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.
The variational approach adopted by the author, together with the comparison between the Mountain Pass level and the optimal anisotropic Sobolev constant, offers an elegant strategy for establishing the existence of positive weak solutions. The results are relevant not only from a theoretical perspective but also for their potential applications in models exhibiting direction-dependent diffusion phenomena.
Congratulations to the author for this valuable contribution to the analysis of nonlinear anisotropic problems.
I sincerely appreciate your positive evaluation of this work and your recognition of the mathematical challenges related to anisotropic quasilinear equations with critical Sobolev growth. Your remarks concerning the interplay between anisotropic diffusion, Finsler geometry, and coercive potentials are highly appreciated.
I am also grateful for your acknowledgment of the variational methodology based on the Mountain Pass framework and the anisotropic Sobolev inequalities. Your encouraging words motivate me to continue exploring nonlinear anisotropic problems and their potential applications.
Thank you again for your valuable feedback and support.
2)-What is the role of the anisotropic Sobolev constant in your proof?
The main difficulty comes from the critical Sobolev growth and the unboundedness of the domain �, which cause a loss of compactness. As a result, minimizing sequences may escape to infinity, preventing the direct application of variational methods. The coercive external potential acts as a confining mechanism: since it becomes large at infinity, it penalizes functions spreading far away from the origin. This restores compactness of the relevant embeddings and allows us to apply the Mountain Pass theorem and compare the Mountain Pass level with the optimal anisotropic Sobolev constant, leading to the existence of a positive weak solution.
2. Answer:
The optimal anisotropic Sobolev constant provides a critical threshold for the energy level. By showing that the Mountain Pass level lies below this threshold, we exclude concentration phenomena and recover the compactness needed for the Palais–Smale condition. This is a key step in proving the existence of a nontrivial positive weak solution.
The interplay between the anisotropic Finsler structure and the critical growth nonlinearity is handled carefully, and the use of Mountain Pass techniques combined with anisotropic Sobolev estimates leads to meaningful existence results. This contribution enriches the current literature on anisotropic nonlinear elliptic equations and opens perspectives for further investigations concerning multiplicity and concentration phenomena.
