Sobolev spaces with variable exponents on complete manifolds

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Journal of Functional Analysis www.elsevier.com/locate/jfa

Sobolev spaces with variable exponents on complete manifolds Michał Gaczkowski a , Przemysław Górka a,∗ , Daniel J. Pons b,∗ a

Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland b Facultad de Ciencias Exactas, Departamento de Matemáticas, Universidad Andres Bello, República 220, Santiago, Chile

a r t i c l e

i n f o

Article history: Received 29 May 2015 Accepted 10 September 2015 Available online xxxx Communicated by L. Gross MSC: 46E35 53B21

a b s t r a c t We study variable exponent function spaces on complete noncompact 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 nontrivial solutions to non-homogeneous q(x)-Laplace equations. © 2015 Elsevier Inc. All rights reserved.

Keywords: Lebesgue spaces with variable exponents Sobolev spaces Riemannian manifolds

1. Introduction In Euclidean domains, the theory of Lebesgue–Sobolev and Hölder spaces with variable exponents has applications in non-linear elastic mechanics [28], electrorheological fluids * Corresponding authors. E-mail addresses: [email protected] (P. Górka), [email protected], [email protected] (D.J. Pons). http://dx.doi.org/10.1016/j.jfa.2015.09.008 0022-1236/© 2015 Elsevier Inc. All rights reserved.

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[24], and image restoration [20], for example. In those areas of science, the exponential growth of quantities induced by observables is modified by spatial inhomogeneities of the material, fluid and image, respectively, or by external perturbations, usually modeled as multiplicative and additive factors. In improved models for such phenomena, the exponents describing the growth are allowed to vary in space according to some law, see [20,24]. As a consequence, Lebesgue–Sobolev spaces with variable exponents have been studied in depth during the last decade (see the surveys [8,25]). In Euclidean space, Sobolev type inequalities with variable exponents have been delivered [6,10,15]. This theory has been extended to metric measure spaces [12,16], and to Riemannian manifolds, see [13]. In geometric and global analysis, classic Sobolev and Hölder spaces on Riemannian manifolds have been studied for more than fifty years [23,2,17]. They have been used to obtain isoperimetric type inequalities [17], and in the Yamabe problem for conformal metrics with prescribed scalar curvature [27], among other topics (see [2] for a survey). In this article we deal with Sobolev and Hölder spaces with variable exponents on complete non-compact Riemannian manifolds. The applications mentioned above suggest that this setup should be useful for similar problems with an additional geometric flavor. We organize this work in several sections. In Section 2 we give definitions and results that will be used later on. Section 3 is a brief introduction to variable Sobolev and Hölder spaces on Riemannian manifolds. Our main contributions begin in Section 4. In Section 4.1, assuming continuity of the exponents and bounds on the geometry of the manifold, we obtain embeddings between Sobolev spaces. Then in Section 4.2, using stronger bounds on the geometry of the space, and also stronger hypothesis on the exponents, namely log-Hölder continuity, we obtain the embedding between Sobolev spaces with critical exponents, and embed Sobolev spaces in Hölder spaces. In Section 5 we consider H-invariant Sobolev spaces, where H is a subgroup of the group of isometries of the manifold (M, g); this leads to compact embeddings. A brief discussion of the conditions that H and g should satisfy provides a huge family of examples. Finally, in Section 6, we show the existence of non-trivial H-invariant solutions to a non-linear elliptic equation involving the q(x)-Laplacian. 2. Preliminaries 2.1. Variable exponent Lebesgue spaces We recall some facts and notation about variable exponent Lebesgue and Sobolev spaces. Most of the properties of these spaces can be found in [7] and [18]. Let (Ω, μ) be a σ-finite, complete measure space. By a variable exponent we mean a bounded measurable function q : Ω → [1, ∞]. We denote by P(Ω) the set of variable exponents on Ω. Given q in P(Ω) and A ⊂ Ω, define

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+ qA := ess sup q(x), x∈A

3

− qA := ess inf q(x), x∈A

and if A = Ω, we simply write q + , q − . The variable exponent, or generalized Lebesgue space Lq(·) (Ω), is the vector space of all measurable functions u : Ω → R for which the functional  q(x) ρq(·) (u) := |u(x)| dμ(x) Ω

is finite. The functional ρq(·) is convex, and is sometimes called a convex-modular [18]. Lq(·) (Ω) is a Banach space with respect to the Luxemburg–Minkowski type norm  u  uLq(·) := inf t > 0 | ρq(·) ≤1 . t Generalized Lebesgue spaces are a special type of Musielak–Orlicz spaces. In particular, when the exponent q(x) is a constant equal to q for every x, then Lq(·) (Ω) is the ordinary Lebesgue space. If r, p and q are elements in P(Ω) satisfying 1 1 1 = + r(x) p(x) q(x) except on a set of measure zero, we have Hölder’s inequality (see [7]) uvLr(·) ≤ 2 uLp(·) vLq(·) . In particular if r = 1, then p(x) is the conjugate exponent of q(x), namely q  (x) := The space Lq(·) (Ω) is reflexive if and only if 1 < q − ≤ q + < ∞, and then the

q(x) q(x)−1 .



dual of Lq(·) (Ω) is Lq (·) (Ω). To compare the functionals  Lq(·) and ρq(·) ( ), one has the relations     − + − + min ρq(·) (u)1/q , ρq(·) (u)1/q ≤ uLq(·) ≤ max ρq(·) (u)1/q , ρq(·) (u)1/q

(1)

between norm and convex-modular. In particular, the unit ball property follows: uLq(·) ≤ 1 if and only if ρq(·) (u) ≤ 1. Using such a strategy, one can compare the norms induced by different exponents, say p and q such that p(x) ≤ q(x) almost everywhere on Ω, see [18]: Theorem 2.1. Let p and q belong to P(Ω), with p(x) ≤ q(x) except on a set of measure zero. If μ(Ω) < ∞, then uLp(·) ≤ (μ(Ω) + 1)uLq(·) .

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2.2. Variable exponent Sobolev and Hölder spaces in the Euclidean setting Let Ln be the n-dimensional Lebesgue measure, and let Ω be a Lebesgue measurable q(·) subset of Rn . The variable exponent Sobolev space Lk (Ω) is the vector space of all q(·) functions f ∈ L (Ω) for which the distributional derivatives of order less than k + 1 q(·) belong to Lq(·) (Ω). Lk (Ω) is a Banach space with the norm  uLq(·) := Dα uLq(·) . k

|α|≤k q(·)

For more properties of the spaces Lq(·) (Ω) and Lk (Ω), we refer to [18]. As in the classic case, one defines the Hölder space of variable exponent C 0,α(·)(Ω), where now α : Ω →]0, 1] is a measurable function: Given a bounded continuous function u one introduces |u(x) − u(y)| , α(x) x=y∈Ω |x − y|

[u]α(·) := sup

to note that [·]α(·) is a seminorm; hence C 0,α(·) (Ω) is the vector space of all bounded continuous functions u for which [u]α(·) is finite. This is a Banach space with respect to the norm uC 0,α(·) := u∞ + [u]α(·) . However, some basic properties of the standard Lebesgue spaces are not valid in the variable exponent case. For instance, Zhikov [29] observed that smooth functions, in q(·) q(·) general, are not dense in Lk (Ω). The smooth functions are dense in Lk (Ω) under the global log-Hölder continuity condition.1 A function s : Ω → R is said to be globally log-Hölder continuous if: 1. It is locally log-Hölder continuous, namely there exists a constant c1 , called a local log-Hölder constant, such that |s(x) − s(y)| ≤ c1 / log(e + 1/|x − y|) for every x and y in Ω, and 2. It satisfies the log-Hölder decay condition, namely there exists a real number s∞ and a constant c2 , called a log-Hölder decay constant, such that |s(x) − s∞ | ≤ c2 / log(e + |x|) for every x in Ω. The maximum between c1 and c2 is called a global log-Hölder constant for s. 1 Unfortunately, there are different names for this condition; even worse, sometimes the same name is used for similar but inequivalent conditions: We follow [7].

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From a metric perspective, given L > 0, consider the function mL : R+ → R+ given by mL (t) := L/ log(e + 1/t). One verifies that mL is increasing, concave, with mL (0) = 0, hence distL (x, y) := mL (|x − y|) =

L = L dist1 (x, y) log(e + 1/|x − y|)

(2)

is a distance on Rn . We infer that the Lipschitz constant for s with respect to dist1 , namely Lipdist1 (s) := sup x=y

|s(x) − s(y)| , dist1 (x, y)

(3)

is the lowest possible value for the local log-Hölder constant of s, and note that s is locally log-Hölder continuous on Ω if and only if s is a Lipschitz function on the metric space (Ω, dist1 ). Definition 2.1. We use the following nomenclature: 1. Say that q ∈ P log (Ω) if q ∈ P(Ω) and 1/q is locally log-Hölder continuous. Denote by clog (q) the local log-Hölder constant of 1/q. log 2. Say that q ∈ Pglob (Ω) if q ∈ P(Ω) and 1/q is globally log-Hölder continuous. Denote by clog,glob (q) the global log-Hölder constant of 1/q. Assuming global log-Hölder continuity, the following Sobolev embeddings hold: log Theorem 2.2. Assume that q ∈ Pglob (Rn ). It follows that:

i) ([6,15]) If q + < n and p(x) =

nq(x) n−q(x) ,

then

q(·)

L1 (Rn ) → Lp(·) (Rn ). ii) If n < q − and α(x) = 1 −

n q(x) ,

then

q(·)

L1 (Rn ) → C 0,α(·) (Rn ). Moreover, the constant of the embeddings depends only on n, clog (q), q + , q − . The proof of ii) when Ω is a bounded domain has been given by Alemida and Samko [1]. The proof when Ω is Rn is given below:

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Proof. Given x ∈ Rn and R > 0, denote by QR (x) the cube of Rn with sides parallel to the coordinate axes, having center at x and side R. Then whenever y ∈ Rn , we have that x and y belong to QR (x) if R = 2|x − y|, thus following the proof of Morrey’s estimate (see e.g. [19]) we have  |u(y) − − u dLn | ≤ QR (x)

− qQ R (x)

1−

− qQ −n R (x)

R

n − QR (x)

q

∇u

L

− QR (x) (Q (x)) R

q

,

(4)

   where −Ω u dLn := Ω u dLn / Ω dLn denotes the average of u over Ω. Observe that if x and y are such that |x − y| < 1/2, then R < 1. Under such restrictions, Hölder’s inequality, estimate (1), and the assumption q ∈ P log (Rn ) together with (4) give  n q+ R1− q(x) ∇uLq(·) (Rn ) , |u(y) − − u dLn | ≤ C − (5) q −n QR (x)

where the constant C depends on n and clog (q). Thus if x and y are such that |x − y| < 1/2, from (5) we get |u(x) − u(y)|

sup

n

|x−y|< 12

|x − y|1− q(x)

1−

n

q + 2 q+ ∇uLq(·) (Rn ) , ≤2C q− − n

or |u(x) − u(y)|

sup

n

|x−y|< 12

|x − y|1− q(x)

≤ D ∇uLq(·) (Rn ) ,

(6)

where D = D(n, clog (q), q + , q − ) depends on n, clog (q), q + and q − . From (6) we infer that q(·) if u is in L1 (Rn ), then u is in C 0,α(·) (Rn ), provided that u is bounded. Finally, we show that u is bounded. Choose x in Rn , and note that using (5)    n n n |u(x)| ≤ − u dL + u(x) − − u dL ≤ − u dL + D ∇uLq(·) (Rn ) , Q1 (x) Q1 (x) Q1 (x) with

 − u dLn ≤ E uLq(·) (Rn ) Q1 (x)

thanks to Hölder’s inequality, where E depends on n and clog (q). It follows that uL∞ (Rn ) ≤ F uLq(·) (Rn ) , 1

where D + E = F = F (n, clog (q), q + , q − ).

(7)

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From estimates (6) and (7) the claim follows. 2 3. Variable exponent Sobolev spaces on manifolds We recall some basic material on Riemannian geometry (see e.g. [2,17]). Let (M, g) be a smooth Riemannian n-manifold, and let ∇ be the Levi-Civita connection. If u is a smooth function on M , then ∇k u denotes the k-th covariant derivative of u. In local coordinates, the pointwise norm of ∇k u is given by

 |∇k u|2 = g i1 j1 · · · g ik jk ∇k u i

1 ···ik





∇k u

j1 ···jk

,

where Einstein’s convention is used. Given {Ωi , φi } an atlas of M , with {ηi } a partition of unity subordinate to {Ωi , φi }, the integral of u : M → R over M with respect to the volume element induced by g is given by  u dVg := M

  i

  dLn . ηi det g u ◦ φ−1 i

φi (Ωi )

To define variable Sobolev spaces, given a variable exponent q in P(M ) and a natural number k, introduce   q(·) Ck (M ) := u ∈ C ∞ (M ) | ∀j 0 ≤ j ≤ k |∇j u| ∈ Lq(·) (M ) . q(·)

On Ck (M ) define the norm uLq(·) := k

k 

|∇j u|Lq(·) ,

j=0

to obtain the variable Sobolev spaces in the following way. q(·)

q(·)

Definition 3.1. The Sobolev space Lk (M ) is the completion of Ck (M ) with respect to q(·) q(·) the norm ·Lq(·) . If Ω is a subset of M , then Lk,0 (Ω) is the completion of Ck (M )∩C0 (Ω) k with respect to  · Lq(·) , where C0 (Ω) denotes the vector space of continuous functions k whose support is a compact subset of Ω. To define variable Hölder spaces on Riemannian manifolds, we recall the notion of geodesic distance. Given a rectifiable curve γ : [a, b] → M , the length of γ is given by  b  dγ dγ , dt. l(γ) = g dt dt a

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For a pair of points x, y ∈ M , the distance dg (x, y) between x and y is the infimum of the length of all rectifiable curves that start from x and end at y. Thus for a fixed function α : M →]0, 1] and a bounded continuous function u, we have [u]α(·) := sup

x,y∈M

|u(x) − u(y)| . dg (x, y)α(x)

It follows that [·]α(·) is a seminorm. Using the previous ingredients, on denotes by C k,α(·) (M ) the space of functions u : M → R of class C k for which both the norm uC k :=

k 

sup |∇j u(x)|

j=0 x∈M

  and the seminorm |∇k u| α(·) are finite: This is a Banach space with respect to the norm   uC k,α(·) := uC k + |∇k u| α(·) . Finally, the notion of log-Hölder continuity on Riemannian manifolds is given: Definition 3.2. A function s : M → R is log-Hölder continuous if there exists a constant c such that for every pair of points {x, y} in M we have the inequality |s(x) − s(y)| ≤

c   . 1 log e + dg (x,y)

We will consider log-Hölder continuous variable exponents, namely those q in P(M ) such that 1/q is log-Hölder continuous; we denote by P log (M ) the set of those exponents. The relation between P log (M ) and P log (Rn ) is the following: Lemma 3.1. Let q ∈ P log (M ), and let (Ω, φ) be a chart such that 1 δij ≤ gij ≤ 2δij 2 as bilinear forms, where δij is the delta Kronecker symbol. Then q ◦ φ−1 ∈ P log (φ(Ω)). 4. Continuous embeddings We refer to the books of Aubin [2] and Hebey [17] for the details of the following discussion. Modulo the best constants appearing in the estimates, the classic Sobolev embeddings with non-variable exponents for compact Riemannian manifolds2 are the same 2

All the manifolds that we consider are smooth.

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as those obtained for functions of compact support in the Euclidean setting. To achieve those results, one carefully picks a finite atlas together with a smooth partition of unity, where the Riemannian metric tensor is uniformly bounded as a bilinear form. On the other hand, for complete3 non-compact manifolds, the classic Sobolev embeddings are more delicate. For the Sobolev estimates to hold on complete manifolds, a necessary condition is a uniform lower bound for the volume of balls. If in addition a lower bound for the Ricci tensor of g is known, i.e. there exists a constant λ such that Rc(g) ≥ λ(n − 1)g , where n is the dimension of M , then the uniform lower bound on the volume of balls is also sufficient. Examples where the Sobolev estimates fail are constructed on warped products, namely manifolds of the form (M = R × N, g), with g(x, y) = dx2 ⊕ φ(x)2 gN (y), where (N, gN ) is a compact Riemannian (n − 1)-manifold whose Ricci tensor is bounded from below by a multiple of gN , and φ : R → R is smooth. If one chooses φ to have exponential decay along the R factor in M , the lower bound on Rc(g) is achieved, but as we go far away from {0} × N , the volume of balls tends to zero. Using Bishop–Gromov’s comparison, the hypothesis Rc(g) ≥ λ(n − 1)g on the Ricci tensor gives, whenever R is at most the diameter of (V, g) and t ≤ 1, a bound |BR (x)|g ≤

R |λ |B |BtR (x)|g  |B tR |λ

(8)

for the volume |BR (x)|g of balls of radius R centered at some point x in terms of the r |λ is the volume of any ball of radius r in volume of smaller concentric balls, where |B the simply connected space of dimension n whose sectional curvature is everywhere equal R |λ /|B  to λ. Note that an explicit upper bound for the quotient |B tR |λ can be given, also  that limt→0 |BtR (x)|g /|BtR |λ = 1. To get a non-trivial lower bound for the volume of balls, the hypothesis of positive injectivity radius can be used. Using both hypothesis, coverings of M by balls, having appropriate properties, are built. Those coverings are used as the charts of an atlas, for which a convenient partition of unity is constructed. Note that the positivity of the injectivity radius is stronger than the positivity of the volume of balls (see [5]). If one has a lower bound for the injectivity radius Inj(g) ≥ i for some i > 0, then there exists some positive RH = RH (n, λ, i), known as the harmonic radius, allowing the construction of charts (BRH (x), φx ) around every point x in M , in such a way that the metric tensor is controlled as a bilinear form, in the sense of Lemma 3.1. 3 Namely Riemannian manifolds for which the distance induced by the Riemannian metric makes them complete metric spaces.

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We will use the following conditions on (M, g), depending on the context: Definition 4.1. We say that the n-manifold (M, g) has property Bvol (λ, ϑ) if its geometry is bounded in the following sense: • Rc(g) ≥ λ(n − 1)g for some λ, and • There exists some ϑ > 0 such that |B1 (x)|g ≥ ϑ for every x in M . Definition 4.2. We say that the n-manifold (M, g) has property Binj (λ, i) if its geometry is bounded in the following sense: • Rc(g) ≥ λ(n − 1)g for some λ, and • There exists some i > 0 such that Inj(g) ≥ i. In the continuous embeddings of Lq1 (M ) into another space, when q is constant, the geometric assumptions on the n-manifold (M, g) depend on whether q < n or n < q, and on the tools used. Indeed: 1. When q < n, property Bvol (λ, ϑ) is assumed (see Theorem 4.1). 2. When n < q, usually the stronger property Binj (λ, i) is assumed (see Theorem 4.3), although approximating the manifold (M, g) by discrete graphs (see [4], and the comments in [17]), the same result can be obtained assuming property Bvol (λ, ϑ). Our extension of those results will use classic geometric analysis, and will depend on the type of continuity on the (non-constant) exponents. The dichotomy q < n or n < q cannot be always established, although there are similitudes (see Theorems 4.2 and 4.4 below). 4.1. Embeddings for continuous exponents When q < n and p are constant, the Sobolev embeddings of type Lq1 (M ) → Lp (M ) for complete n-manifolds can be summarized as follows: Theorem 4.1. (See [2,17].) Let (M, g) be a complete Riemannian n-manifold. Then: n

1. If the embedding L11 (M ) → L n−1 (M ) holds, then whenever the real numbers p and q satisfy 1≤q
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q ≤ p ≤ q∗ =

11

nq , n−q

the embedding Lq1 (M ) → Lp (M ) also holds. 2. If the n-manifold (M, g) has property Bvol (λ, ϑ) for some (λ, ϑ), then L11 (M ) → n L n−1 (M ) holds. The next lemma is the point of departure for Theorem 4.1. The proof uses, among other results, a Riemannian isoperimetric inequality and Federer’s coarea formula. The reader can find the details in [17]. Lemma 4.1. Assume that for some (λ, ϑ) the complete n-manifold (M, g) has property Bvol (λ, ϑ). Then there exist positive constants δ0 = δ0 (n, λ, ϑ) and A = A(n, λ, ϑ), such that uLn/(n−1) ≤ A|∇u|L1 whenever x ∈ M and u ∈ D(BR (x)) for R ≤ δ0 , where D(Ω) is the algebra of smooth functions of compact support in the set Ω. n

In Theorem 4.1, the extension of L11 (M ) → L n−1 (M ) to Lq1 (M ) → Lp (M ) whenever nq 1 ≤ q < n and p = n−q is based on standard density arguments for the elements of D(M ) and some manipulation of the exponents involved. Similar arguments give the following extension of Lemma 4.1. Lemma 4.2. Assume that the complete n-manifold (M, g) has property Bvol (λ, ϑ) for some (λ, ϑ). Then, using the constants δ0 = δ0 (n, λ, ϑ) and A = A(n, λ, ϑ) of Lemma 4.1 we have, if R ≤ δ0 , if x ∈ M , if 1 ≤ q < n, and if u ∈ Lq1,0 (BR (x)), the estimate uLp ≤ A p |∇u|Lq , where 1/p = 1/q − 1/n. Our next result gives an extension of the embedding Lq1 (M ) → Lp (M ), under suitable hypothesis, to the case when q and p are variable exponents. To abbreviate, if p and q belong to P(M ), we write q  p if there exists an > 0 such that for every x in M one has q(x) + ≤ p(x). Theorem 4.2. Assume that for some (λ, ϑ) the complete n-manifold (M, g) has property q(·) Bvol (λ, ϑ). Let q ∈ P(M ) be uniformly continuous, with q + < n. Then L1 (M ) → nq Lp(·) (M ) for every p ∈ P(M ) such that q  p  q ∗ = n−q . In fact, for uLq(·) 1 sufficiently small we have the estimate

 ρp(·) (u) ≤ G ρq(·) (u) + ρq(·) (|∇u|) , (9) where the constant G depends on n, λ, ϑ, p and q.

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q(·)

Remark 1. It is obvious that L1 (M ) → Lq(·) (M ). Hence if both p and q are constant exponents, Theorem 4.2 gives Theorem 4.1, except when p = q ∗ . The case p = q ∗ is implicitly contained in Theorem 4.2 (see initial stage in its proof), and explicitly in Theorem 4.4 (Section 4.2), where (M, g) must satisfy the stronger Binj (λ, i) property. Proof of Theorem 4.2. Since q is uniformly continuous, due to the hypothesis on the exponents there exists a δ = δ(p, q) larger than zero such that for every x in M  p+ Bδ (x)

≤

nq n−q

− = Bδ (x)

− n qB δ (x)

+ qB ≤ p− Bδ (x) . δ (x)

and

− n − qB δ (x)

Given R > 0, we can choose a countable set of points forming an R2 -net, say {xi }, such that dg (xi , xj ) > R2 whenever i = j, and such that the balls of radius R2 centered at those points, namely {B R (xi )}, give a covering of M . Every point of M is contained 2 in a subset of R-balls centered at the points in the R2 -net, and it is not difficult to see that the maximal cardinality of those subsets has a finite upper bound N = N (n, λ, R): This follows from Bishop–Gromov’s estimate (8). Let R = R(δ, δ0 ) := min{δ, δ0 }, where δ0 is from Lemma 4.1. Choose an R2 -net {xi } on M , to construct a partition of unity subordinated to the covering {BR (xi )} in the following way (see [17]). Consider the function ρ : R+ → [0, 1] given by ⎧ ⎪ ⎨1 ρ(t) := 3 − ⎪ ⎩ 0

if t < 4t R

if

R 2

R 2

≤t<

if t ≥

3R 4

3R 4

,

to note that αi (x) := ρ(dg (x, xi )) is a Lipschitz function with compact support in the  ball B 3R (xi ) ⊂ BR (xi ). If ηi := αi / k αk , then {ηi } is a Lipschitz partition of unity 4 subordinated to {BR (xi )}, hence there exists some H = H(N, R) such that |∇ηi | ≤ H almost everywhere, where N is the upper bound for the multiplicity of the covering {BR (xi )} constructed from the R2 -net {xi }. We observe that ηi is in L11,0 (BR (xi )). Fix some xi in the R2 -net for M . To abbreviate, whenever r ∈ P(M ), we identify − + rB with ri+ , and rB with ri− . Due to the assumptions on the exponents, qi− < n, R (xi ) R (xi ) − + qi ≤ pi , and there exists a real number pi such that p+ i ≤ pi =

n qi− . n − qi−

Since spt ηi u ⊂ BR (xi ), Lemma 4.2 provides the inequality ηi uLpi ≤ A pi |∇(ηi u)| where A is the constant from Lemma 4.1.

−

Lqi

,

(10)

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

13

When t → 0 Bishop–Gromov’s estimate provides the bound 

R |λ < ∞ , dVg = |BR (xi )|g ≤ |B

(11)

BR (xi )

and using Theorem 2.1 (Section 2.1) together with (11) in (10) one has R |λ )2 |∇(ηi u)|Lq(·) . ηi uLp(·) ≤ A pi (1 + |B Since pi ≤

+ nqM + n−qM

=

nq + n−q +

=: C(n, q) = C we obtain ηi uLp(·) ≤ D |∇(ηi u)|Lq(·)

(12)

R |λ )2 has been for each i, where the abbreviation D = D(n, λ, ϑ, p, q) := A C (1 + |B used. Given r in P(M ) and a sequence of positive numbers {si } larger than one, where i is the index used in the R-net {xi } and in the partition of unity {ηi }, we define the space Lr(·),si (M ) as the set of functions u : M → R such that the convex functional ρsi ,r(·) (u) :=



ηi usLir(·)

i

is finite, and endow it with the norm4  u  uLr(·),si := inf t > 0 | ρsi ,r(·) ≤1 . t q(·)

To conclude that L1 (M ) → Lp(·) (M ), we choose an adequate r in P(M ) and a sequence {si } such that q(·)

L1 (M ) → Lr(·),si (M ) → Lp(·) (M ) . We have two steps. + q(·) Step 1. We prove that L1 (M ) → Lp(·),qi (M ). Let u be such that uLq(·) ≤ 1. In 1 particular ηi uLq(·) = ηi uLq(·) + |∇(ηi u)|Lq(·) ≤ 1 1

for each i. Using the relations between norm and convex-modular (see Section 2.1) we see that 4

It is not difficult to check that Lr(·),si (M ) is a Banach space.

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

14



q+

|∇(ηi u)|Liq(·) ≤ ρq(·) (|∇(ηi u)|) =

| ∇ηi u + ηi ∇u |q(x) dVg , BR (xi )

with

  q(x) | ∇ηi u + ηi ∇u |q(x) ≤ 2q(x)−1 |∇ηi |q(x) |u|q(x) + ηi |∇u|q(x) ,

hence 

q+

|∇(ηi u)|Liq(·) ≤ 2q

   +  − N max H q , H q ρq(·) (u) + ρq(·) (|∇u|) ,

−1

+

i

thus 

 ρq(·) (u) + ρq(·) (|∇u|) ,

q+

|∇(ηi u)|Liq(·) ≤ E

(13)

i

  + + − where E = E(q, N, H) := 2q −1 1 + N max{H q , H q } , N and H being the constants mentioned before. Combining (13) with (12) we get 

q+

ηi uLip(·) ≤

i



q+

+

+

Dqi |∇(ηi u)|Liq(·) ≤ (D + 1)q E



 ρq(·) (u) + ρq(·) (|∇u|) ,

i

or



ρq+ ,p(·) (u) ≤ F i

 ρq(·) (u) + ρq(·) (|∇u|) ,

+

where F = F (n, λ, ϑ, p, q) := (D + 1)q E. + Step 2. We prove that Lp(x),qi (M ) → Lp(·) (M ). Let u be such that ρqi+ ,p(·) (u) ≤ 1. In particular ηi uLp(·) ≤ 1 for every i. Since qi+ ≤ p− i , it follows that p−

q+

ηi uLip(·) ≤ ηi uLip(·) ≤ 1 , thus the inequality  ρp(·)



 +

≤ Np

ηi u

−1



i

ρp(·) (ηi u)

i

yields  ρp(·) (u) = ρp(·)



 ηi u

+

≤ Np

−1



i

+

≤ Np

−1



ρp(·) (ηi u)

i p−

+

ηi uLip(·) ≤ N p

−1

ρq+ ,p(·) (u) i

i

recalling the relations between norm and convex-modular once more.

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

15

Finally, putting step 1 and step 2 together, we obtain estimate (9) with constant + G = F N p −1 . 2 4.2. Embeddings for log-Hölder continuous exponents When n < q and α ∈ ]0, 1] are constant, the Sobolev embeddings of type Lq1 (M ) → C (M ) for complete n-manifolds can be summarized as follows: 0,α

Theorem 4.3. (See [17].) Assume that for some (λ, i) the complete n-manifold (M, g) has property Binj (λ, i). If n < q and α = 1 − n/q, then Lq1 (M ) → C 0,α (M ). The next result extends Theorem 4.3 to variable exponents when n < q − . It also q(·) extends the result of Theorem 4.2, namely the embedding L1 (M ) → Lp(·) (M ) for nq(x) q + < n, to the case when p(x) = q(x)∗ := n−q(x) . To prove those results, we need suitable hypothesis on the geometry of (M, g), and on the continuity of the exponent q. Theorem 4.4. Assume that the complete n-manifold (M, g) has property Binj (λ, i) for some (λ, i), and that the exponent q belongs to P log (M ). i) If q + < n and p(x) =

nq(x) n−q(x) ,

then q(·)

L1 (M ) → Lp(·) (M ). ii) If n < q − and α(x) = 1 −

n q(x) ,

then

q(·)

L1 (M ) → C 0,α(·) (M ). The following lemma is substantial in the proof of Theorem 4.4. Given x in M , we obtain estimates in a chart centered at x, such that the metric tensor is controlled as a bilinear form, in the sense of Lemma 3.1. An important issue in those estimates is that the constant in the embeddings will not depend on the point x. Lemma 4.3. Suppose that q belongs to P log (M ), and let R be strictly larger than 0. Fix x ∈ M , and let (B3R (x), φx ) be a chart such that φx (x) = 0, with 1 δij ≤ gij ≤ 2δij . 2 i) If q + < n then we have the embedding q(·)

L1,0 (BR (x)) → Lp(·) (M ) ,

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

16

where p(x) = that for every

nq(x) n−q(x) . In fact, there q(·) u in L1,0 (BR (x)) we

exists a constant D = D(n, clog (q), q + , q − ) such have

uLp(·) ≤ D uLq(·) . 1

ii) If n < q − , then we have the embedding q(·)

L1 (BR (x)) → C 0,α(·) (BR (x)) , n + − q(·) . In fact, there exists a constant D = D(n, clog (q), q , q ) q(·) in L1,0 (BR (x)) with uLq(·) small enough, we have 1

where α(·) = 1 − that for every u

such

1

uC 0,α ≤ D (ρq(·) (u) + ρq(·) (|∇u|)) q+ . Remark 2. For every pair {y, y  } in BR (x) the minimal geodesic between them is contained in the ball B3R (x). Hence the inequalities √ 1 √ dg (y, y  ) ≤ |φx (y) − φx (y  )| ≤ 2 dg (y, y  ) 2 hold therein. q(·)

Proof of Lemma 4.3. Let u belong to L1,0 (BR (x)). Then we have the following elementary estimates: 

+ q◦φ−1 u ◦ φ−1 x (z) dLn (z) ≤ 2 n2 ρq(·) (u) ≤ 2 n+q 2 ρq(·) (u) , x

(14)

+ q◦φ−1 ∇(u ◦ φ−1 x (z) dLn (z) ≤ 2 n+q 2 ρq(·) (|∇u|) , x )

(15)

φx (BR (x))



φx (BR (x))

and ρp(·) (u) ≤ 2



n 2

p◦φ−1 u ◦ φ−1 x (z) dLn (z) . x

φx (BR (x))

For instance, (15) follows from  q

|∇u| (y) dVg (y)

ρq(·) (|∇u|) = BR (x)



= φx (BR (x))



q −1 −1 2 ◦φx det gij g kj Dk (u ◦ φ−1 (z) dLn (z) x )Dj (u ◦ φx )

(16)

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

≥ 2−



n+q + 2

17

q◦φ−1 x (z) dLn (z) . ∇(u ◦ φ−1 x )

φx (BR (x)) −1 n Our first aim is to extend u ◦ φ−1 x and q ◦ φx to all R . n We extend the function u ◦ φ−1 x to all R by setting it equal to 0 outside φx (BR (x)), and denote the extended function by u ¯. Observe that φx (BR (x))  B2R (0) ⊂ Rn . n To extend q ◦ φ−1 x to all R in such a way that the extension does not depend on the point x, we proceed as follows. Denote by q˜ the following extension of q ◦ φ−1 x to the region Ω := φx (BR (x)) ∪ (Rn \ B4R (0)):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1/˜ q (z) :=



L/ log e + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1/q ◦ φ−1 x (z) for

z ∈ φx (BR (x))

1/q + for

|z| = 4R

1 |z|−4R



+ 1/q + for 4R < |z| < 1/q − for

|z| ≥

1

Lq + q − e q+ −q−

+ 4R −e

1

Lq + q − e q+ −q−

+ 4R, −e

where       1 1 , 1− log(e + 2R) . L = max 2clog (q), 2 log e + R n Using Definition 2.1 in Section 2.2, we will see that clog (˜ q ) = L and that 1/˜ q is globally log-Hölder continuous in the region Ω, with   clog,glob (˜ q ) = max L, F (L, q + , q − ) , where  −

F (L, q , q ) := sup log (e + |z|) +



1 1 L − +− 1 q− q log(e + |z|−4R )

 : 4R ≤ |z| ≤ α + 4R

Lq + q −

using the abbreviation α := 1/(e q+ −q− − e). Note that the supremum involved in F (L, q + , q − ) is in fact a maximum. We proceed in two steps: q is locally log-Hölder continuous with clog (˜ q ) = L. Indeed: Step 1. We verify that 1/˜ • If both z and z  belong to φx (BR (x)), then 1 2c (q) 1 L ≤ .  log  q˜(z) − q˜(z  ) ≤ 1 1 log e + |z−z | log e + |z−z |

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

• If z ∈ φx (BR (x)) and z  ∈ Rn \ B4R (0), then

 1 1 2 log e + R L 1   ≤ . q˜(z) − q˜(z  ) ≤ 2 ≤ 1 1 log e + |z−z log e + |z−z | | • If z, z  ∈ Bα+4R (0) \ B4R (0), let z  = 4R z  /|z  |: We have 1 L 1  q˜(z) − q˜(z  ) = log e +

L  −  1 1 log e + |z |−4R |z|−4R

≤ |distL (z, z  ) − distL (z  , z  )| ≤ distL (z, z  ) =

L  , 1 log e + |z−z |

where the properties of distL given by (2) in Section 2.2 have been used. • Consider now z ∈ Bα+4R (0) \ B4R (0) and z  ∈ Rn \ Bα+4R (0). First note that 1 L 1   , ≤ ++ − 1 q q log e + |z |−4R hence 0≤ ≤

1 1 1 1 L   − = −− +− 1 q˜(z  ) q˜(z) q q log e + |z|−4R 

L

log e +

1 |z  |−4R

−



L

log e +

1 |z|−4R

≤



L

log e +

1 |z−z  |

.

• Now fix z ∈ Bα+4R (0) \ B4R (0), and set z  = 4R z/|z|: Then 1 L L 1 = .   − = 1 1 q˜(z) q˜(z  ) log e + |z|−4R log e + |z−z | Following Definition 2.1, we see that q˜ belongs to P log (Ω) with clog (˜ q ) = L. q satisfies the log-Hölder decay condition, with q˜∞ = q− . Step 2. We check that 1/˜ Indeed: • If z ∈ φx (BR (x)), then   

1 1 − n1 log (e + R) 1 1 L ≤ . q˜(z) − q − ≤ 1 − n ≤ log (e + |z|) log (e + |z|)

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

19

• If z ∈ Bα+4R (0) \ B4R (0), then   L log (e + |z|) q1− − q1+ − 1 log(e+ |z|−4R ) 1 1 F (L, q + , q − ) ≤ . q˜(z) − q − = log (e + |z|) log (e + |z|) Hence 1/˜ q satisfies the log-Hölder decay condition in Ω with constant max{L, + − F (L, q , q )}. log The conclusion of both steps and Definition 2.1 guarantee that q˜ is in Pglob (Ω), where   clog,glob (˜ q ) = max L, F (L, q + , q − ) = clog,glob (˜ q )(R, clog (q), q + , q − ) does not depend on x. Finally, Proposition 4.1.7 from [7] ensures that we can extend q˜ log to q¯ ∈ Pglob (Rn ) in such a way that clog,glob (¯ q ) = clog,glob (˜ q ), q¯+ = q + and q¯− = q − . We return to items i) and ii) of the lemma. − n+q

q(·)

+

i) Scaling u ∈ L1,0 (BR (x)) in such a way that uLq(·) (BR (x)) ≤ 2 2q− ≤ 1, both 1 uLq(·) (BR (x)) and ∇uLq(·) (BR (x)) are not larger than such a number. Using the relations between norm and convex modular, together with estimate (14) and (15), we observe that ¯ uLq(·) ¯ (Rn ) ≤ 2

n+q + 2q +

1

ρq(·) (u) q+

(17)

and ∇¯ uLq(·) ¯ (Rn ) ≤ 2

n+q + 2q +

1

ρq(·) (|∇u|) q+ .

(18)

To prove (17), we use the chain of inequalities 1≥2

n+q + 2q +

= (2

q− +

uLq q(·) (BR (x)) ≥ 2

n+q + 2

n+q + 2q +

1

1

ρq(·) (u) q−

q− q+

1

ρq(·) (u)) q+ ≥ ρq¯(·) (¯ u) q+ ≥ ¯ uLq(·) ¯ (Rn ) ,

where 



q◦φ−1 u ◦ φ−1 x (z) dLn (z) x

|¯ u| (y) dL (y) = q¯

ρq¯(·) (¯ u) =

n

Rn

φx (BR (x))

by the construction of u and q. Inequality (18) is similar. 1 Moreover, the concavity of t → t q+ ensures that 1

1

ρq(·) (u) q+ + ρq(·) (|∇u|) q+ ≤ 2

q + −1 q+

1

(ρq(·) (u) + ρq(·) (|∇u|)) q+ ,

(19)

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

hence from (17) and (18) we obtain ¯ uLq(·) ≤2 ¯ (Rn )

n+3q + −2 2q +

1

1

(ρq(·) (u) + ρq(·) (|∇u|)) q+ .

(20)

On the other hand, by i) of Theorem 2.2 we have ¯ uLp(·) uLq(·) , ¯ ¯ (Rn ) ≤ C ¯ (Rn ) 1

where p¯(·) =

n¯ q (·) n−¯ q (·) ,

and C = C(n, clog (¯ q ), q¯+ , q¯− ). Combining with (20) we get

¯ uLp(·) ¯ (Rn ) ≤ C 2

n+3q + −2 2q +

1

(ρq(·) (u) + ρq(·) (|∇u|)) q+ .

(21)

Scaling u again if needed, so that ¯ uLp(·) ¯ (Rn ) ≤ 1, inequality (16) gives ¯ uLp(·) ¯ (Rn ) ≥ 2

− 2pn−

1

ρp(·) (u) p− ,

and using (21), we obtain p−

ρp(·) (u) ≤ D (ρq(·) (u) + ρq(·) (|∇u|)) q+ , where the constant D depends only on n, clog (q), q + and q − . This leads to estimate i). √ ii) Since |z − z  | ≤ 2 dg (φ−1 (z), φ−1 (z  )) for z, z  in φx (BR (x)), we have [u]α(·)(BR (x)) + uL∞ (BR (x)) = uC 0,α (BR (x)) ≤

√

2 ¯ uC 0,α¯ (Rn ) .

Scaling u as in i), inequalities (17)–(18) together with ii) in Theorem 2.2 give uC 0,α (BR (x)) ≤ C 2

n+4q + −2 2q +

1

(ρq(·) (u) + ρq(·) (|∇u|)) q+ ,

where C = C(n, clog (q), q + , q − ), proving the assertion in ii).

2

Proof of Theorem 4.4. As mentioned in the beginning of Section 4, the hypothesis Binj (λ, i) on the geometry of (M, g) ensures that there exists some RH > 0, called the harmonic radius, depending only on n, λ and i, such that around every point x in M we have a chart (BRH (x), φx ) satisfying the initial hypothesis of Lemma 4.3. i) Since q  p, and q as well as p are uniformly continuous, there exists some δ > 0 such + RH that for every x ∈ M we have qB ≤ p− Bδ (x) . Then, for R = min{δ, 3 } the conclusions δ (x) of i) in Lemma 4.3 are valid. We proceed analogously to the proof of Theorem 4.2. The details are left to the reader. ii) For every fixed point x ∈ M denote by ηx : M → R any Lipschitz function satisfying the following properties:

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

spt ηx ⊂ B RH (x), 3

ηx |B RH (x) ≡ 1,

∇ηx L∞ (M ) ≤

6

21

12 RH

Let (BRH (x), φx ) be a chart satisfying the assumptions of Lemma 4.3. Scaling u, so that ηx u is small enough in the sense of ii) in Lemma 4.3, we get uL∞ (M ) = sup |u(y)| ≤ sup y∈M

1

sup

x∈M y∈B RH (x)

|ηx u(y)| ≤ D (ρq(·) (ηx u) + ρq(·) (|∇(ηx u)|)) q+ ,

3

with +

ρq(·) (ηx u) + ρq(·) (|∇(ηx u)|) ≤ 2q (1 +

12 q+ ) (ρq(·) (u) + ρq(·) (|∇u|)) , RH

leading to uL∞ (M ) ≤ 2 D (1 +

1 12 ) (ρq(·) (u) + ρq(·) (|∇u|)) q+ . RH

Next, pick x, y ∈ M such that dg (x, y) ≤ |u(x) − u(y)| dg (x, y)

n 1− q(y)

=

|ηx u(x) − ηx u(y)| dg (x, y)

n 1− q(y)

RH 6 .

≤

(22)

Since ηx (x) = ηx (y) = 1, we have sup

y,y  ∈B RH (x)

|ηx u(y) − ηx u(y  )| n

dg (y, y  )1− q(y)

,

3

hence Lemma 4.3 yields |u(x) − u(y)| dg (x, y)

n 1− q(y)



12 ≤2 D 1+ RH



1

(ρq(·) (u) + ρq(·) (|∇u|)) q+ .

Finally, gathering (22) with (23) completes the proof of Theorem 4.4.

(23)

2

5. Compact embeddings Given a Riemannian manifold (M, g), consider Iso(M, g), the group of global isometries of (M, g). Certainly Iso(M, g) is a subgroup of Diff(M ), the group of global diffeomorphisms of M . Moreover, Iso(M, g) is a Lie group acting differentiably on (M, g) (see [22]). In what follows H will be a fixed subgroup of Iso(M, g). As usual, the orbit of x ∈ M under the action of H is the set H(x) = {h(x) | h ∈ H}. Given x in M and R > 0 consider the following quantity: M (x, R) = sup {card{xi }i∈I | xi ∈ H(x), BR (xi ) ∩ BR (xj ) = ∅ for i = j} . M (x, R) gives the lowest upper bound for the number of non-overlapping R-balls in the orbit of x. On the other hand, for every q in P(M ) that is H-invariant, we denote by q(·) q(·) L1,H (M ) the subspace of L1 (M ) consisting of H-invariant functions. With those definitions we can state the main result of this section.

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M. Gaczkowski et al. / Journal of Functional Analysis ••• (••••) •••–•••

Theorem 5.1. Assume that the complete n-manifold (M, g) has property Bvol (λ, ϑ) for some (λ, ϑ), and suppose that given R > 0, we have that lim

inf

T →∞ x∈M \BT (o)

M (x, R) = ∞ ,

where o is any fixed point of M . Let q ∈ P(M ) be uniformly continuous, H-invariant, and satisfying 1 < q − ≤ q + < n. Then, for every p ∈ P(M ) such that q  p  q ∗ , we have the compact embedding q(·)

L1,H (M ) →→ Lp(·) (M ) . Remark 3. We make some comments on the hypothesis of Theorem 5.1. 1. The growth of M (x, R) as x goes to infinity requires that: (a) H must be an infinite group, either discrete or uncountable. As explained below, q(·) the non-triviality of L1,H (M ) requires H to be a compact Lie group. (b) To be able to pack an infinite number of balls of a given radius R > 0 whose centers lie in H(x), the metric g must be such that the maximal distance between two elements in H(x) is unbounded, since M is finite dimensional and g satisfies property Bvol (λ, ϑ). q(·) 2. If L1,H (M ) is non-trivial, then H must be endowed with a bounded Haar measure, say m(H). This can be seen, at least formally, as follows. Consider the projection π : M → M/H from M to the space of orbits of H in M . Consider the set of well behaved fibers of the projection, namely the principal orbits. The volume of each of the well behaved fibers of the projection has a volume that is proportional to m(H). q(·) If u belongs to L1,H (M ), then |u|q is a well defined function on M/H. Since the preimage of the principal orbits of is an open dense submanifold of M (see [3]), one can bound ρq(·) (u) < ∞ from below using Federer’s coarea formula by a multiple of m(H), hence m(H) must be bounded. Hence necessary and sufficient conditions for the non-triviality of Theorem 5.1 are: 1. H is a non-trivial compact Lie subgroup of Iso(M, g). 2. In addition to property Bvol (λ, ϑ), g must be such that the diameter of H(x) diverges when x goes to infinity. A simple example is the action of SO(n) on (Rn ≡ R+ × S n−1 , g) fixing the origin and preserving the S n−1 ’s, where g = dr2 ⊕ r2 dΩ2 is the Euclidean metric on Rn in spherical coordinates. A generalization of the previous example is the following picture: H is a compact Lie group, (Y, gY ) is a homogeneous space for H, and (X, gX ) is a complete non-compact Riemannian manifold; if H acts on M = X × Y leaving all the elements of X fixed so that H(x, y) = {x} × Y whenever (x, y) is in X × Y , then in the warped

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23

product metric g(x, y) = gX (x) ⊕ φ(x)gY (y) for M we demand φ to have a strictly positive lower bound (so the volume of balls has a lower bound),5 and to diverge when x goes to infinity in X in such a way that a uniform lower bound for Rc(g) in terms of g is possible (see [3] for sufficient computations). For the proof of Theorem 5.1 we need the following lemma, whose proof, given below, relies on methods of Fan, Zhao and Zhao (see [11]). Lemma 5.1. Let q ∈ P(M ) be uniformly continuous, with 1 < q − ≤ q + < n. If {vn } is a q(·) bounded sequence in L1 (M ) such that for a given R > 0  |vn |q(x) dVg (x) = 0,

lim sup

n→∞ y∈M BR (y)

then vn → 0 in Lp(·) (M ), for p satisfying the assumptions of Theorem 5.1. Proof of Theorem 5.1. Since (M, g) has property Bvol (λ, ϑ), by Theorem 4.2 we have q(·) L1,H (M ) → Lp(·) (M ). We take a bounded sequence un ∈ L1,H (M ); since 1 < q − , by reflexivity we can assume that there exists a subsequence, that we denote by the same symbols, namely q(·) {un }, that weakly converges to some u in L1,H (M ), thus the sequence {vn := un − u} q(·)

q(·)

weakly converges to 0 in L1,H (M ), with ρq(·) (vn ) ≤ C for some C. We will prove that  |vn |q(x) dVg (x) = 0

lim sup

n→∞ y∈M BR (y)

for a given R, and Lemma 5.1 will imply that {un } converges to u in Lp(·) (M ). Since the Riemannian measure dVg is H-invariant, for each y1 ∈ M and for every y2 ∈ H(y1 ) 

 |vn (x)|

q(x)

BR (y1 )

|vn (x)|q(x) dVg (x),

dVg (x) = BR (y2 )

therefore for every y in M  |vn (x)|q(x) dVg (x) ≤ ρq(·) (vn ) ≤ C .

M (y, R)

(24)

BR (y)

Choose now a base point o in M . Estimate (24) and the assumptions on M (y, R) ensure that for each > 0 there exists some finite T such that for every n 5

In fact this requirement gives a positive injectivity radius.

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24

 sup y∈M \BT (o) BR (y)

|vn (x)|q(x) dVg (x) ≤ C/

inf

y∈M \BT (o)

M (y, R) ≤ .

(25)

¯T +R (o). Given any r > 0 Consider the closed ball of radius T + R around o, namely B  the number of r-balls needed to cover BT +R (o) is not larger than the largest value of |BT +R (o)|g /|Br/2 (x)|g as x varies in BT +R (o). Thanks to Bishop–Gromov’s estimate   (8), such a number of balls has the finite upper bound |B T +R |λ /|Br/2 |λ . In other words, ¯T +R (o) endowed with the metric dg is a complete and totally bounded subset of the B 

complete metric space (M, dg ), and therefore it is compact. Arguments given in [13,14] for compact manifolds, and in [18] for bounded subsets of Rn , allow us to infer that we have the compact embedding q(·) ¯ q(·) ¯ L1 (B (BT +R (o)). T +R (o)) →→ L

(26)

Next, setting k = 1/k and considering the associated Tk in (25), using (26) and Cantor’s diagonal argument we construct a subsequence vn such that  |vn (x)|q(x) dVg (x) = 0.

lim sup

n→∞ y∈M BR (y)

Finally, Lemma 5.1 finishes the proof of Theorem 5.1.

2

In the rest of this section we will prove Lemma 5.1. Using Hölder’s inequality we get the next interpolation result (see [11] for similar considerations). Lemma 5.2. Let Ω be an open subset of M , and let q, p and β be elements of P(Ω) ∩L∞ (Ω) such that qpβ

in Ω.

If u ∈ Lq(·) (Ω) ∩ Lβ(·) (Ω), then u ∈ Lp(·) (Ω) with  |u|p(x) dVg ≤ 2 |u|p1 Lm(·) (Ω) |u|p2 Lm (·) (Ω) , Ω

where p1 (x) =

q(x)(β(x) − p(x)) , β(x) − q(x)

p2 (x) =

β(x)(p(x) − q(x)) , β(x) − q(x)

β(x) − q(x) , β(x) − p(x)

m (x) =

β(x) − q(x) . p(x) − q(x)

m(x) =

Now we can prove the following lemma.

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25

Lemma 5.3. Let q, R, and {vn } satisfy the hypotheses of Lemma 5.1. If p ∈ P(M ) ∩ L∞ (M ) is such that q  p  q ∗ , then  lim sup |vn |p(x) dVg = 0. n→∞ y∈M

BR (y)

Proof. Let β ∈ P(M ) ∩ L∞ (M ) be such that p  β  q ∗ . By Theorem 4.2 we have q(·) L1 (M ) → Lβ(·) (M ). Therefore the sequence {vn } is bounded in Lβ(·) (M ), and there exists c1 > 0 such that  sup |vn |β(x) dVg ≤ c1 . (27) n

M

Using the same notation as in Lemma 5.2, for a fixed y ∈ M and for q  p  β we have  |vn |p(x) dVg ≤ 2 |vn |p1 Lm(·) (BR (y)) |vn |p2 Lm (·) (BR (y)) . (28) BR (y)

Moreover, by (27) and the definition of p2 , there exists some c2 such that sup |vn |p2 Lm (·) (BR (y)) ≤ c2 .

(29)

n

Fix y in M and consider the sequence t(y)n := |vn |p1 Lm(·) (BR (y)) . The definitions of q, m and convex-modular functionals yield 

|vn |q(x) m(x)

BR (y)

t(y)n

dVg = 1.

According to the hypothesis for {vn }, for sufficiently large n we have that t(y)n < 1, hence  + t(y)m ≤ |vn |q(x) dVg , n BR (y)

thus using (28) together with (29) we infer that  1 |vn |p(x) dVg ≤ 2 t(y)n |vn |p2 Lm (·) (BR (y)) ≤ 2 τnm+ c2 , BR (y)

 |vn |q(x) dVg . Taking the limit when n goes to infinity, we finish the

where τn := sup y∈M BR (y)

proof of Lemma 5.3.

2

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26

Proof of Lemma 5.1. To conclude the proof of this lemma, we will need further restrictions on the auxiliary exponent β, in addition to q  p  β  q ∗ appearing in Lemma 5.3 and Lemma 5.2. Since q  p, consider ζ := inf x∈M (p(x) − q(x)), and let be a positive number satisfying ≤ ζ 2 /2q + and p(x) +  q ∗ (x) for every x in M . Choose β given by β(x) := p(x) + , and note that being q uniformly continuous, there exists some R > 0 + such that for every y in M we have p− BR (y) ≥ qBR (y) + ζ/2. Hence m (x) = 1 +

ζ β(x) − p(x) ζ =1+ ≤1+ ≤1+ + ≤1+ + , p(x) − q(x) p(x) − q(x) ζ 2q 2qBR (y)

therefore under such restrictions on β the estimate p− BR (y) + m+ qB R (y)

≥

+ + ζ/2 qB R (y)

=1

+ + (1 + ζ/2qB ) qB R (y) R (y)

(30)

follows. Assume now that vn Lq(x) (M ) ≤ 1, therefore 1



 |vn |

q(x)

dVg +

BR (y)

|∇vn |q(x) dVg ≤ 1

BR (y)

for every R > 0 and every y in M . We first note that uLβ(·) (BR (y)) ≤ D0 uLq(·) (B2R (y)) ,

(31)

1

where D0 does not depend on y. In order to check (31), we will use the decreasing function ρ : R+ → [0, 1] defined as ⎧ if t < R ⎪ ⎨1 3 ρ(t) := 3 − 2t if R ≤ t < 2 R ⎪ ⎩ 0 if t ≥ 32 R , to build the truncation function αy (x) := ρ(dg (x, y)) around y. Observe that uLβ(·) (BR (y)) ≤ uαy Lβ(·) (B2R (y)) = uαy Lβ(·) (M ) and uαy Lq(·) (M ) = uαy Lq(·) (B2R (y)) ≤ 2 uLq(·) (B2R (y)) 1

1

1

because |∇αy | ≤ 2, to bound the last term in the first line with the first term in the second line using Theorem 4.2 since q  p  β  q ∗ , obtaining

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27

uLβ(·) (BR (y)) ≤ D0 uLq(·) (B2R (y)) , 1

as claimed. Next, using the same notation as in the proof of Lemma 5.3, we have 1

t(y)n ≤ τnm+ .

(32)

In a similar way, define r(y)n := |vn |p2 Lm (·) (BR (y)) ,

s(y)n := vn Lβ(·) (BR (y)) .

The basic properties of convex-modular functionals say that 



|vn |β(x)

|vn |β(x)

dVg = 1, m (x)

BR (y)

β(x)

r(y)n

s(y)n

BR (y)

dVg = 1.

According to Lemma 5.3, we have  |vn |β(x) dVg = 0.

lim sup

n→∞ y∈M BR (y)

Hence for n large enough r(y)n < 1 and s(y)n < 1. Taking into account inequality (31) and recalling that p  β, we get +

r(y)m n



− βB

≤

|vn |β(x) dVg ≤ s(y)n

R (y)

p− B

≤ s(y)n

R (y)

 p− BR (y) ≤ D0 vn Lq(·) (B2R (y)) , 1

BR (y)

and it follows that r(y)n ≤ C0 vn 

p− B

/m+ R (y) q(·) L1 (B2R (y))

.

(33)

Since vn Lq(·) (M ) ≤ 1, the relations between norm and convex-modular imply that 1

+ qB

+ qB



2R 2R vn Lq(·) + ∇vn Lq(·) ≤ (B2R (y)) (B2R (y)) (y)

 |vn |q(x) dVg +

(y)

B2R (y)

|∇vn |q(x) dVg ≤ 1.

B2R (y)

Moreover, since for every q + ≥ 1 +

vn q q(·) L1

we obtain

(B2R (y))

≤ 2q

+

−1



+

+

vn qLq(·) (B2R (y)) + ∇vn qLq(·) (B2R (y))

 ,

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28

⎛ vn 

p− B

/m+ R (y) q(·) L1 (B2R (y))

⎜ ≤ C1 ⎝

⎞





⎟ |∇vn |q(x) dVg ⎠

|vn |q(x) dVg + B2R (y)

− p BR (y) + m+ q BR (y)

,

B2R (y)

or ⎛ vn 

p− B

+ (y) /m

R q(·)

L1

(B2R (y))

⎜ ≤ C1 ⎝

⎞





⎟ |∇vn |q(x) dVg ⎠ ,

|vn |q(x) dVg + B2R (y)

(34)

B2R (y)

due to inequality (30). Combining (32)–(34) and Lemma 5.2 we conclude that  |vn |p(x) dVg ≤ 2 t(y)n r(y)n BR (y)

⎛



1 ⎜ ≤ 2 τnm+ C0 C1 ⎝

⎞



⎟ |∇vn |q(x) dVg ⎠ . (35)

|vn |q(x) dVg + B2R (y)

B2R (y)

As in Theorem 4.2, consider a sequence of points {xi } forming an R-net in M , so that every point in M is at most in N = N (n, λ, 2R) of the balls {B2R (xi )}. Thus from (35) we get  |vn |p(x) dVg ≤

  i=1

M

≤

|vn |p(x) dVg

BR (xi )



1 m+

2 τn

⎛ ⎜ C0 C1 ⎝

i=1

⎛ 1 m+

≤ 2 τn



C0 C1 N ⎝

 |vn |q(x) dVg +

B2R (xi )



B2R (xi )

 |vn |q(x) dVg +

M

⎞ ⎟ |∇vn |q(x) dVg ⎠ ⎞

M

The hypothesis of Lemma 5.1 says that τn → 0 as n → ∞, so we obtain  |vn |p(x) dVg = 0,

lim

n→∞ M

as claimed.

2

1

|∇vn |q(x) dVg ⎠ ≤ 2 τnm+ C0 C1 N.

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29

6. Application In this section we apply the main results of Section 5 to study the existence of weak solutions of the non-linear elliptic equation −Δq(x) u(x) + |u(x)|q(x)−2 u(x) = f (x, u(x))

(36)

on (M, g), where Δq(x) u(x) := div(|∇u(x)|q(x)−2 ∇u(x)) is the q(x)-Laplacian on (M, g). We note that in [9] a similar problem has been studied on bounded domains in Rn , but without considering the semilinear term |u(x)|q(x)−2 u(x). Multiplication of (36) by a test function leads to the usual definition of weak solution: q(·) We say that u ∈ L1 (M ) is a weak solution of (36) if for every φ ∈ D(M ) we have  M



|∇u(x)|q(x)−2 g(∇u(x), ∇φ(x)) + |u(x)|q(x)−2 u(x)φ(x)

 − f (x, u(x))φ(x) dVg (x) = 0.

(37)

We will assume certain symmetry or invariance of the entities in (36). Recall the framework of Section 5: We have a Lie subgroup H of Iso(M, g), and given x in M and R > 0, the number M (x, R) is the lowest upper bound for the number of disjoint R-balls in the H-orbit of x. Considering Remark 3 to avoid trivial statements, we have the main result of this section: Theorem 6.1. Let H be a non-trivial compact Lie subgroup of Iso(M, g). Assume that the complete n-manifold (M, g) has property Bvol (λ, ϑ), and lim

inf

T →∞ x∈M \BT (o)

M (x, R) = ∞,

where R > 0 and o is any fixed point in M . Let q and p belong to P(M ) be H-invariant, with q uniformly continuous, and such that 1 < q − ≤ q + < n , q  p  q ∗ , and q + < p− . Let f : M × R → R be a continuous function satisfying: (i) For each t ∈ R the function f (·, t) : M → R is H-invariant, (ii) There exists some c1 > 0 such that for each t ∈ R we have the bound |f (x, t)| ≤ c1 |t|p(x)−1 ,

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30

(iii) There exists some θ > q + and some A > 0 such that for each |t| > A we have 0 < F (x, t) ≤ f (x, t) t/θ , where F (x, t) :=

t 0

f (x, s) ds.

Then equation (36) has an H-invariant non-trivial weak solution, in the sense that (37) holds for every φ in DH (M ). We begin with the following observation.  Lemma 6.1. If u ∈ L1H (M ) is such that M u(x)ψ(x) dVg (x) = 0 for every ψ ∈ DH (M ),  then for every φ ∈ D(M ) we also have M u(x)φ(x) dVg (x) = 0. Proof. Let m be a Haar measure on H, then for φ ∈ D(M ) define  Φ(x) :=

φ(hx) dm(h). H

It follows that Φ is an H-invariant smooth function, and Φ is in DH (M ) since spt Φ is compact. Indeed, denoting by σ : H × M → M the continuous representation of H on Diff(M ) so that σ(h, x) = hx, we have spt(Φ) ⊂ σ(H × spt(φ)). Since H is a subgroup of Iso(M, g), for every function v on M and every h in H 

 v(x) dVg (x) = M

v(hx) dVg (x). M

Normalizing m, Fubini’s theorem yields 

 u(x)φ(x) dVg (x) = M

H

 = M

 = M

with

 M

⎛ ⎝ ⎛ ⎝



⎞ u(x)φ(x) dVg (x)⎠ dm(h)

M

 H

⎞ u(hx)φ(hx) dm(h)⎠ dVg (x) ⎛

u(x) ⎝



⎞ φ(hx) dm(h)⎠ dVg (x) =

H

u(x)Φ(x) dVg (x) = 0 by hypothesis. 2

 u(x)Φ(x) dVg (x) , M

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31

Remark 4. The lemma ensures that if a solution of (37) from Theorem 6.1 is sufficiently regular, then (37) holds for every φ in D(M ). Proof of Theorem 6.1. Consider the functional    1  q(x) q(x) J[u] := + |u(x)| dVg (x) − F (x, u(x)) dVg (x) =: I[u] − K[u]. |∇u(x)| q(x) M

M q(·)

q(·)

Let DJ = DI −DK : L1 (M ) → Hom(L1 (M ), R) denote the differential of J = I −K, so that    v, DI[u] = |∇u(x)|q(x)−2 g(∇u(x), ∇v(x)) + |u(x)|q(x)−2 u(x)v(x) dVg (x) M

(38) and  v, DK[u] =

f (x, u(x))v(x) dVg (x).

(39)

M

We will prove the existence of non-trivial critical points of J. For this purpose we will use the Mountain Pass theorem, see for instance [26]. We proceed in two steps: Step 1. We prove that the functional J provides a Mountain Pass geometry over q(·) L1 (M ). We have the elementary inequalities   1

1

ρq(·) (|∇u|) + ρq(·) (u) ≤ I[u] ≤ − ρq(·) (|∇u|) + ρq(·) (u) q+ q

(40)

and  |u(x)|θ dVg (x) ≤ K[u] ≤

c2 |u|>A

c1 ρp(·) (u) θ

for some c2 > 0. • First we note that the hypothesis on f (x, u(x)) ensure that J[0] = 0. • Using (40)–(41) we have J[u] ≥

 c1 1

ρp(·) (u) , ρq(·) (|∇u|) + ρq(·) (u) − + q θ

where for uLq(·) sufficiently small 1

+

+

ρq(·) (|∇u|) + ρq(·) (u) ≥ 21−q uq q(·) L1

(41)

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32

and −

−

−

ρp(·) (u) ≤ upLp(·) ≤ Gp up q(·) , L1

q(·)

G being the constant of the embedding L1 (M ) → Lp(·) (M ) that appears in Theorem 4.2. Hence if uLq(·) = r is small enough, then J[u] has the lower bound 1

−

+

21−q q+ c1 Gp p− r . J[u] ≥ r − q+ θ Since p− > q + , it follows that J[u] > 0 whenever uLq(·) = r is sufficiently small. 1 • Using (40)–(41) again we get  1

J[u] ≤ − ρq(·) (|∇u|) + ρq(·) (u) − c2 q

 |u(x)|θ dVg (x), |u|>A

hence fixing u = 0 and choosing t > 1 we infer  tq

J[tu] ≤ − ρq(·) (|∇u|) + ρq(·) (u) − tθ c2 q +

with

 |tu|>A

 |u(x)|θ dVg (x), |tu|>A

|u(x)|θ dVg (x) → ρθ (u) as t → ∞. Since θ > q + , we deduce that q(·)

J[tu] → −∞ as t → ∞. The existence of some v ∈ L1 (M ) with vLq(·) > r 1 and with J[v] < 0 follows. q(·)

We conclude that the graph of the functional J over L1,H (M ) has a Mountain Pass geometry in a neighborhood of 0. Step 2. We prove that the functional J satisfies the Palais–Smale condition. To achieve q(·) this, let {un } be a sequence in L1,H (M ) such that the associated sequence of real numbers q(·)

{J[un ]} is bounded, and such that DJ[un ] → 0 in Hom(L1 (M ), R) as n → ∞. Then q(·) we must prove that {un } contains a subsequence that converges in L1,H (M ). • First we check that J is of type C 1 . This follows from (38), (39), and the hypothesis on f and F . • The sequence {un } certainly satisfies: i) J[un ] ≤ L for some L > 0, ii) un , DJ[un ] ≤ θun Lq(·) for n large enough. 1

q(·)

We prove that the sequence is bounded in L1 (M ): Assume that in the sequence of numbers {un Lq(·) } some elements are much larger than 1, otherwise there is 1 nothing to prove. Then using (38)–(40) and the hypothesis on F , we have

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33

( 1' L + un Lq(·) ≥ J[un ] − un , DJ[un ] 1 θ    1 1

≥ ρq(·) (|∇u|) + ρq(·) (u) − + q θ   

 1

 F x, un (x) − f x, un (x) un (x) dVg (x) − θ M

 − − 1 1 21−q un q q(·) ≥ − L1 q+ θ   

 1

 F x, un (x) − f x, un (x) un (x) dVg (x) − θ 

 ≥

|un |
1 1 − + q θ



−

−

21−q un q q(·) − c(f, A) , L1

−

−

since for un Lq(·) large enough ρq(·) (|∇un |) + ρq(·) (un ) ≥ 21−q un q q(·) . Hence the L1

1

q(·)

sequence {un } is bounded in L1 (M ). q(·) • Since {un } is bounded in L1 (M ), Theorem 5.1 entails that there exists a subsequence of {un }, that we also denote by {un }, that converges in Lp(·) (M ). We will q(·) prove that {un } is Cauchy in L1,H (M ), i.e. that lim ρq(·) (um − un ) = 0 =

m,n→∞

lim ρq(·) (|∇(um − un )|) ,

m,n→∞

to finish the proof of the theorem. Consider the functionals    I1 [u] := |∇u(x)|q(x) + |u(x)|q(x) dVg (x)

(42)

q(x)<2

and  I2 [u] :=



 |∇u(x)|q(x) + |u(x)|q(x) dVg (x)

(43)

q(x)≥2 q(·)

on L1 (M ), and note that ρq(·) (um − un ) + ρq(·) (|∇(um − un )|) = I1 [um − un ] + I2 [um − un ] .

(44)

Consider also the inequalities |ξ − η|q ≤

2q−q 2 q 2  q−2 (|ξ| ξ − |η|q−2 η) · (ξ − η) 2 (|ξ| + |η|) 2 q−1

for 1 ≤ q ≤ 2, (45)

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and |ξ − η|q ≤ 2q (|ξ|q−2 ξ − |η|q−2 η) · (ξ − η)

for q ≥ 2 ,

(46)

where ξ and η belong to Rn , see [21], noting that (|ξ|q−2 ξ − |η|q−2 η) · (ξ − η) ≥ 0 for every q. From (43), (46) and (38), we immediately get 2−q I2 [um − un ] ≤ um − un , DI[um ] − DI[un ] . +

(47)

In a similar way, using (42) and (45), we see that q− − 1 I1 [um − un ] 2  q ≤ g(|∇um |q−2 ∇um − |∇un |q−2 ∇un , ∇(um − un )) 2 q<2 2q−q 2

× (|∇um | + |∇un |) 2 dVg (x)  q 2q−q 2 (|um |q−2 um − |un |q−2 un )(um − un ) 2 (|um | + |un |) 2 dVg (x) . + q<2

(48) To go further, observe that if a and b are positive functions on M , then by Hölder’s inequality 

q

a2 b

2q−q 2 2

q

≤ 2 1q<2 a 2 

2

Lq

1q<2 b

2q−q 2 2



2

L 2−q

,

q<2

where 1A is the indicator function on the subset A of M . Moreover, since q

1q<2 a 2 

2 Lq

≤ max {ρ1 (a), ρ1 (a)

q− 2

}

and 1q<2 b

2q−q 2 2



2 L 2−q

≤ max{ρq (b)

2−q − 2

, 1} ,

we get 

q

a2 b

2q−q 2 2

≤ 2 max{ρ1 (a), ρ1 (a)

q<2

Thus using (49) twice in (48) we infer that

q− 2

} max{ρq (b)

2−q − 2

, 1} .

(49)

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(q − − 1) I1 [um − un ]

 ≤ P (um − un , DI[um ] − DI[un ]) Q um Lq1 , un Lq1

(50)

for some constructible continuous functions P (X) and Q (Y, Z), with P (0) = 0. Since the sequence {un Lq1 } is bounded, from (50), (47) and (44) we conclude that if lim

m,n→∞

um − un , DI[um ] − DI[un ] = 0 ,

then we have finished step 2. To obtain such a limit, recall that DJ = DI − DK, hence um − un , DI[um ] − DI[un ] = um − un , DJ[um ] − DJ[un ] + um − un , DK[um ] − DK[un ] . We estimate um − un , DK[um ] − DK[un ] ≤ G Nf [um ] − Nf [un ]Hom(Lp ,R) um − un Lq1 , where the relation between the norm of operators DK[um ] − DK[un ]Hom(Lq1 ,R) ≤ G Nf [um ] − Nf [un ]Hom(Lp ,R) q(·)

has been used, G being the constant of the embedding L1 (M ) → Lp(·) (M ), see Theorem 4.2. Here p(·)

Nf : Lp(·) (M ) → L p(·)−1 (M ) = Hom(Lp(·) (M ), R) is the Nemytskii operator induced by f , namely Nf [u](x) := f (x, u(x)), see [10]. It follows that um − un , DI[um ] − DI[un ]

 ≤ DJ[um ] − DJ[un ]Hom(Lq1 ,R) + G Nf [um ] − Nf [un ]Hom(Lp ,R) × (um Lq1 + un Lq1 ). Using the continuity of the Nemytskii map, the convergence of {un} in Lp(·) , the boundedness of {un Lq1 }, and the hypothesis on DJ[un ] as n → ∞, we obtain the desired conclusion. 2

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Acknowledgments P.G. and D.J.P. thank the members of the Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, for their hospitality while part of this work was done. References [1] A. Almeida, S. Samko, Pointwise inequalities in variable Sobolev spaces and applications, Z. Anal. Anwend. 26 (2007) 179–193. [2] T. Aubin, Nonlinear Analysis on Manifolds, Monge–Ampére Equations, Springer-Verlag, 1982. [3] A. Besse, Einstein Manifolds, Springer-Verlag, 1987. [4] T. Coulhon, Espaces de Lipschitz et inégalités de Poincaré, J. Funct. Anal. 136 (1996) 81–113. [5] C. Croke, H. Karcher, Volume of small balls on open manifolds: lower bounds and examples, Trans. Amer. Math. Soc. 309 (1988) 753–762. [6] L. Diening, Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces Lp(·) and W k,p(·) , Math. Nachr. 263 (2004) 31–43. ziˆ cka, Lebesgue and Sobolev Spaces with Variable Ex[7] L. Diening, P. Harjulehto, P. Hästö, M. Růˆ ponents, Lecture Notes in Mathematics, Springer, Heidelberg, 2011. [8] L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: Proc. Int. Conference Function Spaces, Differential Operators and Nonlinear Analysis, Milovy, Czech Rep., May 28–June 4, 2004, Math. Inst. Acad. Sci. Czech Rep., Prague, 2005, pp. 38–58. [9] X. Fan, Q. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843–1852. [10] X. Fan, D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001) 424–446. [11] X. Fan, Y. Zhao, D. Zhao, Compact imbedding theorems with symmetry of Strauss–Lions type for the space W 1,p(x) (Ω), J. Math. Anal. Appl. 255 (1) (2001) 333–348. [12] T. Futamura, Y. Mizuta, T. Shimomura, Sobolev embeddings for variable exponent Riesz potentials on metric spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006) 495–522. [13] M. Gaczkowski, P. Górka, Sobolev spaces with variable exponents on Riemannian manifolds, Nonlinear Anal. 92 (2013) 47–59. [14] M. Gaczkowski, P. Górka, Variable Hajłasz–Sobolev spaces on compact metric spaces, Math. Slovaca (2015), in press. [15] P. Harjulehto, P. Hästö, Sobolev inequalities for variable exponents attaining the values 1 and n, Publ. Mat. 52 (2008) 347–367. [16] P. Harjulehto, P. Hästö, V. Latvala, Sobolev embeddings in metric measure spaces with variable dimension, Math. Z. 254 (2006) 591–609. [17] E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes, vol. 5, American Mathematical Society, 2000. [18] O. Kováčik, J. Rákosnik, On spaces Lp(x) and W k,p(x) , Czechoslovak Math. J. 41 (1991) 592–618. [19] G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Mathematics, AMS, 2009. [20] F. Li, Z. Li, L. Pi, Variable exponent functionals in image restoration, Appl. Math. Comput. 216 (3) (2010) 870–882. [21] P. Lindqvist, Notes on the p-Laplace equation, Lecture notes, University of Jyväskylä, 2005. [22] S.B. Myers, N. Steenrod, The groups of isometries of a Riemannian manifold, Ann. of Math. 40 (1939) 400–416. [23] R.S. Palais, Seminar on the Atiyah–Singer Index Theorem. With Contributions by M.F. Atiyah, A. Borel, E.E. Floyd, R.T. Seeley, W. Shih and R. Solovay, Annals of Mathematics Studies, vol. 57, Princeton University Press, Princeton, N.J., 1965. [24] M. Růˆ ziˆ cka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000. [25] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec. Funct. 16 (2005) 461–482. [26] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990.

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37

[27] N.S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa (3) 22 (1968) 265–274. [28] V.V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Izv. Akad. Nauk SSSR Ser. Mat. 50 (4) (1986) 675–710. [29] V.V. Zhikov, On Lavrientiev’s phenomenon, Russ. J. Math. Phys. 3 (1995) 249–269.