Harnack type inequality for positive solution of some integral equation

Abstract: In this paper, we establish some Harnack type inequalities satisfied by positive solutions of nonlocal inhomogeneous equations arising in the description of various phenomena ranging from population dynamics to micro-magnetism. For regular domains, we also derive an inequality up to the boundary. The main difficulty in such context lies in a precise control of the solutions outside a compact set and the existence of local uniform estimates. We overcome this problem by proving a contraction result which makes the $L^1$ norms of the solutions on two compact sets $\o_1\subset\subset\o_2$ equivalent. We also construct the principal positive eigenfunctions associated to particular nonlocal operators by using the corresponding Harnack type inequalities.