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On a p(x)-biharmonic Kirchhoff type problem with indefinite weight and no flux boundary condition

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Abstract

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions.

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References

  1. Afrouzi, G.A., Mirzapour, M., Chung, N.T.: Existence and multiplicity of solutions for Kirchhoff type problems involving \(p(x)\)-Biharmonic operators. Z. Anal. Anwend. 33(3), 289–303 (2014)

    Article  MathSciNet  Google Scholar 

  2. Afrouzi, G.A., Mirzapour, M., Rӑdulescu, V.D.: Nonlocal fourth-order Kirchhoff systems with variable growth: low and high energy solutions. Collect. Math. 67(2), 207–223 (2016)

    Article  MathSciNet  Google Scholar 

  3. Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348(1), 305–330 (1996)

    Article  MathSciNet  Google Scholar 

  4. Ayoujil, A., El Amrouss, A.R.: Continuous spectrum of a fourth-order nonhomogeneous elliptic equation with variable exponent. Electron. J. Differ. Equ. 24, 12 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Baraket, S., Rӑdulescu, V.D.: Combined effects of concave-convex nonlinearities in a fourth-order problem with variable exponent. Adv. Nonlinear Stud. 16(3), 409–419 (2016)

    Article  MathSciNet  Google Scholar 

  6. Boureanu, M.M.: Fourth order problems with Leray–Lions type operators in variable exponent spaces. Discrete Contin. Dyn. Syst. Ser. S 12(2), 231–243 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Boureanu, M.M., Rӑdulescu, V.D., Repovš, D.: On a p(.) biharmonic problem with no-flux boundary condition. Comput. Math. Appl. 72(9), 2505–2515 (2016)

    Article  MathSciNet  Google Scholar 

  8. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)

    Article  MathSciNet  Google Scholar 

  9. Chung, N.T., Toan, H.Q.: On a class of fractional Laplacian problems with variable exponents and indefinite weights. Collect. Math. 71(2), 223–237 (2020)

    Article  MathSciNet  Google Scholar 

  10. Corrêa, F.J.S.A., Figueiredo, G.M.: On a p-Kirchhoff equation via Krasnoselskiiʼs genus. Appl. Math. Lett. 22(6), 819–822 (2009)

    Article  MathSciNet  Google Scholar 

  11. Costa, D.G.: An invitation to Variational Methods in Differential Equations. Brikhäuser, Boston (2007)

    Book  Google Scholar 

  12. Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue spaces. Applied and Numerical Harmonic Analysis, Brikhäuser/Springer, Heidelberg, Foundations and harmonic analysis. In (2013)

    Book  Google Scholar 

  13. Dai, G., Ma, R.: Solutions for a \(p(x)\)-Kirchhoff type equation with Neumann boundary data. Nonlinear Anal. Real World Appl. 12(5), 2666–2680 (2011)

    Article  MathSciNet  Google Scholar 

  14. Danet, C.P.: Two maximum principles for a nonlinear fourth order equation from thin plate theory. Electron. J. Qual. Theory Differ. Equ. 31, 9 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Darhouche, O.: Existence and multiplicity results for a class of Kirchhoff type problems involving \(p(x)\)-biharmonic operator. Bol. Soc. Parana. Mat. 37(2), 23–33 (2019)

    Article  MathSciNet  Google Scholar 

  16. Diening, L.: Maximal function on generalized Lebesgue spaces \(L^{p(x)}\). Math. Inequal. Appl. 7(2), 245–253 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture notes in mathematics, vol. 2017. Springer, Heidelberg (2011)

    Book  Google Scholar 

  18. Dreher, M.: The Kirchhoff equation for the p-Laplacian. Rend. Semin. Mat. Univ. Politec. Torino 64(2), 217–238 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Edmunds, D.E., Rákosník, J.: Sobolev embeddings with variable exponent. Studia Math. 143(3), 267–293 (2000)

    Article  MathSciNet  Google Scholar 

  20. El Amrouss, A.R., Ourraoui, A.: Existence of solutions for a boundary problem involving \(p(x)-\)biharmonic operator. Bol. Soc. Parana. Mat. 31(1), 179–192 (2013)

    Article  MathSciNet  Google Scholar 

  21. El Khalil, A., Laghzal, M., Morchid Alaoui, M.D., Touzani, A.: Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential. Adv. Nonlinear Anal. 9(1), 1130–1144 (2020)

    Article  MathSciNet  Google Scholar 

  22. Fan, X.L.: Solutions for \(P(x)-\)Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 312(2), 464–477 (2005)

    Article  MathSciNet  Google Scholar 

  23. Fan, X.L., Han, X.: Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in RN. Nonlinear Anal. 59(1–2), 173–188 (2004)

    MathSciNet  MATH  Google Scholar 

  24. Fan, X.L., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263(2), 424–446 (2001)

    Article  MathSciNet  Google Scholar 

  25. Ferrero, A., Warnault, G.: On solutions of second and fourth order elliptic equations with power-type nonlinearities. Nonlinear Anal. 70(8), 2889–2902 (2009)

    Article  MathSciNet  Google Scholar 

  26. Fragnelli, G.: Positive periodic solutions for a system of anisotropic parabolic equations. J. Math. Anal. Appl. 367(1), 204–228 (2010)

    Article  MathSciNet  Google Scholar 

  27. Jabri, Y.: The mountain pass theorem. Variants, generalizations and some applications. In: Encyclopedia of Mathematics and its Applications, vol. 95, Cambridge University Press, Cambridge (2003)

  28. Kefi, K., Rădulescu, V.D.: Small perturbations of nonlocal biharmonic problems with variable exponent and competing nonlinearities. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29(3), 439–463 (2018)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  30. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslovak Math. J. 41(4), 592–618 (1991)

    Article  MathSciNet  Google Scholar 

  31. Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Penha and Medeiros (eds.) Proceedings of international symposium on continuum mechanics and partial differential equations, Rio de Janeiro 1977. Math. Stud. North Holland, vol. 30, pp 284−346 (1978)

  32. Liu, Q.: Existence of three solutions for p(x)-Laplacian equations. Nonlinear Anal. 68(7), 2119–2127 (2008)

    Article  MathSciNet  Google Scholar 

  33. Matei, P.: Nemytskij operators in Lebesgue spaces with a variable exponent. Rom. J. Math. Comput. Sci. 3(2), 109–118 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Miao, Q.: Multiple solutions for nonlocal elliptic systems involving \(p(x)\)-biharmonic operator. Mathematics 7(8), 756 (2019)

    Article  Google Scholar 

  35. Myers, T.G.: Thin films with high surface tension. SIAM Rev. 40(3), 441–462 (1998)

    Article  MathSciNet  Google Scholar 

  36. Pucci, P., Rădulescu, V.D.: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital. 3(3), 543–582 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Rădulescu, V.D., Repovš, D.D.: Partial differential equations with variable exponents. Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL (2015)

    Google Scholar 

  38. Růžička, M.: Electrorheological fluids: modeling and mathematical theory. Lecture notes in mathematics, vol. 1748. Springer, Berlin (2000)

    Book  Google Scholar 

  39. Xiang, M., Rădulescu, V.D., Zhang, B.: Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58, 57 (2019)

    Article  MathSciNet  Google Scholar 

  40. Xiang, M., Zhang, B., Rădulescu, V.D.: Super linear Schrdinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent. Adv. Nonlinear Anal. 9(1), 690–709 (2020)

    Article  MathSciNet  Google Scholar 

  41. Zang, A., Fu, Y.: Interpolation inequalities for derivatives in variable exponent Lebesgue Sobolev spaces. Nonlinear Anal. 69(10), 3629–3636 (2008)

    Article  MathSciNet  Google Scholar 

  42. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory(in Russian). Izv. Akad. Nauk SSSR Ser. Mat 50(4), 675–710 (1986)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their careful reading and valuable suggestions on this article.

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Correspondence to Mohammed Filali.

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Talbi, M., Filali, M., Soualhine, k. et al. On a p(x)-biharmonic Kirchhoff type problem with indefinite weight and no flux boundary condition. Collect. Math. 73, 237–252 (2022). https://doi.org/10.1007/s13348-021-00316-7

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  • DOI: https://doi.org/10.1007/s13348-021-00316-7

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