On an integral equation of Lieb

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 $\mathbb{R}^n$.