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We deal with anisotropic singular perturbation problems in some weighted spaces of Sobolev type. The perturbed problems are elliptic, semi-linear and non-local. Using a variational approach, we establish the existence of their solutions as critical points of some-functionals. Besides, we study the asymptotic behaviour of these solutions with respect to the parameter of perturbation  ε  and, as, we obtain existence results for some non-standard integro-differential problems. A non-local eigenvalue problem related to the considered problems is also investigated and used to carry out this study.

Anisotropic non-local problems: asymptotic behaviour and existence results