Abstract
In this paper, we study the following integral equationu(x)=∫Rnup(y)(1+|y|α)|x−y|n−αdy,u>0inRn, where n≥3, α∈(0,n), and p>0. When α=2, the equation is associated with the classical Matukuma equation−Δu=up(x)1+|x|2,u>0inRn. We investigate qualitative properties of two kinds of positive solutions. One is the finite total mass solution, and the other is the finite energy solution. We investigate the qualitative properties of these solutions, including existence, radial symmetry, integrability and asymptotic behavior at infinity. Here we use the method of moving planes in integral forms introduced by Chen-Li-Ou to prove the radial symmetry, and develop a new comparing principle in integral forms to estimate the asymptotic rates. In addition, we consider the relation between finite total mass solutions and finite energy solutions, and in this process the Pohozaev identity in integral forms plays a key role.
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