Recently, several works have been undertaken in an 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) Lu=u(P)+lambda mu in a bounded domain Omega with homogeneous boundary or exterior Dirichlet condition, where p > 1 and lambda > 0. The operator 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 L, the source term u(p) and the datum mu yields substantial difficulties and reveals the distinctive feature of the problem. We develop a 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 展开▼
