Moving sphere approach to a general weighted integral equation

TL;DR

This work studies positive solutions of the weighted integral equation $u(x)= \int_{\\mathbf{R}^n} |x-y|^p f(|y|,u(y)) dy$ in $\\mathbf{R}^n\\setminus\\{0\\}$ with $p>0$ and $n\\ge 3$, situating it within the context of GJMS operators and fractional Laplacians. It develops a moving-spheres (Kelvin transform) approach in integral form to derive symmetry, assuming a structural condition on $f$ (Condition-F1). The main result shows that any positive solution $u \in C^1(\\mathbf{R}^n\\setminus\\{0\\})$ is radially symmetric about the origin and monotone increasing with respect to the origin. This contributes to Liouville-type classifications and asymptotic analyses for related nonlocal and higher-order elliptic problems and connects integral-equation symmetry to conformal-geometric frameworks.

Abstract

Let $p$ be positive and $n \geq 3$ be an integer. Let $f(\cdot,\cdot): \mathbf{R}_+\times \mathbf{R}_+\to \mathbf{R}_+$ be a continuous function. In this paper, we are concerned with positive solutions to the following integral equation \[ u(x)= \int_{\mathbf{R}^n} |x-y|^p f(|y|,u(y)) dy \quad\text{in }\mathbf{R}^n\setminus\{\textbf{0}\}. \] By imposing some suitable conditions on $f$, we obtain the radially symmetry property of positive solutions to the above equation by using the method of moving spheres in integral form.

Figures (1)

• Figure 1: Inversion in the method of moving spheres.

Theorems & Definitions (10)

• Theorem 1.1
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• proof : Proof of Proposition \ref{['LeM=002']}
• proof