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Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

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Abstract

This paper is concerned with the following Kirchhoff-type equations:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$

where \(a>0,~b,~\mu \ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2,3+2\alpha )\), the potential V(x) may be unbounded from below and \(\phi |u|^{p-2}u\) is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system.

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References

  1. Liu, S., Guo, S., Zhang, Z.: Existence of ground state solutions for the Schrödinger–Poisson systems. Appl. Math. Comput. 244, 312–323 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Sun, J., Wu, T.: On the nonlinear Schrödinger–Poisson systems with sign-changing potential. Z. Angew. Math. Phys. 66(4), 1649–1669 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Jeanjean, L., Tanaka, K.: A positive solution for a nonlinear Schrödinger–Poisson system on \(\mathbb{R}^{N}\). Indiana Univ. Math. J. 54, 443–464 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jiang, Y., Zhou, H.: Schrödinger–Poisson system with steep potential well. J. Differ. Equ. 251, 582–608 (2011)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Tang, X., Chen, S.: Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equat. 54, 110 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. **e, W., Chen, H., Shi, H.: Ground state solutions for the nonlinear Schrödinger–Poisson systems with sum of periodic and vanishing potentials. Math. Meth. Appl. Sci. 41(1), 144–158 (2018)

    Article  MATH  Google Scholar 

  9. Xu, L., Chen, H.: Multipicity of small negative-energy solutions for a class of nonlinear Schrödinger–Poisson systems. Appl. Math. Comput. 243, 817–824 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Che, G., Chen, H.: Multiple solutions for the Schröinger equations with sign-changing potential and Hartree nonlinearity. Appl. Math. Lett. 81, 21–26 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sun, J., Chen, H., Juan, J.: On ground state solutions for some non-autonomous Schrodinger–Poisson systems. J. Differ. Equ. 252, 3365–3380 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zou, W.: Variant fountain theorem and their applications. Manuscripta Math. 104, 343–358 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Kirchhoff, G.: Mechanik. Teubner (1883)

  15. Sun, J., Wu, T.: Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J. Differ. Equ. 256, 1771–1792 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lü, D.: A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal. 99, 35–48 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Xu, L., Chen, H.: Nontrivial solutions for Kirchhoff-type problems with a parameter. J. Math. Anal. App. 433, 455–472 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, J., Tang, X., Zhang, W.: Existence of multiple solutions of Kirchhoff type equation with sign-changing potential. Appl. Math. Comput. 242, 491–499 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \(\mathbb{R}^{N}\). Nonlinear Anal. RWA. 12, 1278–1287 (2011)

    Article  MATH  Google Scholar 

  20. Liu, Z., Guo, S.: Existence of positive ground state solutions for Kirchhoff type problems. Nonlinear Anal. 120, 1–13 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Che, G., Chen, H.: Existence and multiplicity of systems of Kirchhoff-type equations with general potentials. Math. Meth. Appl. Sci. 40, 775–785 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, S., Tang, X.: Ground state sign-changing solutions for a class of Schrödinger–Poisson type problems in \(\mathbb{R}^{3}\). Z. Angew. Math. Phys. 67(102), 1–18 (2016)

    Google Scholar 

  23. Zhao, G., Zhu, X., Li, Y.: Existence of infinitely many solutions to a class of Kirchhoff–Schrodingr–Poissson system. Appl. Math. Comput. 256, 572–581 (2015)

    MathSciNet  Google Scholar 

  24. Kajikiya, R.: A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J. Funct. Anal. 225, 352–370 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, F., Cao, J., Zhu, X.: Existence and concentration of sign-changing solutions to Kirchhoff-type system with Hartree-type nonlinearity. J. Math. Anal. Appl. 418, 60–80 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Reed, M., Simon, B.: Methods of modern mathematical physics II: fourier analysis, self-adjointness. Academic Press, New York, London (1975)

    MATH  Google Scholar 

  27. Reed, M., Simon, B.: Methods of modern mathematical physics IV: analysis of operators. Academic Press, New York, London (1978)

    MATH  Google Scholar 

  28. Zhang, Q., Xu, B.: Multiple solutions for Schrödinger–Poisson systems with indefinite potential and combined nonlinearity. J. Math. Anal. Appl. 455, 1668–1687 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, Z., Su, J., Weth, T.: Compactness results for Schrödinger equations with asymptotically linear terms. J. Differ. Equ. 231, 501–512 (2006)

    Article  MATH  Google Scholar 

  31. Yang, M., Wei, Y.: Existence and multiplicity of solutions for nonlinear Schrödinger equation with magnetic field and Hartree-type nonlinearities. J. Math. Anal. Appl. 403, 680–694 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. (N.S.) 7, 447–526 (1982)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Guofeng Che & Haibo Chen

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  1. Guofeng Che
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  2. Haibo Chen
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Correspondence to Haibo Chen.

Additional information

This work is partially supported by National Natural Science Foundation of China 11671403, by the Fundamental Research Funds for the Central Universities of Central South University 2017zzts058 and by the Mathematics and Interdisciplinary Sciences Project of CSU.

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Che, G., Chen, H. Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity. Mediterr. J. Math. 15, 131 (2018). https://doi.org/10.1007/s00009-018-1170-4

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  • DOI: https://doi.org/10.1007/s00009-018-1170-4

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