Study of Kirchhoff curvature problem with $\psi$-Hilfer derivative and $p(\cdot)$-Laplacian Operator


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Authors: E. ARHRRABI AND H. EL-HOUARI

DOI: 10.46793/KgJMat2608.1353A

Abstract:

The paper focuses on the existence and multiplicity of weak solutions to nonlinear Kirchhoff-type equations involving ψ-Hilfer derivatives with p()-Laplacian operators and Dirichlet boundary conditions. Through the application of a critical point approach, along with genus theory and variational techniques, we establish the existence and multiplicity results within appropriate fractional ψ-Hilfer derivative spaces. Our novel main results contribute to the advancement of the literature on differential equations involving fractional ψ-Hilfer generalized curvature phenomena.



Keywords:

Generalized ψ-Hilfer derivative, Kirchhoff problem, curvature phenomena, critical point theorem, genus theory, variational approach.



References:

[1]   E. Arhrrabi and H. El-Houari, On a class of generalized capillarity phenomena involving fractional ψ-Hilfer derivative with p()-Laplacian operator, Kragujevac J. Math. (to appear).

[2]   E. Arhrrabi and H. El-Houari, Study of double phase-choquard problem in generalized ψ-Hilfer fractional derivative spaces with p-Laplacian operator, Kragujevac J. Math. (to appear).

[3]   E. Arhrrabi, H. El-Houari and J. V. da C. Sousa, On a class of capillarity phenomenon with logarithmic nonlinearity involving ????()-Laplacian operator, Comp. Appl. Math. 43(6) (2024), Article ID 344. https://doi.org/10.1007/s40314-024-02863-8

[4]   E. Arhrrabi and H. El-Houari, Fractional Sobolev space: Study of Kirchhoff-Schrödinger systems with singular nonlinearity, CUBO (2024), 407–430. https://cubo.ufro.cl/index.php/cubo/article/view/3818

[5]   E. Arhrrabi and H. El-Houari, On a class of generalized capillarity system involving fractional ψ-Hilfer derivative with p()-Laplacian operator, Math. Methods Appl. Sci. (2024). https://doi.org/10.1002/mma.10495

[6]   M. Avci, B. Cekic and R. A. Mashiyev, Existence and multiplicity of the solutions of the p(x)-Kirchhoff type equation via genus theory, Math. Methods Appl. Sci. 34(14) (2011), Article ID 1751. https://doi.org/10.1002/mma.1485

[7]   N. T. Chung, Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities, Complex Var. Elliptic Equ. 58(12) (2013), 1637–1646. https://www.tandfonline.com/doi/abs/10.1080/17476933.2012.701289

[8]   L. Cueto-Felgueroso and R. Juanes, Macroscopic phase-field model of partial wetting: bubbles in a capillary tube, Phys. Rev. Lett. 108(14) (2012), Article ID 144502. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.144502

[9]   M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, 1964. https://cir.nii.ac.jp/crid/1370567920009303426

[10]   D. C. Clark and D. Gilbarg, A variant of the Ljusternik-Schnirelman theory, Indiana Univ. Math. J. 22(1) (1972), 65–74. https://www.jstor.org/stable/24890396

[11]   E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(5) (2012), 521–573. https://www.sciencedirect.com/science/article/pii/S0007449711001254

[12]   D. Edmunds and J. Rakonik, Sobolev embeddings with variable exponent, Studia Math. 3(143) (2000), 267–293. https://www.infona.pl/resource/bwmeta1.element.bwnjournal-article-smv143i3p267bwm

[13]   A. Elhoussain and E. H. Hamza, Fractional Sobolev space with variable exponents: Study of Kirchhoff problem by Berkovits degree theory, Nonlinear Stud. 31(4) (2024), 1135–1147. https://www.nonlinearstudies.com/index.php/nonlinear/article/view/3494

[14]   A. Elhoussain and E. H. Hamza, A singular ψ-Hilfer generalized fractional differential system problems with p()-Laplacian operator, Math. Sci. (N.Y.) (2024). 1–16. https://link.springer.com/article/10.1007/s10958-024-07408-2

[15]   H. El-Houari, L. S. Chadli and H. Moussa, Existence of a solution to a nonlocal Schrodinger system problem in fractional modular spaces, Adv. Oper. Theory 7(1) (2022), 1–30. https://doi.org/10.1007/s43036-021-00166-x

[16]   H. El-Houari, L. S. Chadli and H. Moussa, Existence of solution to M-Kirchhoff system type, 7th International Conference on Optimization and Applications (ICOA), IEEE, (2022), 1–6. https://doi.org/10.1109/ICOA51614.2021.9442669

[17]   H. El-Houari, L. S. Chadli and H. Moussa, Multiple solutions in fractional Orlicz-Sobolev spaces for a class of nonlocal Kirchhoff systems, Filomat 38(8) (2024), 2857–2875. https://doi.org/10.2298/FIL2408857E

[18]   E. H. Hamza, A. Elhoussain and J. V. D. D. C. Sousa, On a class of Kirchhoff problems with nonlocal terms and logarithmic nonlinearity, J. Pseudo-Differ. Oper. Appl. 15(3) (2024), 52 pages. https://link.springer.com/article/10.1007/s11868-024-00624-z

[19]   X. Fan, Q. Zhang and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302(2) (2005), 306–317. https://www.sciencedirect.com/science/article/pii/S0022247X03008606

[20]   X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263(2) (2001), 424–446.

[21]   A. Ghanmi and K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, Fract. Differ. Calc. 6(2) (2016), 201–217. https://dx.doi.org/10.7153/fdc-06-13

[22]   A. Ghanmi and Z. Zhang, Nehari manifold and multiplicity results for a class of fractional boundary value problems with p-Laplacian, Bull. Korean Math. Soc. 56(5) (2019), 1297–1314. https://koreascience.kr/article/JAKO201928463078719.page

[23]   A. Ghanmi, M. Kratou, K. Saoudi and D. D. Repovs, Nonlocal p-Kirchhoff equations with singular and critical nonlinearity terms, Asymptot. Anal. 131(1) (2022), 125–143. https://journals.sagepub.com/doi/full/10.3233/ASY-221769

[24]   A. Ghanmi, Existence of nonnegative solutions for a class of fractional p-Laplacian problems, Nonlinear Stud. 22(3) (2015), 373–379.

[25]   S. Heidarkhani, A. Ghobadi and M. Avci, Multiple solutions for a class of p(x)-Kirchhoff-type equations, Appl. Math. E-Notes 22 (2022), 160–168.

[26]   S. Heidarkhani, G. A. Afrouzi and S. Moradi, Variational approaches to p(x)-Laplacian-like problems with Neumann condition originated from a capillary phenomena, Int. J. Math. Math. Sci. 19(2) (2018), 189–203.

[27]   A. E. Herr, J. I. Molho, J. G. Santiago, M. G. Mungal, T. W. Kenny and M. Garguilo, Electroosmotic capillary flow with nonuniform zeta potential, Analytical Chemistry 72(5) (2000), 1053–1057.

[28]   G. Kirchhoff, Vorlesungen Huber. Mechanik, Leipzig, Teubner, 1883.

[29]   V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Springer Science and Business Media, 1999.

[30]   O. Kovacik and J. Rakosnik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41(116) (1991) 592–618. http://dml.cz/dmlcz/102493

[31]   M. Kratou, Ground state solutions of p-Laplacian singular Kirchhoff problem involving a Riemann-Liouville fractional derivative, Filomat 33(7) (2019), 2073–2088. https://doi.org/10.2298/FIL1907073K

[32]   E. E. Miller and R. D. Miller, Physical theory for capillary flow phenomena, J. Appl. Phys. 27(4) (1956), 324–332. https://doi.org/10.1063/1.1722370

[33]   N. Nyamoradi, Y. Zhou, B. Ahmad and A. Alsaedi, Variational methods for Kirchhoff type problems with tempered fractional derivative, Electron. J. Differ. Equ. 34 (2018), 13 pages.

[34]   M. M. Rodrigues, Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators, Mediterr. J. Math. 9 (2012), 211–223. https://link.springer.com/article/10.1007/s00009-011-0115-y

[35]   J. Simon, Regularite de la Solution d’une Equation non Lineaire Dans N, Journées d’Analyse Non Linéaire, Springer, Berlin, Heidelberg, 205–227.

[36]   J. V. D. C. Sousa, and E. C. de Oliveira, On the stability of a hyperbolic fractional partial differential equation, Differ. Equ. Dyn. Syst. 31(1) (2023). 31–52. https://link.springer.com/article/10.1007/s12591-019-00499-3

[37]   J. V. D. C. Sousa, K. D. Kucche and J. J. Nieto, Existence and multiplicity of solutions for fractional k(ξ)-Kirchhoff-type equation, Qual. Theory Dyn. Syst. 23(1) (2024), Article ID 27. https://link.springer.com/article/10.1007/s12346-023-00877-x

[38]   J. Sousa, K. B. Lima and L. S. Tavares, Existence of solutions for a singular double phase problem involving a ψ-Hilfer fractional operator via Nehari manifold, Qual. Theory Dyn. Syst. 22(3) (2023). 1–26. https://link.springer.com/article/10.1007/s12346-023-00794-z

[39]   J. V. D. C. Sousa and E. C. De Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 60 (2018). 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005

[40]    J. V. da C. Sousa, Existence of nontrivial solutions to fractional Kirchhoff double phase problems, Comput. Appl. Math. 43(2) (2024), Article ID 93. https://link.springer.com/article/10.1007/s40314-024-02599-5

[41]   J. V. D. C. Sousa, D. S. Oliveira and L. S. Tavares, Solutions of the mean curvature equation with the Nehari manifold, Comput. Appl. Math. 43(1) (2024), ARticle ID 24. https://link.springer.com/article/10.1007/s40314-023-02534-0

[42]   J. V. D. C. Sousa, D. S. Oliveira and P. R. Agarwal, Existence and multiplicity for fractional Dirichlet problem with γ(ξ)-Laplacian equation and Nehari manifold, Appl. Anal. Disc. Math. 17(2) (2023). 480–495. https://www.jstor.org/stable/27281422

[43]   J. V. D. C. Sousa, A. Elhoussain, E. H. Hamza and L. S. Tavares, Basic results for fractional anisotropic spaces and applications, J. Pseudo-Differ. Oper. Appl. 15(4) (2024), Article ID 71. https://link.springer.com/article/10.1007/s11868-024-00641-y

[44]   H. M. Srivastava and V. da Costa Sousa, Multiplicity of solutions for fractional-order differential equations via the κ(x)-Laplacian operator and the Genus theory, Fractal and Fractional 6(9) (2022), Article ID 481. https://doi.org/10.3390/fractalfract6090481

[45]   E. H. Van Brummelen, M. Shokrpour-Roudbari and G. J. Van Zwieten, Elasto-capillarity simulations based on the Navier-Stokes-Cahn-Hilliard equations, in: Advances in Computational Fluid-Structure Interaction and Flow Simulation: New Methods and Challenging Computations, 2016, 451–462. https://link.springer.com/chapter/10.1007/978-3-319-40827-9_35

[46]   Z. Yucedag, M. Avci and R. Mashiyev, On an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition, Math. Model. Anal. 17(2) (2012). 161–170. https://doi.org/10.3846/13926292.2012.655788

[47]   Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, 2023.