Qualitative Analysis Of Positive Solutions Of A Class Of Quasilinear Elliptic Equations With Hardy Potential

In the paper,we consider the following quasilinear elliptic equation with Hardy potential p-1.where N > p > 1,? >-p,? (?)RN(N ? 3)is a domain and 0 ? ?.We will mainly study uniqueness and asymptotic behavior of positive solutions.Firstly,we show the existence of maximal positive solution and minimal positive solution,and then obtain the estimate of positive solutions of the equation(P).Secondly,we will prove the following conclusions:(i)Suppose that ? = BR(0)and ? ? 0.Then(P)with Dirichlet boundary condition has a unique positive solution;(ii)Suppose that ? = RN,? ? 0 and ? >N(s-p+1)-sp/p-1 hold.Then(P)has a unique positive solution;(iii)Suppose that ? is a bounded smooth domain,? ? 0 and ? >N(q-p+1)-pq/p-1.Then any positive solution u of(P)has the same blow-up rate i.e.,and the equation(P)with boundary condition u = (?) ? 0 on ?? has a unique positive solution.Finally,by using the similar arguments,we obtain the uniqueness of positive solutions to quasilinear equation with a multi-singular potential.