Anisotropic non-local problems: asymptotic behaviour and existence results

Abstract

ABSTRACT 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.