  We solve the nonlinear Dirichlet problem (uniquely) for functions with
prescribed asymptotic singularities at a finite number of points, and with
arbitrary continuous boundary data, on a domain in euclidean space. The main
results apply, in particular, to subequations with a Riesz characteristic $p
\geq 2$. In this case it is shown that, without requiring uniform ellipticity,
the Dirichlet problem can be solved uniquely for arbitrary continuous boundary
data with singularities asymptotic to the Riesz kernel: $\Theta_j K_p(x -
x_j)$, where $K_p(x) = - {1\over|x|^{p-2}}$ for $p>2$ and $K_2(x) = \log |x|$,
at any prescribed finite set of points $x_1,...,x_k$ in the domain and any
finite set of positive real numbers $\Theta_1,..., \Theta_k$. This sharpens a
previous result of the authors concerning the discreteness of high-density sets
of subsolutions.
  Uniqueness and existence results are also established for finite-type
singularities such as $\Theta_j |x - x_j|^{2-p}$ for $1\leq p<2$.
  The main results apply similarly with prescribed singularities asymptotic to
the fundamental solutions of Armstrong-Sirakov-Smart (in the uniformly elliptic
case).
