On a complete parametric Sturm-Liouville problem with sign changing coefficients

Eleonora Amoroso; Giuseppina D'Aguì; Valeria Morabito; Eleonora Amoroso; Giuseppina D'Aguì; Valeria Morabito

doi:10.3934/math.2024316

AIMS Mathematics

2024, Volume 9, Issue 3: 6499-6512. doi: 10.3934/math.2024316

Previous Article Next Article

Research article

On a complete parametric Sturm-Liouville problem with sign changing coefficients

Department of Engineering, University of Messina, 98166 Messina, Italy

Received: 02 January 2024 Revised: 01 February 2024 Accepted: 04 February 2024 Published: 06 February 2024
MSC : 34B10

In this paper we study a complete second order differential equation of Sturm-Liouville type under Dirichlet boundary condition and where the variable coefficients are allowed to be sign changing. Through critical point theory, we obtain the existence of two nontrivial generalized solutions by requiring a specific growth on the nonlinearity. Moreover, the solutions turn out to be nonnegative and with opposite energy sign.
- critical point theory,
- Sturm-Liouville equations,
- nonlinear differential problems,
- ordinary differential equations,
- multiple results
Citation: Eleonora Amoroso, Giuseppina D'Aguì, Valeria Morabito. On a complete parametric Sturm-Liouville problem with sign changing coefficients[J]. AIMS Mathematics, 2024, 9(3): 6499-6512. doi: 10.3934/math.2024316

Related Papers:

Abstract

In this paper we study a complete second order differential equation of Sturm-Liouville type under Dirichlet boundary condition and where the variable coefficients are allowed to be sign changing. Through critical point theory, we obtain the existence of two nontrivial generalized solutions by requiring a specific growth on the nonlinearity. Moreover, the solutions turn out to be nonnegative and with opposite energy sign.

References

[1]	W. O. Amrein, A. M. Hinz, D. B. Pearson, Sturm-Liouville Theory: Past and Present, Birkhäuser Verlag Basel/Switzerland, 2005. https://doi.org/10.1007/3-7643-7359-8
[2]	A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex non linearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519–543. https://doi.org/10.1006/jfan.1994.1078 doi: 10.1006/jfan.1994.1078
[3]	E. Amoroso, G. Bonanno, G. D'Aguì, S. De Caro, S. Foti, D. O'Regan, et al., Second order differential equations for the power converters dynamical performance analysis, Math. Methods Appl. Sci., 45 (2022), 5573–5591. https://doi.org/10.1002/mma.8127 doi: 10.1002/mma.8127
[4]	E. Amoroso, G. Bonanno, G. D'Aguí, S. Foti, Multiple solutions for nonlinear Sturm-Liouville differential equations with possibly negative variable coefficients, Nonlinear Anal. Real World Appl., 69 (2023), 103730. https://doi.org/10.1016/j.nonrwa.2022.103730 doi: 10.1016/j.nonrwa.2022.103730
[5]	P. B. Bailey, J. Billingham, R. J. Cooper, W. N. Everitt, A. C. King, Q. Kong, et al., On some eigenvalue problems in fuel-cell dynamics, Proc. R. Soc. Lond. (A), 459 (2003), 241–261. https://doi.org/10.1098/rspa.2002.1058 doi: 10.1098/rspa.2002.1058
[6]	G. Bonanno, G. D'Aguì, A Neumann boundary value problem for the Sturm-Liouville equation, Appl. Math. Comput., 208 (2009), 318–327. https://doi.org/10.1016/j.amc.2008.12.029 doi: 10.1016/j.amc.2008.12.029
[7]	G. Bonanno, G. D'Aguì, Two non-zero solutions for elliptic Dirichlet problems, Z. Anal. Anwend., 35 (2016), 449–464. https://doi.org/10.4171/zaa/1573 doi: 10.4171/zaa/1573
[8]	G. Bonanno, S. Heidarkhani, D. O'Regan, Nontrivial solutions for Sturm-Liouville systems via a local minimum theorem for functionals, Bull. Aust. Math. Soc., 89 (2014), 8–18. https://doi.org/10.1017/S000497271300035X doi: 10.1017/S000497271300035X
[9]	J. P. Boyd, Sturm-Liouville eigenproblems with an interior pole, J. Math. Phys., 22 (1981), 1575–1590. https://doi.org/10.1063/1.525100 doi: 10.1063/1.525100
[10]	H. Brézis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011. https://doi.org/10.1007/978-0-387-70914-7
[11]	J. R. Graef, S. Heidarkhani, L. Kong, Infinitely many solutions for systems of Sturm-Liouville boundary value problems, Results Math., 66 (2014), 327–341. https://doi.org/10.1007/s00025-014-0379-1 doi: 10.1007/s00025-014-0379-1
[12]	J. R. Graef, S. Heidarkhani, L. Kong, Nontrivial solutions for systems of Sturm-Liouville boundary value problems, Differ. Equ. Appl., 6 (2014), 255–265. https://doi.org/10.7153/dea-06-12 doi: 10.7153/dea-06-12
[13]	J. R. Graef, S. Heidarkhani, L. Kong, Multiple solutions for systems of Sturm- Liouville boundary value problems, Mediterr. J. Math., 13 (2016), 1625–1640. https://doi.org/10.1007/s00009-015-0595-2 doi: 10.1007/s00009-015-0595-2
[14]	S. Heidarkhani, On a class of systems of n Neumann two-point boundary value Sturm-Liouville type equations, Bull. Iran. Math. Soc., 39 (2013), 821–840.
[15]	M. S. Homer, Boundary value problems for the Laplace tidal wave equation, Proc. R. Soc. Lond. (A), 428 (1990), 157–180. https://doi.org/10.1098/rspa.1990.0029 doi: 10.1098/rspa.1990.0029
[16]	Z. Li, XB. Shu, T. Miao, The existence of solutions for Sturm–Liouville differential equation with random impulses and boundary value problems, Bound. Value Probl., 2021 (2021), Article number 97. https://doi.org/10.1186/s13661-021-01574-x doi: 10.1186/s13661-021-01574-x
[17]	J. Sun, H. Chen, Variational method to the impulsive equation with Neumann boundary conditions, Bound. Value Probl., 2009 (2019), Article number 316812. https://doi.org/10.1155/2009/316812 doi: 10.1155/2009/316812
[18]	Y. Tian, W. Ge, Multiple solutions of impulsive Sturm–Liouville boundary value problem via lower and upper solutions and variational methods, J. Math. Anal. Appl., 387 (2012), 475–489. https://doi.org/10.1016/j.jmaa.2011.08.042 doi: 10.1016/j.jmaa.2011.08.042

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)