Existence and Multiplicity of Solutions for Fractional ()-Kirchhoff-Type Equation

  1. J. Vanterler da C. Sousa 1
  2. Kishor D. Kucche 2
  3. Juan J. Nieto 3
  1. 1 Universidade Estadual do Maranhão
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    Universidade Estadual do Maranhão

    São Luís, Brasil

    ROR https://ror.org/04ja5n907

  2. 2 Shivaji University
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    Shivaji University

    Kolhāpur, India

    ROR https://ror.org/01bsn4x02

  3. 3 Universidade de Santiago de Compostela
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    Universidade de Santiago de Compostela

    Santiago de Compostela, España

    ROR https://ror.org/030eybx10

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Revista:
Qualitative theory of dynamical systems

ISSN: 1575-5460

Ano de publicación: 2024

Volume: 23

Número: 1

Tipo: Artigo

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Outras publicacións en: Qualitative theory of dynamical systems

Resumo

In this paper, we aim to tackle the questions of existence and multiplicity of solutions to a new class of κ(ξ )-Kirchhoff-type equation utilizing a variational approach. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of space ψ-fractional Hμ,ν; ψ κ(ξ ) (). In this sense, we present a few special cases and remark on the outcomes explored.

Referencias bibliográficas

  • 1. Ambrosio, V., Isernia, T.: Concentration phenomena for a fractional Schrödinger–Kirchhoff type equation. Math. Meth. Appl. Sci. 41(2), 615–645 (2018)
  • 2. Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348(1), 305–330 (1996)
  • 3. Correa, F.J.S.A., Nascimento, R.G.: On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition. Math. Comput. Modell. 49(3–4), 598–604 (2009)
  • 4. Correa, F.J.S.A., Figueiredo, G.M.: On an elliptic equation of p-Kirchhoff type via variational methods. Bull. Australian Math. Soc. 74(2), 263–277 (2006)
  • 5. da Costa Sousa, J.V., Capelas de Oliveira, E.: On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)
  • 6. da Costa Sousa, J.V., Zuo, J., Donal O.: The Nehari manifold for a ψ-Hilfer fractional p-Laplacian. Appl. Anal. 1–31 (2021)
  • 7. da Costa Sousa, J.V.: Existence and uniqueness of solutions for the fractional differential equations with p-Laplacian in Hν,ηψ p . J. Appl. Anal. Comput. 12(2), 622–661 (2022)
  • 8. da Costa Sousa, J.V., Ledesma, C.T., Pigossi, M., Zuo, J.: Nehari manifold for weighted singular fractional p-Laplace equations. Bull. Braz. Math. Soc. 1–31 (2022)
  • 9. da Costa Sousa, J.V.: Nehari manifold and bifurcation for a ψ-Hilfer fractional p-Laplacian. Math. Meth. Appl. Sci. (2021). https://doi.org/10.1002/mma.7296
  • 10. da Costa Sousa, J.V., Leandro, S., Tavares, C.E., Ledesma, T.: A variational approach for a problem involving a ψ-Hilfer fractional operator. J. Appl. Anal. Comput. 11(3), 1610–1630 (2021)
  • 11. Dai, G., Hao, R.: Existence of solutions for a p(x)-Kirchhoff-type equation. J. Math. Anal. Appl. 359(1), 275–284 (2009)
  • 12. Dai, G., Liu, D.: Infinitely many positive solutions for a p(x)-Kirchhoff-type equation. J. Math. Anal. Appl. 359(2), 704–710 (2009)
  • 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)
  • 14. Diening, L., Hasto, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces. In: FSDONA04 Proceedings, pp. 38–58 (2004)
  • 15. Ezati, R., Nyamoradi, N.: Existence and multiplicity of solutions to a ψ-Hilfer fractional p-Laplacian equations. Asian-Eur. J. Math. 2350045 (2022)
  • 16. Ezati, R., Nyamoradi, N.: Existence of solutions to a Kirchhoff ψ-Hilfer fractional p-Laplacian equations. Math. Meth. Appl. Sci. 44(17), 12909–12920 (2021)
  • 17. Fan, X.L., Zhao, D.: On the generalized Orlicz–Sobolev space W,k,p(x)(). J. Gansu. Educ. College 12(1), 1–6 (1998)
  • 18. Fan, X.L., Zhao, D.: On the spaces L p(x) and Wm,p(x). J. Math. Anal. Appl. 263, 424–446 (2001)
  • 19. Fan, X.L., Shen, J.S., Zhao, D.: Sobolev embedding theorems for spaces Wk,p(x)(). J. Math. Anal. Appl. 262, 749–760 (2001)
  • 20. Fan, X.: On the sub-supersolution method for p(x)-Laplacian equations. J. Math. Anal. Appl. 330(1), 665–682 (2007)
  • 21. Fan, X.: On nonlocal p(x)-Laplacian Dirichlet problems. Nonlinear Anal. Theory Methods Appl. 72(7–8), 3314–3323 (2010)
  • 22. Fan, X.-L., Zhang, Q.-H.: Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Anal. Theory Methods Appl. 52(8), 1843–1852 (2003)
  • 23. Fiscella, A., Pucci, P.: p-fractional Kirchhoff equations involving critical nonlinearities. Nonlinear Anal. Real World Appl. 35, 350–378 (2017)
  • 24. He, X., Zou, W.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. Theory Methods Appl. 70(3), 1407–1414 (2009)
  • 25. Mingqi, X., R˘adulescu, V.D., Zhang, B.: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58(2), 1–27 (2019)
  • 26. Mingqi, X., R˘adulescu, V.D., Zhang, B.: Combined effects for fractional Schrödinger-Kirchhoff systems with critical nonlinearities. ESAIM Control Opt. Calc. Var. 24(3), 1249–1273 (2018)
  • 27. Naimen, D.: The critical problem of Kirchhoff type elliptic equations in dimension four. J. Differ. Equ. 257(4), 1168–1193 (2014)
  • 28. Ourraoui, A.: On an elliptic equation of p-Kirchhoff type with convection term. Comptes Rendus. Mathématique 354(3), 253–256 (2016)
  • 29. Pucci, P., Xiang, M., Zhang, B.: Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN ”. Calc. Var. Partial Differ. Equ. 54(3), 2785–2806 (2015)
  • 30. Pucci, P., Xiang, M., Zhang, B.: Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv. Nonlinear Anal. 5(1), 27–55 (2016)
  • 31. Srivastava, H.M., da Costa Sousa, J.V.: Multiplicity of solutions for fractional-order differential equations via the κ(x)-Laplacian operator and the genus theory. Fractal Fract. 6(9), 481 (2022)
  • 32. Tang, X.-H., Cheng, B.: Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J. Differ. Equ. 261(4), 2384–2402 (2016)
  • 33. Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
  • 34. Xiang, M., Zhang, B., R˘adulescu, V.D.: Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity 29(10), 3186 (2016)
  • 35. Zeider, E.: Nonlinear Functional Analysis and its Applications, II=B: Nonlinear Monotone Operators. Springer, New York (1990)
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