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Existence and Multiplicity of Bound State Solutions to a Kirchhoff Type Equation with a General Nonlinearity

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Abstract

In this paper, we consider the following Kirchhoff type equation

$$\begin{aligned} -\left( a+ b\int _{{\mathbb {R}}^3}|\nabla u|^2\right) \Delta {u} +V(x)u=f(u),\,\,x\in {\mathbb {R}}^3, \end{aligned}$$

where \(a,b>0\) and \(f\in C({\mathbb {R}},{\mathbb {R}})\), and the potential \(V\in C^1({\mathbb {R}}^3,{\mathbb {R}})\) is positive, bounded and satisfies suitable decay assumptions. By using a perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the Ambrosetti-Rabinowitz condition with \(\mu >4\) nor any monotonicity assumption on f is required. Moreover, the potential V may not be radially symmetry or coercive. As a prototype, the nonlinear term involves the power-type nonlinearity \(f(u) =|u|^{p-2}u\) for \(p\in (2, 6)\).

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References

  1. Alves, C., Figueiredo, G.: Nonliear perturbations of peiodic Krichhoff equation in \({\mathbb{R}}^{N}\). Nonlinear Anal. 75, 2750–2759 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, C., Corrêa, F., Figueiredo, G.: On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2, 409–417 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Azzollini, A.: The elliptic Kirchhoff equation in \({\mathbb{R}}^N\) perturbed by a local nonlinearity. Differ. Int. Equ. 25, 543–554 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Cassani, D., Liu, Z., Tarsi, C., Zhang, J.: Multiplicity of sign-changing solutions for Kirchhoff-type equations. Nonlinear Anal. 186, 145–161 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cavalcanti, M., Cavalcanti, V., Soriano, J.: 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 

  9. Cerami, G., Devillanova, G., Solimini, S.: Infinitely many bound states for some nonlinear scalar field equations. Calc. Var. Partial Differ. Equ. 23, 139–168 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, S., Tang, X.: 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. de Figueiredo, D.G., Lions, P., Nussbaum, R.: A priori estimates and existence of positive solutions of semilinear elliptic equations. J. Math. Pures Appl. 9, 41–63 (1982)

    MathSciNet  MATH  Google Scholar 

  13. 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 

  14. Figueiredo, G., Ikoma, N., Junior, J.: 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 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  18. He, X., Zou, W.: Ground state solutions for a class of fractional Kirchhoff equations with critical growth. Sci. China Math. 62, 853–890 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jin, J., Wu, X.: Infinitely many radial solutions for Kirchhoff-type problems in \({\mathbb{R}}^N\). J. Math. Anal. Appl. 369, 564–574 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kato, T.: Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  22. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Liang, Z., Li, F., Shi, J.: Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. Ann. Inst. H. Poincare Anal. Non Linéaire 31, 155–167 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lions, J.: On some questions in boundary value problems of mathematical physics. North-Holland Math. Stud. 30, 284–346 (1978)

    Article  MathSciNet  Google Scholar 

  26. Liu, Z., Guo, S.: Existence of positive ground state solutions for Kirchhoff type problems. Nonlinear Anal. 120, 1–13 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, Z., Siciliano, G.: A perturbation approach for the Schrödinger-Born-Infeld system: solutions in the subcritical and critical case. J. Math. Anal. Appl. 503, 125326 (2021)

    Article  MATH  Google Scholar 

  28. Liu, J., Wang, Z.-Q.: Multiple solutions for quasilinear elliptic equations with a finite potential well. J. Differ. Equ. 257, 2874–2899 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, Z., Wang, Z.-Q., Zhang, J.: Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system. Annali di Matematica 195, 775–794 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, Z., Squassina, M., Zhang, J.: Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension. Nonlinear Differ. Equ. Appl. 24, 50 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, Z., Ouyang, Z., Zhang, J.: Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in \(R_2\). Nonlinearity 32, 3082–3111 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu, Z., Zhang, Z., Huang, S.: Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation. J. Differ. Equ. 266, 5912–5941 (2019)

    Article  MATH  Google Scholar 

  33. Liu, Z., Lou, Y., Zhang, J.: A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity. Annali di Matematica (2021). https://doi.org/10.1007/s10231-021-01155-w

  34. Ma, T., Rivera, J.: Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett. 16, 243–248 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Rabinowitz, P.: Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. Math. 65, 1 (1986)

    Article  MathSciNet  Google Scholar 

  39. 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 

  40. Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sun, J., Wu, T.: Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains. Proc. R. Soc. Edinburgh Sect. A 146, 435–448 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sun, J., Li, L., Cencelj, M., Gabrovšek, B.: Infinitely many sign-changing solutions for Kirchhoff type problems in \({\mathbb{R}}^3\). Nonlinear Anal. 186, 33–54 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tang, X., Chen, S.: Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equ. 56, 1 (2017). https://doi.org/10.1007/s00526-017-1214-9

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. 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 

  46. Willem, M.: Minimax Theorem. Birkhäuser, Boston (1996)

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  48. Xie, W., Chen, H.: On ground state solutions for the nonlinear Kirchhoff type problems with a general critical nonlinearity. Topol. Methods Nonlinear Anal. 53, 518–545 (2019)

    MathSciNet  Google Scholar 

  49. Xie, Q., Ma, S., Zhang, X.: Positive ground state solutions for some non-autonomous Kirchhoff type problems. Rocky Mountain J. Math 47, 329–350 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  50. 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  MathSciNet  MATH  Google Scholar 

  51. Zhang, Y., Tang, X., Qin, D.: Infinitely many solutions for Kirchhoff problems with lack of compactness. Nonlinear Anal. 197, 111856 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Center for Mathematical Sciences, China University of Geosciences, Wuhan, 430074, Hubei, People’s Republic of China

    Zhisu Liu

  2. School of Mathematics, Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Haijun Luo

  3. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, People’s Republic of China

    Jianjun Zhang

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  1. Zhisu Liu
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  2. Haijun Luo
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Correspondence to Jianjun Zhang.

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Zhisu Liu was supported by the NSFC (Grant No. 11701267), the Hunan Natural Science Excellent Youth Fund (Grant No. 2020JJ3029) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan, Grant No. CUG2106211; CUGST2). H. Luo is supported by National Natural Science Foundation of China, (Grant No. 11901182), by Natural Science Foundation of Hunan Province, (Grant No. 2021JJ40033), and by the Fundamental Research Funds for the Central Universities, (Grant No. 531118010205). J. Zhang was supported by NSFC (Grant No. 11871123) and Team Building Project for Graduate Tutors in Chongqing (Grant No. JDDSTD201802)

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Liu, Z., Luo, H. & Zhang, J. Existence and Multiplicity of Bound State Solutions to a Kirchhoff Type Equation with a General Nonlinearity. J Geom Anal 32, 125 (2022). https://doi.org/10.1007/s12220-021-00849-0

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