| Abstract: |
| We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem $D_{0+}^\alpha u + h(t)f(u) = 0$ for $tin(0,1)$; $u(0)=u(1)=0$, where $D_{0+}^\alpha$ is the Riemann-Liouville fractional derivative of order $\alpha\in(1,2]$. Firstly, by characterizing the first eigenvalue $\lambda_1(\alpha)$ of the associated eigenvalue problem, we establish the existence of positive solutions for both sublinear and superlinear cases relative to $\lambda_1(\alpha)$, thereby extending previously known results. Secondly, we address the uniqueness of these solutions. In the sublinear case, we impose certain monotonicity conditions on $f$, while for the superlinear case, we assume a specific condition on $h$ to ensure uniqueness at $\alpha=2$. For values of $\alpha$ near $2$, uniqueness is established by leveraging the non-degeneracy of the unique solution. Finally, we apply this methodology to H\`{e}non-type problems to demonstrate the existence of at least three positive solutions. This is a joint work with Inbo Sim (University of Ulsan). |
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