On an integral equation of Lieb

TL;DR

This paper proves the existence of multiple solutions to a specific non-linear convolution integral equation related to Lieb's problem, and explores related orthogonality relations and solution regularity.

Contribution

It establishes the existence of at least two non-equivalent solutions to a class of weakly singular convolution integral equations, answering a question posed by Lieb.

Findings

Existence of at least two non-equivalent solutions.

Orthogonality relations among differential forms related to convolution operators.

Discussion on the regularity of solutions over open subsets.

Abstract

We prove that the weakly singular, non-linear convolution integral equation $\int_{\mathbb{R}^n}|x-y|^{-\lambda}f(y)dy=f(x)^{p-1}$, where $0<\lambda<n$, and $p=2n/(2n-\lambda)$ has at least two non-equivalent solutions. This answers a problem of Elliott Lieb. We also prove certain orthogonality relations among linear differential forms with constant coefficients related to the corresponding type of convolution operators. Finally, we discuss the regularity of the solutions of such non-linear integral equations over not necessarily bounded open subsets of $\mathbb{R}^n$.