Integrability and asymptotics of positive solutions of a γ-Laplace system

In this paper, we use the potential analysis to study the properties of the positive solutions of a γ  -Laplace system in RnRn−div(|∇u|γ−2∇u)=upvq,−div(|∇v|γ−2∇v)=vpuq. Here 1<γ⩽21<γ⩽2, p,q>0p,q>0 satisfy the critical condition p+q=γ⁎−1p+q=γ⁎−1. First, the positive solutions u and v   satisfy an integral system involving the Wolff potentials. We then use the method of regularity lifting to obtain an optimal integrability for this Wolff type integral system. Different from the case of γ=2γ=2, it is more difficult to handle the asymptotics since u and v have not radial structures. We overcome this difficulty by a new method and obtain the decay rates of u and v   as |x|→∞|x|→∞. We believe that this new method is appropriate to deal with the asymptotics of other decaying solutions without the radial structures.