2021/01/11 by Huynh, Phuoc-Truong, Nguyen, Phuoc-Tai · 1 citation
#35B33 #35B65 #35D30 #35J08 #35J61 #35R06 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2101.03941
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 = up +λμ 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 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 λ^* such that the multiplicity holds for 1