Singular measure as principal eigenfunction of some nonlocal operators

In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution (lambda, phi) of a nonlocal operator: integral(Omega) K(x, y)phi(y) dy + a(x)phi(x) = -lambda phi(x), where Omega subset of R-n is a bounded domain, K is a nonnegative kernel and a is continuous. We prove that for the generalised principal eigenvalue lambda(p) := sup{lambda is an element of R vertical bar there exists phi is an element of C(Omega), phi > 0 so that L-Omega[phi]+a(x)phi + lambda phi <= 0} there exists always a solution (d mu, lambda(p)) of the problem in the space of positive measure. When d mu, is absolutely continuous with respect to the Lebesgue measure, d(mu) = phi(p) (x)dx is called the principal eigenfunction associated with lambda(p),. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem