Global branching for semilinear fractional Laplace with sublinear nonlinearity

TL;DR

This paper studies positive solutions to a fractional elliptic problem with sublinear nonlinearity, identifying critical thresholds and establishing existence, nonexistence, and multiplicity results using variational and topological methods.

Contribution

It extends classical results to the nonlocal fractional setting, providing new existence and multiplicity results for sublinear fractional elliptic equations.

Findings

Identified a critical threshold for the parameter λ affecting solution existence

Proved the existence of at least two positive solutions using linking theorem

Extended previous local results to the fractional nonlocal framework

Abstract

This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_{\lambda}^s)$. We begin by establishing several a priori estimates that provide regularity results and describe the qualitative behavior of solutions. A critical threshold level for the parameter $\lambda$ is identified, which plays a crucial role in determining the existence or nonexistence of solutions. The sub and supersolution method is employed to obtain a weak solution. Furthermore, we establish a relation between the local minimizers of $\mathcal{D}^{s,2}(\mathbb{R}^N)$ versus $C(\mathbb{R}^N; 1+|x|^{N-2s})$. Combining these results with the Classical Linking Theorem, we demonstrate the existence of at least two distinct positive weak solutions to $(P_{\lambda}^s)$. This work extends the results of Yang, Abrantes, Ubilla, and Zhou (J. Differential Equations, 416:159-189, 2025) to the nonlocal setting, i.e., when $s \in (0,1)$. Several technical challenges arise in this framework, such as the lack of a standard comparison principle in $\mathbb{R}^N$ in the fractional setting.