Semilinear nonlocal elliptic equations with source term and measure data

Abstract

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) \(\mathbb{L}u=u^{P}+\delta\mu\) in a bounded domain Ω with homogeneous boundary or exterior Dirichlet condition, where p > 1 and λ > 0. The operator \(\mathbb{L}\) belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum μ is taken in the optimal weighted measure space. The interplay between the operator \(\mathbb{L}\), the source term up and the datum μ 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 λ* such that the multiplicity holds for 1 < p < p* and 0 <λ < λ*, the uniqueness holds for 1 < p < p* and λ = λ*, and the nonexistence holds in other cases. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.