Three nonnegative solutions for Sturm-Liouville BVP and application to the complete Sturm-Liouville equations

Zhongqian Wang; Xuejun Zhang; Mingliang Song; Zhongqian Wang; Xuejun Zhang; Mingliang Song

doi:10.3934/math.2023330

AIMS Mathematics

2023, Volume 8, Issue 3: 6543-6558. doi: 10.3934/math.2023330

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Research article

Three nonnegative solutions for Sturm-Liouville BVP and application to the complete Sturm-Liouville equations

School of Mathematical Sciences, Jiangsu Second Normal University, Nanjing 210013, China

Received: 18 September 2022 Revised: 24 December 2022 Accepted: 26 December 2022 Published: 05 January 2023
MSC : 34B15, 34B24, 35B38, 58E05, 70H05

The main purpose of this manuscript is to investigate the Sturm-Liouville BVP for non-autonomous Lagrangian systems. Under the suitable assumptions, we establish an existence theorem for three nonnegative solutions via Bonanno-Candito's three critical point theory. As an application in the complete Sturm-Liouville equations with Sturm-Liouville BVC, we get an existence theorem of three nonnegative solutions. Meanwhile, we give three examples to show the correctness of our results.
- Sturm-Liouville BVC,
- Lagrangian systems,
- critical points,
- index theory,
- complete Sturm-Liouville equations
Citation: Zhongqian Wang, Xuejun Zhang, Mingliang Song. Three nonnegative solutions for Sturm-Liouville BVP and application to the complete Sturm-Liouville equations[J]. AIMS Mathematics, 2023, 8(3): 6543-6558. doi: 10.3934/math.2023330

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Abstract

The main purpose of this manuscript is to investigate the Sturm-Liouville BVP for non-autonomous Lagrangian systems. Under the suitable assumptions, we establish an existence theorem for three nonnegative solutions via Bonanno-Candito's three critical point theory. As an application in the complete Sturm-Liouville equations with Sturm-Liouville BVC, we get an existence theorem of three nonnegative solutions. Meanwhile, we give three examples to show the correctness of our results.

References

[1]	D. Averna, G. Bonanno, A three critical point theorem and its applications to the ordinary Dirichlet problem, Topol. Method. Nonl. An., 22 (2003), 93–103.
[2]	D. Averna, N. Giovannelli, E. Tornatore, Existence of three solutions for a mixed boundary value problem with the Sturm-Liouville equation, B. Korean Math. Soc., 49 (2012), 1213–1222. https://doi.org/10.4134/BKMS.2012.49.6.1213 doi: 10.4134/BKMS.2012.49.6.1213
[3]	G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. Theor., 75 (2012), 2992–3007. https://doi.org/10.1016/j.na.2011.12.003 doi: 10.1016/j.na.2011.12.003
[4]	G. Bonanno, P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differ. Eq., 244 (2008), 3031–3059. https://doi.org/10.1016/j.jde.2008.02.025 doi: 10.1016/j.jde.2008.02.025
[5]	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
[6]	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
[7]	G. Bonanno, A. Iannizzotto, M. Marras, Two positive solutions for superlinear Neumann problems with a complete Sturm-Liouville operator, J. Convex Anal., 25 (2018), 421–434.
[8]	Y. Chen, Y. Dong, Y. Shan, Existence of solutions for sublinear or superlinear operator equations, Sci. China Math., 58 (2015), 1653–1664. https://doi.org/10.1007/s11425-014-4966-0 doi: 10.1007/s11425-014-4966-0
[9]	G. D'Aguì, Existence results for a mixed boundary value problem with Sturm-Liouville equation, Adv. Pure Appl. Math., 2 (2011), 237–248. https://doi.org/10.1515/apam.2010.043 doi: 10.1515/apam.2010.043
[10]	G. D'Aguì, A. Sciammetta, E. Tornatore, Two non-zero solutions for Sturm-Liouville equations with mixed boundary conditions, Nonlinear Anal. Real, 47 (2019), 324–331. https://doi.org/10.1016/j.nonrwa.2018.11.002 doi: 10.1016/j.nonrwa.2018.11.002
[11]	Y. J. Dong, Index theory for linear self-adjoint operator equations and nontrivial solutions for asymptotically linear operator equations, Calc. Var., 38 (2010), 75–109. https://doi.org/10.1007/s00526-009-0279-5 doi: 10.1007/s00526-009-0279-5
[12]	Y. J. Dong, Index theory for Hamiltonian systems and multiple solutions problems, Beijing: Science Press, 2014.
[13]	H. L. Gao, R. Y. Ma, Multiple positive solutions for a class of Neumann problems, Electron. J. Qual. Differ. Equ., 48 (2015), 1–7.
[14]	Q. Y. Li, F. Cong, Z. Li, J. Lv, Multiplicity of positive solutions of superlinear semi-positone singular Neumann problems, Bound. Value Probl., 217 (2014), 1–11.
[15]	Z. L. Li, Existence of positive solutions of superlinear second-order Neumann boundary value problem, Nonlinear Anal. Theory, 72 (2010), 3216–3221. https://doi.org/10.1016/j.na.2009.12.021 doi: 10.1016/j.na.2009.12.021
[16]	M. L. Song, Existence of solutions for subquadratic convex or $B$-concave operator equations and applications to second order Hamiltonian systems, Electron. J. Qual. Differ. Equ, 49 (2020), 1–19.
[17]	J. P. Sun, W. T. Li, Multiple positive solutions to second-order Neumann boundary value problems, Appl. Math. Comput., 146 (2003), 187–194. https://doi.org/10.1016/S0096-3003(02)00535-0 doi: 10.1016/S0096-3003(02)00535-0
[18]	J. P. Sun, W. T. Li, S. S. Cheng, Three positive solutions for second-order Neumann boundary value problems, Appl. Math. Lett., 17 (2004), 1079–1084. https://doi.org/10.1016/j.aml.2004.07.012 doi: 10.1016/j.aml.2004.07.012
[19]	Q. L. Yao, Multiple positive solutions of nonlinear Neumann problems with time and space singularities, Appl. Math. Let., 25 (2012), 93–98. https://doi.org/10.1016/j.aml.2011.06.001 doi: 10.1016/j.aml.2011.06.001
[20]	Y. W. Zhang, H. X. Li, Positive solutions of a second-order Neumann boundary value problem with a parameter, Bull. Aust. Math. Soc., 86 (2012), 244–253. https://doi.org/10.1017/S0004972712000159 doi: 10.1017/S0004972712000159
[21]	E. Zeidler, Nonlinear functional analysis and its applications, New York: Springer, 1985.

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