R E S E A R C H On a new fractional Sobolev space with variable exponent on complete manifolds

We present the theory of a new fractional Sobolev space in complete manifolds with variable exponent. As a result, we investigate some of our new space’s qualitative properties, such as completeness, reﬂexivity, separability, and density. We also show that continuous and compact embedding results are valid. We apply the conclusions of this study to the variational analysis of a class of fractional p ( z , · )-Laplacian problems involving potentials with vanishing behavior at inﬁnity as an application.


Introduction
Let (M, g) be a smooth complete compact Riemannian n-manifold. The present paper is devoted to proving some qualitative properties of a new fractional Sobolev space with variable exponent in complete manifolds, as well as to studying the existence of weak solutions to the following problem as an application: (P) ⎧ ⎨ ⎩ (g ) s p(z,·) u(z) + V(z)|u(z)| q(z)-2 u = h(z, u(z)) in Q, u| ∂Q = 0, where Q ⊂ M is an open bounded set with a smooth boundary ∂Q, s ∈ (0, 1), p ∈ C(M × M, (1; ∞)) with sp(z, y) < n, we assume that p is symmetric and satisfies the following conditions: also q : M → (1, ∞) satisfies 1 < q -≤ q + < p -≤ p + < +∞, where q + = sup z∈M q(z), q -= inf z∈M q(z), and functions h, V satisfy some suitable conditions (see Sect. 4). This type of operator has a significant role in many fields in mathematics, e.g., calculus of variations and partial differential equations, and it has also been used in a variety of physical and engineering contexts, e.g., fluid filtration in porous media, constrained heating, elastoplasticity, image processing, optimal control, financial mathematics, and elsewhere, see [8,18,37] and the references therein.
In recent years, wide research has been done on fractional partial differential equations with variable growth. For example, Bahrouni and Rădulescu [7] developed some qualitative properties on the fractional Sobolev space W s,q(z),p(z,y) (Q) for s ∈ (0, 1) and Q being a bounded domain in R n with a Lipschitz boundary. Moreover, they studied the existence of solutions to the following problem: where Lu(z) = p.v. Q |u(z) -u(y)| p(z,y)-2 (u(z) -u(y)) |z -y| n+sp(z,y) dy, λ > 0, and 1 < r(z) < p -= min (z,y)∈Q×Q p(z, y). Bahrouni [6] continued the study of this class of fractional Sobolev spaces with variable exponent and the related nonlocal operator. More precisely, he proved a variant of the comparison principle for (p(z) ) s . He gave a general principle of sub-supersolution method for the following problem: where Q is a smooth open bounded domain, n ≥ 3, s ∈ (0, 1), p, f are continuous functions, and f satisfies the following assumption: where r ∈ C(R n , R) and 1 < r(z) < p * (z) = np(z,z) n-sp(z,z) , ∀z ∈ R n . Kaufmann, Rossi, and Vidal [32] proved a compact embedding theorem for fractional Sobolev spaces with variable exponents into variable exponent Lebesgue space and, as an application, they showed the existence and uniqueness of solutions to the following fractional p(z, y)-Laplacian equation: In [31] the authors refined the fractional Sobolev spaces with variable exponents given in [6,7,32] and established fundamental embeddings of this space. In addition, they gave a sufficient condition for the exponent p(·, ·) on R n × R n for the iteration argument of De Giorgi type and proved global boundedness of weak solutions to the problem (P 1 ). Readers may refer to [1, 4, 5, 12-14, 19, 21-23, 27, 33, 34, 36, 38, 42] and the references therein for more ideas and techniques developed to guarantee the existence of weak solutions for a class of nonlocal fractional problems with variable exponents. When p(·, ·) = p = constant, we quote, for example, the relevant work of Vázquez [41], see also [2, 9-11, 15, 17, 20, 35] and the references therein. Various techniques have been proposed in the literature in order to recover the compactness in several circumstances. We refer to Tang and Cheng [40], who proposed a new approach to restore the compactness of Palais-Smale sequences, and to Tang and Chen [39], who introduced an original method to recover the compactness of minimizing sequences. A related approach has been developed by Chen and Tang [16] in the framework of Cerami sequences.
Before discussing our main results, we give a review of equations involving the fractional p-Laplace operator on Riemannian manifolds. As far as we know, there is only the work of Guo, Zhang, and Zhang [29] who proved the existence of solutions to the following p-Laplacian equations with homogeneous Dirichlet boundary conditions: where sp < n with s ∈ (0, 1), p ∈ (1; ∞), (g ) s p is the fractional p-Laplacian on Riemannian manifolds, (M, g) is a compact Riemannian n-manifold, Q is an open bounded subset of M with a smooth boundary ∂Q, and f is a Carathéodory function satisfying the Ambrosetti-Rabinowitz-type condition.
The motivation of this paper was, on the one hand, the work of Fu and Guo [24] who introduced the variable exponent function spaces on Riemannian manifolds in 2012, followed by Gaczkowski and Górka [25] who in 2013 examined the above space in the case of compact manifolds, and Guo [28] who in 2015 discussed the properties of the Nemytsky operator and obtained the existence of weak solutions for Dirichlet problems of nonhomogeneous p(m)-harmonic equations. Finally, in 2016 Gaczkowski, Górka, and Pons [26] studied the variable exponent function spaces on complete noncompact Riemannian manifolds. Furthermore, they proved the continuous embeddings results between Sobolev and Hölder function spaces, using classic assumptions on the geometry. In addition, they established the compact embeddings of H-invariant Sobolev spaces, where H is a compact Lie subgroup of the manifold group of isometries, and, as an application, they showed the existence of weak solutions to nonhomogeneous q(z)-Laplace equations. For further background, we recommend that readers consult [1,12] and the references therein. On the other hand, we were also motivated by the work of Guo, Zhang, and Zhang [29] who established the theory of fractional Sobolev spaces on Riemannian manifolds.
The novelty of our work is in extending Sobolev spaces with variable exponents to cover the fractional case with complete manifolds. We prove some qualitative properties of this new space. Next, we study the existence of solutions to some nonlocal problems involving potentials allowed for vanishing behavior at infinity. However, the main difficulty is presented by the fact that the p(z)-Laplacian operator has a more complicated nonlinearity than the p-Laplacian operator. For example, it is nonhomogeneous. To the best of our knowledge, there is no known result along this line.
The outline of the paper is as follows. In Sect. 2, we collect the pertinent properties and notations of Lebesgue spaces with variable exponents and Sobolev-Orlicz spaces with variable exponents on a complete manifold. Moreover, we show the relation between the norm and the modular. In Sect. 3, we study the completeness, reflexivity, separability, and density of our new space. Furthermore, we prove a continuous and compact embedding theorem of this space into variable exponent Lebesgue spaces. In Sect. 4, we deal with a fractional p(z)-Laplacian problem involving potentials allowed for vanishing behavior at infinity as an application.

Preliminaries
In this section, we review some definitions and properties of spaces W is an open subset of R n , and W 1,q(z) 0 (M), which are known as the Sobolev spaces with variable exponents and the Sobolev spaces with variable exponents on a complete manifold, respectively. For more background, we refer to [1,3,12,21,26,28,30] and the references therein.

Sobolev spaces with variable exponents
We define real numbers q + and qas follows:

Definition 2.1 ([21])
We define the Lebesgue space with variable exponent L q(·) (Q) as follows: and endow it with the Luxemburg norm ) is a separable Banach space, and uniformly convex for 1 < q -≤ q + < +∞, hence reflexive.

Sobolev spaces with variable exponents on complete manifolds
Let (M, g) be a smooth complete compact Riemannian n-manifold. We begin by recalling some background, more can be found in [1,3,26,28,30].
Now, we define a natural positive Radon measure.
where dv g = (det(g ij )) 1 2 dz is the Riemannian volume element on (M, g), g ij are the components of the Riemannian metric g in the chart, and dz is the Lebesgue volume element of R n .
Next, we define the Sobolev spaces L q(·) with |D k u| being the norm of the kth covariant derivative of u, defined in local coordinates by and, for a pair of points z, y ∈ M, we define the distance d g (z, y) between z and y by Let P log (M) be the set of log-Hölder continuous real functions on M, which is linked to P log (R n ) by the following proposition: 3,26]) Given q ∈ P log (M), let (Q, φ) be a chart such that as bilinear forms, where δ ij is the Kronecker delta symbol. Then q • φ -1 ∈ P log (φ(Q)).

Definition 2.6 ([3])
If the Ricci tensor of g, denoted by Rc(g), satisfies Rc(g) ≥ λ(n -1)g, for some λ and for all z ∈ M, ∃v > 0 such that |B 1 (z)| g ≥ v, where B 1 (z) are balls of radius 1 centered at some point z in terms of the volume of smaller concentric balls, then we say that the n-manifold (M, g) has property B vol (λ, v). .
We now prove the following proposition.

Proposition 2.5
If u, u k ∈ L q(z) (M) and k ∈ N, then the following assertions are equivalent: It is now easy to observe that u k → u a.e. on M. Thus |u k | q(z) → |u| q(z) on M and the integrals of the functions |u k -u| q(z) are absolutely equicontinuous on M, and since |u k | q(z) ≤ 2 q + -1 |u k -u| q(z) + |u| q(z) , the integrals of the |u k | q(z) are also absolutely equicontinuous on M, so, by the Vitali convergence theorem, we obtain that Conversely, if u k → u on M, we can deduce that |u k -u| q(z) → 0 on M, and using the same techniques as in the above proof, and due to the fact that and lim k→+∞ q(·) (u k ) = q(·) (u), we obtain that lim k→+∞ q(·) (u k -u) = 0.
Remark 2.2 The following relation will be used to compare the functionals · L q(·) (M) and q(·) (·):

Fractional Sobolev space with variable exponent on a complete manifold
On a complete manifold, we introduce in this section a new fractional Sobolev space with variable exponent and state our mains results. For s ∈ (0, 1), we introduce the variable exponent Sobolev fractional space on a complete manifold as follows: W s,p(z,y) (M) = u : M → R : u ∈ Lp (z) (M) such as M×M |u(z) -u(y)| p(z,y) (d g (z, y)) n+sp(z,y) dv g (z) dv g (y) < ∞, for some λ > 0 . Consequently, The modular p(·,·) has the following properties.

Lemmas
In this part, we will go through some of our new fractional space's qualitative lemmas.

M×M
|u(z) -u(y)| p(z,y) (d g (z, y)) n+sp(z,y) dv g (z) dv g (y) y)) n+sp(z,y) dv g (z) dv g (y) |u(z) -u(y)| p(z,y) (d g (z, y)) n+sp(z,y) dv g (z) dv g (y) y)) n+sp(z,y) dv g (z) dv g (y) As a result of the Minkowski inequality (see Theorem 2.13 in [15]) and inequality (27) in [2], we obtain that Proof Consider the following real-valued function: Let ϕ ∈ C ∞ (M) ∩ W s,p(z,y) (M), and let y be a fixed point of M such that ϕ ν (α) = ϕ(α)f (d g (y, α))), where d g is the Riemannian distance associated to g and ν ∈ N. We can easily see that ϕ ν (α) ∈ W s,p(z,y) (M) for ν ∈ N. Then, since M is a compact Riemannian n-manifolds, it can be covered by a finite number of charts (Q k , φ k ) k=1,...,m . Let η k be a smooth partition of unity subordinate to the covering Q k . We can see that h = η k ϕ ν • φ -1 k ∈ W s,p(z,y) (φ k (Q k )).
So, by Lemma 3.2 in [7], we can extract a subsequence h t ∈ C ∞ (R ) such that h t → h strongly in W s,p(z,y) (φ k (Q k )) as t → ∞. Thus, h t • φ k ∈ C ∞ (M) and h t • φ k converge strongly to η k ϕ ν in W s,p(z,y) (M) as t → ∞.
Remark 3.2 We can also prove the previous lemma, without assuming condition (2), by using the following method: For u ∈ C ∞ 0 (M), we need to prove that M×M |u(z) -u(y)| p(z,y) (d g (z, y)) n+sp(z,y) dv g (z) dv g (y) < ∞.
Thus Hence u(z) -u(y) p(z,y) ≤ 2 p + -1 u Therefore, according to [30], we obtain where (η s ) is a smooth partition of unity subordinate of the covering B z k (r) for any k, and B z k (r) denotes the Euclidean ball of R n with center z k and radius r. Then we deduce that, for u ∈ C ∞ 0 (M), M×M |u(z) -u(y)| p(z,y) (d g (z, y)) n+sp(z,y) dv g (z) dv g (y) < ∞.
Thus u ∈ W s,p(z,y) (M). Now, we will extend an embedding result between W 1,p(z,y) (M) and W s,p(z,y) (M) to manifolds.
Thus, we can conclude that W s,p(z,y) (M) ⊂ L q(z) (M), and the embedding is continuous and compact.