Home Ground state solutions of Schrödinger system with fractional p-Laplacian
Article
Licensed
Unlicensed Requires Authentication

Ground state solutions of Schrödinger system with fractional p-Laplacian

  • Yan Qiao , Fangqi Chen EMAIL logo and Yukun An
Published/Copyright: October 6, 2022

Abstract

This article deals with a class of nonlinear fractional p-Laplacian Schr o ̈ dinger coupled system with critical and subcritical nonlinear terms. Firstly, the existence of a nonnegative ground state solution of the system is proved by the Nehari manifold method and the Ekeland’s variational principle. In addition, through the Ljusternik–Schnirelmann theory, we link the number of solutions to the topology of the set in which the potentials in the system reach their minimum values.

PACS (2010): 35J10; 35J50; 35J60

Corresponding author: Fangqi Chen, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China, E-mail:

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12172166

Award Identifier / Grant number: 11872201

Funding source: Research Project of the Natural Science of the Jiangsu Higher Education Institutions, China

Award Identifier / Grant number: 22KJB110013

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work is supported by the National Natural Science Foundation of China (Grant NO.: 12172166 and 11872201) and the Research Project of the Natural Science of the Jiangsu Higher Education Institutions (Grant NO.: 22KJB110013).

  3. Conflict of interest statement: This work does not have any conflicts of interest.

References

[1] K. Adriouch and A. Hamidi, “The Nehari manifold for systems of nonlinear elliptic equations,” Nonlinear Anal., vol. 64, pp. 2149–2167, 2006. https://doi.org/10.1016/j.na.2005.06.003.Search in Google Scholar

[2] C. Alves, D. Filho, and M. Souto, “On systems of elliptic equations involving subcritical or critical Sobolev exponents,” Nonlinear Anal., vol. 42, pp. 771–787, 2000. https://doi.org/10.1016/s0362-546x(99)00121-2.Search in Google Scholar

[3] B. Barrios, E. Colorado, A. Pablo, and U. Sánchez, “On some critical problems for the fractional Laplacian operator,” J. Differ. Equ., vol. 252, pp. 6133–6162, 2012.10.1016/j.jde.2012.02.023Search in Google Scholar

[4] Y. Hua and X. Yu, “On the ground state solution for a critical fractional Laplacian equation,” Nonlinear Anal., vol. 87, pp. 116–125, 2013. https://doi.org/10.1016/j.na.2013.04.005.Search in Google Scholar

[5] C. Chen, J. Bao, and H. Song, “Multiple solutions for a class of fractional (p, q)-Laplacian system in RN${\mathbb{R}}^{N}$,” J. Math. Phys., vol. 59, p. 031505, 2018. https://doi.org/10.1063/1.5027564.Search in Google Scholar

[6] L. Zhang, B. Ahmad, G. Wang, and X. Ren, “Radial symmetry of solutions for fractional p-Laplacian system,” Nonlinear Anal., vol. 196, p. 111801, 2020. https://doi.org/10.1016/j.na.2020.111801.Search in Google Scholar

[7] X. Chang and Y. Sato, “Multiplicity of localized solutions of nonlinear Schrö$\ddot{o}$dinger systems for infinite attractive case,” J. Math. Anal. Appl., vol. 491, pp. 1–17, 2020. https://doi.org/10.1016/j.jmaa.2020.124358.Search in Google Scholar

[8] J. Kang and C. Tang, “Ground state radial sign-changing solutions for a gauged nonlinear Schrö$\ddot{o}$dinger equation involving critical growth,” Commun. Pure Appl. Anal., vol. 19, no. 11, pp. 5239–5252, 2020. https://doi.org/10.3934/cpaa.2020235.Search in Google Scholar

[9] L. Lu, “L2 normalized solutions for nonlinear Schrö$\ddot{o}$dinger systems in R3${\mathbb{R}}^{3}$,” Nonlinear Anal., vol. 191, pp. 1–19, 2020. https://doi.org/10.1016/j.na.2019.111621.Search in Google Scholar

[10] X. Shang and J. Zhang, “Ground states for fractional Schrö$\ddot{o}$dinger equations with critical growth,” Nonlinearity, vol. 27, pp. 187–207, 2014. https://doi.org/10.1088/0951-7715/27/2/187.Search in Google Scholar

[11] S. Li, Y. Ding, and Y. Chen, “Concentrating standing waves for the fractional Schrö$\ddot{o}$dinger equation with critical nonlinearities,” Bound. Value Probl., vol. 2015, no. 1, pp. 1–26, 2015.10.1186/s13661-015-0507-1Search in Google Scholar

[12] S. Chen, J. Liu, and Z. Wang, “Localized nodal solutions for a critical nonlinear Schrö$\ddot{o}$dinger equation,” J. Funct. Anal., vol. 277, pp. 594–640, 2019, https://doi.org/10.1016/j.jfa.2018.10.027.Search in Google Scholar

[13] Y. Ao and W. Zou, “Ground states for a class of quasilinear elliptic systems with critical exponent,” Nonlinear Anal., vol. 181, pp. 222–248, 2019. https://doi.org/10.1016/j.na.2018.11.015.Search in Google Scholar

[14] H. Qiu and M. Xiang, “Existence of solutions for fractional p-Laplacian problems via Leray–Schauder’s nonlinear alternative,” Bound. Value Probl., vol. 2016, no. 1, p. 83, 2016. https://doi.org/10.1186/s13661-016-0593-8.Search in Google Scholar

[15] J. Zhao, X. Liu, and J. Liu, “Infintely many sign-changing solutions for system of p-Laplace equations in RN${\mathbb{R}}^{N}$,” Nonlinear Anal., vol. 182, pp. 113–142, 2019. https://doi.org/10.1016/j.na.2018.12.005.Search in Google Scholar

[16] V. Ambrosio, “Nontrivial solutions for a fractional p-Laplacian problem via Rabier Theorem,” Complex Var. Elliptic Equ., vol. 62, no. 6, pp. 838–847, 2017. https://doi.org/10.1080/17476933.2016.1245725.Search in Google Scholar

[17] P. Garain and T. Mukherjee, “Quasilinear nonlocal elliptic problems with variable singular exponent,” Commun. Pure Appl. Anal., vol. 19, no. 11, pp. 5059–5075, 2020.10.3934/cpaa.2020226Search in Google Scholar

[18] T. Mukherjee and K. Sreenadh, “On Dirichlet problem for fractional p-Laplacian with singular non-linearity,” Adv. Nonlinear Anal., vol. 8, pp. 52–72, 2019. https://doi.org/10.1515/anona-2016-0100.Search in Google Scholar

[19] M. Zhen and B. Zhang, “A different approach to ground state solutions for p-Laplacian system with critical exponent,” Appl. Math. Lett., vol. 111, pp. 1–8, 2021. https://doi.org/10.1016/j.aml.2020.106593.Search in Google Scholar

[20] M. Zhen and B. Zhang, “The Nehari manifold for fractional p-Laplacian system involving concave-convex nonlinearities and sign-changing weight functions,” Complex Var. Elliptic Equ., vol. 66, no. 10, pp. 1731–1754, 2020. https://doi.org/10.1080/17476933.2020.1779237.Search in Google Scholar

[21] E. Di, 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.Search in Google Scholar

[22] P. Felmer, A. Quaas, and J. Tan, “Positive solutions of the nonlinear Schrö$\ddot{o}$dinger equation with the fractional Laplacian,” Proc. R. Soc. Edinb. A: Math., vol. 76, pp. 1–12. 2006.Search in Google Scholar

[23] G. Palatucci and A. Pisante, “Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces,” Calc. Var., vol. 50, pp. 799–829, 2014. https://doi.org/10.1007/s00526-013-0656-y.Search in Google Scholar

[24] R. Servadei and E. Valdinoci, “The Brezis–Nirenberg result for the fractional Laplacian,” Trans. Am. Math. Soc., vol. 367, no. 1, pp. 67–102, 2015. https://doi.org/10.1090/s0002-9947-2014-05884-4.Search in Google Scholar

[25] V. Ambrosio, “Multiplicity of solutions for fractional Schrö$\ddot{o}$dinger systems in RN${\mathbb{R}}^{N}$,” Complex Var. Elliptic Equ., vol. 65, no. 5, pp. 856–885, 2020. https://doi.org/10.1080/17476933.2019.1631290.Search in Google Scholar

[26] I. Ekeland, “On the variational principle,” J. Math. Anal. Appl., vol. 47, pp. 324–353, 1974. https://doi.org/10.1016/0022-247x(74)90025-0.Search in Google Scholar

[27] H. Brezis 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/10.1090/s0002-9939-1983-0699419-3.Search in Google Scholar

[28] X. He and W. Zou, “Existence and concentration behavior of positive solutions for a Kirch–Hoff equation in R3${\mathbb{R}}^{3}$,” J. Differ. Eqs., vol. 252, pp. 1813–1834, 2012, https://doi.org/10.1016/j.jde.2011.08.035.Search in Google Scholar

[29] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge, Cambridge University Press, 1993.10.1017/CBO9780511551703Search in Google Scholar

Received: 2022-03-18
Accepted: 2022-09-18
Published Online: 2022-10-06

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Article
  3. Hybrid solitary wave solutions of the Camassa–Holm equation
  4. Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions
  5. A Chebyshev collocation method for solving the non-linear variable-order fractional Bagley–Torvik differential equation
  6. Higher order codimension bifurcations in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect
  7. Numerical modeling of the dam-break flood over natural rivers on movable beds
  8. A class of piecewise fractional functional differential equations with impulsive
  9. Lie symmetry analysis for two-phase flow with mass transfer
  10. Asymptotic behavior for stochastic plate equations with memory in unbounded domains
  11. Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay
  12. A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity
  13. Ground state solutions of Schrödinger system with fractional p-Laplacian
  14. Bifurcation analysis of a new stochastic traffic flow model
  15. An uncertainty measure based on Pearson correlation as well as a multiscale generalized Shannon-based entropy with financial market applications
  16. Hilfer fractional stochastic evolution equations on infinite interval
  17. Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths
  18. Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations
  19. Theoretical assessment of the impact of awareness programs on cholera transmission dynamic
  20. Theoretical and numerical analysis of a prey–predator model (3-species) in the frame of generalized Mittag-Leffler law
  21. Controllability discussion for fractional stochastic Volterra–Fredholm integro-differential systems of order 1 < r < 2
  22. Shehu transform on time-fractional Schrödinger equations – an analytical approach
  23. A (2 + 1)-dimensional variable-coefficients extension of the Date–Jimbo–Kashiwara–Miwa equation: Lie symmetry analysis, optimal system and exact solutions
  24. Mathematical model of fluid flow in a double constricted tapered tube with permeable boundary
Downloaded on 11.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2022-0112/html
Scroll to top button