Abstract
In this article, we present some Sobolev-type inequalities on compact Riemannian manifolds with boundary, the data and the functions being invariant under the action of a compact subgroup of the isometry group. We investigate the best constants for the Sobolev, trace Sobolev, Nash, and trace Nash inequalities. By developing particular geometric properties of the manifold as well as of the solid torus, we can calculate the precise values of the best constants in the presented Sobolev-type inequalities. We apply these results to solve nonlinear elliptic, type Dirichlet and Neumann, PDEs of upper critical Sobolev exponent.
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