Semilinear nonlocal elliptic equations with source term and measure d...

In this paper, we study the Dirichlet problem for superlinear equation

Lu + u^p = λµ in Ω

with homogeneous boundary or exterior Dirichlet condition, where Ω is a bounded domain, p > 1 and λ > 0. The operator 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.  Our technique is based on a fine analysis on the Green’s kernel, which enables us to construct a theory for semilinear equation in measure frameworks. 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. 

This is a joint work with Phuoc-Tai Nguyen.