Radially symmetric functions as fixed points of some logarithmic operators (Q1275365)

Consider the operator \[ Nu(x)=\int_{\mathbb{R}^2} f(u(y))\log\frac{c}{| x-y| }dy \] on \(C(\mathbb{R}^2)\), where \(f\in C((0,\infty); \mathbb{R}_+)\cap C^1((0,\infty))\) with \(\text{supp}(f)\subset [0,\infty)\). The author shows that if \(Nu=u\), with \(u\in C(\mathbb{R}^2)\) and \(\text{supp}(u^+)\) is a compact set, then there exists \(x_0\in \mathbb{R}^2\) such that \(u(x)\) depends only on \(| x-x_0| \) for all \(x\in \mathbb{R}^2\).