  The existence problem is solved, and global pointwise estimates of solutions
are obtained for quasilinear and Hessian equations of Lane-Emden type,
including the following two model problems: $ -\Delta_p u = u^q + \mu$ and
$F_k[-u] = u^q + \mu$, $u > 0$, on $R^n$, or on a bounded domain $\Omega$. Here
$\Delta_p$ is the p-Laplacian, and $F_k[u]$ is the $k$-Hessian defined as the
sum of $k\times k$ principal minors of the Hessian matrix $D^2 u$; $\mu$ is a
nonnegative measurable function (or measure) on $\Omega$.
  The solvability of these classes of equations in the renormalized (entropy)
or viscosity sense has been an open problem even for good data $\mu \in L^s
(\Omega)$, $s>1$. Such results are deduced from our existence criteria with the
sharp exponents $s = n(q-p+1)/pq$ for the first equation, and $s = n(q-k)/2kq$
for the second one. Furthermore, a complete characterization of removable
singularities is given.
  Our methods are based on systematic use of Wolff's potentials, dyadic models,
and nonlinear trace inequalities. We make use of recent advances in potential
theory and PDE due to Kilpelainen and Maly, Trudinger and Wang, and Labutin.
This enables us to treat singular solutions, nonlocal operators, and
distributed singularities, and develop the theory simultaneously for
quasilinear equations and equations of Monge-Ampere type.
