Implicit equations Involving The  p-Biharmonic Operator

Eugenio Cabanillas Lapa; Gabriel Huaraca Pardo; Ronald Huayhua Huayhua

doi:10.14244/lajm.v4i1.38

Authors

Eugenio Cabanillas Lapa Instituto de Investigación- Facultad de Cienias Matematicas- UNMSM https://orcid.org/0000-0002-8941-4394
Gabriel Huaraca Pardo Instituto de Investigación- Facultad de Ciencias Matematicas- UNMSM
Ronald Huayhua Huayhua Instituto de Investigación- Facultad de Cienias Matematicas- UNMSM

DOI:

https://doi.org/10.14244/lajm.v4i1.38

Keywords:

p-biharmonic operator, Implicit Elliptic problems, Krasnoselskii Theorem

Abstract

In this research, we will study the existence of weak solutions for a class of implicit elliptic equations involving the p-biharmonic operator. Using a Krasnoselskii-Schaefer type theorem we establish our result, extending and complementing those obtained by R. Precup, 2020, and P.C. Carriao et al., 2009.

References

[1] Precup R. Implicit elliptic equations via Krasnoselskii–Schaefer type theorems. Electronic Journal of Qualitative Theory of Differential Equations. 2020;2020(87):1-9. DOI: https://doi.org/10.14232/ejqtde.2020.1.87

[2] Bonanno G, Di Bella B. A boundary value problem for fourth-order elastic beam equations. Journal of Mathematical Analysis and Applications. 2008;343(2):1166-76. DOI: https://doi.org/10.1016/j.jmaa.2008.01.049

[3] Lazer AC, McKenna PJ. Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. Siam Review. 1990;32(4):537-78. DOI: https://doi.org/10.1137/1032120

[4] Micheletti AM, Pistoia A. Multiplicity results for a fourth-order semilinear elliptic problem. Nonlinear Analysis: Theory, Methods & Applications. 1998;31(7):895-908. DOI: https://doi.org/10.1016/S0362-546X(97)00446-X

[5] Carl S, Heikkil¨ä S. Discontinuous implicit elliptic boundary value problems. Differential Integral Equations. 1998;11:823-34. DOI: https://doi.org/10.57262/die/1367329478

[6] Cubiotti P. Existence results for highly discontinuous implicit elliptic equations. Atti della Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche, Matematiche e Naturali. 2022;100(1):5.

[7] Dacorogna B, Tanteri C. Implicit partial differential equations and the constraints of nonlinear elasticity. Journal de mathématiques pures et appliquées. 2002;81(4):311-41. DOI: https://doi.org/10.1016/S0021-7824(01)01235-1

[8] Marano SA. Implicit elliptic differential equations. Set-Valued Analysis. 1994;2:545-58. DOI: https://doi.org/10.1007/BF01033071

[9] Dacorogna B, Marcellini P. Implicit partial differential equations. vol. 37. Springer Science & Business Media; 2012.

[10] El Khalil A, Kellati S, Touzani A. On the principal frequency curve of the p-biharmonic operator. Arab Journal of Mathematical Sciences. 2011;17(2):89-99. DOI: https://doi.org/10.1016/j.ajmsc.2011.01.002

[11] Sun J, Wu Tf. Existence of nontrivial solutions for a biharmonic equation with p-Laplacian and singular sign-changing potential. Applied Mathematics Letters. 2017;66:61-7. DOI: https://doi.org/10.1016/j.aml.2016.11.001

[12] Sun J, Wu Tf. The Nehari manifold of biharmonic equations with p-Laplacian and singular potential. Applied Mathematics Letters. 2019;88:156-63. DOI: https://doi.org/10.1016/j.aml.2018.08.025

[13] Wang W. p-biharmonic equation with Hardy–Sobolev exponent and without the Ambrosetti–Rabinowitz condition. Electronic Journal of Qualitative Theory of Differential Equations. 2020;2020(42):1-16. DOI: https://doi.org/10.14232/ejqtde.2020.1.42

[14] Wang W, Zhao P. Nonuniformly nonlinear elliptic equations of p-biharmonic type. Journal of mathematical analysis and applications. 2008;348(2):730-8. DOI: https://doi.org/10.1016/j.jmaa.2008.07.068

[15] Marino G, Paratore A. Implicit equations involving the p-Laplace operator. Mediterranean Journal of Mathematics. 2021;18:1-20. DOI: https://doi.org/10.1007/s00009-021-01713-9

[16] Colasuonno F, Pucci P. Multiplicity of solutions for p (x)-polyharmonic elliptic Kirchhoff equations. Nonlinear Analysis: Theory, Methods & Applications. 2011;74(17):5962-74. DOI: https://doi.org/10.1016/j.na.2011.05.073

[17] Autuori G, Colasuonno F, Pucci P. On the existence of stationary solutions for higher-order p-Kirchhoff problems. Communications in Contemporary Mathematics. 2014;16(05):1450002. DOI: https://doi.org/10.1142/S0219199714500023

[18] Gao H, Li Y, Zhang B. A fixed point theorem of Krasnoselskii-Schaefer type and its applications in control and periodicity of integral equations. Fixed Point Theory. 2011;12(4):91-112.

[19] Adams R, Fournier JJ. Sobolev spaces. vol. 140. Elsevier; 2003.

[20] El Khalil A, Kellati S, Touzani A. On the spectrum of the-biharmonic operator. Electronic Journal of Differential Equations (EJDE)[electronic only]. 2002;2002:161-70.

[21] Gazzola F, Grunau HC, Sweers G. Optimal Sobolev and Hardy–Rellich constants under Navier boundary conditions. Annali di Matematica Pura ed Applicata. 2010;189:475-86. DOI: https://doi.org/10.1007/s10231-009-0118-5

[22] Carriao PC, Faria LF, Miyagaki OH. A biharmonic elliptic problem with dependence on the gradient and the Laplacian. Electronic Journal of Differential Equations (EJDE)[electronic only]. 2009;2009:Paper-No.