  Recently, several works have been carried out in attempt to develop a theory
for linear or sublinear elliptic equations involving a general class of
nonlocal operators characterized by mild assumptions on the associated Green
kernel. In this paper, we study the Dirichlet problem for superlinear equation
(E) ${\mathbb L} u = u^p +\lambda \mu$ in a bounded domain $\Omega$ with
homogeneous boundary or exterior Dirichlet condition, where $p>1$ and
$\lambda>0$. The operator ${\mathbb L}$ belongs to a class of nonlocal
operators including typical types of fractional Laplacians and the datum $\mu$
is taken in the optimal weighted measure space. The interplay between the
operator ${\mathbb L}$, the source term $u^p$ and the datum $\mu$ yields
substantial difficulties and reveals the distinctive feature of the problem. We
develop a new unifying technique based on a fine analysis on the Green kernel,
which enables us to construct a theory for semilinear equation (E) in measure
frameworks. A main thrust of the paper is to provide a fairly complete
description of positive solutions to the Dirichlet problem for (E). In
particular, we show that there exist a critical exponent $p^*$ and a threshold
value $\lambda^*$ such that the multiplicity holds for $1<p<p^*$ and
$0<\lambda<\lambda^*$, the uniqueness holds for $1<p<p^*$ and
$\lambda=\lambda^*$, and the nonexistence holds in other cases. Various types
of nonlocal operator are discussed to exemplify the wide applicability of our
theory.
