| Citation: |
Omar Djidel, Salah Boulaaras, Mokhtar Boumaaza. NONLOCAL FRACTIONAL P$(Z, Y)$-LAPLACIAN PROBLEMS WITH VARIABLE EXPONENTS: EXISTENCE RESULTS AND A NUMERICAL ILLUSTRATION[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1136-1182. doi: 10.11948/20260073
|
NONLOCAL FRACTIONAL P$(Z, Y)$-LAPLACIAN PROBLEMS WITH VARIABLE EXPONENTS: EXISTENCE RESULTS AND A NUMERICAL ILLUSTRATION
-
Abstract
In this work, we study the existence of weak solutions for a class of nonlocal problems driven by the fractional $ \mathbf{p}(z, y) $-Laplacian operator with spatially variable exponents. The considered model includes a nonlinear source term depending simultaneously on local interactions and a global integral over the domain, which introduces significant analytical difficulties. Specifically, we examine the problem:
$ \begin{equation*} (P)\left\{ \begin{array}{ll} \left( \int_{\Omega} \mathrm{h}(z,\mathrm{w})\,dz\right)^{\beta(z)} (-\Delta_{\mathbf{p}(z,y)})^{s}\mathrm{w} = \big(\mathrm{h}(z,\mathrm{w})\big)^{\alpha(z)}, & \text{in } \Omega, \\[1.0ex] \mathrm{w}=0, & \text{on } \mathbb{R}^{N} \setminus \Omega,\\[0.5ex] \mathrm{w}>0, & \text{in } \Omega, \end{array} \right. \end{equation*} $
where $ \Omega \subset \mathbb{R}^{N} $ is a bounded domain, $ (-\Delta_{\mathbf{p}(z, y)})^{s} $ denotes the fractional $ \mathbf{p}(z, y) $-Laplacian, and $ \alpha(z) $, $ \beta(z) $ are spatially varying exponents. By employing fixed-point arguments, specifically Tychonoff's and Schauder's fixed-point theorems, we establish two existence results for weak solutions. To complement the theoretical analysis, we also present a two-dimensional numerical example of a simplified fractional model ($ \mathbf{p}=2 $) featuring a nonlinear and nonlocal source inspired by the analytical structure of the studied problem.
-
-
References
[1] E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 2002, 164, 213–259. doi: 10.1007/s00205-002-0208-7 [2] R. P. Agarwal, E. Azroul, N. Kamali and M. Shimi, On a class of $p(x, \cdot)$-integro-differential Kirchhoff-type problem with singular kernel, J. Appl. Anal. Comput., 2025, 15(1), 333–353. [3] D. Arcoya, T. Leonori and A. Primo, Existence of solutions for semilinear nonlocal elliptic problems via a Bolzano theorem, Acta Appl. Math., 2013, 127, 87–104. doi: 10.1007/s10440-012-9792-1 [4] E. Azroul, A. Benkirane and M. Shimi, General fractional Sobolev space with variable exponent and applications to nonlocal problems, Adv. Oper. Theory, 2020, 5, 1512–1540. doi: 10.1007/s43036-020-00062-w [5] E. Azroul, A. Benkirane, M. Shimi and M. Srati, On a class of fractional $p(x)$-Kirchhoff type problems, Appl. Anal., 2021, 100, 383–402. doi: 10.1080/00036811.2019.1603372 CrossRef $p(x)$-Kirchhoff type problems" target="_blank">Google Scholar
[6] E. Azroul, N. Kamali, M. A. Ragusa and M. Shimi, Multiple solutions for anisotropic nonlocal problems with variable exponents, Fract. Calc. Appl. Anal., 2025, 28(5), 2333–2356. doi: 10.1007/s13540-025-00433-1 [7] A. Bahrouni and V. D. Rădulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S, 2018, 11(3), 379–389. doi: 10.3934/dcdss.2018021 [8] G. M. Bisci, V. D. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016. [9] S. Boulaaras, Optimal shape design for two-dimensional fractional heat equations with source terms: Long-time behavior and $\Gamma$-convergence, J. Nonlinear Math. Phys., 2026. DOI: 10.1007/s44198-026-00458-x. CrossRef $\Gamma$-convergence" target="_blank">Google Scholar
[10] S. Boulaaras and R. Guefaifia, Global existence and blow-up phenomena for a hyperbolic $p$-Laplacian equation with logarithmic nonlinearity and nonlinear boundary conditions, J. Appl. Anal. Comput., 2026, 16(6), 2887–2912. [11] S. Boulaaras, R. Guefaifia, B. Cherif and T. Radwan, Existence result for a Kirchhoff elliptic system involving $p$-Laplacian operator with variable parameters and additive right-hand side via sub- and supersolution methods, AIMS Math., 2021, 6(3), 2315–2329. [12] F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Aust. Math. Soc., 2006, 74(2), 263–277. doi: 10.1017/S000497270003570X CrossRef $p$-Kirchhoff type via variational methods" target="_blank">Google Scholar
[13] F. J. S. A. Corrêa and G. M. Figueiredo, On a $p$-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Lett., 2009, 22, 819–822. doi: 10.1016/j.aml.2008.06.042 CrossRef $p$-Kirchhoff equation via Krasnoselskii's genus" target="_blank">Google Scholar
[14] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1998, 15(4), 493–516. [15] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Heidelberg, 2011. [16] O. Djidel, K. Biroud, M. -S. Rezaoui, S. Boulaaras and R. Guefaifia, A study of the nonlocal solution of $p$-Laplacian fractional elliptic problems via approximation methods, Bound. Value Probl., 2024, Article 169. $p$-Laplacian fractional elliptic problems via approximation methods" target="_blank">Google Scholar
[17] X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl., 2001, 262, 749–760. doi: 10.1006/jmaa.2001.7618 CrossRef $W^{k, p(x)}(\Omega)$" target="_blank">Google Scholar
[18] X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl., 2001, 263, 424–446. doi: 10.1006/jmaa.2000.7617 CrossRef $L^{p(x)}(\Omega)$ and
$W^{k, p(x)}(\Omega)$ [19] R. Guefaifia and S. Boulaaras, A qualitative study on the existence of positive solutions for local and nonlocal elliptic integro-differential quasilinear systems, Bound. Value Probl., 2024, Article 159. [20] R. Guefaifia, S. Boulaaras and F. Kamache, Existence of weak solutions for a new class of fractional boundary value impulsive systems with Riemann–Liouville derivatives, J. Integral Equations Appl., 2021, 33(3), 301–313. doi: 10.1216/jie.2021.33.301 [21] R. Guefaifia, G. C. G. dos Santos, T. Bouali, R. Jan, S. Boulaaras and A. Alharbi, Sub-super solutions method combined with Schauder's fixed point for existence of positive weak solutions for anisotropic non-local elliptic systems, Mathematics, 2022, 10, Article 4479. doi: 10.3390/math10234479 [22] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 2014. [23] H. El-Houari, E. Arhrrabi and H. Moussa, Study of Schrödinger–Choquard problem with $p(\cdot)$-Laplacian operator, J. Appl. Anal. Comput., 2026, 16(1), 479–494. $p(\cdot)$-Laplacian operator" target="_blank">Google Scholar
[24] A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var., 2016, 9(2), 101–125. doi: 10.1515/acv-2014-0024 CrossRef $p$-Laplacian problems via Morse theory" target="_blank">Google Scholar
[25] F. Kamache, R. Guefaifia, S. Boulaaras and A. Alharbi, Existence of weak solutions for a new class of fractional $p$-Laplacian boundary value systems, Mathematics, 2020, 8, Article 475. doi: 10.3390/math8040475 CrossRef $p$-Laplacian boundary value systems" target="_blank">Google Scholar
[26] U. Kaufmann, J. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians, Electron. J. Qual. Theory Differ. Equ., 2017, 76, 1–10. [27] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math. J., 1991, 41, 592–618. doi: 10.21136/CMJ.1991.102493 CrossRef $L^{p(x)}$ and
$W^{k, p(x)}$ [28] D. Liu, On a $p$-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Anal., 2010, 72, 302–308. doi: 10.1016/j.na.2009.06.052 CrossRef $p$-Kirchhoff equation via fountain theorem and dual fountain theorem" target="_blank">Google Scholar
[29] M. Louchaich and K. Saoudi, On a weak solutions for a singular problems involving the fractional $p(\cdot)$-Laplacian type operators, Complex Var. Elliptic Equ., 2026, 71(5), 1043–1063. doi: 10.1080/17476933.2025.2509318 CrossRef $p(\cdot)$-Laplacian type operators" target="_blank">Google Scholar
[30] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 2012, 136(5), 521–573. doi: 10.1016/j.bulsci.2011.12.004 [31] R. Pei, On ground state of fractional $p$-Kirchhoff equation involving subcritical and critical exponential growth, J. Appl. Anal. Comput., 2024, 14(5), 2653–2672. [32] K. Perera, M. Squassina and Y. Yang, Bifurcation and multiplicity results for critical fractional $p$-Laplacian problems, Math. Nachr., 2016, 289(2–3), 332–342. doi: 10.1002/mana.201400259 CrossRef $p$-Laplacian problems" target="_blank">Google Scholar
[33] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1980. [34] A. Yahiaoui, M. -S. Rezaoui, O. Djidel, R. Guefaifia and S. Boulaaras, Kirchhoff-type problems with the non-local fractional $\mathrm{d}(z, \cdot)$-Laplacian operator, J. Inequal. Appl., 2025, Article 43. $\mathrm{d}(z, \cdot)$-Laplacian operator" target="_blank">Google Scholar
[35] E. Zaouche, Nontrivial weak solutions for nonlocal nonhomogeneous elliptic problems, Appl. Anal., 2022, 101(4), 1261–1270. doi: 10.1080/00036811.2020.1778674 [36] C. Zhang and X. Zhang, Renormalized solutions for the fractional $p(x)$-Laplacian equation with $L^{1}$ data, 2017. arXiv: 1708.04481. -
-
Figures(5) / Tables(5)
Export File
Citation
Omar Djidel, Salah Boulaaras, Mokhtar Boumaaza. NONLOCAL FRACTIONAL P$(Z, Y)$-LAPLACIAN PROBLEMS WITH VARIABLE EXPONENTS: EXISTENCE RESULTS AND A NUMERICAL ILLUSTRATION[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1136-1182. doi: 10.11948/20260073
Format
Content
DownLoad:
-
Figure 1.
Principal representations of the computed solution: (a) Three-dimensional surface plot of
$ \mathrm{w}(x, y) $ -
Figure 2.
Comparison between the solution and the spatial excitation: (a) Heat map of
$ \mathrm{w}(x, y) $ $ z_c=(0.5, 0.5) $ -
Figure 3.
Cross-sectional and radial behavior of the converged solution: (a) Central horizontal and vertical profiles; (b) radially averaged profile with respect to
$ z_c=(0.5, 0.5) $ -
Figure 4.
Convergence diagnostics for the iterative solution procedure: (a) Relative Picard increment and nonlinear residual; (b) number of PCG iterations required at each Picard step.
-
Figure 5.
Sensitivity of the numerical solution to the nonlocal parameters: (a) Influence of the fractional order
$ s $ $ R=0.25 $ $ R $ $ s=0.45 $