Anisotropic non-local problems: asymptotic behaviour and existence results

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 $C^1$-functionals. Besides, we study the asymptotic behaviour of these solutions with respect to the parameter of perturbation ε and, as $ϵ\to 0$, 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.

Keywords:

• Anisotropic singular perturbations
• non-local problems
• eigenvalues and eigenfunctions
• critical points

AMS Subject Classifications:

• 35B25
• 35B40
• 45K05
• 35J20
• 47A75

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by General Direction of Scientific Research and Technological Development (Algerian Ministry of Higher Education and Scientific Research) [C00L03UN280120180007].