Global branching for semilinear fractional Laplace with sublinear nonlinearity
Jefferson Abrantes, Rohit Kumar, Abhishek Sarkar
TL;DR
The paper analyzes a nonlocal semilinear problem driven by the fractional Laplacian with a sublinear nonlinearity on $\mathbb{R}^N$. It develops a robust variational framework, leveraging a weighted eigenvalue problem and sub-/supersolution techniques to identify a positive threshold $\Lambda_s$ that delineates existence versus nonexistence: a positive weak solution exists for $\lambda>\Lambda_s$ and none for $\lambda<\Lambda_s$, with potential attainment at $\lambda=\Lambda_s$ under extra conditions. For large $\lambda$, the authors establish multiplicity by combining a local minimizer in a decay-weighted space with a linking argument to obtain a second positive weak solution, ensuring $v_\lambda<\tilde v_\lambda$. The results extend known local Laplacian findings to the nonlocal setting with $s\in(0,1)$ and address technical challenges such as the lack of a standard comparison principle in $\mathbb{R}^N$, providing new insights into nonlocal diffusion models and their solution structure.
Abstract
This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_λ^s)$. 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 $λ$ 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 $(P_λ^s)$. 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.