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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2501.05810 |
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| _version_ | 1866917210502463488 |
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| author | Sim, Inbo Tanaka, Satoshi |
| author_facet | Sim, Inbo Tanaka, Satoshi |
| contents | We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P}
\left\{
\begin{array}{l}
D_{0+}^αu + h(t)f(u) = 0, \quad 0<t<1, \\[1ex]
u(0)=u(1)=0,
\end{array}
\right. \end{align} where $D_{0+}^α$ is the Riemann-Liouville fractional derivative of order $α\in(1,2]$. Firstly, by considering the first eigenvalue $λ_1(α)$ of the corresponding eigenvalue problem, we establish the existence of positive solutions for both sublinear and superlinear cases involving $λ_1(α)$, thereby extending existing results in the literature. In addition, we address the issue of non-existence, which reinforces the sharpness of both hypotheses. Secondly, we demonstrate the uniqueness of positive solutions. For the sublinear case, we impose certain monotonicity conditions on $f$. For the superlinear case, we assume that $h$ satisfies a specific condition to ensure the uniqueness of positive solutions when $α=2.$ Near $α=2,$ we prove uniqueness by leveraging the non-degeneracy of the unique solution, which represents a novel approach to studying fractional-order differential equations. Finally, we apply this methodology to establish the multiple existence of at least three positive solutions for Hénon-type problems, which is also a new contribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05810 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive solutions for fractional-order boundary value problems with or without dependence of integer-order ones Sim, Inbo Tanaka, Satoshi Analysis of PDEs 34A08, 34A12, 34B18 We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P} \left\{ \begin{array}{l} D_{0+}^αu + h(t)f(u) = 0, \quad 0<t<1, \\[1ex] u(0)=u(1)=0, \end{array} \right. \end{align} where $D_{0+}^α$ is the Riemann-Liouville fractional derivative of order $α\in(1,2]$. Firstly, by considering the first eigenvalue $λ_1(α)$ of the corresponding eigenvalue problem, we establish the existence of positive solutions for both sublinear and superlinear cases involving $λ_1(α)$, thereby extending existing results in the literature. In addition, we address the issue of non-existence, which reinforces the sharpness of both hypotheses. Secondly, we demonstrate the uniqueness of positive solutions. For the sublinear case, we impose certain monotonicity conditions on $f$. For the superlinear case, we assume that $h$ satisfies a specific condition to ensure the uniqueness of positive solutions when $α=2.$ Near $α=2,$ we prove uniqueness by leveraging the non-degeneracy of the unique solution, which represents a novel approach to studying fractional-order differential equations. Finally, we apply this methodology to establish the multiple existence of at least three positive solutions for Hénon-type problems, which is also a new contribution. |
| title | Positive solutions for fractional-order boundary value problems with or without dependence of integer-order ones |
| topic | Analysis of PDEs 34A08, 34A12, 34B18 |
| url | https://arxiv.org/abs/2501.05810 |