Advertisement

Standing waves for a class of Kirchhoff type problems in \({\mathbb {R}^3}\) involving critical Sobolev exponents

  • Yi He
  • Gongbao LiEmail author
Article

Abstract

We are concerned with the following Kirchhoff type equation with critical nonlinearity:
$$\begin{aligned} \left\{ \begin{array}{ll} - \Bigl ( {{\varepsilon ^2}a + \varepsilon b\int _{{\mathbb {R}^3}} {{{| {\nabla u} |}^2}} } \Bigr )\Delta u + V(x)u = \lambda {| u |^{p - 2}}u + {| u |^4}u{\text { in }}{\mathbb {R}^3}, \\ u > 0,u \in {H^1}({\mathbb {R}^3}), \\ \end{array} \right. \end{aligned}$$
where \(\varepsilon \) is a small positive parameter, \(a,b>0\), \(\lambda > 0\), \(2 < p \le 4\). Under certain assumptions on the potential V, we construct a family of positive solutions \({u_\varepsilon } \in {H^1}({\mathbb {R}^3})\) which concentrates around a local minimum of V as \(\varepsilon \rightarrow 0\). Although, critical growth Kirchhoff type problem
$$\begin{aligned} \left\{ \begin{array}{ll} - \Bigl ( {{\varepsilon ^2}a + \varepsilon b\int _{{\mathbb {R}^3}} {{{| {\nabla u} |}^2}} } \Bigr )\Delta u + V(x)u = f(u)+{u^5}{\text { in }}{\mathbb {R}^3}, \\ u > 0,u \in {H^1}({\mathbb {R}^3}) \\ \end{array} \right. \end{aligned}$$
has been studied in e.g. He et al. [18], where the assumption for f(u) is that \(f(u) \sim |u{|^{p - 2}}u\) with \(4 < p < 6\) and satisfies the Ambrosetti-Rabinowitz condition which forces the boundedness of any Palais-Smale sequence of the corresponding energy functional of the equation. As \(g(u): = \lambda {| u |^{p - 2}}u + {| u |^4u}\) with \(2<p\le 4\) does not satisfy the Ambrosetti-Rabinowitz condition (\(\exists \mu > 4, 0 < \mu \int _0^u {g(s)ds \le g(u)u}\)), the boundedness of Palais–Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function \(g(s)/{s^3}\) is not increasing for \(s > 0\) prevents us from using the Nehari manifold directly as usual. Our result extends the main result in He et al. [18] concerning the existence and concentration of positive solutions to the case where \(f(u) \sim |u{|^{p - 2}}u\) with \(4 < p < 6\).

Mathematics Subject Classification

Primary 35J20 35J60 35J92 

Notes

Acknowledgments

The authors would like to express their sincere gratitude to the referee for all insightful comments and valuable suggestions, based on which the paper was revised.

References

  1. 1.
    Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical points theory and applications. J. Funct. Anal. 14, 349–381 (1973)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Benci, V., Cerami, G.: Existence of positive solutions of the equation \( - \Delta u + a(x)u = {u^{\frac{{N + 2}}{{N - 2}}}}\) in \({\mathbb{R}^N}\). J. Funct. Anal. 88, 90–117 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Berestycki, H., Lions, P.L.: Nonlinear scalar field equations, I existence of a ground state. Arch. Rational Mech. Anal. 82, 313–345 (1983)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Berestycki, H., Lions, P.L.: Nonlinear scalar field equations, II existence of infinitely many solutions. Arch. Rational Mech. Anal. 82, 347–375 (1983)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Byeon, J., Jeanjean, L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Rational Mech. Anal. 185, 185–200 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Byeon, J., Wang, Z.Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ. 18, 207–219 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)zbMATHCrossRefGoogle Scholar
  10. 10.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Cingolani, S., Lazzo, N.: Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations. Topol. Methods Nonlinear Anal. 10, 1–13 (1997)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Figueiredo, G.M., Ikoma, N., Santos, J.R.: Junior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Rational Mech. Anal. 213, 931–979 (2014)zbMATHCrossRefGoogle Scholar
  13. 13.
    Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Gui, C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Commun. Partial Differ. Equ. 21, 787–820 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn, vol. 224. Grundlehren Math. Wiss., Springer, Berlin (1983)Google Scholar
  16. 16.
    Hirata, J., Ikoma, N., Tanaka, K.: Nonlinear scalar field equations in \({\mathbb{R}^N}\): mountain pass and symmetric mountain pass approaches. Topol. Methods Nonlinear Anal. 35, 253–276 (2010)zbMATHMathSciNetGoogle Scholar
  17. 17.
    He, Y., Li, G.: The existence and concentration of weak solutions to a class of \(p\)-Laplacian type problems in unbounded domains. Sci. China Math. 57, 1927–1952 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    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, 441–468 (2014)MathSciNetGoogle Scholar
  19. 19.
    He, X., Zou, W.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    He, X., Zou, W.: Existence and concentration behavior of positive solutions for a kirchhoff equation in \(\mathbb{R}^3\). J. Differ. Equ. 2, 1813–1834 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jeanjean, L.: On the existence of bounded Palais-Smale sequences and application to a Landsman–Lazer-type problem set on \({\mathbb{R}^N}\). Proc. Roy. Soc. Edingburgh Sect. A Math. 129, 787–809 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)Google Scholar
  23. 23.
    Li, G.: Some properties of weak solutions of nonlinear scalar field equations. Ann. Acad. Sci. Fenn. A I Math. 15, 27–36 (1990)zbMATHGoogle Scholar
  24. 24.
    Lions, P.L.: The concentration–compactness principle in the calculus of variations, the locally compact case, part 2. Ann. Inst. H. Poincaré Anal. Non. Linéaire 2, 223–283 (1984)Google Scholar
  25. 25.
    Lions, P.L.: The concentration–compactness principle in the calculus of variations, the limit case, part 1. Rev. Mat. H. Iberoamericano 1(1), 145–201 (1985)zbMATHCrossRefGoogle Scholar
  26. 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., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, North-Holland Math. Stud. vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346Google Scholar
  27. 27.
    Liu, W., He, X.: Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 39, 473–487 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Li, G., Ye, H.: Existence of positive solutions for nonlinear Kirchhoff type equations in \({\mathbb{R}^3}\) with critical Sobolev exponent. Math. Meth. Appl. Sci. 37, 2570–2584 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Ma, T.F., Munoz, J.E.: Rivera, positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett. 16, 243–248 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Oh, Y.G.: Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class \({( V )_a}\). Commun. Partial Differ. Equ. 13, 1499–1519 (1988)zbMATHCrossRefGoogle Scholar
  32. 32.
    Oh, Y.G.: Corrections to existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class \({( V )_a}\). Commun. Partial Differ. Equ. 14, 833–834 (1989)zbMATHGoogle Scholar
  33. 33.
    Oh, Y.G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131, 223–253 (1990)zbMATHCrossRefGoogle Scholar
  34. 34.
    del Pino, M., Felmer, P.L.: Local mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4, 121–137 (1996)zbMATHCrossRefGoogle Scholar
  35. 35.
    Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35, 681–703 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Perera, K., Zhang, Z.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Rabinowitz, P.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Ramos, M., Wang, Z.Q., Willem, M.: Positive Solutions for Elliptic Equations with Critical Growth in Unbounded Domains, Calculus of Variations and Differential Equations. Chapman & Hall/CRC Press, Boca Raton (2000)Google Scholar
  39. 39.
    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153, 229–244 (1993)zbMATHCrossRefGoogle Scholar
  41. 41.
    Willem, M.: Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston Inc, Boston (1996)Google Scholar
  42. 42.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Zhang, J., Chen, Z., Zou, W.: Standing waves for nonlinear Schrödinger equations involving critical growth, arXiv:1209.3074v1 (2012)
  44. 44.
    Zhu, X., Yang, J.: Regularity for quasilinear elliptic equations in involving critical Sobolev exponent. System Sci. Math. 9, 47–52 (1989)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and StatisticsCentral China Normal UniversityWuhanPeople’s Republic of China

Personalised recommendations