Harnack type inequality for positive solution of some integral equation

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.