Skip to main content
SpringerLink
Log in

Homoclinic Solutions for p(t)-Laplacian–Hamiltonian Systems Without Coercive Conditions

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

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. Admas, R.A.: Sobolev Spaces. Academic Press, New York (1975)

  2. Alves C.O., Carrião P.C., Miyagaki O.H.: Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. Appl. Math. Lett 16(5), 639–642 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Antontsev S.N., Shmarev S.I.: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. 60, 515–545 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Antontsev S.N., Rodrigues J.F.: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 52, 19–36 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caldiroli P., Montecchiari P.: Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Commun. Appl. Nonlinear Anal. 1(2), 97–129 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Chen, P., Tang, X.H., Agarwal, R.P.: Infinitely many homoclinic solutions for nonautonomous p(t)-Laplacian–Hamiltonian systems. Comput. Math. Appl. 62, 131–141 (2012)

  8. Chen Y., Levine S., Rao M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Coti Zelati V., Rabinowitz P.H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4(4), 693–727 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Daouas A.: Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption. Nonlinear Anal. 74(11), 3407–3418 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Diening, L., Harjulehto, L., Hästö, P., Rŭžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)

  12. Ding Y.H.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25(11), 1095–1113 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fan X.L., Zhao D.: On the space \({L^{p(x)}(\Omega)}\) and \({W^{m,p(x)}(\Omega)}\). J. Math. Anal. Appl 263, 424–446 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan X.L., Zhao Y.Z., Zhao D.: Compact embeddings theorems with symmetry of Strauss–Lions type for the space \({W^{m,p(x)}(\Omega)}\). J. Math. Anal. Appl. 255, 333–348 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fan X.L., Zhang Q.H.: Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fan X.L., Han X.Y.: Existence and multiplicity of solutions for p(x)-Laplacian equations in \({\mathbb{R}^N}\). Nonlinear Anal. 59, 173–188 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Fan X.L., Deng S.G.: Remarks on Ricceris variational principle and applications to the p(x)-Laplacian equations. Nonlinear Anal. 67, 3064–3075 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Halsey T.C.: Electrorheological fluids. Science 258, 761–766 (1992)

    Article  Google Scholar 

  19. Izydorek M., Janczewska J.: Homoclinic solutions for a class of the second order Hamiltonian systems. J. Differ. Equ. 219(2), 375–389 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)

  21. Kefi K.: p(x)-Laplacian with indefinite weight. Proc. Am. Math. Soc. 139, 4351–4360 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Korman P., Lazer A.C.: Homoclinic orbits for a class of symmetric Hamiltonian systems. Electron. J. Differ. Equ. 01, 1–10 (1994)

    MathSciNet  MATH  Google Scholar 

  23. Lv Y., Tang C.L.: Existence of even homoclinic orbits for a class of Hamiltonian systems. Nonlinear Anal. 67(7), 2189–2198 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lv X., Lu S.P., Yan P.: Existence of homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal. 72(1), 390–398 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lv X., Jiang J.F.: Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials. Nonlinear Anal. Real World Appl. 13(3), 1152–1158 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ma, Y.H.: Homoclinic orbits for second-order p(t) Hamiltonian system. Ph.D. thesis, Lanzhou University (2005)

  27. Manásevich R., Mawhin J.: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differ. Equ. 145(2), 367–393 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Marcellini P.: Regularity and existence of solutions of elliptic equations with (p,q)-growth conditions. J. Differ. Equ. 90, 1–30 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mihălescu M., Rădulesu V.: Multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. R. Soc. A 462, 2625–2641 (2006)

    Article  MathSciNet  Google Scholar 

  30. Musielak, J.: Orlicz spaces and modular spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)

  31. Omana W., Willem M.: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ. 5(5), 1115–1120 (1992)

    MathSciNet  MATH  Google Scholar 

  32. Qin B., Chen P.: Existence and multiplicity of homoclinic solutions for p(t)-Laplacian systems with subquadratic potentials. Electron. J. Differ. Equ. 2014(111), 1–10 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence (1986)

  34. Rabinowitz P.H.: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb. Sect. A. 114(1–2), 33–38 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rabinowitz P.H., Tanaka K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206(3), 473–499 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ružička, M.: Electrorheological Fluids: Modelling and Mathematical Theory. Springer, Berlin (2000)

  37. Salvatore A.: Homoclinic orbits for a special class of nonautonomous Hamiltonian systems. Nonlinear Anal. 30(8), 4849–4857 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Salvatore A.: On the existence of homoclinic orbits for a second-order Hamiltonian system. Differ. Integral Equ. 10, 381–392 (1997)

    MathSciNet  MATH  Google Scholar 

  39. Shi, X.B., Zhang, Q.F., Zhang, Q.M.: Existence of homoclinic orbits for a class of p-Laplacian systems in a weighted Sobolev space. Bound. Value Prob. 2013, 137 (2013)

  40. Sun J.T., Chen H.B., Nieto J.J.: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 373(1), 20–29 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tang X.H., Lin X.Y.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Proc. R. Soc. Edinb. Sect. A. 141(5), 1103–1119 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wan L.L., Tang C.L.: Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete Cont. Dyn. Syst. Ser. B 15(1), 255–271 (2011)

    MathSciNet  MATH  Google Scholar 

  43. Wang J., Xu J.X., Zhang F.B.: Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Commun. Pure. Appl. Anal. 10(1), 269–286 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Yang M.H., Han Z.Q.: The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials. Nonlinear Anal. Real World Appl. 12(5), 2742–2751 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang Q.Y., Liu C.G.: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal. 72(2), 894–903 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang Z.H., Yuan R.: Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems. Nonlinear Anal. 71(9), 4125–4130 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhikov V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR. Izv. 29, 33–36 (1987)

    Article  MATH  Google Scholar 

  48. Zou W.M., Li S.J.: Infinitely many homoclinic orbits for the second-order Hamiltonian systems. Appl. Math. Lett. 16(8), 1283–1287 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China

    Ziheng Zhang

  2. Institute for mathematical sciences, Renmin university of China, Beijing, 100872, China

    Tian Xiang

  3. Department of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Rong Yuan

Authors
  1. Ziheng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tian Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rong Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ziheng Zhang.

Additional information

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11101304, 11031002, 11371058), RFDP (Grant No. 20110003110004), and the Grant of Beijing Education Committee Key Project (Grant No. KZ20130028031).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Z., Xiang, T. & Yuan, R. Homoclinic Solutions for p(t)-Laplacian–Hamiltonian Systems Without Coercive Conditions. Mediterr. J. Math. 13, 1589–1611 (2016). https://doi.org/10.1007/s00009-015-0580-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-015-0580-9

Mathematics Subject Classification

Keywords

Search

Navigation