A LOGISTIC EQUATION WITH REFUGE AND NONLOCAL DIFFUSION

In this work we consider the nonlocal stationary nonlinear problem (J * u)(x) - u(x) = - lambda u(x) + a(x)u(p)(x) in a domain Omega, with the Dirichlet boundary condition u(x) = 0 in R-N \ Omega and p > 1. The kernel J involved in the convolution (J * u)(x) = integral N-R 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 Omega(0) of Omega. 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 lambda.