  We show that the Sobolev embedding is compact on punctured manifolds with
conical singularities. On the other hand, we find the Sobolev inequality does
not hold on punctured manifolds with Poincar\'{e} like metric, on which one has
Poincar\'{e} inequality. Applying the results to the Laplace's equation on the
singular manifolds, we obtain the existences of the solution in both cases. In
conical singularity case, we prove further that the solution can be extended to
the singular points and it is H\"{o}lder continuous. However, the solution can
not be continuously extended to the singular points in Poincar\'{e} like metric
case. Moreover, on singular manifolds with conical singularities, we obtain the
existence and regularity result of nontrivial nonnegative solutions for the
semilinear elliptic equation with subcritical exponent.
