Existence and non-existence results for a semilinear fractional Neumann problem

TL;DR

This paper derives bounds for solutions of a nonlocal fractional Neumann problem and establishes conditions for the existence or non-existence of solutions, extending previous analysis to new parameter ranges.

Contribution

It provides new a priori estimates and existence results for radial solutions in fractional Neumann problems, especially for the case 0<s≤1/2.

Findings

Non-existence of non-constant solutions under certain conditions.

Existence of radial, radially non-decreasing solutions for supercritical nonlinearities.

Extension of previous results to the case 0<s≤1/2.

Abstract

We establish a priori $L^\infty$-estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion coefficient and on the nonlinearity. Moreover, we prove an existence result for radial, radially non-decreasing solutions in the case of a possible supercritical nonlinearity, extending to the case $0<s\le 1/2$ the analysis started in [7].