Publication details

Semilinear nonlocal elliptic equations with source term and measure data


HUYNH Phuoc-Truong NGUYEN Phuoc-Tai

Year of publication 2023
Type Article in Periodical
Magazine / Source Journal d'Analyse Mathématique
MU Faculty or unit

Faculty of Science

Keywords nonlocal elliptic equations; integro-differential operators; weak-dual solutions; measure data; Green function; mountain pass theorem
Description 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=uP+?µ in a bounded domain ? with homogeneous boundary or exterior Dirichlet condition, where p > 1 and ? > 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. The interplay between the operator 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 ?* such that the multiplicity holds for 1 < p < p* and 0 <? < ?*, the uniqueness holds for 1 < p < p* and ? = ?*, and the nonexistence holds in other cases. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.
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