SpringerLink
Log in

Cylindrical Solutions and Ground State Solutions to Weighted Kirchhoff Equations

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

Abstract

We analyze the existence and non-existence of cylindrical solutions as well as ground state solutions to a class of Kirchhoff-type problems with critical Sobolev exponent, subcritical Sobolev exponent, and critical Sobolev–Hardy exponent. To the best of our knowledge, this is the first time to study the existence of cylindrically symmetric solutions to the Kirchhoff equations in the literature.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348(1), 305–330 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Badiale, M., Tarantello, G.: A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163(4), 259–293 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertin, G.: Dynamics of Galaxies. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  4. D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108(2), 247–262 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deng, Y., Peng, S., Shuai, W.: Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in \({\mathbb{R} }^3\). J. Funct. Anal. 269, 3500–3527 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Erbe, L., Jia, B., Zhang, Q.: Homoclinic solutions of discrete nonlinear systems via variational method. J. Appl. Anal. Comput. 9, 271–294 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Fang, X.D.: Bound state solutions for some non-autonomous asymptotically cubic Schrödinger–Poisson systems. Z. Angew. Math. Phys. 70, 50 (2019)

    Article  MATH  Google Scholar 

  8. Fang, X.D., Szulkin, A.: Multiple solutions for a quasilinear Schrödinger equation. J. Differ. Equ. 254, 2015–2032 (2013)

    Article  MATH  Google Scholar 

  9. Figueiredo, G.M., Ikoma, N., Santos Jünior, J.R.: Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Ration. Mech. Anal. 213, 931–979 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Figueiredo, G.M., Nascimento, R.G.: Existence of a nodal solution with minimal energy for a Kirchhoff equation. Math. Nachr. 288, 48–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. He, X.M., Zou, W.M.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({R^3}\). J. Differ. Equ. 252, 1813–1834 (2012)

  12. Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

    MATH  Google Scholar 

  13. Li, G., Ye, H.: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb{R} }^3\). J. Differ. Equ. 257, 566–600 (2014)

    Article  MATH  Google Scholar 

  14. Lin, G., Yu, J.: Homoclinic solutions of periodic discrete Schrödinger equations with local superquadratic conditions. SIAM J. Math. Anal. 54, 1966–2005 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin, G., Yu, J.: Existence of a ground-state and infinitely many homoclinic solutions for a periodic discrete system with sign-changing mixed nonlinearities. J. Geom. Anal. 32, 127 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, G., Yu, J., Zhou, Z.: Homoclinic solutions of discrete nonlinear Schrödinger equations with partially sublinear nonlinearities. Electron. J. Differ. Equ. 2019(96), 1–14 (2019)

    MATH  Google Scholar 

  17. Lin, G., Zhou, Z., Yu, J.: Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials. J. Dyn. Differ. Equ. 32, 527–555 (2020)

    Article  MATH  Google Scholar 

  18. Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Development in Continuum Mechanics and Partial Differential Equations. North-Holland Mathematics Studies, vol. 30, pp. 284–346. North-Holland, Amsterdam (1978)

  19. Lions, J.L.: The concentration–compactness principle in the Calculus of Variations. The locally compact case, I. Ann. Inst. H. Poincare Anal. NonLinaire 1(2), 109–145 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mukherjee, T., Pucci, P., **ang, M.: Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems. Discrete Contin. Dyn. Syst. 42, 163–187 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ni, W.M.: On the elliptic equation \(\triangle u +K(|x|)u^{\frac{n+2}{n-2}}=0\), its generalizations and applications in geometry. Indiana Univ. Math. J. 31, 439–529 (1982)

    Google Scholar 

  22. Pucci, P., Temperini, L.: Existence for fractional \((p, q)\) systems with critical and Hardy terms in \({\mathbb{R}}^3\). Nonlinear Anal. 211, 112477 (2021)

    Article  MATH  Google Scholar 

  23. Pucci, P., Temperini, L.: On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces. Math. Eng. 5(1), 1–21 (2023)

    Article  MathSciNet  Google Scholar 

  24. Shen, Z., Han, Z.: Existence of nontrivial solutions for a class of Kirchhoff equation with indefinite and 3-linear nonlinearity. Appl. Math. Lett. 103, 7p (2020)

    Article  MathSciNet  Google Scholar 

  25. Shen, Z., Yu, J.: On positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb{R} }^3\). Appl. Anal. 99, 2137–2149 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shen, Z., Yu, J.: Multiple solutions for weighted Kirchhoff equations involving critical Hardy–Sobolev exponent. Adv. Nonlinear Anal. 10, 673–683 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tang, X.H., Chen, S.T.: Ground state solutions of Nehari–Poho\(\check{z}\)aev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equ. 56, 110–134 (2017)

    Article  Google Scholar 

  28. Tang, X.H., Cheng, B.T.: Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J. Differ. Equ. 261, 2384–2402 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, J., Tian, L., Xu, J., Zhang, F.: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253, 2314–2351 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. **e, Q., Ma, S., Zhang, X.: Bound state solutions of Kirchhoff type problems with critical exponent. J. Differ. Equ. 261, 890–924 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. **e, Q., Yu, J.: Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension. Commun. Pure Appl. Anal. 18, 129–158 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yuan, Z., Yu, J.: Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Commun. Pure Appl. Anal. 19, 391–405 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, Q.: Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Commun. Pure Appl. Anal. 18, 425–434 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11701114).

Author information

Authors and Affiliations

  1. School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou, China

    Zupei Shen

  2. Center for Applied Mathematics, Guangzhou University, Guangzhou, China

    Jianshe Yu

Authors
  1. Zupei Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianshe Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianshe Yu.

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 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

Shen, Z., Yu, J. Cylindrical Solutions and Ground State Solutions to Weighted Kirchhoff Equations. J Geom Anal 32, 260 (2022). https://doi.org/10.1007/s12220-022-00995-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-00995-z

Keywords

Mathematics Subject Classification

Search

Navigation