A unified treatment of degenerate nonlocal elliptic problems

TL;DR

This paper introduces a comprehensive framework for analyzing a wide range of nonlocal elliptic problems, including classical and nonlinear cases, providing existence, non-existence, and multiplicity results with sharp parameter thresholds.

Contribution

It unifies the treatment of various nonlocal elliptic problems and develops new methods for analyzing solution sets, including a direct approach for powerlike nonlinearities.

Findings

Established existence and multiplicity results for positive solutions.

Identified sharp parameter thresholds for solution existence.

Provided a direct homogeneity-based approach for powerlike nonlinearities.

Abstract

We develop a unified framework for a broad class of nonlocal elliptic problems, encompassing a wide spectrum of nonlocal terms, including the classical Kirchhoff and Carrier-type equations as particular cases, and nonlinearities having sublinear or asymptotically linear growth. By combining the study of a suitable auxiliary problem and fixed-point techniques with careful parameter analysis, we establish existence, non-existence, and multiplicity results for positive solutions. Our method reveals sharp parameter thresholds and provides a comprehensive description of the solution set. Finally, for powerlike nonlinearities (including superlinear and singular ones) we provide a more direct approach, based on homogeneity.