TY - JOUR T1 - EXISTENCE OF SOLUTIONS FOR NAVIER PROBLEM INVOLVING $(p(.),q(.))$-LAPLACIAN AND $(p(.),q(.))$-BIHARMONIC OPERATORS AU - El Katit, Abdessamad AU - El Amrouss, Abdelrachid AU - Kissi, Fouad PY - 2025 DA - September Y2 - 2025 JF - TWMS Journal of Applied and Engineering Mathematics JO - JAEM PB - Işık University Press WT - DergiPark SN - 2146-1147 SP - 2192 EP - 2201 VL - 15 IS - 9 LA - en AB - In the present paper, we are interested in the study of nonlinear problem driven by $(p(.),q(.))$-Laplacian and $(p(.),q(.))$-Biharmonic operators subject to Navier boundary conditions. By means of variational method and critical point theory, we establish the existence of at least one solution and infinitely many solutions under some suitable assumptions. KW - (p(.) KW - q(.))-Laplacian Operator KW - (p(.) KW - q(.))-Biharmonic Operator KW - Navier boundary problem KW - variational method CR - Ambrosetti, A., Rabinowitz, P.H., (1973), Dual variational methods in critical points theory and applications, J. Funct. Anal., 14, pp. 349-381. CR - Allaoui, M., El Amrouss, A.R., Ourraoui, A., (2015), Infinitely many solutions for a nonlinear Navier boundary systems involving $(p (x), q (x)) $-biharmonic, Bol. Socied. Paran. Mat., 33 (1), pp. 157-170. CR - Ayoujil, A., El Amrouss, A. R., (2009), On the spectrum of a fourth order elliptic equation with variable exponent, Nonl. Anal.: Theo., Meth. Appl., 71 (10), pp. 4916-4926. CR - Azorero, J. G., Alonso, I. P., (1998), Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equ., 144 (2), pp. 441-476. CR - Bobkov, V., Tanaka, M., (2015), On positive solutions for $(p, q)$-Laplace equations with two parameters, Calc. Var. Part. Diff. Equ., 54, pp. 3277-3301. CR - Cavalheiro, A.C., (2018), Existence results for Navier problems with degenerated $(p, q)$-Laplacian and $(p, q)$-biharmonic operators, Resul. Nonl. Anal., 1 (2), pp. 74-87. CR - Chen, Y., Levine, S., Rao, M., (2006), Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (4), pp. 1383-1406. CR - Diening, L., (2002), Theoretical and Numerical Results for Electrorheological Fluids, PhD. thesis. CR - Fan, X., Zhao, D., (2001), On the spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2), pp. 424-446. CR - Ho, K., Sim, I., (2015), Existence and multiplicity of solutions for degenerate $ p (x) $-Laplace equations involving concave-convex type nonlinearities with two parameters, Taiw. J. Math., 19 (5), pp. 1469-1493. CR - Ruzicka, M., (2000), Electrorheological Fluids Modeling and Mathematical Theory, Springer-Verlag, Berlin. CR - Willem, M., (1996), Minimax Theorems, Birkhauser, Boston. CR - Zhikov, V. V. E., (1987), Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR-Izv., 29 (1), pp. 33. CR - Moujane, N., El Ouaarabi, M., Allalou, C., (2023), Study of some elliptic system of $(p(x); q(x))$-Kirchhoff type with convection, J. Ell. Parab. Equ., 9 (2), pp. 687-704. CR - El Ouaarabi, M., Allalou, C., Melliani, S., (2023), Weak solutions for double phase problem driven by the $(p (x), q (x))$-Laplacian operator under Dirichlet boundary conditions, Bol. Soci. Para. Mat., 41, pp. 1-14. CR - Ouaarabi, M. E., Allalou, C., Melliani, S., (2023), Existence Result for a Double Phase Problem Involving the $(p (x), q (x))$-Laplacian Operator, Math. Slov., 73 (4), pp. 969-982. CR - Moujane, N., El, O. M., Allalou, C., (2023), Elliptic Kirchhoff-type system with two convections terms and under Dirichlet boundary conditions, Filo., 37 (28), pp. 9693-9707. CR - Allalou, M., El Ouaarabi, M., Raji, A., (2024), On a Class of $p (z)$-Biharmonic Kirchhoff Type Problems with Indefinite Weight and No-Flow Boundary Condition, Iran. J. Sci., pp. 1-10. CR - Arora, R., Shmarev, S., (2022), Double-phase parabolic equations with variable growth and nonlinear sources, Adv. Nonl. Anal., 12 (1), pp. 304-335. CR - Cai, L., Zhang, F., (2024), Normalized solutions for the double-phase problem with nonlocal reaction, Adv. Nonl. Anal., 13 (1), pp. 20240026. CR - Leonardi, S., Papageorgiou, N. S., (2023), Positive solutions for a class of singular $(p, q)$-equations, Adv. Non. Anal., 12 (1), 20220300. CR - Liu, J., Pucci, P., (2023), Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition, Adv. Nonl. Anal., 12 (1), pp. 20220292. CR - Xiang, M., Ma, Y., Yang, M., (2024), Normalized homoclinic solutions of discrete nonlocal double phase problems, Bull. Math. Sci., 14 (02), pp. 2450003. UR - https://dergipark.org.tr/en/pub/twmsjaem/issue//1792032 L1 - https://dergipark.org.tr/en/download/article-file/5278421 ER -