In this article, we are interested in the following non-linear Schrödinger equation with non-local regional diffusion , where ϵ > 0, 0 < α < 1, and is a variational version of the regional Laplacian, whose range of scope is a ball with radius ρϵ(x) = ρ(ϵx) > 0, where ρ is a continuous function. We give general conditions on ρ and f which assure the existence and multiplicity of solution for the cited problem.
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