Abstract
Given a complete non-compact Riemannian manifold (M, g) with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries G of (M, g) that characterizes the coerciveness of G in the sense of Skrzypczak and Tintarev (Arch Math 101(3): 259–268, 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments.
Highlights
Introduction and main resultsCompact Sobolev embeddings turn out to be fundamental tools in the study of variational problems, being frequently used to study the existence of solutions to elliptic equations, see e.g. Willem [45]
If ⊆ Rd is an open set with sufficiently smooth boundary in the Euclidean space Rd, it is well known that the Sobolev space W 1,p( ) can be continuously embedded into the Lebesgue space Lq ( ), assuming the parameters p and q verify the range properties: (i) pd d−p if p d, and (iii)
In the light of their works, our purpose is twofold; namely, we provide an alternative characterization of the properties described by Skrzypczak and Tintarev [41,44] by using the expansion of geodesic balls and state the compact Sobolev embeddings of isometry-invariant Sobolev
Summary
Compact Sobolev embeddings turn out to be fundamental tools in the study of variational problems, being frequently used to study the existence of solutions to elliptic equations, see e.g. Willem [45]. [18], as well as Kristály and Rudas [30] In spite of such examples, it turns out that similar compactness results to Theorems 1.1 and 1.2 can be established on a subclass of Finsler manifolds, namely on Randers spaces with finite reversibility constant. (y, ρ) denotes the maximal number of mutually disjoint geodesic Finsler balls with radius ρ on the orbit OGy. Theorem 1.3 Let (M, F) be a d-dimensional Randers space endowed with the Finsler metric (1.2), such that (M, g) is either a Hadamard manifold or a Riemannian manifold with bounded geometry. We prove that on the Finslerian Funk model (Bd (1), F),—which is a non-compact Finsler manifold of Randers-type, having infinite reversibility constant,— the space WF1,p(Bd (1)) cannot be continuously embedded into Lq (Bd (1)) for every d-admissible pair ( p, q), no further compact embedding can be expected.
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