The direct moving sphere for fractional Laplace equation

TL;DR

This paper develops a direct moving spheres method to establish Liouville-type theorems for fractional elliptic equations with general nonlinearities, avoiding Lipschitz conditions and integral representations.

Contribution

It introduces a novel direct moving spheres approach that relaxes previous assumptions, applicable to broader non-local operators without integral solutions.

Findings

Proves Liouville-type theorems under weaker conditions.

Provides an alternative proof for fractional Lane-Emden equations.

Method applicable to general non-local operators.

Abstract

This paper works on the direct method of moving spheres and establishes a Liouville-type theorem for the fractional elliptic equation \[ (-\Delta)^{\alpha/2} u =f(u) ~~~~~~ \text{in } \mathbb{R}^{n} \] with general non-linearity. One of the key improvement over the previous work is that we do not require the usual Lipschitz condition. In fact, we only assume the structural condition that $f(t) t^{- \frac{n+\alpha}{n-\alpha}}$ is monotonically decreasing. This differs from the usual approach such as Chen-Li-Li (Adv. Math. 2017), which needed the Lipschitz condition on $f$, or Chen-Li-Zhang (J. Funct. Anal. 2017), which relied on both the structural condition and the monotonicity of $f$. We also use the direct moving spheres method to give an alternative proof for the Liouville-type theorem of the fractional Lane-Emden equation in a half space. Similarly, our proof does not depend on the integral representation of solutions compared to existing ones. The methods developed here should also apply to problems involving more general non-local operators, especially if no equivalent integral equations exist.