Asymptotic Behavior At The Singularities Of Solutions Of Elliptic Equations On Singular Riemannian Manifolds

In this dissertation,we mainly study the asymptotic behavior at singularities of solutions of elliptic equations on singular Riemannian manifolds.It is seen that ellip-tic equations with conical metrics are equivalent to degenerate elliptic equations with singularity at x=0.We present the sharp characterization of the asymptotic behavior at the singularities of nonnegative solutions of degenerate elliptic equations with sin-gular nonlinearities and obtain the corresponding Liouville theorems.Furthermore,we establish expansions of these nonnegative solutions up to arbitrary orders.This disseration contains two parts.In the first part,we show that the Sobolev em-beddings at isolated singularities and co-dimension 2 singularities are both compact on Riemannian manifolds with conical metrics.Applying the compact embedding to ellip-tic equations on singular manifolds,we obtain the existence and regularity of solutions.As contrast,we find the Sobolev equality does not hold on Riemannian manifolds with Poincare like metrics,on which one has the Poincare inequality.In the second part,we study the asymptotic expansions at conical singularities of nonnegative solutions of elliptic equations.Firstly,we prove the Bocher theorems at conical singularities of nonnegative solutions for homogeneous and nonhomogeneous second order elliptic equations and then generalize the results to the fourth and higher order equations.In the last of this disseration,we establish expansions up to arbitary orders at conical singularities of nonnegative solutions of second order semilinear ellip-tic equations,including subcritical and critical equations,which generalizes the earlier works by Caffarelli,Gidas and Spruck to conical metrics.