Today we study a beautiful version of the method of moving planes. This method just invented around 2003 by W.X. Chen, C.M. Li and B. Ou and published in Comm. Pure Appl. Math. around 2006 (for the paper, please go here). In that paper, the author proved among other things that every positive regular solution of the integral equation
is radially symmetric and monotone about some point and therefore assumes the form
with some constant and for some and . These solutions in case and have the following shape.
This integral equation is also closely related to the following family of semilinear PDEs
.