Sobolev spaces with variable exponents on complete manifolds

We study variable exponent function spaces on complete non-compact Riemannian manifolds. Using classic assumptions on the geometry, continuous embeddings between Sobolev and Hölder function spaces are obtained. We prove compact embeddings of H-invariant Sobolev spaces, where H   is a compact Lie subgroup of the group of isometries of the manifold. As an application, we prove the existence of non-trivial solutions to non-homogeneous q(x)q(x)-Laplace equations.