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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


On an Elliptic Equation of $p(x)$-Kirchhoff Type with Convection Term and Singular Weights

Discontinuity, Nonlinearity, and Complexity 14(2) (2025) 269--278 | DOI:10.5890/DNC.2025.06.002

Ayoub Zaki$^{1}$, Mustapha Haddaoui$^{2}$, Walid Bochiha$^{1}$

$^{1}$ LAMAO Laboratory, Department of Mathematics, FSO, University of Mohamed Premier, Oujda, Morocco

$^{2}$ ROALI Team, LMIMA Laboratory,FST-Erachidia, Moulay Ismail University of Meknes, Meknes, Morocco

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Abstract

In this paper we will use new compact embedding and by Galerkin's approach we prove the existence of at least one solution to a p(x)-Kirchhoff problem with convection term and singular weights.

References

  1. [1]  Dràbek, P. and Hernàndez, J. (2018), Quasilinear eigenvalue problems with singular weights for the p-Laplacian, Annali di Matematica Pura ed Applicata, 198, 1069-1086.
  2. [2]  Zaki, A., Hamydy, A., and Tsouli, N. (2022), Existence of solution for a p(x)-Kirchhoff type with singular weights, Advanced Studies: Euro-Tbilisi Mathematical Journal, special Issue (10), 191–202.
  3. [3]  Diening, L., Harjulehto, P., Hästö, P., and Ruzicka, M. (2011), Lebesgue and Sobolev Spaces with Variable Exponents, Berlin: Springer-Verlag.
  4. [4]  Fan, X.L. and Zhao, D. On the spaces $ L^{p(x)}(\Omega ) $ and $ W^{m,p(x)}(\Omega ) $, Journal of Mathematical Analysis and Applications, 263, 424-446.
  5. [5]  Kovacik, O. and Rakosnik, J. On spaces $ L^{p(x)}(\Omega ) $ and $ W^{k,p(x)}(\Omega ) $, Czechoslovak Mathematical Journal, 41(116), 592-618.
  6. [6]  Kesavan, S. (1989), Topics in Functional Analysis and Applications, John Wiley. Sons, New York.
  7. [7]  Alves, C.O. and Correa, F.J.S. (2015), A sub-supersolution approach for a quasilinear Kirchhoff equation, Journal of Mathematical Physics, 56(5), 051501.
  8. [8]  Li, Y., Li, F., and Shi, J. (2012), Existence of a positive solution to Kirchhoff type problems without compactness conditions, Journal of Differential Equations, 253, 2285-2294.
  9. [9]  Chen, C.,kuo, Y., and Wu, T. (2011), The Nehari manifold for a kirchhoff type problem involving sign-changing weight functions, Journal Differential Equations, 250, 1876-1908.
  10. [10]  Autuori, G., Fiscella, A., and Pucci, P. (2015), Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity, Nonlinear Analysis, 125, 699-714.
  11. [11]  He, X. and Zou, W.(2009), Infinitely many positive solutions for kirchhoff-type problems, Nonlinear Analysis: Theory, Methods $\&$ Applications, 70(3), 1407-1414.
  12. [12]  Cheng, X. and Dai, G. (2015), Positive solutions for p-Kirchhoff type problems on $ \mathbb{R}^{n} $, Mathematical Methods in the Applied Sciences, 38(12), 2650-2662.
  13. [13]  Liu, D.C. (2010), On a P-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Analysis, 72, 302-308.
  14. [14]  Liu, C., Wang, J., and Gao, Q. (2013), Existence of nontrivial solutions for p-Kirchhoff type equations, Boundary Value Problems, 2013, 279.
  15. [15]  Fan, X.L. and Zhang, Q.H. (2003), Existence of solutions for $ p(x)-$Laplacian Dirichlet problems, Nonlinear Analysis: Theory, Methods $\&$ Applications, 52(8), 1843-1852.
  16. [16]  Motreanu, D. and Tornatore, E. (2021), Quasilinear Dirichlet problems with degenerated $p$-Laplacian and convection term, Mathematics, 9, 139, https://doi.org/10.3390/math9020139.
  17. [17]  Ourraoui, A. (2015), On an elliptic equation of p-Kirchhoff type with convection term, Comptes Rendus. Mathématique, 354(3), 253-256.