Study of nonlinear anisotropic elliptic problems with non-local boundary conditions in weighted variable exponent Sobolev spaces

Abstract

This study investigates the existence of weak solutions for nonlinear anisotropic elliptic equations characterized by non-local boundary conditions within anisotropic weighted variable exponent Sobolev spaces. By employing variational methods and compact embedding theorems tailored to anisotropic Sobolev spaces, the research focuses on understanding the impact of anisotropy, non-locality, and weighted structures on the solution behavior. We establish sufficient conditions for the existence of solutions under various boundary conditions. These results deepen the understanding of anisotropic elliptic problems by highlighting the role of weighted structures and variable exponents in the interaction between anisotropy and non-locality. The study also explores non-local boundary conditions, which may include integrals of the unknown function over parts of the domain or non-local operators, often encountered in applications such as well modeling in 3D stratified petroleum reservoirs with arbitrary geometries. This work provides a solid theoretical foundation for broader applications in engineering and physics.