwaz, “A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation,” Math. Methods Appl. Sci., vol. 42, no. 18, pp. 6277–6283, 2019, https://doi.org/10.1002/mma.5721.10.1002/mma.5721Search in Google Scholar

[32] M. S. Osman, H. Rezazadeh, and M. Eslami, “Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity,” Nonlinear Eng., vol. 8, no. 1, pp. 559–567, 2019, https://doi.org/10.1515/nleng-2018-0163.10.1515/nleng-2018-0163Search in Google Scholar

[33] H. I. Abdel-Gawad and M. S. Osman, “Exact solutions of the Korteweg-de Vries equation with space and time dependent coefficients by the extended unified method,” Indian J. Pure Appl. Math., vol. 45, no. 1, pp. 1–11, February 2014, https://doi.org/10.1007/s13226-014-0047-x.10.1007/s13226-014-0047-xSearch in Google Scholar

[34] J. Ahmad, N. Raza, and M. S. Osman, “Multi-solitons of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported graphene sheets,” Commun. Theor. Phys., vol. 71, no. 4, pp. 362–366, 2019, https://doi.org/10.1088/0253-6102/71/4/362.10.1088/0253-6102/71/4/362Search in Google Scholar

[35] D. Lu, M. S. Osman, M. M. A. Khater, R. A. M. Attia, and D. Baleanu, “Analytical and numerical simulations for the kinetics of phase separation in iron (Fe-Cr-X (X=Mo, Cu)) based on ternary alloys,” Physica A, vol. 537, 2020, Art no. 122634, https://doi.org/10.1016/j.physa.2019.122634.10.1016/j.physa.2019.122634Search in Google Scholar

[36] V. Senthil Kumar, H. Rezazadeh, M. Eslami, F. Izadi, and M. S. Osman, “Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity,” Int. J. Algorithm. Comput. Math., vol. 5, no. 5, p. 127, 2019, https://doi.org/10.1007/s40819-019-0710-3.10.1007/s40819-019-0710-3Search in Google Scholar

[37] B. Ghanbari, M. S. Osman, and D. Baleanu, “Generalized exponential rational function method for extended Zakharov Kuzetsov equation with conformable derivative,” Mod. Phys. Lett., vol. 34, no. 20, 2019, Art no. 1950155, https://doi.org/10.1142/s0217732319501554.10.1142/S0217732319501554Search in Google Scholar

[38] L. Wang, V. D. Rǎdulescu, and B. Zhang, “Infinitely many solutions for fractional Kirchhoff–Schrödinger–Poisson systems,” J. Math. Phys., vol. 60, 2019, Art no. 011506, https://doi.org/10.1143/JPSJ.50.3262.10.1143/JPSJ.50.3262Search in Google Scholar

[39] J. Zhang, J. M. do Ó, and M. Squassina, “Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity,” Adv. Nonlinear Stud., vol. 16, pp. 15–30, 2016, https://doi.org/10.1515/ans-2015-5024.10.1515/ans-2015-5024Search in Google Scholar

[40] K. Teng, “Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent,” J. Differ. Equ., vol. 261, pp. 3061–3106, 2016, https://doi.org/10.1016/j.jde.2016.05.022.10.1016/j.jde.2016.05.022Search in Google Scholar

[41] Z. Wei, “Existence of Infinitely Many Solutions for the Fractional Schrödinger-Maxwell equations,” Mathematics, 2015.Search in Google Scholar

[42] G. Che, H. Chen, H. Shi, and Z. Wang, “Existence of nontrivial solutions for fractional Schrödinger–Poisson system with sign-changing potentials,” Math. Methods Appl. Sci., vol. 41, pp. 5050–5064, 2018, https://doi.org/10.1002/mma.4951.10.1002/mma.4951Search in Google Scholar

[43] L. Shen and X. Yao, “Least energy solutions for a class of fractional Schrödinger–Poisson systems,” J. Math. Phys., vol. 59, no. 8, p. 21, 2018, Art no. 081501, https://doi.org/10.1063/1.5047663.10.1063/1.5047663Search in Google Scholar

[44] M. Xiang and F. Wang, “Fractional Schrödinger–Poisson–Kirchhoff type systems involving critical nonlinearities,” Nonlinear Anal., vol. 164, pp. 1–26, 2017, https://doi.org/10.1016/j.na.2017.07.012.10.1016/j.na.2017.07.012Search in Google Scholar

[45] V. Ambrosio, “An existence result for a fractional Kirchhoff–Schrödinger–Poisson system,” Z. Angew. Math. Phys., vol. 69, no. 2, p. 13, 2018, Art no. 30, https://doi.org/10.1007/s00033-018-0921-1.10.1007/978-3-030-60220-8_14Search in Google Scholar

[46] G. Molica Bisci, “Sequences of weak solutions for fractional equations,” Math. Res. Lett., vol. 21, pp. 1–13, 2014, https://doi.org/10.4310/mrl.2014.v21.n2.a3.10.4310/MRL.2014.v21.n2.a3Search in Google Scholar

[47] M. Xiang, B. Zhang, and X. Guo, “Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem,” Nonlinear Anal., vol. 120, pp. 299–313, 2015, https://doi.org/10.1016/j.na.2015.03.015.10.1016/j.na.2015.03.015Search in Google Scholar

[48] Z. Binlin, G. Molica Bisci, and R. Servadei, “Superlinear nonlocal fractional problems with infinitely many solutions,” Nonlinearity, vol. 28, pp. 2247–2264, 2015, https://doi.org/10.1088/0951-7715/28/7/2247.10.1088/0951-7715/28/7/2247Search in Google Scholar

[49] X. Mingqi, G. Molica Bisci, G. Tian, and B. Zhang, “Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p–Laplacian,” Nonlinearity, vol. 29, pp. 357–374, 2016, https://doi.org/10.1088/0951-7715/29/2/357.10.1088/0951-7715/29/2/357Search in Google Scholar

[50] P. Pucci, M. Xiang, and B. Zhang, “Existence and multiplicity of entire solutions for fractional p–Kirchhoff equations,” Adv. Nonlinear Anal., vol. 5, pp. 27–55, 2016, https://doi.org/10.1515/anona-2015-0102.10.1515/anona-2015-0102Search in Google Scholar

[51] A. Fiscella, “Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator,” Differ. Integr. Equ., vol. 29, pp. 513–530, 2016.10.57262/die/1457536889Search in Google Scholar

[52] L. Wang and B. Zhang, “Infinitely many solutions for Schrödinger–Kirchhoff type equations involving the fractional p–Laplacian and critical exponent,” Electron J. Differ. Equ., vol. 2016, pp. 1–18, 2016.10.1186/s13662-016-0828-0Search in Google Scholar

[53] N. Nyamoradi and L. Zaidan, “Existence and multiplicity of solutions for fractional p–Laplacian Schrödinger-Kirchhoff type equations,” Complex Var. Elliptic Equ., vol. 63, pp. 346–359, 2018, https://doi.org/10.1080/17476933.2017.1310851.10.1080/17476933.2017.1310851Search in Google Scholar

[54] M. Xiang, B. Zhang, and V. Rădulescu, “Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p–Laplacian,” Nonlinearity, vol. 29, pp. 3186–3205, 2016, https://doi.org/10.1088/0951-7715/29/10/3186.10.1088/0951-7715/29/10/3186Search in Google Scholar

[55] M. Willem, Minimax Theorems, Berlin, Birkh a user, 1996.10.1007/978-1-4612-4146-1Search in Google Scholar

[56] H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proc. Am. Math. Soc., vol. 88, pp. 486–490, 1983, https://doi.org/10.2307/2044999.10.1007/978-3-642-55925-9_42Search in Google Scholar

[57] V. Moroz and J. Van Schaftingen, “Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics,” J. Funct. Anal., vol. 256, pp. 153–184, 2013, https://doi.org/10.1016/j.jfa.2013.04.007.10.1016/j.jfa.2013.04.007Search in Google Scholar

Received: 2019-08-13

Accepted: 2020-08-08

Published Online: 2020-09-25

Published in Print: 2021-08-26

Ground state solutions for nonlinear fractional Kirchhoff–Schrödinger–Poisson systems

Abstract

Acknowledgments

References

Journal and Issue

Articles in the same Issue