Direct Method of Scaling Spheres for the Laplacian and Fractional Laplacian Equations with Hardy-Henon Type Nonlinearity

TL;DR

This paper develops a direct method of scaling spheres to establish Liouville-type theorems for fractional Laplacian equations with Hardy-Henon nonlinearities, extending previous results to more general settings.

Contribution

It introduces a novel application of the direct scaling spheres method to fractional Laplacian equations with Hardy-Henon type nonlinearities, broadening the scope of Liouville-type theorems.

Findings

Derived Liouville-type theorems for fractional Laplacian equations with Hardy-Henon nonlinearities.

Extended previous results to cover equations with nonlinearity depending on both position and solution.

Provided new insights into symmetry and nonexistence of solutions for these equations.

Abstract

In this paper, we focus on the partial differential equation \begin{equation*} (-\Delta)^\frac{\alpha}{2} u(x)=f(x,u(x))\;\;\;\;\text{ in }\mathbb{R}^n, \end{equation*} where $0<\alpha\leq 2$. By the direct method of scaling spheres investigated by Dai and Qin (\cite{dai2023liouville}, \textit{International Mathematics Research Notices, 2023}), we derive a Liouville-type theorem. This mildly extends the previous researches on Liouville-type theorem for the semi-linear equation $ (-\Delta)^\frac{\alpha}{2} u(x)=f(u(x))$ where the nonlinearity $f$ depends solely on the solution $u(x)$, and covers the Liouville-type theorem for Hardy-H\'enon equations $(-\Delta)^\frac{\alpha}{2} u(x)=|x|^au^p(x)$.