Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

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., 2013). Furthermore, under these conditions, compact Sobolev-type embeddings \`a 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.


Introduction and main results
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]. More precisely, they are used for proving essential properties of the energy functionals associated with the studied problems (such as sequential lower semicontinuity or the Palais-Smale condition), in order to apply certain minimization and/or minimax arguments.
If Ω ⊆ R d is an open set with sufficiently smooth boundary in the Euclidean space R d , it is well known that the Sobolev space W 1,p (Ω) can be continuously embedded into the Lebesgue space L q (Ω), assuming the parameters p and q verify the range properties: (i) p ≤ q ≤ p * := pd d−p if p < d; (ii) q ∈ [p, +∞) if p = d, and (iii) q = +∞ if p > d. On one hand, when Ω is bounded, due to the Rellich-Kondrachov theorem, the previous embeddings are all compact injections, see Brezis [8]. On the other hand, when Ω is unbounded, the aforementioned compactness need not hold, see Adams and Fournier [1]; for instance, if Ω = R d , the dilation and translation of functions preclude such compactness phenomena. However, symmetries may recover compactness; indeed, it was proved by Berestycki-Lions (see Berestycki and Lions [5], Lions [32], and also Cho and Ozawa [10], Ebihara and Schonbek [16], Strauss [43], and Willem [45]) that if p ≤ d then the embedding W 1,p rad (R d ) ֒→ L q (R d ) is compact whenever p < q < p * , where W 1,p rad (R d ) stands for the subspace of radially symmetric functions of W 1,p (R d ), i.e.
where O(d) is the orthogonal group in R d . In the case of Morrey-Sobolev embeddings, it turns out that W 1,p rad (R d ) can be also compactly embedded into L ∞ (R d ) when 2 ≤ d < p < +∞, see Kristály [25]. Geometrically, Berestycki-Lions' compactness is based on a careful estimate of the functions at infinity. One first observes that the maximal number of mutually disjoint balls having a fixed radius and centered on the orbit {ξx : ξ ∈ O(d)} tends to infinity whenever |x| → ∞; this phenomenon is similar to the maximal number of disjoint patches with fixed diameter on a balloon with continuous expansion. Now, the latter expansiveness property of the balls combined with the invariance of the Lebesgue measure w.r.t. translations implies that the radially symmetric functions rapidly decay to zero at infinity; this fact is crucial to recovering compactness of Sobolev embeddings on unbounded domains, see e.g. Ebihara and Schonbek [16], Kristály [25] and Willem [45]; moreover, this argument is in full concordance with the initial approach of Strauss [43].
Notice that a Berestycki-Lions-type theorem has been established on Riemannian manifolds by Hebey and Vaugon [23], see also Hebey [22,Theorems 9.5 & 9.6]. More precisely, if G is a compact subgroup of the group of global isometries of the complete Riemannian manifold (M, g), then (under additional assumptions on the geometry of (M, g) and on the orbits under the action of G) the embedding W 1,p G (M ) ֒→ L q (M ) is compact, where W 1,p G (M ) denotes the set of G-invariant functions of W 1,p g (M ). Berestycki-Lions-type compactness results have been extended to non-compact metric measure spaces as well, see Górka [21], and generalized to Lebesgue-Sobolev spaces W 1,p(·) G (M ) in the setting of complete Riemannian manifolds, see Gaczkowski, Górka and Pons [19] and Skrzypczak [40].
Skrzypczak and Tintarev [41,44] identified general geometric conditions that are behind the compactness of Sobolev embeddings of the type W 1,p G (M ) ֒→ L q (M ) for certain ranges of p and q; their studies deeply depend on the curvature of the Riemannian manifold. 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 functions to Lebesgue spaces for the full admissible range of parameters. Given d ∈ N with d ≥ 2, we say that (p, q) ∈ (1, ∞) × (1, ∞] is a d-admissible pair whenever Since G is a subgroup of isometries, W 1,p G (M ) turns out to be a closed subspace of W 1,p g (M ). We say that a continuous action of a group G on a complete Riemannian manifold M is coercive (see Tintarev [44,Definition 7.10.8] We notice that the equivalence between (i) and (ii) in Theorem 1.1 is proved by Skrzypczak and Tintarev [41,Proposition 3.1], from which they conclude the compactness of the embedding W 1,p G (M ) ֒→ L q (M ) for the admissible case (S); for a similar result in the case (MT), see Kristály [27]. Accordingly, our purpose in Theorem 1.1 is to characterize their geometric properties by our expansion condition (EC) G , by applying a careful constructive argument based on the Rauch comparison principle, complementing also the admissible range of parameters in the Morrey-case (M).
Our next result concerns Riemannian manifolds with bounded geometry (i.e., complete non-compact Riemannian manifolds with Ricci curvature bounded from below having positive injectivity radius): Theorem 1.2. Let (M, g) be a d-dimensional Riemannian manifolds with bounded geometry, and let G be a compact connected subgroup of Isom g (M ). Then the following statements are equivalent: is compact for some d-admissible pair (p, q). In Theorem 1.2, the equivalence between (i) and the compactness of the embedding W 1,p G (M ) ֒→ L q (M ) for every d-admissible pair (p, q) in (S) is well known by Tintarev [44,Theorem 7.10.12]; in addition, Gaczkowski, Górka and Pons [21,19] proved that a slightly stronger form of (EC) G implies (iii) in the (S) admissible case by using a Strauss-type argument. Thus, the novelty of Theorem 1.2 is the equivalence of our expansion condition (EC) G not only with the coerciveness of G but also with the validity of the compact embeddings in the full range of d-admissible pairs (p, q).
Our next aim is to study similar compactness results on non-compact Finsler manifolds. We notice that in non-Riemannian Finsler settings the situation may change dramatically; indeed, there exist non-compact Finsler-Hadamard manifolds (M, F ) such that the Sobolev space W 1,p F (M ) over (M, F ) is not even a vector space, see Farkas, Kristály and Varga [18], as well as Kristály and Rudas [30]. In spite of such examples, it turns out that similar compactness results to Theorems 1.1 & 1.2 can be established on a subclass of Finsler manifolds, namely on Randers spaces with finite reversibility constant.
Randers spaces are specific non-reversible Finsler structures which are deduced as the solution of the Zermelo navigation problem. In fact, a Randers metric shows up as a suitable perturbation of a Riemannian metric; more precisely, a Randers metric on a manifold M is a Finsler structure F : T M → R defined as F (x, y) = g x (y, y) + β x (y), (x, y) ∈ T M, (1.2) where g is a Riemannian metric and β x is a 1-form on M . For further use, let β g (x) := g * x (β x , β x ) for every x ∈ M, where g * is the co-metric of g.
In order to state our result on Randers spaces, we emphasize that if F is given by (1.2), then the isometry group of (M, F ) is a closed subgroup of the isometry group of the Riemannian manifold (M, g), see Deng [12,Proposition 7.1]. As usual, W 1,p F,G (M ) stands for the subspace of G-invariant functions of W 1,p F (M ), where G is a subgroup of Isom F (M ), while m F (y, ρ) denotes the maximal number of mutually disjoint geodesic Finsler balls with radius ρ on the orbit O y G . 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. Let G be a compact connected subgroup of Isom F (M ) such that m F (y, ρ) → ∞ as d F (x 0 , y) → ∞ for some x 0 ∈ M and ρ > 0. If sup x∈M β g (x) < 1, then for every d-admissible pair (p, q) the embedding In fact, assumption sup x∈M β g (x) < 1 in Theorem 1.3 is equivalent to the finiteness of the reversibility constant of (M, F ) (see Section 5). Furthermore, Example 5.1 shows that this assumption is indispensable. Indeed, we prove that on the Finslerian Funk model (B d (1), F ), -which is a non-compact Finsler manifold of Randers-type, having infinite reversibility constant,-the space W 1,p F (B d (1)) cannot be continuously embedded into L q (B d (1)) for every d-admissible pair (p, q), thus no further compact embedding can be expected.
In the sequel, we provide an application of Theorem 1.3 in the admissible case (M); we notice that applications in the admissible cases (S) and (MT) can be found in Gaczkowski, Górka and Pons [19] and Kristály [27], respectively. Accordingly, in the last part of the paper we consider the following elliptic equation on the d-dimensional Randers space (M, F ) endowed with the metric (1.2), namely where ∆ F,p is the  If (x i ) denotes the local coordinate system on a coordinate neighbourhood of x ∈ M , and the local components of the differential of u are denoted by u i = ∂u ∂x i , then the local components of the gradient ∇ g u are u i = g ij u j . Here, g ij are the local components of g −1 = (g ij ) −1 . In particular, for every x 0 ∈ M one has the eikonal equation When no confusion arises, if X, Y ∈ T x M , we simply write |X| and X, Y instead of the norm |X| x and inner product g x (X, Y ) = X, Y x , respectively.
The L p (M ) norm of ∇ g u : M → T M is given by The space W 1,p g (M ) is the completion of C ∞ 0 (M ) with respect to the norm u p be the volume of the ball with radius ρ > 0 in the d-dimensional space form (i.e., either the hyperbolic space with sectional curvature c when c < 0, or the Euclidean space when c = 0), where and ω d is the volume of the unit d-dimensional Euclidean ball. Note that for every x ∈ M , we have The notation K ≤ c means that the sectional curvature is bounded from above by c at any point and direction. The Bishop-Gromov volume comparison principle states that if (M, g) be a d-dimensional Hadamard manifold with K ≤ c ≤ 0 and x ∈ M fixed, then the function , ρ > 0 is non-decreasing; in particular, from (2.2) one has If equality holds in (2.3) for all x ∈ M and ρ > 0, then K ≡ c; for further details, see Shen [38].
In a similar way, if the Ricci curvature of (M, g) is bounded from below by (n − 1)c (with c ≤ 0), then , ρ > 0 is non-increasing; moreover, by (2.2) one has Let G be a compact connected subgroup of Isom g (M ), and let O x G = {ξx : ξ ∈ G} be the orbit of the element x ∈ M . The action of G on where D is smooth enough, u being of class C 2 in D and having only non-degenerate critical points in D. Based on classical Morse theory and density arguments, in the sequel we shall consider test functions u ∈ C (M ) in order to handle generic Sobolev inequalities.
Let u ∈ C (M ) and Ω ⊂ supp(u) ⊂ M be an open set. Similarly to Druet, Hebey, and Vaugon [15], we may associate to the restriction of u to Ω, namely u| Ω , its Euclidean rearrangement function u * : B e (0, R Ω ) → [0, ∞), which is radially symmetric, non-increasing in |x|, and for every t ≥ inf Ω u is defined by In the sequel, we state the most important properties of this rearrangement which are crucial in the proof of Theorem 1.1; the proof relies on suitable application of the co-area formula combined with the weak form of the isoperimetric inequality on Hadamard manifolds (for a similar proof, see Druet, Hebey, and Vaugon [15], and Kristály [28]).
Lemma 2.2. Let (M, g) be a d-dimensional complete Riemannian manifold. If R > 0, then the embedding W 1,p g (B g (y, R)) ֒→ L q (B g (y, R)) is compact for every y ∈ M and every d-admissible pair (p, q).
Thus, we are in the position to use Hebey [22, Theorem 1.2]; therefore, for every ε > 0 there exists a harmonic radius r H > 0, such that for every z ∈ B g (y, R) one can find a harmonic coordinate chart ϕ z : B g (z, r H ) → R d such that ϕ z (z) = 0 and the components (g jl ) of g in this chart satisfy as bilinear forms. Therefore, it follows that thus U z j j=1,L is a finite covering of B g (y, R).
First observe that for any j ∈ {1, . . . , L} and u ∈ W 1,p g (B g (y, R)), on account of (2.9), we have that We first focus on the (S) admissible case. Observe that Now, by the euclidean Sobolev inequality (see Brezis [8,Corollary 9.14]), for every j ∈ {1, . . . , L} there exists a constant C S,j such that Therefore, by (2.10), (2.11) and (2.12) we have that which proves the validity of the continuous Sobolev embedding W 1,p g (B g (y, R)) ֒→ L q (B g (y, R)) in the (S) case. Now we prove that the previous embedding is compact. To do this, let {u n } n be a bounded sequence in W 1,p g (B g (y, R)), and denoteũ j n = u n | Uz j for every j ∈ {1, . . . , L}. Using (2.10), we have that for every j, the sequenceũ j . By the Rellich-Kondrachov theorem one gets that there exists a subsequence of {ũ j n } n which is a Cauchy sequence in L q (Ω z j ). Let {u m } m be a subsequence of {u n } n such that for any j, {ũ j m } m is a Cauchy sequence in L q (Ω z j ). Thus, applying (2.11), for any m 1 , m 2 we have that hence {u m } m is a Cauchy sequence in L q (B g (y, R)), which proves the claim. One can prove the (MT) admissible case analogously, replacing (2.12) with the euclidean Sobolev inequality when p = d.
Finally, in the (M) case, we have that (2.14) Again, by Brezis [8, Corollary 9.14], for each j ∈ {1, . . . , L} there exists a constant C 0,j such that thus this inequality together with (2.10) and (2.14) yields that which proves again that the continuous embedding holds. Now we prove that this injection is compact. To do this, consider a bounded set A ⊂ W 1,p g (B g (y, R)) , i.e. there exists M > 0 such that u p ≤ M for all u ∈ A. From the previous inequality and (2.15) it follows that there exists C 2 > 0 such that u C 0 (Bg (y,R)) ≤ M C 2 for all u ∈ A. Thus by Ascoli's Theorem (see Aubin [3, Theorem 3.15]), we get that A is precompact in C 0 (B g (y, R)), which concludes the proof. (ii) ⇒ (iii) Without loss of any generality, it is enough to prove that m(γ(t), ρ) → ∞ as t → ∞ for every unit speed geodesic γ : [0, ∞) → M emanating from x 0 = γ(0), i.e., γ(t) = exp x 0 (ty) for some y ∈ T x 0 M with |y| gx 0 = 1, where g x 0 and | · | gx 0 denote the inner product and norm on T x 0 M induced by the metric g.
We notice that O contains infinitely many elements for every is a connected submanifold of M whose dimension is at least 1; if its dimension would be 0 for some t 0 > 0, by which is a contradiction. Therefore, cardO for every i ∈ N and t > 0; the latter statement immediately follows from the fact that ξ i ∈ G, i ∈ N are isometries, thus t → (ξ i • γ)(t), are also geodesics of unit speed emanating from x 0 .
Let us transplant the geodesic balls , we have nothing to prove. Otherwise, consider the geodesic triangle uniquely determined by the points x 0 , ξ i γ(t) andz, respectively. Since (M, g) is a Hadamard manifold, the Rauch comparison principle (see e.g. do Carmo [14, Proposition 2.5, p. 218]) implies that ,z) < ρ, which concludes the proof of (3.1).
Since the geodesics ξ i • γ are mutually different for any i ∈ N, the angle between any two vectors exp −1 x 0 (ξ i γ(t)) ⊂ T x 0 M are positive and it does not depend on the value of t > 0. Let α ij ∈ (0, π] be the angle between v i := exp −1 x 0 (ξ i γ(t)) and v j := exp −1 x 0 (ξ j γ(t)), i = j. Geometrically, the semilines τ → τ v i ⊂ T x 0 M , τ > 0, move away in T x 0 M from each other, independently of t > 0. Accordingly, it turns out that larger values of t > 0 imply more mutually disjoint balls of the form B t i (ρ). More precisely, if we definẽ To prove this, for every n ≥ 2, let Let t 1 = 0. By the latter definition, it turns out thatm(t, ρ) ≥ n whenever t ≥ t n . Let us observe that the sequence {t n } n is non-decreasing and lim n→∞ t n = +∞. The former statement is trivial, while the limit follows from the fact that the sequence of w i : has a convergent subsequence, say {w i k } k ; in particular, the sequence of angles {α i k i k+1 } k converges to 0, which implies the validity of the required limit. Now, let {t n k } k be a strictly increasing subsequence of {t n } n with t n 1 = t 1 = 0, and let f : for every s ∈ [k, k + 1), k ∈ N. It is clear that f is strictly increasing and lim s→∞ f −1 (s) = +∞. By the above construction, for every t > 0, there exists a unique k ∈ N such that t n k ≤ t < t n k+1 .
On the other hand, by (3.1) and the fact that exp x 0 is a diffeomorphism, it turns out that Therefore, we have that m(γ(t), ρ) ≥m(t, ρ), (3.2) and the aforementioned limit concludes the proof.
(iii) ⇒ (ii) Let us assume that the set Fix G (M ) is not a singleton, i.e. there exists x 0 , x 1 ∈ Fix G (M ) such that δ := d g (x 0 , x 1 ) > 0. Since M is a Hadamard manifold, there exists a unique minimal geodesic γ : R → M , parametrized by arc-length, and passing throughout the points x 0 and x 1 . Let It is clear that γ(t 2 ) = ξx 2 and due to the fact that x 0 , x 1 ∈ Fix G (M ), it turns out that γ(t i ) = ξx i = x i , i ∈ {0, 1}. Therefore, by the uniqueness of the geodesic between x 0 and x 1 , it follows that γ(t) = γ(t) for every t ∈ [t 0 , t 1 ]. Since Riemannian manifolds are non-branching spaces, it follows in fact that γ ≡ γ, thus ξx 2 = x 2 ; by the arbitrariness of ξ ∈ G we obtain that x 2 ∈ Fix G (M ) and d g (x 0 , x 2 ) = d g (x 0 , x 1 ) + d g (x 1 , x 2 ) = 2δ. By repeating this argument, one can construct a sequence of point {x n } n ⊂ M such that x n ∈ Fix G (M ) and d g (x 0 , x n ) = nδ, n ∈ N. In particular, d g (x 0 , x n ) → ∞ as n → ∞ and since x n ∈ Fix G (M ) for every n ∈ N, it follows that m(x n , ρ) = 1, which is a contradiction.
(ii) ⇒ compact embeddings. First of all, the compactness of embeddings W 1,p G (M ) ֒→ L q (M ) in the admissible cases (S) and (MT) follow by Skrzypczak and Tintarev [41]. It remains to consider the admissible case (M), i.e. to prove the compactness of W 1,p G (M ) ֒→ L ∞ (M ) whenever p > d. To complete this, we first claim that for every ρ > 0 fixed, one has inf y∈M S(y, ρ) −1 > 0, (3.3) where S(y, ρ) is the embedding constant defined by the embedding W 1,p g (B g (y, ρ)) ֒→ C 0 (B g (y, ρ)), see Lemma 2.2. It is clear that S(y, ρ) > 0 can be considered for non-negative and non-zero functions.
To prove (3.3), for y ∈ M arbitrarily fixed, let u ∈ W 1,p g (B g (y, ρ)) \ {0} be non-negative. By Lemma 2.1/(ii) it turns out that Bg(y,ρ) where u * : B e (0,ρ y ) → [0, ∞) denotes the Euclidean rearrangement of u; in particular, we have Since the latter value does not depend on y ∈ M , we conclude the proof of (3.3). Now, let {u n } n ⊂ W 1,p G (M ) be a bounded sequence and ρ > 0 be an arbitrarily fixed number. Then, up to a subsequence, u n ⇀ u in W 1,p G (M ). Since G is a subgroup of Isom g (M ), for every ξ 1 , ξ 2 ∈ G, by a change of variables, one has u n − u W 1,p g (Bg (ξ 1 y,ρ)) = u n − u W 1,p g (Bg (ξ 2 y,ρ)) . Therefore, on account of the definition of m(y, ρ) (see (1.1)), we have that m(y, ρ) .
By using Lemma 2.2 and the latter inequality, we obtain . According to (ii) and relation (3.3) we have that On the other hand, u n ⇀ u in W 1,p G (M ), thus by the Rellich-Kondrachov-type result (see Lemma 2.2) it follows that u n → u in C 0 B(y, R ε ) , hence there exists n ε ∈ N such that u n − u C 0 (B(y,Rε)) < ε for all n ≥ n ε .  2) can be viewed as a comparison of the maximal number of mutually disjoint geodesic balls with radius ρ on (M, g) and the Euclidean space, respectively. In fact,m(t, ρ) is related to the particular inner product given by g x 0 , which is equivalent to the usual Euclidean metric. This comparison result can be efficiently applied for every Hadamard manifold. In particular, in the usual Euclidean space R d , a simple covering argument shows that m(t, ρ) = ω V −1 cap (2ρ/t) as t → ∞, 1 where V cap (r) denotes the area of the spherical cap of radius r > 0 on the unit (d − 1)-dimensional sphere. For instance, when d = 3, we havem(t, ρ) = ω sin −2 (ρ/t) as t → ∞.

Proof of Theorem 1.2
(i) ⇒ (ii) Let us assume by contradiction that (EC) G fails, i.e. there exist K ∈ N and a sequence {x n } n ⊂ M such that m(x n , ρ) ≤ K for every n ∈ N and d g (x 0 , x n ) → ∞ as n → ∞.
This particular choice clearly shows that B g (γ(t j ), ρ) are situated in some concentric annuli with the same width; more precisely, Beside of the latter property, by d g (ξ i 0 x n , ξ 1 x n ) ≥ 2k n ρ we also have that Combining all these constructions, it follows that the balls B g (γ(0), ρ) = B g (ξ 1 x n , ρ), B g (γ(t 1 ), ρ)..., B g (γ(t kn−1 ), ρ) and B g (γ(1), ρ) = B g (ξ i 0 x n , ρ) are mutually disjoint sets, whose centers belong to Imγ ⊂ O xn G . Since the number of these balls is k n + 1, this contradicts again the maximality of k n = m(x n , ρ).
(ii) ⇒ (iii) We shall focus first on the Morrey-case (M), i.e., we assume that p > d and q = ∞; then we discuss the cases (S) and (MT).
Similarly to (3.3), we are going to prove that for every fixed ρ > 0 one has where S(y, ρ) is the embedding constant in W 1,p g (B g (y, ρ)) ֒→ C 0 (B g (y, ρ)), see Lemma 2.2. We have that for any ε > 0 there exists r H > 0 depending only on ε, d, K and i 0 , which satisfies the following property: for any y ∈ M there exists a harmonic coordinate chart ϕ : B g (y, r H ) → R d , such that ϕ(y) = 0, and the components (g jl ) of g in this chart satisfy as bilinear forms. Fix ρ < r H , then it is obvious that On the other hand, combining (2.10) with (4.3), we have that S(y, ρ) −1 = inf u∈W 1,p g (Bg (y,ρ)) Bg(y,ρ) Let f * : Ω * y → [0, ∞) be the symmetric decreasing rearrangement of the function f (see Lieb and Loss [31, Section 3.3]), thus Vol e (Ω y ) = Vol e (Ω * y ) and .
meaning that inf y∈M S(y, ρ) −1 > 0, which concludes the proof of (4.1). Now, let {u n } n ⊂ W 1,p G (M ) be a bounded sequence and ρ > 0 be an arbitrarily fixed number. Then, up to a subsequence, u n ⇀ u in W 1,p G (M ). By using Lemma 2.2, we obtain Due to the validity of (EC) G and relation (4.1) we have that lim dg (x 0 ,y)→∞ S(y, ρ) m(y, ρ) = 0, thus for every ε > 0 there exists R ε > 0 such that sup dg(x 0 ,y)≥Rε u n − u C 0 (Bg(y,ρ)) ≤ ε 2 for every n ∈ N. Since u n ⇀ u in W 1,p G (M ), by the Rellich-Kondrachov-type result (see Lemma 2.2) it follows that u n → u in C 0 B(y, R ε ) , hence there exists n ε ∈ N such that u n − u C 0 (B(y,Rε)) < ε for all n ≥ n ε . Let us fix an arbitrary d-admissible pair (p, q) from (S) or (MT). A suitable modification of the above argument, based on Lemma 2.1/(i), implies that S(y, ρ) −1 := inf u∈W 1,p g (Bg (y,ρ)) Bg(y,ρ) The latter inequality together with the validity of (EC) G implies that The rest is analogous as before, by using the Rellich-Kondrachov compactness result W 1,p (B g (y, R)) ֒→ L q (B g (y, R)) for any R > 0 fixed.
We follow the argument presented in Skrzypczak and Tintarev [42,Theorem 4.3] (see also Tintarev [44,Theorem 7.10.12]); in fact, the admissible case (S) is exactly the one proved in Tintarev [44]. Since the case (MT) can be similarly discussed as (S), we restrict our proof to the remaining admissible case (M).
Suppose that G is not coercive, thus there exists R > 0 and a discrete sequence of x n ∈ M , such that O xn G ⊂ B g (x n , R) and d g (x 0 , x n ) → ∞ as n → ∞. Let r ∈ (0, inj (M,g) ) and let us replace {x n } n with a renumbered subsequence such that distance between any two terms in the sequence will be greater than 2(R + r). We define a sequence of functions {f n } n by where the Haar measure of G is normalized to the value 1, and u + = max{0, u}. It is easy to see that f n ∈ W 1,p G (M ) for every n ∈ N and any fixed p ∈ (1, ∞); indeed, since the support of f n is a subset of B g (ξ −1 x n , r), by an elementary computation (with (2.1) in hand) and the volume-estimate (2.4), it follows that f n W 1,p g (M ) ≤ C(p, r, d), where C(p, r, d) > 0 does not independent on n. On one hand, since the supports of the functions f n are disjoint sets, we have that On the other hand, Since (M, g) is a Riemannian manifold with bounded geometry, then Vol g is doubling on (M, g), thus whereC(r, R, d) > 0 does not independent on n. Thus, {f n } n is not a Cauchy sequence in L ∞ (M ), a contradiction.
The following theorem is related to the results obtained in Hebey and Vaugon [23]: Let (M, g) be a d-dimensional complete non-compact Riemannian manifold with Ricci curvature bounded from below having positive injectivity radius, and let G be a compact connected subgroup of Isom g (M ) such that Fix(G) = {x 0 } for some x 0 ∈ M and ρ > 0 be small enough. Assume that there exists κ = κ(G, d) > 0 such that for every y ∈ M with d g (x 0 , y) ≥ 1, one has where l = l(y) = dim O y G ≥ 1. Then the embedding W 1,p G (M ) ֒→ L ∞ (M ) is compact for every p > d. Proof. Let y ∈ M be arbitrarily fixed such that d g (x 0 , y) ≥ 1, and consider the elements ξ i ∈ G, i = 1, . . . , m(y, ρ) which appear in the definition of m(y, ρ) in (1.1). Let also l = l(y) = dim O y G . Notice that by the connectedness of G, we have l ≥ 1. We claim that for every k > 2 (independent of y). To see this, it is sufficient to prove that Indeed, if the contrary holds, then B g (x, ρ) ∩ B g (ξ i y, ρ) = ∅, i = 1, . . . , m(y, ρ), thus B g (x, ρ) is a new ball in the definition of m(y, ρ), contradicting the maximality of m(y, ρ). Therefore, d g (x, ξ i 0 y) < 2ρ, which means that x ∈ B g (ξ i 0 y, kρ) ∩ O y G for every k > 2, which proves (4.6). We also notice that since Fix Therefore, a slight modification of Gallot, Hulin, and Lafontaine [20,Theorem 3.98] gives that for every i = 1, . . . , m(y, ρ), ) as ρ → 0, whenever k > 2 is kept small (e.g. k = 3). To see this, we explore that exp ξ i y : T ξ i y M → M is a local diffeomorphism at 0 ∈ T ξ i y M with d(exp ξ i y ) 0 = id, while for small ρ > 0 one has exp −1 ξ i y (B g (ξ i y, kρ) ∩ O y G ) = B e (0, kρ) ∩ exp −1 ξ i y (O y G ), and 0 < H l (exp −1 ξ i y (O y G )) < ∞. Now, if we fix ρ ∈ (0, 1) from the usual range (see Gallot, Hulin, and Lafontaine [20]), it follows by (4.6) that H l (O y G ) ≤ m(y, ρ)k l+1 ω l ρ l . Hypothesis (H) and the latter estimate imply that κ · d g (x 0 , y) ≤ m(y, ρ)k l+1 ω l ρ l . By using this inequality, one can obtain an l = l(y)-independent estimate, namely Letting d g (x 0 , y) → ∞ immediately implies that m(y, ρ) → ∞. The rest of the proof is similar to the last part of the proof of Theorem 1.2.
In the sequel, we provide two examples where hypothesis (H) holds.
The special linear group SL(d) leaves P(d, R) 1 invariant and acts transitively on it. Moreover, for every σ ∈ SL(d), the map [σ] : P(d, R) 1 → P(d, R) 1 defined by [σ](X) = σXσ t , is an isometry, where σ t denotes the transpose of σ. If G = SO(d), we can prove that Fix P(d,R) 1 (G) = {I d }, where I d is the identity matrix; for more details, see Kristály [27]. On the other hand, the metric function on P(d, R) is given by d P (X, Y ) = Tr ln 2 X − 1 2 Y X − 1 2 , see Kristály [26].
For simplicity, fix d = 2, and consider the following positive-definite symmetric matrix X = a b b c , a, c > 0, and ac − b 2 = 1. Thus One can see that where λ 1 , λ 2 are the positive eigenvalues of the matrix X. Since √ a 2 + 2b 2 + c 2 = λ 2 1 + λ 2 2 , by using a Bernoulli-type inequality, it turns out that H 1 O X G ≥ κd P (I 2 , X), with κ := π, which proves the validity of (H). and Now, let |y 1 | + · · · + |y k | = c ≥ 1. By the scaling y i := cz i , one has |z 1 | + · · · + |z k | = 1 and Note that the continuous function (z 1 , . . . , z k ) → k i=1 |z i | d i −1 attains its minimum on the simplex |z 1 | + · · · + |z k | = 1, and this minimum is strictly positive, say m G > 0 (otherwise, if m G = 0, we would have all variables equal to zero, which is a contradiction). Summing up, it follows that H l (O y G ) ≥ 2πcm G = 2πm G (|y 1 | + · · · + |y k |) ≥ 2πm G |y|, thus G satisfies the assumption in (H). where g is a Riemannian metric on M , β x is a 1-form on M , and we assume that

Sobolev-type embeddings on Randers spaces
Here, the co-metric g * x can be identified by the inverse of the symmetric, positive definite matrix g x . The pair (M, F ) is called a Randers space, which is a typical Finsler manifold, i.e. the following properties hold: see Bao, Chern, and Shen [4]. Clearly, the Randers metric F is symmetric, i.e. F (x, −y) = F (x, y) for every (x, y) ∈ T M, if and only if β = 0 (which means that (M, F ) = (M, g) is the original Riemannian manifold).
Let σ : [0, r] → M be a piecewise C ∞ curve. The value L F (σ) = r 0 F (σ(t),σ(t)) dt denotes the integral length of σ. For x 1 , x 2 ∈ M , denote by Λ(x 1 , x 2 ) the set of all piecewise C ∞ curves σ : [0, r] → M such that σ(0) = x 1 and σ(r) = x 2 . Define the distance function d F : One clearly has that d F (x 1 , x 2 ) = 0 if and only if x 1 = x 2 , and that d F verifies the triangle inequality. The Hausdorff volume form dV F on the Randers space (M, F ) is given by where dv g denotes the canonical Riemannian volume form induced by g on M .

(5.4)
Let u : M → R be a differentiable function in the distributional sense. The gradient of u is defined by where Du(x) ∈ T * x M denotes the (distributional) derivative of u at x ∈ M and J * is the Legendre transform given by In local coordinates, one has In general, note that u → ∇ F u is not linear. If x 0 ∈ M is fixed, then due to Ohta and Sturm [34], one has Let X be a vector field on M . In a local coordinate system (x i ) the divergence is defined by div(X) = 1 .
The Finsler p-Laplace operator is defined by while the Green theorem reads as: for every v ∈ C ∞ 0 (M ), We notice that the reversibility constant associated with F (see (5.1)) is given by , (5.9) see Rademacher [35] and Zhao and Yuan [47]. Note that r F ≥ 1 (possibly, r F = +∞), and r F = 1 if and only if (M, F ) is Riemannian. Analogously, the uniformity constant of F is defined by the number and measures how far F and F * are from Riemannian structures, see Egloff [17]. Note that l F ∈ [0, 1], and l F = 1 if and only if (M, F ) is Riemannian, i.e. β = 0.

5.2.
Embedding results on Randers spaces: the influence of reversibility.
where g is a Riemannian metric such that (M, g) is either a Hadamard manifold or a Riemannian manifold with bounded geometry. Let In this case, the volume form on (M, F ) is given by (5.3), one has that Next, by using the definition of the polar transform of F , see (5.4), we get that On the other hand, Combining (5.11), (5.12) and (5.13), one gets that . (5.14) Thus, by the continuous embedding on the Riemannian manifold, we have that where (p, q) is any d-admissible pair. For the compact embedding, let G be a compact connected subgroup of Isom F (M ), such that m F (y, ρ) → ∞ as d F (x 0 , y) → ∞ for some x 0 ∈ M . According to Deng [12, Proposition 7.1], G is a closed subgroup of the isometry group of the Riemannian manifold (M, g).
On the other hand, since (1 − a)d g (x 0 , y) ≤ d F (x 0 , y) ≤ (1 + a)d g (x 0 , y), we have that if m F (y, ρ) → ∞ as d F (x 0 , y) → ∞, then m y, ρ (1+a) → ∞ as d g (x 0 , y) → ∞. Now let {u n } n be a bounded sequence in W 1,p F,G (M ). From (5.14), it follows that {u n } n is bounded in W 1,p G (M ), thus by Theorems 1.1 & 1.2 (condition (EC) G implies the compact embedding), there exists a subsequence {u n k } k which converges strongly to a function u in L q (M ), where (p, q) is any d-admissible pair. This concludes the proof.
We emphasize that Theorem 1.3 is sharp in the following sense: if we consider the d-dimensional Finslerian Funk model (B d (1), F ), which is a non-compact Finsler manifold of Randers-type having constant flag curvature − 1 4 , then we can construct a function u ∈ W 1,p (1)), for any (p, q) d-admissible pair. As it turns out, (B d (1), F ) is a non-reversible Finsler manifold with sup x∈M β g (x) = 1, i.e. r F = ∞, see Kristály and Rudas [30]. Therefore, the continuous embeddings of Sobolev spaces do not necessarily hold on Randers spaces having infinite reversibility constant. Details are provided in the next example.
where t is a parameter. A direct calculation yields that By applying (5.7), we have that F * (x, Dd F (0, x)) = 1 for a.e. x ∈ B d (1), thus Therefore, since dV F (x) = dx, we have that where B denotes the Euler-Beta function.

5.3.
Application: Multiple solutions for an elliptic PDE on Randers spaces. In order to prove Theorem 1.4, we recall an abstract tool, which is the following critical point result of Bonanno [7] (which is actually a refinement of a general principle of Ricceri [37,36]): , Theorem 2.1). Let X be a separable and reflexive real Banach space, and let Φ, J : X → R be two continuously Gâteaux differentiable functionals, such that Φ(u) ≥ 0 for every u ∈ X. Assume that there exist u 0 , u 1 ∈ X and ρ > 0 such that Further, put and assume that the functional Φ − λJ is sequentially weakly lower semicontinuous, satisfies the Palais-Smale condition and Then there exists an open interval Λ ⊂ [0, a] and a number µ > 0 such that for each λ ∈ Λ, the equation Φ ′ (u) − λJ ′ (u) = 0 admits at least three solutions in X having norm less than µ.
For every λ > 0 we define the energy functional associated with problem (P λ ) as Since (M, F ) is a Randers space with a := sup x∈M β g (x) < 1, the reversibility constant r F is finite, thus W 1,p F (M ) is a separable and reflexive Banach space, see Farkas, Kristály, and Varga [18]. Having in our mind Theorem 1.3, we restrict the energy functional to the space W 1,p F,G (M ). For simplicity, in the following we denote E λ = E λ | W 1,p F,G (M ) , Φ = Φ 0 | W 1,p F,G (M ) , and J = J 0 | W 1,p F,G (M ) . In the sequel we prove that the energy functional E λ is G-invariant. Note that the G-invariance of the energy functional is an important tool in proving our theorem.
Lemma 5.4. For every λ ≥ 0 the functional E λ is sequentially weakly lower semicontinuous.
Proof. As a norm-type function, Φ is sequentially weakly lower semicontinuous, therefore it suffices to prove that J is sequentially weakly continuous. To this end, consider a sequence {u n } n in W 1,p F,G (M ) which converges weakly to u ∈ W 1,p F,G (M ), and suppose that J(u n ) ✟ ✟ →J(u n ) as n → ∞.
Thus, there exist ε > 0 and a subsequence of {u n } n , denoted again by {u n } n , such that u n → u in L ∞ (M ) and 0 < ε ≤ |J(u n ) − J(u)|, for every n ∈ N.
Note that the last term tends to 0, which provides a contradiction.

(5.21)
We claim that In Theorem 5.1 we choose u 1 = u s 0 ,R,r and u 0 = 0, and observe that the hypotheses (1) and (2)