Existence of nodal solutions of nonlinear Lidstone boundary value problems

Meng Yan; Tingting Zhang; Meng Yan; Tingting Zhang

doi:10.3934/era.2024256

Electronic Research Archive

2024, Volume 32, Issue 9: 5542-5556. doi: 10.3934/era.2024256

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Research article

Existence of nodal solutions of nonlinear Lidstone boundary value problems

Meng Yan ^,,
Tingting Zhang

School of Mathematics and Statistics, Xidian University, Xi'an 710071, China

Received: 18 August 2024 Revised: 13 September 2024 Accepted: 23 September 2024 Published: 26 September 2024

We investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem
$ \begin{align} \left\{\begin{array}{ll} (-1)^m (u^{(2m)}(t)+c u^{(2m-2)}(t)) = \lambda a(t)f(u), \; \; \ \ \ t\in (0, r), \\ u^{(2i)}(0) = u^{(2i)}(r) = 0, \ \ i = 0, 1, \cdots, m-1, \end{array} \right.~~(P) \end{align} $
where $ \lambda > 0 $ is a parameter, $ c $ is a constant, $ m\geq1 $ is an integer, $ a :[0, r]\rightarrow [0, \infty) $ is continuous with $ a\not\equiv0 $ on the subinterval within $ [0, r] $, and $ f: \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function. We analyze the spectrum structure of the corresponding linear eigenvalue problem via the disconjugacy theory and Elias's spectrum theory. As applications of our spectrum results, we show that problem $ (P) $ has nodal solutions under some suitable conditions. The bifurcation technique is used to obtain the main results of this paper.
- Lidstone,
- nodal solutions,
- spectrum,
- bifurcation,
- disconjugacy theory
Citation: Meng Yan, Tingting Zhang. Existence of nodal solutions of nonlinear Lidstone boundary value problems[J]. Electronic Research Archive, 2024, 32(9): 5542-5556. doi: 10.3934/era.2024256

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Abstract

We investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem

$ \begin{align} \left\{\begin{array}{ll} (-1)^m (u^{(2m)}(t)+c u^{(2m-2)}(t)) = \lambda a(t)f(u), \; \; \ \ \ t\in (0, r), \\ u^{(2i)}(0) = u^{(2i)}(r) = 0, \ \ i = 0, 1, \cdots, m-1, \end{array} \right.~~(P) \end{align} $

where $ \lambda > 0 $ is a parameter, $ c $ is a constant, $ m\geq1 $ is an integer, $ a :[0, r]\rightarrow [0, \infty) $ is continuous with $ a\not\equiv0 $ on the subinterval within $ [0, r] $, and $ f: \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function. We analyze the spectrum structure of the corresponding linear eigenvalue problem via the disconjugacy theory and Elias's spectrum theory. As applications of our spectrum results, we show that problem $ (P) $ has nodal solutions under some suitable conditions. The bifurcation technique is used to obtain the main results of this paper.

References

[1]	J. P. Keener, Principles of Applied Mathematics: Transformation and Approximation, Addison-Wesley, Redwood City, CA, 1988.
[2]	J. Henderson, H. Wang, Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl., 208 (1997), 252–259. https://doi.org/10.1006/jmaa.1997.5334 doi: 10.1006/jmaa.1997.5334
[3]	Y. X. Li, Positive solutions for second order boundary value problems with derivative terms, Math. Nachr., 289 (2016), 2058–2068. https://doi.org/10.1002/mana.201500040 doi: 10.1002/mana.201500040
[4]	F. Y. Li, Y. J. Zhang, Multiple symmetric nonnegative solutions of second-order ordinary differential equations, Appl. Math. Lett., 17 (2004), 261–267. https://doi.org/10.1016/S0893-9659(04)90061-4 doi: 10.1016/S0893-9659(04)90061-4
[5]	R. Y. Ma, B. Thompson, Nodal solutions for nonlinear eigenvalue problems, Nonlinear Anal., 59 (2004), 707–718. https://doi.org/10.1016/j.na.2004.07.030 doi: 10.1016/j.na.2004.07.030
[6]	R. Y. Ma, Nodal solutions for singular nonlinear eigenvalue problems, Nonlinear Anal., 66 (2007), 1417–1427. https://doi.org/10.1016/j.na.2006.01.028 doi: 10.1016/j.na.2006.01.028
[7]	G. W. Dai, X. L. Han, Global bifurcation and nodal solutions for fourth-order problems with sign-changing weight, Appl. Math. Comput., 219 (2013), 9399–9407. https://doi.org/10.1016/j.amc.2013.03.103 doi: 10.1016/j.amc.2013.03.103
[8]	R. Y. Ma, Nodal solutions of boundary value problems of fourth-order ordinary dierential equations, J. Math. Anal. Appl., 319 (2006), 424–434. https://doi.org/10.1016/j.jmaa.2005.06.045 doi: 10.1016/j.jmaa.2005.06.045
[9]	A. Cabada, J. Cid, L. Sanchez, Positivity and lower and upper solutions for fourth order boundary value problems, Nonlinear Anal., 67 (2007), 1599–1612. https://doi.org/10.1016/j.na.2006.08.002 doi: 10.1016/j.na.2006.08.002
[10]	A. C. Lazer, P. J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181 (1994), 648–655. https://doi.org/10.1006/jmaa.1994.1049 doi: 10.1006/jmaa.1994.1049
[11]	Z. B. Bai, W. G. Ge, Solutions of 2$n$th Lidstone boundary value problems and dependence on higher order derivatives, J. Math. Anal. Appl., 279 (2003), 442–450. https://doi.org/10.1016/S0022-247X(03)00011-8 doi: 10.1016/S0022-247X(03)00011-8
[12]	E. Cetin, F. S. Topal, R. Agarwal, Existence of positive solutions for Lidstone boundary value problems on time scales, Bound. Value Probl., 31 (2023). https://doi.org/10.1186/s13661-023-01707-4
[13]	J. Graef, B. Yang, Positive solutions of the complementary Lidstone boundary value problem, Rocky Mountain J. Math., 51 (2021), 139–147. https://doi.org/10.1216/rmj.2021.51.139 doi: 10.1216/rmj.2021.51.139
[14]	J. Neugebauer, A. Wingo, Positive solutions for a fractional boundary value problem with Lidstone like boundary conditions, Kragujevac J. Math., 48 (2024), 309–322. https://doi.org/10.46793/kgjmat2402.309n doi: 10.46793/kgjmat2402.309n
[15]	P. K. Palamides, Positive solutions for higher-order Lidstone boundary value problems, A new approach via Sperner's lemma, Comput. Math. Appl., 42 (2001), 75–89. https://doi.org/10.1016/S0898-1221(01)00132-8 doi: 10.1016/S0898-1221(01)00132-8
[16]	Q. L. Yao, On the positive solutions of Lidstone boundary value problems, Appl. Math. Comput., 137 (2003), 477–485. https://doi.org/10.1016/S0096-3003(02)00152-2 doi: 10.1016/S0096-3003(02)00152-2
[17]	C. J. Yuan, X. D. Wen, D. Q. Jiang, Existence and uniqueness of positive solution for nonlinear singular 2$m$th-order continuous and discrete Lidstone boundary value problems, Acta Math. Sci., 31 (2011), 281–291. https://doi.org/10.1016/S0252-9602(11)60228-2 doi: 10.1016/S0252-9602(11)60228-2
[18]	H. T. Li, Y. S. Liu, D. O'Regan, Global behaviour of the components of nodal solutions for Lidstone boundary value problems, Appl. Anal., 88 (2009), 1173–1182. https://doi.org/10.1080/00036810903156180 doi: 10.1080/00036810903156180
[19]	J. Xu, X. L. Han, Existence of nodal solutions for Lidstone eigenvalue problems, Nonlinear Anal., 67 (2007), 3350–3356. https://doi.org/10.1016/j.na.2006.10.017 doi: 10.1016/j.na.2006.10.017
[20]	B. Rynne, Global bifurcation for 2$m$th-order boundary value problems and infinitely many solutions of superlinear problems, J. Differ. Equations, 188 (2003), 461–472. https://doi.org/10.1016/S0022-0396(02)00146-8 doi: 10.1016/S0022-0396(02)00146-8
[21]	U. Elias, Eigenvalue problems for the equation $Ly +\lambda p(x)y = 0$, J. Differ. Equations, 29 (1978), 28–57. https://doi.org/10.1016/0022-0396(78)90039-6 doi: 10.1016/0022-0396(78)90039-6
[22]	W. Coppel, Disconjugacy, Lecture Notes in Mathematics, Springer-Verlag, Berlin New York, 1971.
[23]	R. Y. Ma, H. Y. Wang, M. Elsanosi, Spectrum of a linear fourth-order differential operator and its applications, Math. Nachr., 286 (2013), 1805–1819. https://doi.org/10.1002/mana.201200288 doi: 10.1002/mana.201200288
[24]	R. Y. Ma, Y. Q. Lu, Disconjugacy and extremal solutions of nonlinear third-order equations, Commun. Pure Appl. Anal., 13 (2014), 1223–1236. https://doi.org/10.3934/cpaa.2014.13.1223 doi: 10.3934/cpaa.2014.13.1223
[25]	P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487–513. https://doi.org/10.1016/0022-1236(71)90030-9 doi: 10.1016/0022-1236(71)90030-9
[26]	E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1974), 1069–1076. https://doi.org/10.2307/24890776 doi: 10.2307/24890776
[27]	Y. K. An, R. Y. Liu, Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation, Nonlinear Anal., 68 (2008), 3325–3331. https://doi.org/10.1016/j.na.2007.03.028 doi: 10.1016/j.na.2007.03.028
[28]	H. Gu, T. Q. An, Infinitely many solutions for a class of fourth-order partially sublinear elliptic problem, Bound. Value Probl., 1 (2017). https://doi.org/10.1186/s13661-016-0733-1
[29]	E. D. Silva, T. R. Cavalcante, Multiplicity of solutions to fourth-order superlinear elliptic problems under Navier conditions, Electron. J. Differ. Equations, 2017 (2017), 167.
[30]	E. D. Silva, T. R. Cavalcante, Fourth-order elliptic problems involving concave-superlinear nonlinearities, Topol. Methods Nonlinear Anal., 60 (2022), 581–600. https://doi.org/10.12775/TMNA.2022.011 doi: 10.12775/TMNA.2022.011
[31]	E. Kaufmann, A fourth-order iterative boundary value problem with Lidstone boundary conditions, Differ. Equations Appl., 14 (2022), 305–312. https://doi.org/10.7153/dea-2022-14-21 doi: 10.7153/dea-2022-14-21
[32]	Y. H. Long, Q. Q. Zhang, Sign-changing solutions of a discrete fourth-order Lidstone problem with three parameters, J. Appl. Anal. Comput., 12 (2022), 1118–1140. https://doi.org/10.11948/20220148 doi: 10.11948/20220148
[33]	L. T. Wesen, K. G. Yeneblih, Existence of three positive solutions for higher order separated and Lidstone type boundary value problems with $p$-Laplacian, Azerb. J. Math., 11 (2021), 25–38.

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