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Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in \({\mathbb{R}^{3}}\)

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Abstract

In this paper, we deal with the Kirchhoff-type equation

$$\left\{\begin{array}{l@{\quad}l} \displaystyle-\left[1+\int\limits_{{\mathbb{R}}^3} \left(\left|\nabla u\right|^2+V(x)u^2\right){\rm d}x \right]\left[\Delta u+V(x)u\right] = \lambda Q(x) f(u), \quad x\in \ {\mathbb{R}}^3, \quad \quad \quad {(\rm P)_\lambda}\\ u(x)\rightarrow 0, \quad {\rm as} \ |x|\rightarrow \infty , \end{array}\right.$$

where \({\lambda > 0}\), V and Q are radial functions, which can be vanishing or coercive at infinity. With assumptions on f just in a neighborhood of the origin, existence and multiplicity of nontrivial radial solutions are obtained via variational methods. In particular, if f is sublinear and odd near the origin, we obtain infinitely many solutions of \({(\rm P)_\lambda}\) for any \({\lambda > 0}\).

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References

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

  2. Chen J.: Multiple positive solutions to a class of Kirchhoff equation on \({{\mathbb{R}}^3}\) with indefinite nonlinearity. Nonlinear Anal. TMA 96, 134–145 (2014)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Park S.: General decay of a transmission problem for Kirchhoff type wave equations with boundary memory condition. Acta Math. Sci. 34B(5), 1395–1403 (2014)

    Article  Google Scholar 

  5. Ruggieri, M.: Kink Solutions for a Class of Generalized Dissipative Equations. Abstr. Appl. Anal. (2012). doi:10.1155/2012/237135

  6. Ruggieri, M., Valenti, A.: Exact solutions for a nonlinear model of dissipative media. J. Math. Phys. 52, Article ID: 043520 (2011)

  7. Ruggieri, M., Valenti, A.: Symmetries and reduction techniques for dissipative models. J. Math. Phys. 50, Article ID: 063506 (2009)

  8. Ruggieri, M., Valenti, A.: Approximate symmetries in nonlinear viscoelastic media. Bound. Value Problem (2013). doi:10.1186/1687-2770-2013-143

  9. Lions, J.: On some equations in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposioum, Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1997, North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, pp. 284–346 (1978)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Alves C., Corrêa F., Ma T.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Mao A., Zhang Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. TMA 70, 1275–1287 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zhang Z., Perera K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456–463 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sun J., Liu S.: Nontrivial solutions of Kirchhoff type problems. Appl. Math. Lett. 25, 500–504 (2012)

    MATH  MathSciNet  Google Scholar 

  18. Corrêa F., Figueiredo G.: On a elliptic equation of p-Kirchhoff type via variational methods. Bull. Aust. Math. Soc. 74, 263–277 (2006)

    Article  MATH  Google Scholar 

  19. Dreher M.: The Kirchhoff equation for the p-Laplacian. Rend. Semin. Mat. Univ. Politec. Torino 64, 217–238 (2006)

    MATH  MathSciNet  Google Scholar 

  20. Dreher M.: The ware equation for the p-Laplacian. Hokkaido Math. J. 36, 21–52 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Liu D., Zhao P.: Multiple nontrivial solutions to a p-Kirchhoff equation. Nonlinear Anal. TMA 75, 5032–5038 (2012)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  23. Wu X.: Existence of nontrivial solutions and high energy solutions for Schröinger–Kirchhoff-type equations in \({{\mathbb{R}}^N}\). Nonlinear Anal. RWA 12, 1278–287 (2011)

  24. Nie J., Wu X.: Existence and multiplicity of non-trivial solutions for Schröinger–Kirchhoff-type equations with radial potential. Nonlinear Anal. TMA 75, 3470–3479 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Su J., Wang Z.-Q., Willem M.: Weighted Sobolev embedding with unbounded and decaying radial potentials. Commun. Contemp. Math. 9, 571–583 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Su J., Tian R.: Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on \({{\mathbb{R}}^N}\). Proc. Am. Math. Soc. 140, 891–903 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wang Z.-Q.: Nonlinear boundary value problems with concave nonlinearities near the origin. Nonlinear Differ. Equ. Appl. 8, 15–33 (2001)

    Article  MATH  Google Scholar 

  28. Costa D., Wang Z.-Q.: Multiplicity results for a class of superlinear elliptic problems. Proc. Am. Math. Soc. 133, 787–794 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Adams R.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  30. Chen S., Li S.: On a nonlinear elliptic eigenvalue problem. J. Math. Anal. Appl. 307, 691–698 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Liu Z., Wang Z.-Q.: Schrödinger equations with concave and convex nonlinearites. Z. Angew. Math. Phys. 56, 609–629 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Anran Li

  2. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Jiabao Su

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  1. Anran Li
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  2. Jiabao Su
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Correspondence to Anran Li.

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Supported by NSFC11271264, NSFC11171204 and KZ201510028032.

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Li, A., Su, J. Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in \({\mathbb{R}^{3}}\) . Z. Angew. Math. Phys. 66, 3147–3158 (2015). https://doi.org/10.1007/s00033-015-0551-9

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  • DOI: https://doi.org/10.1007/s00033-015-0551-9

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