Skip to main content
SpringerLink
Log in

Ground State Solutions of Nehari-Pohozaev Type for Schrödinger–Poisson–Slater Equation with Zero Mass and Critical Growth

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this article, we study the Schrödinger–Poisson–Slater type equation with the critical growth and zero mass:

$$\begin{aligned} {\left\{ \begin{array}{ll} {-\Delta } u+\phi u=\mu |u|^{p-2}u+u^5, \ \ \ \ x\in {\mathbb {R}}^{3},\\ {-\Delta } \phi =u^2, \ \ \ \ x\in {\mathbb {R}}^{3}, \end{array}\right. } \end{aligned}$$

where \(3<p<6\) and \(\mu >0\). By combining a new perturbation method and the mountain pass theorem, Liu et al. [J. Diff. Eq., 266 (2019), 5912–5941] prove that the above equation has at least one positive ground state solution for \(p \in (4, 6)\) and \(\mu >0\) or \(p \in (3, 4]\) if \(\mu \) is sufficiently large. By using a much simpler method than the ones used in the above mentioned paper, together with subtle estimates and analyses, we obtain better results on the existence for a ground state solution of Nehari-Pohozaev type.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

There are no relevant data in our paper.

References

  1. Agueh, M.: Sharp Gagliardo-Nirenberg inequalities and mass transport theory. J. Dyn. Differ. Equ. 18, 1069–1093 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosetti, A., Ruiz, D.: Multiple bound states for the Schrödinger–Poisson problem. Commun. Contemp. Math. 10, 391–404 (2008)

    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. Cerami, G., Vaira, G.: Positive solutions for some non-autonomous Schrödinger–Poisson systems. J. Differ. Equ. 248, 521–543 (2010)

    Article  MATH  Google Scholar 

  5. Chen, S.T., Fiscella, A., Pucci, P., Tang, X.H.: Semiclassical ground state solutions for critical Schrödinger–Poisson systems with lower perturbations. J. Differ. Equ. 268, 2672–2716 (2020)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Chen, S.T., Tang, X.H.: Axially symmetric solutions for the planar Schrödinger–Poisson system with critical exponential growth. J. Differ. Equ. 269, 9144–9174 (2020)

    Article  MATH  Google Scholar 

  8. Chen, S.T., Tang, X.H.: On the planar Schrödinger–Poisson system with the axially symmetric potentials. J. Differ. Equ. 268, 945–976 (2020)

    Article  MATH  Google Scholar 

  9. Chen, S.T., Tang, X.H.: Berestycki-Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials. Adv. Nonlinear Anal. 9, 496–515 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, S.T., Tang, X.H.: Another look at Schrödinger equations with prescribed mass. J. Differ. Equ. 386, 435–479 (2024)

    Article  MATH  Google Scholar 

  11. Coclite, G.M.: A multiplicity result for the nonlinear Schrödinger–Maxwell equations. Commun. Appl. Anal. 7, 417–423 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Huang, L.R., Rocha, E.M., Chen, J.Q.: Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity. J. Differ. Equ. 255, 2463–2483 (2013)

    Article  MATH  Google Scholar 

  13. Huang, W.N., Tang, X.H.: Semiclassical solutions for the nonlinear Schrödinger–Maxwell equations. J. Math. Anal. Appl. 415, 791–802 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ianni, I., Ruiz, D.: Ground and bound states for a static Schrödinger–Poisson–Slater problem. Commun. Contemp. Math. 14, 1250003 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jeanjean, L.: On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on \(\mathbb{R} ^N\). Proc. Roy. Soc. Edinburgh Sect. A. 129, 787–809 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jeanjean, L., Toland, J.: Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. I Math. 327 23–28 (1998)

  17. Lei, C.Y., Radulescu, V.D., Zhang, B.L.: Ground states of the Schrödinger–Poisson–Slater equation with critical growth. Racsam. Rev. R. Acad. A. 117(3), 128 (2023)

    MATH  Google Scholar 

  18. Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev inequality and related inequalities. Ann. Math. 118, 349–374 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lieb, E.H., Loss, M.: Analysis, graduate studies in mathematics, vol. 14. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  20. Lei, C.Y., Lei, Y.T.: On the existence of ground states of an equation of Schrödinger–Poisson–Slater type. Comptes Rendus Mathématique 359, 219–227 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lei, C.Y., Lei, J., Suo, H.M.: Ground state for the Schrödinger–Poisson–Slater equation involving the Coulomb-Sobolev critical exponent. Adv. Nonlinear Anal. 12, 1–17 (2023)

    MathSciNet  MATH  Google Scholar 

  22. Lions, P.L.: Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109, 33–97 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, Z., Radulescu, V.D., Tang, C., Zhang, J.: Another look at planar Schrödinger–Newton systems. J. Differ. Equ. 328, 65–104 (2022)

    Article  MATH  Google Scholar 

  24. Liu, Z.S., Zhang, Z.T., Huang, S.B.: Existence and nonexistence of positive solutions for a static Schrödinger–Poisson–Slater equation. J. Differ. Equ. 266, 5912–5941 (2019)

    Article  MATH  Google Scholar 

  25. Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Nonlinear analysis-theory and methods. Springer, Switzerland (2019)

    Book  MATH  Google Scholar 

  26. Ruiz, D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ruiz, D.: On the Schrödinger–Poisson–Slater system: behavior of minimizers, radial and nonradial cases. Arch. Ration. Mech. Anal. 198, 349–368 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Seok, J.: On nonlinear Schrödinger–Poisson equations with general potentials. J. Math. Anal. Appl. 401, 672–681 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Tang, X.H., Chen, S.T.: Ground state solutions of Nehari-Pohozaev type for Schrödinger–Poisson problems with general potentials. Disc. Contin. Dyn. Syst. 37, 4973–5002 (2017)

    Article  MATH  Google Scholar 

  31. Wang, X.P., Liao, F.F.: Existence and nonexistence of solutions for Schrödinger–Poisson problems. J. Geom. Anal. 33, 56 (2023)

    Article  MATH  Google Scholar 

  32. Wen, L., Chen, S., Radulescu, V.D.: Axially symmetric solutions of the Schrödinger–Poisson system with zero mass potential in \({\mathbb{R} }^{N}\). Appl. Math. Lett. 104, 106244 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Willem, M.: Minimax theorems. Birkhäuser, Boston (1996)

    Book  MATH  Google Scholar 

  34. Yang, L., Liu, Z.S.: Infinitely many solutions for a zero mass Schrödinger–Poisson–Slater problem with critical growthe. J. Appl. Anal. Comput. 5, 1706–1718 (2019)

    MATH  Google Scholar 

  35. Zheng, T.T., Lei, C.Y., Liao, J.F.: Multiple positive solutions for a Schrödinger–Poisson–Slater equation with critical growth. J. Math. Anal. Appl. 525, 127206 (2023)

    Article  MATH  Google Scholar 

  36. 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  MATH  Google Scholar 

  37. Zhao, L.G., Zhao, F.K.: Positive solutions for Schrödinger–Poisson equations with a critical exponent. Nonlinear Anal. 70, 2150–2164 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Information, Xiangnan University, Chenzhou, 423000, Hunan, People’s Republic of China

    Yu Gu & Fangfang Liao

Authors
  1. Yu Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fangfang Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fangfang Liao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the NNSF (12071395), Scientific Research Fund of Hunan Provincial Education Department(22A0588), and the Natural Science Foundation of Hunan Province(2022JJ30550), Aid Program for Science and Technology Inbovative Research Team in Higher Educational Institutions of Hunan Province(2023), and Chenzhou Applied Mathematics Achievement Transformation Technology Research and Development Center(2022).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gu, Y., Liao, F. Ground State Solutions of Nehari-Pohozaev Type for Schrödinger–Poisson–Slater Equation with Zero Mass and Critical Growth. J Geom Anal 34, 221 (2024). https://doi.org/10.1007/s12220-024-01656-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-024-01656-z

Keywords

Mathematics Subject Classification

Search

Navigation