Critical Anisotropic Quasilinear Equations with External Potentials in IRn

Anisotropic diffusion phenomena arise naturally in several physical models, including crystalline media, non-Newtonian fluids, and nonlinear optics, where the material properties depend directly on direction. In the current work, we investigate a wide class of nonlinear anisotropic quasilinear equations in the whole space IRn, which is driven by a convex Finsler structure and involves a critical Sobolev growth. The critical fact of nonlinearity generates a loss of compactness, which leads to a fundamental difficulty in domains that are unbounded, and it is also closely related to the concentration phenomena.

To be able to model confinement effects and to restore compactness, we introduce an external potential that is assumed to be continuous, strictly positive, and coercive at infinity. This potential can play a role widely analogous to trapping mechanisms in a nonlinear Schrödinger-type model. Within this framework, we develop a method using a variational approach and analyze the associated energy functional. By combining compact embedding arguments with a careful comparison between the Mountain Pass level and the optimal anisotropic Sobolev constant, we prove the existence of at least one positive weak solution.

Our results provide a rigorous analytical framework for anisotropic critical models with external potentials and contribute to the mathematical understanding of direction-dependent nonlinear phenomena in unbounded media.