Qualitative Properties Of Positive Solutions Of Several Types Of Integral Equations

This paper studies some properties of the positive solutions of three types of integral systems with Riesz potentials or Wolff potentials,including the existence and nonexistence of positive solutions,integrability,asymptotic behavior and the connection between integrability and decaying rates,etc.First we prove radial symmetry and monotonicity of positive solutions by the method of moving planes in integral form and the initial integrability.Second we use the regularity lifting lemma to obtain the optimal integrable interval.Based on these results,the asymptotic rates of those positive solutions at infinity is estimated.These results are very helpful for understanding the shape of the solutions.In addition,an iteration and integral estimates are used to study the existence and nonexistence of the positive solutions,and hence we obtain some Liouville-type theorems.Here the Serrin-type condition plays a key role.