INTEGRABILITY AND BOUNDEDNESS OF EXTREMAL FUNCTIONS OF A HARDY-SOBOLEV TYPE INEQUALITY

In this paper, we study the properties of positive solutions of an integral equation in R-n [GRAPHICS] . Such a nonlinear singular equation is related to the study of the best constant of the Hardy-Sobolev type inequality. According to the Newton potential theory, this integral equation is helpful to understand the Henon type partial differential equation when alpha = 2. We use the weighted Hardy-Littlewood-Sobolev inequality to obtain the optimal integrability interval of positive integrable solutions. Namely, if u is an element of L2n/n-alpha-sigma, then u is an element of L2n/n-alpha-sigma(R-n) for all t > n/n-alpha-sigma. Based on this result, we prove that those integrable solutions must be bounded.