Semilinear nonlocal elliptic equations with source term and measure data

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.