Skip to main content
SpringerLink
Log in

Existence and Multiplicity of Solutions for a Class of Kirchhoff–Boussinesq-Type Problems with Logarithmic Growth

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, two problems related to the following class of elliptic Kirchhoff–Boussinesq-type models are analyzed in the subcritical (\(\beta =0\)) and critical (\(\beta =1\)) cases:

$$\begin{aligned} \Delta ^{2} u \!- \!\Delta _p u \!=\! \tau |u|^{q-2} u{\ln |u|}\!+\!\beta |u|^{2_{**}-2}u\ \text{ in } \ \Omega \ \ \text{ and } \ {\Delta u=u=0} \ \text{ on } \ \ \partial \Omega , \end{aligned}$$

where \(\tau >0\), \(2< p< 2^{*}= \frac{2N}{N-2}\) for \( N\ge 3\) and \(2_{**}= \infty \) for \(N=3\), \(N=4\), \(2_{**}= \frac{2N}{N-4}\) for \(N\ge 5\). The first one is concerned with the existence of a nontrivial ground-state solution via variational methods. As for the second problem, we prove the multiplicity of such a solution using the Mountain Pass Theorem.

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

All data that support the findings of this study are included within the article (and any supplementary files).

References

  1. Andrade, D., Jorge Silva, M.A., Ma, T.F.: Exponential stability for a plate equation with \(p\)-Laplacian and memory terms. Math. Methods Appl. Sci. 35, 417–426 (2012)

    Article  MathSciNet  Google Scholar 

  2. Bartsch, T., Pankov, A., Wang, Z.Q.: Nonlinear Schrödiger equations with steep potential well. Commun. Contemp. Math. 3, 549–569 (2001)

    Article  MathSciNet  Google Scholar 

  3. Bartsch, T., Tang, Z.W.: Multibump solutions of nonlinear Schröndiger equations with steep potential well and indefinite potential. Discrete Cont. Dyn. Syst. 33, 7–26 (2013)

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Bernis, F., Garcia Azorero, J., Peral, I.: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differ. Equ. 2, 219–240 (1996)

    MathSciNet  Google Scholar 

  6. Boudjeriou, T.: Global existence and blow-up for the fractional \(p\)-Laplacian with logarithmic nonlinearity. Mediter. J. Math. 17, 162 (2020)

    Article  MathSciNet  Google Scholar 

  7. 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 

  8. Carlos, R.D., Figueiredo, G.M.: On an elliptic Kirchhoff–Boussinesq type problems with exponential growth. Math. Methods Appl. Sci. 47, 397–408 (2024)

    Article  MathSciNet  Google Scholar 

  9. Chueshov, I., Lasiecka, I.: Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff–Boussinesq models. Discrete Contin. Dyn. Syst. 15, 777–809 (2006)

    Article  MathSciNet  Google Scholar 

  10. Chueshov, I., Lasiecka, I.: On global attractor for 2D Kirchhoff–Boussinesq model with supercritical nonlinearity. Commun. Partial Differ. Equ. 36, 67–99 (2011)

    Article  MathSciNet  Google Scholar 

  11. Fiorenza, A., Formica, M.R., Roskovec, T.G., Soudsky, F.: The Detailed proof of classical Gagliardo–Nirenberg interpolation inequality with historical remarks. Z. Anal. Anwend. 2, 217–236 (2021)

    Article  MathSciNet  Google Scholar 

  12. He, Q., Lv, J., Lv, Z., Wu, T.: Existence of nontrivial solutions for critical biharmonic equations with logarithmic term. arXiv preprint arXiv: 2303.07659 (2023)

  13. He, Q., Lv, Z.: Existence and nonexistence of nontrivial solutions for critical biharmonic equations. J. Math. Anal. Appl. 495, 1–29 (2021)

    Article  MathSciNet  Google Scholar 

  14. Kavian, O.: Introduction à la Théorie des Points Critiques. Springer, Berline (1991)

    Google Scholar 

  15. Lagnese J., Lions J. L., Modelling analysis and control of thin plates. Rech. Math. Appl. 6, Masson, Paris, 1988, vi\(+\)177 pp

  16. Lagnese, J.: Boundary stabilization of thin plates. SIAM Stud. Appl. Math. 10, Philadelphia, PA, 1989, viii\(+\)176 pp

  17. Li, Q., Han, Y., Wang, T.: Existence and nonexistence of solutions to a critical biharmonic equation with logarithmic perturbation. J. Differ. Equ. 365, 1–37 (2023)

    Article  MathSciNet  Google Scholar 

  18. Li, Y., Li, G., Tang, C.: Existence and concentration of ground state solutions for Choquard equations involving critical growth and steep potential well. Nonlinear Anal. 200, 111997 (2020)

    Article  MathSciNet  Google Scholar 

  19. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 223–283 (1984)

    Article  MathSciNet  Google Scholar 

  20. Liu, H., Liu, Z., Xiao, Q.: Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity. Appl. Math. Lett. 79, 176–181 (2018)

    Article  MathSciNet  Google Scholar 

  21. Nhan, L.C., Truong, L.X.: Global solution and blow-up for a class of pseudo \(p\)-Laplacian evolution equations with logarithmic nonlinearity. Comput. Math. Appl. 73, 2076–2091 (2017)

    Article  MathSciNet  Google Scholar 

  22. Sun, F., Liu, L., Wu, Y.: Infinitely many sign-changing solutions for a class of biharmonic equation with \(p\)-Laplacian and Neumann boundary condition. Appl. Math. Lett. 73, 128–135 (2017)

    Article  MathSciNet  Google Scholar 

  23. Sun, J., Chu, J., Wu, T.: Existence and multiplicity of nontrivial solutions for some biharmonic equations with \(p\)-Laplacian. J. Differ. Equ. 262, 945–977 (2017)

    Article  MathSciNet  Google Scholar 

  24. Sun, J., Wu, T.: Existence of nontrivial solutions for a biharmonic equation with \(p\)-Laplacian and singular sign-changing potential. Appl. Math. Lett. 66, 61–67 (2017)

    Article  MathSciNet  Google Scholar 

  25. Gao D. Y., Motreanu D.: Handbook of Nonconvex Analysis and Applications, International Press, Somerville, MA, 2010, viii\(+\)680 pp

  26. Yang, T.: On a critical biharmonic system involving p-Laplacian and Hardy potential. Appl. Math. Lett. 121, 107433 (2021)

    Article  MathSciNet  Google Scholar 

  27. Yang, Z.: Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term. J. Differ. Equ. 187, 520–540 (2003)

    Article  MathSciNet  Google Scholar 

  28. Yang, Z.: Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow. Math. Methods Appl. Sci. 32, 1082–1104 (2009)

    Article  MathSciNet  Google Scholar 

  29. Yang, Z.: Global attractors and their Hausdorff dimensions for a class of Kirchhoff models. J. Math. Phys. 51, 032701 (2010)

    Article  MathSciNet  Google Scholar 

  30. Yu, Y., Zhao, Y., Luo, C.: Ground state solution of critical \(p\)-biharmonic equation involving Hardy potential. Bull. Malays. Math. Sci. Soc. 45, 501–512 (2022)

    Article  MathSciNet  Google Scholar 

  31. Zhang, H., Zhou, J.: Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity. Commun. Pure Appl. Anal. 20, 1601–1631 (2021)

    Article  MathSciNet  Google Scholar 

  32. Zhang, J., Li, S.: Multiple nontrivial solutions for some fourth-order semilinear elliptic problems. Nonlinear Anal. 60, 221–230 (2005)

    Article  MathSciNet  Google Scholar 

  33. Zhang, J., Wei, Z.: Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems. Nonlinear Anal. 74, 7474–7485 (2011)

    Article  MathSciNet  Google Scholar 

  34. Zhou, J.: Ground state solution for a fourth-order elliptic equation with logarithmic nonlinearity modeling epitaxial growth. Comput. Math. Appl. 78, 1878–1886 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Funding

The authors would like to express their sincere thanks to the referee for valuable comments. This work was partially supported by CNPq, Capes and FapDF-Brazil, and the China Postdoctoral Science Foundation (Grant No. 2023M741266).

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília-UnB, Brasília, DF, 70910-900, Brazil

    Romulo D. Carlos

  2. Mathematics department, faculty of sciences of Tunis, University of Tunis El Manar, 2092, Tunis, Tunisia

    Lamine Mbarki

  3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Shuang Yang

Authors
  1. Romulo D. Carlos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lamine Mbarki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shuang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

RDC made substantial contributions to the conception or design of the work. RDC, LM and SY initiated the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Shuang Yang.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

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

Carlos, R.D., Mbarki, L. & Yang, S. Existence and Multiplicity of Solutions for a Class of Kirchhoff–Boussinesq-Type Problems with Logarithmic Growth. Mediterr. J. Math. 21, 108 (2024). https://doi.org/10.1007/s00009-024-02649-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-024-02649-6

Keywords

Mathematics Subject Classification

Search

Navigation