A LOGISTIC EQUATION WITH REFUGE AND NONLOCAL DIFFUSION

Abstract. In this work we consider the nonlocal stationary nonlinear problem (J ∗ u)(x) − u(x) = −λu(x) + a(x)up(x) in a domain Ω, with the Dirichlet boundary condition u = 0 in RN \ Ω and p> 1. The kernel J involved in the convolution (J ∗ u)(x) = ∫RN J(x − y)u(y) dy is a smooth, compactly supported nonnegative function with unit integral, while the weight a(x) is assumed to be nonnegative and is allowed to vanish in a smooth subdomain Ω0 of Ω. Both when a(x) is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter λ. 1