Positivity results for a nonlocal elliptic equation

Abstract

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.