Skip to main content
SpringerLink
Log in

Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

This paper is dedicated to studying the following Kirchhoff-type problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\left( a+b\int _{\mathbb {R}^3}|\nabla u|^2\mathrm {d}x\right) \triangle u+V(x)u=f(u), &{} x\in \mathbb {R}^3; \\ u\in H^1(\mathbb {R}^3), \end{array} \right. \end{aligned}$$
(0.1)

where \(a>0,\,b\ge 0\) are two constants, V(x) is differentiable and \(f\in \mathcal {C}(\mathbb {R}, \mathbb {R})\). By introducing some new tricks, we prove that the above problem admits a ground state solution of Nehari–Pohozaev type and a least energy solution under some mild assumptions on V and f. Our results generalize and improve the ones in Guo (J Differ Equ 259:2884–2902, 2015) and Li and Ye (J Differ Equ 257:566–600, 2014) and some other related literature.

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

References

  1. Alves, C.O., Corréa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    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. Bernstein, S.: Sur une class d’źquations fonctionnelles aux dźrivźes partielles. Bull. Acad. Sci. URSS. Sźr. Math. [Izv. Akad. Nauk SSSR] 4, 17–26 (1940)

    Google Scholar 

  5. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation. Adv. Differ. Equ. 6, 701–730 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Chen, J.S., Li, L.: Multiple solutions for the nonhomogeneous Kirchhoff equations on \(\mathbb{R}^N\). Nonlinear Anal. Real World Appl. 14, 1477–1486 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, S.T., Tang, X.H.: Infinitely many solutions for super-quadratic Kirchhoff-type equations with sign-changing potential. Appl. Math. Lett. 67, 40–45 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, C.Y., Kuo, Y.C., Wu, T.F.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876–1908 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng, B.T., Tang, X.H.: Ground state sign-changing solutions for asymptotically 3-linear Kirchhoff-type problems. Complex Var. Elliptic 62, 1093–1116 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheng, B.T., Wu, X.: Existence results of positive solutions of Kirchhoff type problems. Nonlinear Anal. 71, 4883–4892 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chipot, M., Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30(7), 4619–4627 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Deng, Y.B., Peng, S.J., 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 

  14. Ding, Y.H.: Variational Methods for Strongly Indefinite Problems. World Scientific, Singapore (2007)

    Book  MATH  Google Scholar 

  15. Figueiredo, G.M.: Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J. Math. Anal. Appl. 401, 706–713 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. He, X.M., Zou, W.M.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, X.M., Zou, W.M.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^3\). J. Differ. Equ. 2, 1813–1834 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. He, Y., Li, G.B., Peng, S.J.: Concentrating bound states for Kirchhoff type problems in \(\mathbb{R}^3\) involving critical Sobolev exponents. Adv. Nonlinear Stud. 14, 483–510 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. Lei, C.-Y., Liao, J.-F., Tang, C.-L.: Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents. J. Math. Anal. Appl. 421, 521–538 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, G.B., 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  MathSciNet  MATH  Google Scholar 

  24. Li, Y., et al.: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253, 2285–2294 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, Y.H., Li, F.Y., Shi, J.P.: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253, 2285–2294 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposium Inst. Mat., Universidade Federal do Rio de Janeiro, 1997, in: North-Holland Mathematical Studies, vol. 30, pp. 284–346. North-Holland, Amsterdam (1978)

  27. Naimen, D.: The critical problem of Kirchhoff type elliptic equations in dimension four. J. Differ. Equ. 257, 1168–1193 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Naimen, D.: Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent. NoDEA Nonlinear Differ. Equ. Appl. 21, 885–914 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Perera, K., Zhang, Z.T.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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 

  31. Shuai, W.: Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J. Differ. Equ. 259, 1256–1274 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sun, J.J., Tang, C.L.: Existence and multiplicityof solutions for Kirchhoff type equations. Nonlinear Anal. 74, 1212–1222 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tang, X.H., Chen, S.T.: Ground state solutions of Nehari-Pohozaev type for Schrödinger–Poisson problems with general potentials. Disc. Contin. Dyn. Syst.-Ser. A 37(9), 4973–5002 (2017)

    Article  MATH  Google Scholar 

  34. 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 

  35. Tolksdorf, P.: Regularity for some general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)

    Article  MATH  Google Scholar 

  36. Wang, J., Tian, L.X., Xu, J.X., Zhang, F.B.: 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 

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

    Book  MATH  Google Scholar 

  38. Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \(\mathbb{R}^3\). Nonlinear Anal. Real World Appl. 12, 1278–1287 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, W., Tang, X.H., Cheng, B.T.: Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Commun. Pur. Appl. Anal. 15, 2161–2177 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    X. H. Tang & Sitong Chen

Authors
  1. X. H. Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sitong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. H. Tang.

Additional information

Communicated by P. Rabinowitz.

This work is partially supported by the NNFC (No. 11571370) of China.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, X.H., Chen, S. Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials. Calc. Var. 56, 110 (2017). https://doi.org/10.1007/s00526-017-1214-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-017-1214-9

Mathematics Subject Classification

Search

Navigation