Singular measure as principal eigenfunction of some nonlocal operators

Abstract: 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. $$\int_{\O}K(x,y)\phi(y)\, dy +a(x)\phi(x) =-\lambda \phi(x),$$ where $\O\subset\R^n$ is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue $\lambda_p:=\sup {\lambda \in \R \, |\, \exists \, \phi \in C(\O), \phi > 0 \;\text{so that}\; \oplb{\phi}{\O}+ a(x)\phi + \lambda\phi\le 0}$ there exists always a solution $(\mu, \lambda_p)$ of the problem in the space of signed measure. Moreover $\mu$ a positive measure. When $\mu$ is absolutely continuous with respect to the Lebesgue measure, $\mu =\phi_p(x)$ is called the principal eigenfunction associated to $\lambda_p$. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem.