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Journal of Applied Mathematics and Physics 2026

Existence of Solutions for (p₁(x),p₂(x)) -Triharmonic Problem with Navier Boundary Conditions

DOI: 10.4236/jamp.2026.142034, PP. 631-647

Zenghui Li, Qing Miao

Keywords: Variable Exponent Space, -Triharmonic Operator, Navier Boundary Condition, Mountain Pass Theorem, Fountain Theorem

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Abstract:

In this paper, we use the Mountain Pass theorem and the Fountain theorem to study the existence of solutions for the following $(p_{1} (x), p_{2} (x))$ -triharmonic equations: ${\begin{array}{l} ？ Δ_{p_{1} (x)}^{3} u ？ Δ_{p_{2} (x)}^{3} u = f (x, u), & in ？ Ω ？, \\ u = Δ u = Δ^{2} u = 0, & on ？？ Ω ？, \end{array}$ . where the nonlinear term satisfying growth conditions weaker than Ambrosetti-Rabinowitz condition. We establish the existence of weak solutions using critical point theory and variational methods.

References

[1]	Acerbi, E. and Mingione, G. (2002) Regularity Results for Electrorheological Fluids: The Stationary Case. Comptes Rendus. Mathématique, 334, 817-822. https://doi.org/10.1016/s1631-073x(02)02337-3
[2]	Acerbi, E. and Mingione, G. (2002) Regularity Results for Stationary Electro-Rheological Fluids. Archive for Rational Mechanics and Analysis, 164, 213-259. https://doi.org/10.1007/s00205-002-0208-7
[3]	Růžička, M. (2004) Modeling, Mathematical and Numerical Analysis of Electrorheological Fluids. Applications of Mathematics, 49, 565-609. https://doi.org/10.1007/s10492-004-6432-8
[4]	Zhikov, V.V. (1987) Averaging of Functionals of the Calculus of Variations and Elasticity Theory. Mathematics of the USSR-Izvestiya, 29, 33-66. https://doi.org/10.1070/im1987v029n01abeh000958
[5]	Lesnic, D. (2009) On the Boundary Integral Equations for a Two-Dimensional Slowly Rotating Highly Viscous Fluid Flow. Advances in Applied Mathematics and Mechanics, 1, 140-150.
[6]	Elder, K.R., Katakowski, M., Haataja, M. and Grant, M. (2002) Modeling Elasticity in Crystal Growth. Physical Review Letters, 88, Article 245701. https://doi.org/10.1103/physrevlett.88.245701
[7]	Chen, Y., Levine, S. and Rao, M. (2006) Variable Exponent, Linear Growth Functionals in Image Restoration. SIAM Journal on Applied Mathematics, 66, 1383-1406. https://doi.org/10.1137/050624522
[8]	Harjulehto, P., Hästö, P. and Latvala, V. (2008) Minimizers of the Variable Exponent, Non-Uniformly Convex Dirichlet Energy. Journal de Mathématiques Pures et Appliquées, 89, 174-197. https://doi.org/10.1016/j.matpur.2007.10.006
[9]	Levine, S.E. (2005) An Adaptive Variational Model for Image Decomposition. In: Rangarajan, A., Vemuri, B. and Yuille, A.L., Eds., Lecture Notes in Computer Science, Springer, 382-397. https://doi.org/10.1007/11585978_25
[10]	González Castro, G., Ugail, H., Willis, P. and Palmer, I. (2007) A Survey of Partial Differential Equations in Geometric Design. The Visual Computer, 24, 213-225. https://doi.org/10.1007/s00371-007-0190-z
[11]	You, L.H., Comninos, P. and Zhang, J.J. (2004) PDE Blending Surfaces with C2 Continuity. Computers & Graphics, 28, 895-906. https://doi.org/10.1016/j.cag.2004.08.003
[12]	Zhang, J.J. and You, L.H. (2004) Fast Surface Modelling Using a 6th Order PDE. Computer Graphics Forum, 23, 311-320. https://doi.org/10.1111/j.1467-8659.2004.00762.x
[13]	Fan, X. and Zhao, D. (2001) On the Spaces and . Journal of Mathematical Analysis and Applications, 263, 424-446. https://doi.org/10.1006/jmaa.2000.7617
[14]	Kováčik, O. and Rákosník, J. (1991) On Spaces and . Czechoslovak Mathematical Journal, 41, 592-618. https://doi.org/10.21136/cmj.1991.102493
[15]	Fan, X. and Zhang, Q. (2003) Existence of Solutions for-Laplacian Dirichlet Problem. Nonlinear Analysis: Theory, Methods & Applications, 52, 1843-1852. https://doi.org/10.1016/s0362-546x(02)00150-5
[16]	El Amrouss, A., Moradi, F. and Moussaoui, M. (2009) Existence of Solutions for Fourth-Order PDEs with Variable Exponents. Electronic Journal of Differential Equations, 153, 1-13.
[17]	Afrouzi, G.A., Chung, N.T. and Mirzapour, M. (2018) Existence of Solutions for a Class of-Biharmonic Problems without (A-R) Type Conditions. International Journal of Mathematical Analysis, 12, 505-515. https://doi.org/10.12988/ijma.2018.8861
[18]	Ayoujil, A., Belaouidel, H., Berrajaa, M. and Tsouli, N. (2020) On a Class of Elliptic Navier Boundary Value Problems Involving the-Biharmonic Operator. Matematicki Vesnik, 72, 196-206.
[19]	Bartsch, T. (1993) Infinitely Many Solutions of a Symmetric Dirichlet Problem. Nonlinear Analysis: Theory, Methods & Applications, 20, 1205-1216. https://doi.org/10.1016/0362-546x(93)90151-h
[20]	LI, L. and Tang, C. (2013) Existence and Multiplicity of Solutions for a Class of-Biharmonic Equations. Acta Mathematica Scientia, 33, 155-170. https://doi.org/10.1016/s0252-9602(12)60202-1
[21]	Zhao, X. and Miao, Q. (2025) Existence of Solutions for-Triharmonic Problem with Navier Boundary Conditions. Journal of Nonlinear Mathematical Physics, 32, Article No. 11. https://doi.org/10.1007/s44198-024-00247-4
[22]	Belakhdar, A., Belaouidel, H., Filali, M. and Tsouli, N. (2023) Positivity of the Infimum Eigenvalue for the-Triharmonic Operator with Variable Exponents. Mediterranean Journal of Mathematics, 20, Article No. 63. https://doi.org/10.1007/s00009-023-02259-8
[23]	Belakhdar, A., Belaouidel, H., Filali, M. and Tsouli, N. (2022) Existence and Multi-Plicity of Solutions of-Triharmonic Problem. Nonlinear Functional Analysis and Applications, 27, 349-361.
[24]	Rahal, B. (2019) Existence Results of Infinitely Many Solutions for-Kirchhoff Type Triharmonic Operator with Navier Boundary Conditions. Journal of Mathematical Analysis and Applications, 478, 1133-1146. https://doi.org/10.1016/j.jmaa.2019.06.006
[25]	Vetro, C. and Vetro, F. (2020) On Problems Driven by the-Laplace Operator. Mediterranean Journal of Mathematics, 17, Article No. 24. https://doi.org/10.1007/s00009-019-1448-1
[26]	Zhong, Q.P. and Wu, T.T. (2020) Existence and Multiplicity of Solutions for a Class of-Biharmonic Equations. Mathematics in Practice and Theory, 50, 240-249.
[27]	Zhang, S. (2018) Multiple Solutions of Navier Boundary Value Problem for Fourth-Order Elliptic Equation with Variable Exponents. Journal of Shandong University (Natural Science), 53, 32-37.
[28]	Boureanu, M., Rădulescu, V. and Repovš, D. (2016) On a-Biharmonic Problem with No-Flux Boundary Condition. Computers & Mathematics with Applications, 72, 2505-2515. https://doi.org/10.1016/j.camwa.2016.09.017
[29]	Zang, A. and Fu, Y. (2008) Interpolation Inequalities for Derivatives in Variable Exponent Lebesgue-Sobolev Spaces. Nonlinear Analysis: Theory, Methods & Applications, 69, 3629-3636. https://doi.org/10.1016/j.na.2007.10.001
[30]	Galewski, M. (2006) On the Continuity of the Nemytskij Operator between the Spaces and . Journal of Mathematical Analysis and Applications, 13, 261-265. https://doi.org/10.1515/gmj.2006.261

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Existence of Solutions for (p1(x),p2(x)) -Triharmonic Problem with Navier Boundary Conditions

Existence of Solutions for (p₁(x),p₂(x)) -Triharmonic Problem with Navier Boundary Conditions