Existence of nodal solutions of nonlinear Lidstone boundary value problems
We investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem \begin{document}$ \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,...
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AIMS Press
2024-09-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024256 |
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| author | Meng Yan Tingting Zhang |
| author_facet | Meng Yan Tingting Zhang |
| author_sort | Meng Yan |
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| description | We investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem \begin{document}$ \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} $\end{document} 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. |
| format | Article |
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| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-09-01 |
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| spelling | doaj-art-bef708cf0e064a0bbdb5e66a8b49c4b62025-01-23T07:52:42ZengAIMS PressElectronic Research Archive2688-15942024-09-013295542555610.3934/era.2024256Existence of nodal solutions of nonlinear Lidstone boundary value problemsMeng Yan0Tingting Zhang1School of Mathematics and Statistics, Xidian University, Xi'an 710071, ChinaSchool of Mathematics and Statistics, Xidian University, Xi'an 710071, ChinaWe investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem \begin{document}$ \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} $\end{document} 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.https://www.aimspress.com/article/doi/10.3934/era.2024256lidstonenodal solutionsspectrumbifurcationdisconjugacy theory |
| spellingShingle | Meng Yan Tingting Zhang Existence of nodal solutions of nonlinear Lidstone boundary value problems Electronic Research Archive lidstone nodal solutions spectrum bifurcation disconjugacy theory |
| title | Existence of nodal solutions of nonlinear Lidstone boundary value problems |
| title_full | Existence of nodal solutions of nonlinear Lidstone boundary value problems |
| title_fullStr | Existence of nodal solutions of nonlinear Lidstone boundary value problems |
| title_full_unstemmed | Existence of nodal solutions of nonlinear Lidstone boundary value problems |
| title_short | Existence of nodal solutions of nonlinear Lidstone boundary value problems |
| title_sort | existence of nodal solutions of nonlinear lidstone boundary value problems |
| topic | lidstone nodal solutions spectrum bifurcation disconjugacy theory |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024256 |
| work_keys_str_mv | AT mengyan existenceofnodalsolutionsofnonlinearlidstoneboundaryvalueproblems AT tingtingzhang existenceofnodalsolutionsofnonlinearlidstoneboundaryvalueproblems |