Mixed Anisotropic and Nonlocal Sobolev Type Inequalities with Extremal

We establish mixed anisotropic and nonlocal Sobolev type inequalities with an extremal. We show that the extremal function is unique up to a multiplicative constant that is associated with the corresponding mixed anisotropic and nonlocal singular partial differential equation. We prove that such a mixed Sobolev type inequality is necessary and sufficient for the existence of solutions to the associated mixed anisotropic and nonlocal singular partial differential equation. We establish a relation between the purely anisotropic singular equation and mixed anisotropic and nonlocal singular equation.