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In this work we consider the nonlocal stationary nonlinear 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.doi:10.3934/cpaa.2009.8.2037 fatcat:26gzjzajkjbwdfuwlkhqigq3rm