On an integral equation of the form \(u(x)= F(x)+ \int G(x,\xi)u_ +^{(n-2)/2}(\xi)d\xi/ \int u_ +^{(n-2)/2}(\xi)d\xi\) for \(n=2\) and \(n=3\) (Q1893649)

The integral equation \[ u(x)= F(x)+ {\int_{u> 0} G(x, \xi) u^{(n- 2)/2}(\xi)d\xi\over \int_{u> 0} u^{(n- 2)/2} (\xi)d\xi} \] is deduced heuristically, where \(x\in \mathbb{R}^n\), \(\xi\in \mathbb{R}^n\), the function \(G(x, \xi)\) plays the role of the potential of the particle interaction, and the function \(F(x)\) the role of an external potential. The most frequent cases in physics are for \(n= 3\), when the attractive or repulsive potential is of the form \({\alpha\over r}\), where \(\alpha> 0\) is a physical constant, and \(r= \sqrt{(x_1- \xi_1)^2+ (x_2- \xi_2)^2+ (x_3- \xi_3)^2}\) is the distance between particles. Under some conditions, the existence and uniqueness of a continuous solution for the above integral equation are studied.