Asymptotic properties of positive solutions of the Hardy-Sobolev type equations

In this paper, we are concerned with the Lane-Emden type 2m-order PDE with weight(−Δ)mu(x)=|x|σup(x),u>0 in Rn, where n⩾3, p>1, m∈[1,n/2), σ∈(−2m,0], and the more general Hardy-Sobolev type integral equationu(x)=∫Rn|y|σup(y)dy|x−y|n−α, where α∈(0,n), σ∈(−α,0]. If 0n+σn−α, we obtain that the integrable solution u of the integral equation (i.e. u∈Ln(p−1)α+σ(Rn)) is bounded and decays fast with rate n−α. On the other hand, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one α+σp−1. In addition, the classical solution u of the 2m-order PDE satisfies the integral equation with α=2m. Therefore, for the 2m-order PDE, all the decay properties above are still true.