Skip to main content
SpringerLink
Log in

Existence and Concentration Results for the General Kirchhoff-Type Equations

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

Abstract

We consider the following singularly perturbed Kirchhoff-type equations

$$\begin{aligned} -\varepsilon ^2 M\left( \varepsilon ^{2-N}\int _{{\mathbb {R}}^N}|\nabla u|^2 \textrm{d}x\right) \Delta u +V(x)u=|u|^{p-2}u~\hbox {in}~{\mathbb {R}}^N, u\in H^1({\mathbb {R}}^N),N\ge 1, \end{aligned}$$

where \(M\in C([0,\infty ))\) and \(V\in C({\mathbb {R}}^N)\) are given functions. Under very mild assumptions on M, we prove the existence of single-peak or multi-peak solution \(u_\varepsilon \) for above problem, concentrating around topologically stable critical points of V, by a direct corresponding argument. This gives an affirmative answer to an open problem raised by Figueiredo et al. (Arch Ration Mech Anal 213(3):931–979, 2014)

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. Ambrosetti, A., Malchiodi, A.: Concentration phenomena for nonlinear Schrödinger equations: recent results and new perspectives. In: Perspectives in nonlinear partial differential equations. Contemp. Math., vol. 446, pp. 19–30. American Mathematical Society, Providence (2007)

  2. Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140(3), 285–300 (1997)

    Article  MATH  Google Scholar 

  3. Ambrosetti, A., Malchiodi, A., Secchi, S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159(3), 253–271 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambrosetti, A., Malchiodi, A., Ni, W.-M.: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I. Commun. Math. Phys. 235(3), 427–466 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bahri, A., Li, Y., Rey, O.: On a variational problem with lack of compactness: the topological effect of the critical points at infinity. Calc. Var. Partial Differ. Eqn. 3(1), 67–93 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4):313–345 (1983)

  7. Byeon, J.: Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity. Trans. Am. Math. Soc. 362(4), 1981–2001 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Byeon, J., Jeanjean, L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185(2), 185–200 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Byeon, J., Jeanjean, L.: Erratum: standing waves for nonlinear Schrödinger equations with a general nonlinearity [Arch. Ration. Mech. Anal. 185(2), 185–200 (2007); mr2317788]. Arch. Ration. Mech. Anal. 190(3), 549–551 (2008)

  10. Byeon, J., Tanaka, K.: Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. J. Eur. Math. Soc. (JEMS) 15(5), 1859–1899 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. II. Calc. Var. Partial Differ. Equ. 18(2), 207–219 (2003)

    Article  MATH  Google Scholar 

  12. Byeon, J., Jeanjean, L., Tanaka, K.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases. Commun. Partial Differ. Equ. 33(4–6), 1113–1136 (2008)

    Article  MATH  Google Scholar 

  13. Byeon, J., Zhang, J., Zou, W.: Singularly perturbed nonlinear Dirichlet problems involving critical growth. Calc. Var. Partial Differ. Equ. 47(1–2), 65–85 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cao, D., Heinz, H.-P.: Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations. Math. Z. 243(3), 599–642 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cao, D., Peng, S.: Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity. Commun. Partial Differ. Equ. 34(10–12), 1566–1591 (2009)

    Article  MATH  Google Scholar 

  16. Cao, D., Dancer, N.E., Noussair, E.S., Yan, S.: On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discret. Contin. Dyn. Syst. 2(2), 221–236 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cao, D., Peng, S., Yan, S.: Singularly Perturbed Methods for Nonlinear Elliptic Problems. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  18. Cao, D., Li, S., Luo, P.: Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 54(4), 4037–4063 (2015)

    Article  MATH  Google Scholar 

  19. Chen, Yu., Ding, Y.: Multiplicity and concentration for Kirchhoff type equations around topologically critical points in potential. Topol. Methods Nonlinear Anal. 53(1), 183–223 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Cingolani, S., Lazzo, M.: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 160(1), 118–138 (2000)

    Article  MATH  Google Scholar 

  21. Cortázar, C., Elgueta, M., Felmer, P.: Uniqueness of positive solutions of \(\Delta u+f(u)=0\) in \({\textbf{R} }^N, N \ge 3\). Arch. Ration. Mech. Anal. 142(2), 127–141 (1998)

    MATH  Google Scholar 

  22. Dancer, E.N.: Peak solutions without non-degeneracy conditions. J. Differ. Equ. 246(8), 3077–3088 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. d’Avenia, P., Pomponio, A., Ruiz, D.: Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods. J. Funct. Anal. 262(10), 4600–4633 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. del Pino, M., Felmer, P.L.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4(2), 121–137 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. del Pino, M., Felmer, P.L.: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149(1), 245–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Del Pino, M., Felmer, P.L.: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(2):127–149 (1998)

  27. del Pino, M., Felmer, P.: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 324(1), 1–32 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. del Pino, M., Kowalczyk, M., Wei, J.-C.: Concentration on curves for nonlinear Schrödinger equations. Commun. Pure Appl. Math. 60(1), 113–146 (2007)

    Article  MATH  Google Scholar 

  29. Deng, Y., Peng, S., Pi, H.: Bound states with clustered peaks for nonlinear Schrödinger equations with compactly supported potentials. Adv. Nonlinear Stud. 14(2), 463–481 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Figueiredo, G.M.: Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J. Math. Anal. Appl. 401(2), 706–713 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Figueiredo, G.M., Santos Junior, J.R.: Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differ. Integral Equ. 25(9–10), 853–868 (2012)

  32. Figueiredo, G.M., Ikoma, N., Santos Júnior, J.R.: Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Ration. Mech. Anal. 213(3):931–979 (2014)

  33. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. Grossi, M.: On the number of single-peak solutions of the nonlinear Schrödinger equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(3), 261–280 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gui, C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Commun. Partial Differ. Equ. 21(5–6), 787–820 (1996)

    Article  MATH  Google Scholar 

  36. He, Y.: Concentrating bounded states for a class of singularly perturbed Kirchhoff type equations with a general nonlinearity. J. Differ. Equ. 261(11), 6178–6220 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. He, X., Zou, W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({\mathbb{R} }^3\). J. Differ. Equ. 252(2), 1813–1834 (2012)

    Article  MATH  Google Scholar 

  38. He, Y., Li, G., Peng, S.: Concentrating bound states for Kirchhoff type problems in \({\mathbb{R} }^3\) involving critical Sobolev exponents. Adv. Nonlinear Stud. 14(2), 483–510 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Jeanjean, L., Tanaka, K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 21(3), 287–318 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Jeanjean, L., Zhang, J., Zhong, X.: A global branch approach to normalized solutions for the Schrödinger equation. (2021)

  41. Kang, X., Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Equ. 5(7–9), 899–928 (2000)

    MATH  Google Scholar 

  42. Li, Y.Y.: On a singularly perturbed elliptic equation. Adv. Differ. Equ. 2(6), 955–980 (1997)

    MathSciNet  MATH  Google Scholar 

  43. Li, Y., Nirenberg, L.: The Dirichlet problem for singularly perturbed elliptic equations. Commun. Pure Appl. Math. 51(11–12), 1445–1490 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  44. Li, G., Ye, H.: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb{R} }^3\). J. Differ. Equ. 257(2), 566–600 (2014)

    Article  MATH  Google Scholar 

  45. Li, G., Ye, H.: Existence of positive solutions for nonlinear Kirchhoff type problems in \({\mathbb{R} }^3\) with critical Sobolev exponent. Math. Methods Appl. Sci. 37(16), 2570–2584 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Li, G., Luo, P., Peng, S., Wang, C., Xiang, C.-L.: A singularly perturbed Kirchhoff problem revisited. J. Differ. Equ. 268(2), 541–589 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  47. Luo, P., Peng, S., Wang, C., Xiang, C.-L.: Multi-peak positive solutions to a class of Kirchhoff equations. Proc. R. Soc. Edinb. Sect. A 149(4), 1097–1122 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  48. Noussair, E.S., Yan, S.: On positive multipeak solutions of a nonlinear elliptic problem. J. Lond. Math. Soc. (2) 62(1), 213–227 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. Rabinowitz, P.H.: On a class of nonlinear schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  50. Rey, O.: The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89(1), 1–52 (1990)

  51. Tingxi, H., Shuai, W.: Multi-peak solutions to Kirchhoff equations in \({\mathbb{R} }^3\) with general nonlinearity. J. Differ. Equ. 265(8), 3587–3617 (2018)

    Article  MATH  Google Scholar 

  52. Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153(2), 229–244 (1993)

    Article  MATH  Google Scholar 

  53. Wang, J., Tian, L., Junxiang, X., Zhang, F.: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253(7), 2314–2351 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wang, Z., Zeng, X., Zhang, Y.: Multi-peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents. Math. Methods Appl. Sci. 43(8), 5151–5161 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  55. Yong-Geun, O.: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class \((V)_a\). Commun. Partial Differ. Equ. 13(12), 1499–1519 (1988)

    Article  Google Scholar 

  56. Yong-Geun, O.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131(2), 223–253 (1990)

    Article  MATH  Google Scholar 

  57. Zhang, J.: Standing waves with a critical frequency for nonlinear Schrödinger equations involving critical growth. Appl. Math. Lett. 63, 53–58 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhang, J., Zou, W.: Solutions concentrating around the saddle points of the potential for critical Schrödinger equations. Calc. Var. Partial Differ. Equ. 54(4), 4119–4142 (2015)

    Article  MATH  Google Scholar 

  59. Zhang, J., Chen, Z., Zou, W.: Standing waves for nonlinear Schrödinger equations involving critical growth. J. Lond. Math. Soc.(2) 90(3), 827–844 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhang, J., David G, C., do Ó João, M.: Existence and concentration of positive solutions for nonlinear Kirchhoff-type problems with a general critical nonlinearity. Proc. Edinb. Math. Soc. 61(4), 1023–1040 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank Prof. Peng Luo for the valuable discussions when preparing the paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Yinbin Deng & Wei Shuai

  2. South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Xuexiu Zhong

Authors
  1. Yinbin Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Shuai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuexiu Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuexiu Zhong.

Additional information

Publisher's Note

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

The research is partially supported by the NSFC (Nos. 11801581, 12271184, 12071170, 12271196 and 11931012), Guangdong Basic and Applied Basic Research Foundation (2021A1515010034), Guangzhou Basic and Applied Basic Research Foundation (202102020225), and Province Natural Science Fund of Guangdong (2018A030310082).

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

Deng, Y., Shuai, W. & Zhong, X. Existence and Concentration Results for the General Kirchhoff-Type Equations. J Geom Anal 33, 88 (2023). https://doi.org/10.1007/s12220-022-01145-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-01145-1

Keywords

Mathematics Subject Classification

Search

Navigation