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Ground state solution for a class of Schrödinger–Poisson-type systems with partial potential

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Abstract

In this paper, we consider the existence of positive ground state solution for a class of Schrödinger–Poisson-type systems with both nonlinear critical growth and nonlocal critical growth when the coefficient of the potential function is neither coercive nor periodic and asymptotic periodic. The usual concentration–compactness method is invalid; hence, we employ the modified concentration–compactness principle and mountain pass theorem to deal with the system.

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References

  1. Alves, C., Souto, M., Soares, S.: Schrödinger–Poisson equations without Ambrosetti–Rabinowitz condition. J. Math. Anal. Appl. 377, 584–592 (2011)

    Article  MathSciNet  Google Scholar 

  2. Azzollini, A., Pomponio, A.: Ground state solutions for the nonlinear Schrödinger–Maxwell equations. J. Math. Anal. Appl. 345, 90–108 (2008)

    Article  MathSciNet  Google Scholar 

  3. Bartsch, T., Wang, Z.Q.: Existence and multiplicity results for some superlinear elliptic problems on \({\mathbb{R}}^3\). Commun. Partial Differ. Equ. 20(9–10), 1725–1741 (1995)

    Article  Google Scholar 

  4. Bartsch, T., Wang, Z.Q., Willem, M.: The Dirichlet problem for superlinear elliptic equations. In: Stationary Partial Differential Equations. Vol. II, In: Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, pp. 1–55 (2005)

    Google Scholar 

  5. Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283–293 (1998)

    Article  MathSciNet  Google Scholar 

  6. Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)

    Article  MathSciNet  Google Scholar 

  7. Costa, D.G.: On a class of elliptic systems in \({\mathbb{R}}^N\). Electron. J. Differ. Equ. 7, 1–14 (1994)

    Google Scholar 

  8. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed. In: Classics in Mathematics. Springer-Verlag, Berlin (2001)

    Chapter  Google Scholar 

  9. Guo, Z.: Ground states for Kirchhoff equations without compact condition. J. Differ. Equ. 259, 2884–2902 (2015)

    Article  MathSciNet  Google Scholar 

  10. He, X., Zou, W.: Existence and concentration of ground states for Schrödinger–Poisson equations with critical growth. J. Math. Phys. 53, 023702 (2012)

    Article  MathSciNet  Google Scholar 

  11. Li, G., Tang, C.: Existence of a ground state solution for Choquard equation with the upper critical exponent. Comput. Math. Appl. 76, 2635–2647 (2018)

    Article  MathSciNet  Google Scholar 

  12. Liu, H.: Positive solutions of an asymptotically periodic Schrödinger–Poisson system with critical exponent. Nonlinear Anal. RWA 32, 198–212 (2016)

    Article  Google Scholar 

  13. Li, F., Li, Y., Shi, J.: Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent. Commun. Contemp. Math. 16, 1450036 (2014)

    Article  MathSciNet  Google Scholar 

  14. Li, Q., Su, H., Wei, Z.: Existence of infinitely many large solutions for the nonlinear Schrödinger–Maxwell equations. Nonlinear Anal. 72, 4264–4270 (2010)

    Article  MathSciNet  Google Scholar 

  15. Liu, H., Chen, H., Yang, X.: Multiple solutions for superlinear Schrödinger–Poisson system with sign-changing potential and nonlinearity. Comput. Math. Appl. 68, 1982–1990 (2014)

    Article  MathSciNet  Google Scholar 

  16. Liu, Z., Guo, S.: On ground state solutions for the Schrödinger–Poisson equations with critical growth. J. Math. Anal. Appl. 412, 435–448 (2014)

    Article  MathSciNet  Google Scholar 

  17. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1977)

    Article  MathSciNet  Google Scholar 

  18. Omana, W., Willem, M.: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ. 5, 1115–1120 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Pekar, S.: Untersuchungen über Die Elektronentheorie Der Kristalle. Akademie Verlag, Berlin (1954)

    MATH  Google Scholar 

  20. Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 2nd edn. Springer, Berlin (1996)

    MATH  Google Scholar 

  21. Sun, J.: Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations. J. Math. Anal. Appl. 390, 514–522 (2012)

    Article  MathSciNet  Google Scholar 

  22. Sun, J., Ma, S.: Ground state solutions for some Schrödinger–Poisson systems with periodic potentials. J. Differ. Equ. 260, 2119–2149 (2016)

    Article  Google Scholar 

  23. Trudinger, N.S.: On Harnack type inequality and their applications to quasilinear elliptic equations. Commun. Pure Appl. Math. 20, 721–747 (1967)

    Article  Google Scholar 

  24. Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)

    Book  Google Scholar 

  25. Zhang, J.: On the Schrödinger–Poisson equations with a general nonlinearity in the critical growth. Nonlinear Anal. 75, 6391–6401 (2012)

    Article  MathSciNet  Google Scholar 

  26. Zhang, J.: Ground state and multiple solutions for Schrödinger–Poisson equations with critical nonlinearity. J. Math. Anal. Appl. 440, 466–482 (2016)

    Article  MathSciNet  Google Scholar 

  27. Zhang, J., do \(\acute{O}\), J. M., Squassina, M.: Schrödinger–Poisson systems with a general critical nonlinearity. Commun. Contemp. Math. 1650028 (2016)

  28. Zhang, X., Xia, J.: Semi-classical solutions for Schrödinger–Poisson equations with a critical frequency. J. Differ. Equ. 265, 2121–2170 (2018)

    Article  Google Scholar 

  29. Zhang, X., Ma, S., Xie, Q.: Bound state solutions of Schrödinger–Poisson system with critical exponent. Discrete Contin. Dyn. Syst. 37, 605–625 (2017)

    Article  MathSciNet  Google Scholar 

  30. Zhao, L.G., Zhao, F.K.: On the existence of solutions for the Schrödinger–Poisson equations. J. Math. Anal. Appl. 346, 155–169 (2008)

    Article  MathSciNet  Google Scholar 

  31. Zhong, X.J., Tang, C.L.: Ground state sign-changing solutions for a Schrödinger–Poisson system with a critical nonlinearity in \({\mathbb{R}}^3\). Nonlinear Anal. RWA 39, 166–184 (2018)

    Article  Google Scholar 

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Acknowledgements

The author would like to express their sincere gratitude to anonymous referees for his/her constructive comments for improving the quality of this paper.

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  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Xiaojing Feng

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  1. Xiaojing Feng
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Correspondence to Xiaojing Feng.

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Projects supported by National Natural Science Foundation of China (Grant Nos. 11571209, 11671239) and Natural Science Foundation of Shanxi Province (Grant Nos. 201801D211001, 201801D121002, 201801D221012).

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Feng, X. Ground state solution for a class of Schrödinger–Poisson-type systems with partial potential. Z. Angew. Math. Phys. 71, 37 (2020). https://doi.org/10.1007/s00033-020-1254-4

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  • DOI: https://doi.org/10.1007/s00033-020-1254-4

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