Positive solutions for fractional-order boundary value problems with or without dependence of integer-order ones

Abstract

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+}^\alpha u + h(t)f(u) = 0, \quad 0<t<1, \\[1ex] u(0)=u(1)=0, \end{array} \right. \end{align} where $D_{0+}^\alpha$ is the Riemann-Liouville fractional derivative of order $\alpha\in(1,2]$. Firstly, by considering the first eigenvalue $\lambda_1(\alpha)$ of the corresponding eigenvalue problem, we establish the existence of positive solutions for both sublinear and superlinear cases involving $\lambda_1(\alpha)$, 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 $\alpha =2.$ Near $\alpha =2,$ 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 H\'{e}non-type problems, which is also a new contribution.