$ \phi - $Laplacian second-order boundary value problems with parameters

Feliz Minhós; Nuno Oliveira; Feliz Minhós; Nuno Oliveira

doi:10.3934/era.2026126

Electronic Research Archive

2026, Volume 34, Issue 5: 2760-2773. doi: 10.3934/era.2026126

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$ \phi - $Laplacian second-order boundary value problems with parameters

Feliz Minhós ^{1,2
,
,},
Nuno Oliveira ²

1.
Departamento de Matemática, Escola de Ciências e Tecnologia, Universidade de Évora, Portugal
2.
Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada, Universidade de Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal

Received: 08 January 2026 Revised: 15 March 2026 Accepted: 23 March 2026 Published: 30 March 2026

This paper deals with a second-order fully differential equation involving a $ \phi - $Laplacian operator, which generalizes the more common $ p $-Laplacian, with a parameter, applied to a nonlinearity depending on the unknown function and its first derivative. The main result states the problem's solvability for parameter values for which there are lower and upper solutions for the problem. Singular $ \phi - $Laplacians considered with an application to the special relativity field.
- lower and upper solutions,
- Nagumo condition,
- regular and singular $ \phi - $Laplacian equations,
- fixed-point theory
Citation: Feliz Minhós, Nuno Oliveira. $ \phi - $Laplacian second-order boundary value problems with parameters[J]. Electronic Research Archive, 2026, 34(5): 2760-2773. doi: 10.3934/era.2026126

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Abstract

This paper deals with a second-order fully differential equation involving a $ \phi - $Laplacian operator, which generalizes the more common $ p $-Laplacian, with a parameter, applied to a nonlinearity depending on the unknown function and its first derivative. The main result states the problem's solvability for parameter values for which there are lower and upper solutions for the problem. Singular $ \phi - $Laplacians considered with an application to the special relativity field.

References

[1]	Y. Xin, Z. Cheng, Multiple results to $\phi $ -Laplacian singular Liénard equation and applications, J. Fixed Point Theory Appl., 23 (2021), 21. https://doi.org/10.1007/s11784-021-00860-6 doi: 10.1007/s11784-021-00860-6
[2]	T. Jung, Q. H. Choi, Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities, Boundary Value Probl., 2019 (2019), 56. https://doi.org/10.1186/s13661-019-1165-5 doi: 10.1186/s13661-019-1165-5
[3]	X. Wang, On a class of one-dimensional superlinear semipositone p-Laplacian problem, AIMS Math., 8 (2023), 25740–25753. https://doi.org/10.3934/math.20231313 doi: 10.3934/math.20231313
[4]	Y. Y. Yang, Q. R. Wang, Multiple positive solutions for p-Laplacian equations with integral boundary conditions, J. Math. Anal. Appl., 453 (2017), 558–571. https://doi.org/10.1016/j.jmaa.2017.04.013 doi: 10.1016/j.jmaa.2017.04.013
[5]	K. D. Chu, D. Hai, Positive solutions for the one-dimensional Sturm-Liouville superlinear p-Laplacian problem, Electron. J. Differ. Equations, 2018 (2018), 1–14. https://doi.org/10.1051/itmconf/20182002002 doi: 10.1051/itmconf/20182002002
[6]	S. Tanaka, An identity for a quasilinear ODE and its applications to the uniqueness of solutions of BVPs, J. Math. Anal. Appl., 351 (2009), 206–217. https://doi.org/10.1016/j.jmaa.2008.09.069 doi: 10.1016/j.jmaa.2008.09.069
[7]	H. Tair, K. Bachouche, T. Moussaou, A four-point $\phi $ -Laplacian BVPs with first-order derivative dependence, Adv. Oper. Theory, 8 (2023). https://doi.org/10.1007/s43036-022-00227-9 doi: 10.1007/s43036-022-00227-9
[8]	M. Shao, Existence and multiplicity of solutions to $ \mathit{p}$-Laplacian equations on graphs, Rev. Mat. Complutense, 37 (2024), 185–203. https://doi.org/10.1007/s13163-022-00452-z doi: 10.1007/s13163-022-00452-z
[9]	F. Minhós, G. Rodrigues, Impulsive coupled systems with regular and singular $\varphi $-Laplacian and generalized jump conditions, Boundary Value Probl., 2024 (2024), 73. https://doi.org/10.1186/s13661-024-01882-y doi: 10.1186/s13661-024-01882-y
[10]	U. Guarnotta, R. Livrea, S. A. Marano, Some recent results on singular p-Laplacian equations, Demonstr. Math., 55 (2022), 416–428. https://doi.org/10.1515/dema-2022-0031 doi: 10.1515/dema-2022-0031
[11]	U. Guarnotta, R. Livrea, S. A. Marano, Some recent results on singular p-laplacian systems, Discrete Contin. Dyn. Syst., 2022 (2022), 1435–1451. https://doi.org/10.3934/dcdss.2022170
[12]	C. Bereanu, J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi $-Laplacian, J. Differ. Equations, 243 (2007), 536–557. https://doi.org/10.1016/j.jde.2007.05.014
[13]	P. Binding, P. Drábek, Sturm-Liouville theory for the p-Laplacian, Stud. Sci. Math. Hung., 40 (2003), 373–396. https://doi.org/10.1556/SScMath.40.2003.4.1 doi: 10.1556/SScMath.40.2003.4.1
[14]	K. Perera, E. A. B. Silva, On singular p-Laplacian problems, Differ. Integr. Equations, 20 (2007), 105–120. https://doi.org/10.57262/die/1356050283
[15]	A. Cabada, J. Ángel Cid, Heteroclinic solutions for non-autonomous boundary value prob- ´ lems with singular $\phi $ -Laplacian operators, Discrete Contin. Dyn. Syst., 2009 (2009), 118–122. https://doi.org/10.3934/proc.2009.2009.118
[16]	F. Minhós, On heteroclinic solutions for BVPs involving $ \varphi $-Laplacian operators without asymptotic or growth assumptions, Math. Nachr., 2018 (2018), 1–9. https://doi.org/10.1002/mana.201700470
[17]	F. Minhós, Heteroclinic solutions for classical and singular $\varphi $-Laplacian non-autonomous differential equations, Axioms, 8 (2019), 22. https://doi.org/10.3390/axioms8010022
[18]	J. Fialho, F. Minhós, H. Carrasco, Singular and regular second order $\varphi $-Laplacian equations on the half-line with functional boundary conditions, Electron. J. Qual. Theory Differ. Equations, 10 (2017), 1–15. https://doi.org/10.14232/ejqtde.2017.1.10 doi: 10.14232/ejqtde.2017.1.10
[19]	L. Gambera, U. Guarnotta, Existence, uniqueness, and decay results for singular $\phi $-Laplacian systems in $ \mathbb{R} ^{N}$, Nonlinear Differ. Equations Appl., 31 (2024), 111. https://doi.org/10.1007/s00030-024-01001-x
[20]	M. X. Wang, A. Cabada, J. Nieto, Monotone method for nonlinear second order periodic boundary value problems with Carathé odory functions, Ann. Polon. Math., 58 (1993), 221–235.
[21]	E. Zeidler, Nonlinear Functional Analysis and Its Applications: Fixed-Point Theorems, Springer, New York, 1986.
[22]	J. D. Jackson, Classical Electrodynamics, Wiley, New York, 2003.
[23]	R. D. Blandford, K. S. Thorne, Applications of Classical Physics, California Institute of Technology, Pasadena, 2008.
[24]	J. P. Goedbloed, R, Keppens, S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2010.

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