On the nonexistence of positive solution of a nonlinear integral equation (Q2777558)

The authors consider the following nonlinear integral equation NEWLINE\[NEWLINEu(x)= \int_{\mathbb{R}^N} {g(y,u(y))\over|y-x|^\sigma} dy,\quad \text{for all }x \in\mathbb{R}^NNEWLINE\]NEWLINE where \(\sigma\) is a given constant with \(0<\sigma <N\), \(N\geq 2\) and the given function \(g(y,u)\) is a continuous, bounded below by the power function of order \(\alpha\) with respect to the variable \(u\). Using elementary arguments they prove that if \(0\leq \alpha\leq N/ \sigma\), \(N\geq 2\), the above nonlinear integral equation has no positive solution.