Semilinear nonlocal elliptic equations with source term and measure data

Abstract

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 $p^{\ast }$ and a threshold value ${\lambda }^{\ast }$ such that the multiplicity holds for $1<p<p^{\ast }$ and $0<\lambda <{\lambda }^{\ast }$, the uniqueness holds for $1<p<p^{\ast }$ and $\lambda ={\lambda }^{\ast }$, and the nonexistence holds in other cases. Various types of nonlocal operator are discussed to exemplify the wide applicability of our theory.

We have made changes in Subsection 2.2 and section 5, added Appendix and corrected the proof of Theorem 3.3. The paper will appear in Journal d'Analyse Mathematique