  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 $\mathbb{R}^n$.
