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 (λ,ϕ)(λ,ϕ) of a nonlocal operator: ∫ΩK(x,y)ϕ(y)dy+a(x)ϕ(x)=−λϕ(x), where Ω⊂RnΩ⊂Rn is a bounded domain, KK is a nonnegative kernel and aa is continuous. We prove that for the generalised principal eigenvalue λp≔sup{λ∈R∣∃ϕ∈C(Ω),ϕ>0 so that LΩ[ϕ]+a(x)ϕ+λϕ≤0} there exists always a solution (dμ,λp)(dμ,λp) of the problem in the space of positive measure. When dμdμ is absolutely continuous with respect to the Lebesgue measure, dμ=ϕp(x)dxdμ=ϕp(x)dx is called the principal eigenfunction associated with λpλp. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem.