Existence of multiple solutions for a singular $ p(\cdot) $-biharmonic problem with variable exponents

Ramzi Alsaedi; Ramzi Alsaedi

doi:10.3934/math.2025175

AIMS Mathematics

2025, Volume 10, Issue 2: 3779-3796. doi: 10.3934/math.2025175

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Existence of multiple solutions for a singular $ p(\cdot) $-biharmonic problem with variable exponents

Ramzi Alsaedi ^,

Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 28 November 2024 Revised: 12 February 2025 Accepted: 14 February 2025 Published: 25 February 2025
MSC : 35G30, 35J60, 35J20, 35J35

In this work, we study a multiplicity result related to a $ p(\tau) $-Biharmonic equation involving singular and Hardy nonlinearities. More precisely, we use the variational method to develop the analysis of the fibering map in the Nehari manifold sets to prove the existence of two nontrivial solutions for such a problem.
- $ p(\tau) $-biharmonic problem,
- variational method,
- singular equation,
- variable exponents
Citation: Ramzi Alsaedi. Existence of multiple solutions for a singular $ p(\cdot) $-biharmonic problem with variable exponents[J]. AIMS Mathematics, 2025, 10(2): 3779-3796. doi: 10.3934/math.2025175

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Abstract

In this work, we study a multiplicity result related to a $ p(\tau) $-Biharmonic equation involving singular and Hardy nonlinearities. More precisely, we use the variational method to develop the analysis of the fibering map in the Nehari manifold sets to prove the existence of two nontrivial solutions for such a problem.

References

[1]	Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383–1406. https://doi.org/10.1137/050624522 doi: 10.1137/050624522
[2]	Y. Su, B. Liu, Z. Feng, Ground state solution of the thin film epitaxy equation, J. Math. Anal. Appl., 503 (2021), 125357. https://doi.org/10.1016/j.jmaa.2021.125357 doi: 10.1016/j.jmaa.2021.125357
[3]	T. C. Halsey, Electrorheological fluids, Science, 258 (1992), 761–766. https://doi.org/10.1126/science.258.5083.761 doi: 10.1126/science.258.5083.761
[4]	M. Ruzicka, Electrortheological fluids: Modeling and mathematical theory, Berlin: Springer, 2000.
[5]	V. V. Zhikov. Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR-Izvestiya, 29 (1987), 33–66. https://doi.org/10.1070/IM1987v029n01ABEH000958 doi: 10.1070/IM1987v029n01ABEH000958
[6]	G. H. Hardy, Notes on some points in the integral calculus LX, Messenger Math., 54 (1925), 150–156.
[7]	J. Necǎs, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm.-Sci., 16 (1962), 305–326.
[8]	Y. Su, H. Chen, The existence of nontrivial solution for biharmonic equation with sign-changing potential, Math. Method. Appl. Sci., 41 (2018), 6170–6183. https://doi.org/10.1002/mma.5127 doi: 10.1002/mma.5127
[9]	Y. Su, H. Shi, Ground state solution of critical biharmonic equation with Hardy potential and $p$- Laplacian, Appl. Math. Lett., 112 (2021), 106802. https://doi.org/10.1016/j.aml.2020.106802 doi: 10.1016/j.aml.2020.106802
[10]	Y. Su, Z. Feng, Ground state solution to the biharmonic equation, Z. Angew. Math. Phys., 73 (2022), 15. https://doi.org/10.1007/s00033-021-01643-2 doi: 10.1007/s00033-021-01643-2
[11]	R. Alsaedi, Infinitely many solutions for a class of fractional Robin problems with variable exponents, AIMS Math., 6 (2021), 9277–9289. https://doi.org/10.3934/math.2021539 doi: 10.3934/math.2021539
[12]	A. Dhifli, R. Alsaedi, Existence and multiplicity of solutions for a singular problem involving the $p$-biharmonic operator in $\mathbb{R}^N$, J. Math. Anal. Appl., 499 (2021), 125049. https://doi.org/10.1016/J.JMAA.2021.125049 doi: 10.1016/J.JMAA.2021.125049
[13]	C. Ji, W. Wang, On the $p$-biharmonic equation involving concave-convex nonlinearities and sign-changing weight function, Electron. J. Qual. Theo., 2 (2012), 1–17.
[14]	A. Ghanmi, A. Sahbani, Existence results for $p(x)$-biharmonic problems involving a singular and a Hardy type nonlinearities, AIMS Math., 8 (2023), 29892–29909. https://doi.org/10.3934/math.20231528 doi: 10.3934/math.20231528
[15]	R. Alsaedi, A. Dhifli, A. Ghanmi, Low perturbations of $p$-biharmonic equations with competing nonlinearities, Complex Var. Elliptic, 66 (2021), 642–657. https://doi.org/10.1080/17476933.2020.1747057 doi: 10.1080/17476933.2020.1747057
[16]	A. Drissi, A. Ghanmi, D. D. Repovš, Singular $p$-biharmonic problem with the Hardy potential, Nonlinear Anal. Model. Control, 29 (2024), 762–782. https://doi.org/10.15388/namc.2024.29.35410 doi: 10.15388/namc.2024.29.35410
[17]	V. D. Rǎdulescu, D. D. Repovš, Combined effects for non-autonomous singular biharmonic problems, AIMS Math., 13 (2020), 2057–2068. https://doi.org/10.3934/dcdss.2020158 doi: 10.3934/dcdss.2020158
[18]	M. Avci, Existence results for a class of singular $p(x)$-Kirchhoff equations, Complex Var. Elliptic, 2024, 1–32. https://doi.org/10.1080/17476933.2024.2378316 doi: 10.1080/17476933.2024.2378316
[19]	D. D. Repovš, K. Saoudi, The Nehari manifold approach for singular equations involving the $p(x)$-Laplace operator, Complex Var. Elliptic, 68 (2023), 135–149. https://doi.org/10.1080/17476933.2021.1980878 doi: 10.1080/17476933.2021.1980878
[20]	S. Saiedinezhad, M. Ghaemi, The fibering map approach to a quasilinear degenerate $p (x)$-Laplacian equation, B. Iran. Math. Soc., 41 (2015), 1477–1492.
[21]	A. El khalil, M. El Moumni, M. Alaoui, A. Touzani, $p(x)$-Biharmonic operator involving $p(x)$-Hardy's inequality, Georgian Math. J., 27 (2020), 233–247. https://doi.org/10.1515/gmj-2018-0013 doi: 10.1515/gmj-2018-0013
[22]	X. Fan, D. Zhao, On the spaces $ L^{p}(\Omega) $ and $ W^{m, p(x)}(\Omega) $, J. Math. Anal. Appl., 263 (2001), 424–446. https://doi.org/10.1006/jmaa.2000.7617 doi: 10.1006/jmaa.2000.7617
[23]	M. Laghzal, A. El Khalil, M. Alaoui, A. Touzani, Eigencurves of the $p(x)$-biharmonic operator with a Hardy-type term, Moroc. J. Pure Appl. Anal., 6 (2020), 198–209. https://doi.org/10.2478/mjpaa-2020-0015 doi: 10.2478/mjpaa-2020-0015
[24]	R. Chammem, A. Ghanmi, A. Sahbani, Nehari manifold for singular fractional $p(x, \cdot)$-Laplacian problem, Complex Var. Elliptic, 68 (2023), 1603–1625. https://doi.org/10.1080/17476933.2022.2069757 doi: 10.1080/17476933.2022.2069757
[25]	A. Sahbani, Infinitely many solutions for problems involving Laplacian and biharmonic operators, Com. Var. Elliptic, 69 (2023), 2138–2151. https://doi.org/10.1080/17476933.2023.2287007 doi: 10.1080/17476933.2023.2287007
[26]	M. Hsini, N. Irzi, K. Kefi, Existence of solutions for a $p(x)$-biharmonic problem under Neumann boundary conditions, Appl. Anal., 100 (2021), 2188–2199. https://doi.org/10.1080/00036811.2019.1679788 doi: 10.1080/00036811.2019.1679788
[27]	H. Brezis, E. Lieb, A Relation between pointwise convergence of function and convergence of functionals, P. Am. Math. Soc., 88 (1983), 486–490. https://doi.org/10.2307/2044999 doi: 10.2307/2044999

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