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An Existence Result for a Class of Kirchhoff Type Systems via Dynamical Methods

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Abstract

In this paper, by using the so-called dynamical approach we study the existence of nontrivial solutions for a class of Kirchhoff type systems. The presented result is a generalization of previous works Alves and Boudjeriou (Nonlinear Anal. 197:1–17, 2020; Rend. Circ. Mat. Palermo 71(2):611–632, 2022) where the existence of nontrivial solutions for some scalar nonlocal elliptic problems has been studied.

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References

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

    MATH  Google Scholar 

  2. Alves, C.O., Corrêa, F.J.S.A.: A sub-supersolution approach for quasilinear Kirchhoff equation. J. Math. Phys. 56, 051501 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, C.O., Boudjeriou, T.: Existence of solution for a class of nonvariational Kirchhoff type problem via dynamical methots. Nonlinear Anal. 197, 1–17 (2020)

    Article  MATH  Google Scholar 

  4. Alves, C.O., Boudjeriou, T.: Existence of solution for a class of nonlocal problem via dynamical methods. Rend. Circ. Mat. Palermo 71(2), 611–632 (2022). https://doi.org/10.1007/s12215-021-00644-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Alves, C.O., Corrêa, F.J.S.A.: On existence of solutions for a class of problem involving a nonlinear operator. Commun. Appl. Nonlinear Anal. 8, 43–56 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kircchoff type. Comput. Math. Appl. 49, 85–93 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Alves, C.O., Figueiredo, G.M.: Nonlinear perturbations of a periodic Kirchhoff equation in \(\mathbb{R}^{N}\). Nonlinear Anal. 75, 2750–2759 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Afrouzi, G.A., Shakeri, S., Zahmatkesh, H.: Existence results for a class Kirchhoff-type systems with combined nonlinear effects. Ukr. Math. J. 71, 651–662 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Boudjeriou, T.: Existence and non-existence of global solutions for a nonlocal Choquard–Kirchhoff diffusion equations in \(\mathbb{R}^{N}\). Appl. Math. Optim. 84, 695–732 (2021). https://doi.org/10.1007/s00245-021-09783-7

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheng, B., Wu, X., Liu, J.: Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity. NoDEA Nonlinear Differ. Equ. Appl. 19, 521–537 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, C., Song, H., Xiu, Z.: Multiple solutions for p-Kirchhoff equations in \(\mathbb{R}^{N}\). Nonlinear Anal. 86, 146–156 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, S., Li, L.: Multiple solutions for the nonhomogeneous Kirchhoff equation on \(\mathbb{R}^{N}\). Nonlinear Anal., Real World Appl. 14, 1477–1486 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cammaroto, F., Vilasi, L.: On a Schrödinger-Kirchhoff-type equation involving the \(p(x)\)-Laplacian. Nonlinear Anal. 81, 42–53 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chung, N.T.: An existence result for a class of Kirchhoff type systems via sub and supersolutions method. Appl. Math. Lett. 35, 95–101 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cazenave, T., Lions, P.L.: Solutions globales d’équations de la chaleur semi linéaires. Commun. Partial Differ. Equ. 9, 955–978 (1984)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Figueiredo, G.M., Morales, C., Santos Junior, J.R., Suarez, A.: Study of a nonlinear Kirchhoff equation with non-homogeneous material. J. Math. Anal. Appl. 416, 597–608 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Figueiredo, G.M., Santos Junior, J.R.: Multiplicity and concentration behavior of positive solutions for a Schrodinger-Kirchhoff type problem via penalization method. ESAIM Control Optim. Calc. Var. 20, 389–415 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Figueiredo, G.M., Ikoma, N., Santos Junior, J.R.: 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 

  20. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math., vol. 840. Springer, New York (1981)

    MATH  Google Scholar 

  21. Huy, N.B., Quan, B.T.: Positive solutions of logistic equations with dependence on gradient and nonhomogeneous Kirchhof term. J. Math. Anal. 444, 95–109 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. Lions, J.L., Magenes, E.: Problemes aux Limites Non Homogenes et Applications, Vol. I. Dunod, Paris (1968)

    MATH  Google Scholar 

  25. Liang, S., Shi, S.: Existence of multi-bump solutions for a class of Kirchhoff type problems in \(\mathbb{R}^{3}\). J. Math. Phys. 54, 121510 (2013). https://doi.org/10.1063/1.4850835

    Article  MathSciNet  MATH  Google Scholar 

  26. 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  MathSciNet  MATH  Google Scholar 

  27. Liao, Y.-F., Zhang, P., Liu, J., Tang, C.-L.: Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity. J. Math. Anal. Appl. 430, 1124–1148 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, J., Wang, L., Zhao, P.: Positive solutions for a nonlocal problem with convection term and small perturbation. Math. Methods Appl. Sci. 40(3), 720–728 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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 

  30. Ma, T.F.: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal. 63, 1967–1977 (2005)

    Article  Google Scholar 

  31. Naimen, D.: On the Brezis-Nirenberg problem with a Kirchhoff type perturbation. Adv. Nonlinear Stud. 15, 135–156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Naimen, D.: The critical problem of Kirchhoff type elliptic equations in dimension four. J. Differ. Equ. 257, 1168–1193 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Quittner, P.: Boundedness of trajectories of parabolic equations and stationary solutions via dynamical methods. Differ. Integral Equ. 7, 1547–1556 (1994)

    MathSciNet  MATH  Google Scholar 

  34. Quittner, P.: Signed solutions for a semilinear elliptic problem. Differ. Integral Equ. 11, 551–559 (1998)

    MathSciNet  MATH  Google Scholar 

  35. Quittner, P., Souplet, P.: Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, 2nd edn. Birkhäuser, Basel (2019)

    Book  MATH  Google Scholar 

  36. Quittner, P., Souplet, P.: A priori estimates and existence for elliptic systems via bootstrap in weighted. Arch. Ration. Mech. Anal. 174, 49–81 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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 

  38. Wang, L.: On a quasilinear Schrödinger-Kirchhoff-type equation with radial potentials. Nonlinear Anal. 83, 58–68 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, X., Zhang, J.: Non-existence of positive solutions to nonlocal Lane-Emden equations. J. Math. Anal. Appl. 488(1), 124067 (2020) 22 pp.

    Article  MathSciNet  MATH  Google Scholar 

  40. Xiang, M., Zhang, B., Guo, X.: Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem. Nonlinear Anal. 120, 299–313 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to warmly thank the anonymous referee for his/her useful and nice comments that were very important to improve this paper.

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  1. Department of Basic Teaching Institute of Electrical & Electronic Engineering, University of boumerdes, boumerdes, 35000, Algeria

    Tahir Boudjeriou

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  1. Tahir Boudjeriou
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Boudjeriou, T. An Existence Result for a Class of Kirchhoff Type Systems via Dynamical Methods. Acta Appl Math 182, 11 (2022). https://doi.org/10.1007/s10440-022-00546-2

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