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

TL;DR

This work analyzes positive solutions to a fractional-order Dirichlet problem driven by the Riemann-Liouville derivative with order $α∈(1,2]$. By leveraging Green's function, an order topology framework, and a first eigenvalue $λ_1(α)$, it establishes existence, nonexistence, and uniqueness for sublinear and superlinear nonlinearities, and it proves multiplicity results including at least three positive solutions for a Henon-type problem. The authors deploy Schauder fixed-point theory, Leray-Schauder degree, and nondegeneracy arguments, with careful perturbation arguments to extend results to $α$ near $2$. The results advance fractional-order boundary value theory by providing sharp eigenvalue thresholds, persistence of multiplicity under parameter variation, and a novel uniqueness approach near the integer-order limit.

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+}^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.

Theorems & Definitions (65)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1
• Theorem 1.5
• Remark 1.2
• Theorem 1.6
• Proposition 2.1
• Lemma 2.2