The concentration-compactness principles for $W^{s,p(\cdot,\cdot)}(\mathbb{R}^N)$ and application

We obtain a critical imbedding and then, concentration-compactness principles for fractional Sobolev spaces with variable exponents. As an application of these results, we obtain the existence of many solutions for a class of critical nonlocal problems with variable exponents, which is even new for constant exponent case.


Introduction
Nonlocal equations have been modeled for various problems in real fields, for instance, phase transitions, thin obstacle problem, soft thin films, crystal dislocation, stratified materials, anomalous diffusion, semipermeable membranes and flame propagation, material science, ultra-relativistic limits of quantum mechanics, multiple scattering, minimal surfaces, water waves, etc.After the seminal papers by Caffarelli et al. [13][14][15], problems involving fractional p-Laplacian have been intensively studied.On the other hand, various other real fields such as electrorheological fluids and image processing, etc. require partial differential equations with variable exponents (see e.g., [32,33]).Natural solution spaces for those problems are Sobolev spaces with fractional order or variable exponents, which were comprehensively investigated in [17] and [18].
Recently, many authors have been studied the fractional Sobolev spaces with variable exponents and the corresponding nonlocal equations with variable exponents (see e.g., [5,6,25,27]).To the authors' best knowledge, though most properties of the classical fractional Sobolev spaces have been extended to the fractional Sobolev spaces with variable exponents, there have no results for the critical Sobolev type imbedding for these spaces.Consequently, there have no results on nonlocal equations with variable critical growth because the critical Sobolev type imbedding is essential in the study of such critical equations.The critical problem was initially studied in the seminal paper by Brezis-Nirenberg [12], which treated for Laplace equations.Since then there have been extensions of [12] in many directions.Elliptic equations involving critical growth are delicate due to the lack of compactness arising in connection with the variational approach.For such problems, the concentration-compactness principles (the CCPs, for short) introduced by P.L. Lions [30,31] and its variant at infinity [7,9,16] have played a decisive role in showing a minimizing sequence or a Palais-Smale sequence is precompact.By using these CCPs or extending them to the Sobolev spaces with fractional order or variable exponents, many authors have been successful to deal with critical problems involving p-Laplacian or p(•)-Laplacian or fractional p-Laplacian, see e.g., [1,4,8,10,11,[21][22][23][24]26] and references therein.
As we mentioned above, there have no results for the critical Sobolev type imbedding for the fractional Sobolev spaces with variable exponents.Although the usual critical Sobolev immersion theorem holds in the fractional order or variable exponents setting, we do not know this assertion even in fractional Sobolev spaces with variable exponents defined in bounded domain; see [5,6,25,27].Because of this, our first aim of the present paper is to obtain a critical imbedding from fractional Sobolev spaces with variable exponents into Lebesgue spaces with variable exponents.We provide sufficient conditions on the variable exponents such as the log-Hölder type continuity condition to obtain such critical imbedding (Theorem 3.3).Thanks to this critical Sobolev imbedding, inspired by [4,10,11,26,31], we then establish two Lions type concentration-compactness principles for fractional Sobolev spaces with variable exponents, which are our second aim (Theorems 4.1 and 4.2).As an application of these results, we will obtain the existence of many solutions for the following nonlocal problem with variable exponents Lu(x) + |u| p(x,x)−2 u = f (x, u) + λ|u| q(x)−2 u in R N , (1.1) where the operator L is defined as where s ∈ (0, 1), p ∈ C(R N × R N ) is symmetric i.e., p(x, y) = p(y, x) for all (x, y) ∈ R N × R N such that 1 < p − := inf (x,y)∈R N ×R N p(x, y) ≤ p + := sup (x,y)∈R N ×R N p(y, x) < N s ; q ∈ C(R N ) satisfies p(x, x) < q(x) ≤ p * s (x) := N p(x,x) N −sp(x,x) for all x ∈ R N ; λ is a real parameter; and f : R N × R → R is a Carathéodory function of local p + -superlinear and to be specified later.
The main feature of our final consequence in the present paper is to establish the multiplicity result for problem (1.1) under the critical growth condition {x ∈ R N : q(x) = p * s (x)} = φ, originally introduced in [10] for the p(•)-Laplacian case, and some conditions on f different from the related works [1,19,34] (Theorem 5.1).As far as we are aware, there are no existence results about the critical problems in this situation even in the case of constant exponents.
The rest of our paper is organized as follows.In Section 2, we briefly review some properties of the Sobolev spaces with fractional order or variable exponents.In Section 3, we establish a critical Sobolev type imbedding for the fractional Sobolev spaces with variable exponents, which is a key to our arguments.In Section 4 we establish Lions type concentrationcompactness principles for fractional Sobolev spaces with variable exponents.In Section 5, we show the existence of many solutions for a superlinear nonlocal problem with variable exponents using genus theory.In Appendix, we give an auxiliary result, which is used to prove our CCPs.

Variable exponent Lebesgue spaces and fractional Sobolev spaces
In this section, we briefly review the Lebesgue spaces with variable exponents and the classical fractional Sobolev spaces.
Let Ω be a Lipschitz domain in R N .Denote and for h ∈ C + (Ω), denote For p ∈ C + (Ω) and a σ-finite, complete measure µ in Ω, define the variable exponent Lebesgue space L p(•) µ (Ω) as When µ is the Lebesgue measure, we write dx, L p(•) (Ω) and Proposition 2.2.( [28]) Define the modular ρ : Then, we have the following relations between the norm and modular. .
Proposition 2.4.( [17]) Let s ∈ (0, 1) and p ∈ (1, ∞) be such that sp < N. It holds that In this section, we recall the fractional Sobolev spaces with variable exponents that was first introduced in [27], and was then refined in [25].Furthermore, we will obtain a critical Sobolev type imbedding on these spaces.
Let Ω be a bounded Lipschitz domain in R N or Ω = R N .Throughout this article, we assume that (P 1 ) s ∈ (0, 1); p ∈ C(Ω × Ω) is uniformly continuous and symmetric such that In the following, for brevity, we write p(x) instead of p(x, x) and with this notation, p ∈ C + (Ω).Define y)  |x − y| N +sp(x,y) dx dy < +∞ endowed with the norm where M Ω (u) := ´Ω |u| p(x) dx + ´Ω ´Ω |u(x)−u(y)| p(x,y) |x−y| N+sp(x,y) dx dy.Then, W s,p(•,•) (Ω) is a separable reflexive Banach space (see [5,6,27]).On W s,p(•,•) (Ω), we also make use of the following norm In what follows, when Ω is understood, we just write ,Ω , respectively.We also denote the ball in R N centered at z with radius ε by B ε (z) and denote the Lebesgue measure of a set E ⊂ R N by |E|.For brevity, we write B ε and B c ε instead of B ε (0) and R N \ B ε (0), respectively.
Case 1: Ω is a bounded Lipschitz domain.
We cover Ω by ∩ Ω being Lipschitz domains and (3.3) being satisfied for all i ∈ {1, • • • , m}.Fix i ∈ {1, • • • , m} and denote p i := inf (y,z)∈Ω i ×Ω i p(y, z) and q i := sup x∈Ω i q(x).By (3.6) and the choice of ε i , we have From this and Proposition 2.4, we have , and hence, On the other hand, we have Note that by (3.6), we have and so (3.5) is claimed.
Decompose R N by cubes {Q i } i∈N with sides of length ε ∈ (0, 1) and parallel to coordinates axes.By (3.3) and the uniform continuity of q we can choose ε > 0 sufficiently small such that where , and q i := sup Set v = u u s,p .Thus, v s,p = 1 and hence, M R N (v) = 1 in view of Proposition 3.1.This yields M Q i (v) ≤ 1 for all i ∈ N and hence, Here and in the remainder of the proof, C is a positive constant independent of v and i.In order to prove (3.14), we first prove that Indeed, let i ∈ N and consider the measure As in (3.11) we have dµ(x, y). Thus, Meanwhile, invoking Proposition 2.1 we have Combining the last two inequalities and (3.16) we obtain Noting Combining (3.18) with the following estimate : This and (3.12) yield Note that by Proposition 2.1 again, then by (3.13), (3.14) and Proposition 3.1 we have So in any case, |x − y| N +sp(x,y) dx dy .
Summing up for i ∈ N, we obtain The proof is complete.
A proof of Theorem 3.4 can be obtained in a similar fashion to that of [26,Lemma 4.1] and we omit it.

The concentration-compactness principles for
In this section we establish two Lions type concentration-compactness principles for the spaces W s,p(•,•) (R N ).

4.1.
Statements of the concentration-compactness principles.Let M(R N ) be the space of all signed finite Radon measures on R N endowed with the total variation norm.Note that we may identify M(R N ) with the dual of C 0 (R N ), the completion of all continuous functions u : R N → R whose support is compact relative to the supremum norm • ∞ (see, e.g., [20,Section 1.3.3]).
In the rest of this paper, we always assume that the variable exponents p and q satisfy the following assumptions.
(P 2 ) p : R N × R N → R is uniformly continuous and symmetric such that It is clear that if p satisfies (P 2 ), then p(x, x) = p for all x ∈ R N and p satisfies (P 1 ) and (3.2).Hence, by Theorem 3.3, we have On the other hand, by (P 2 ) we have that for any u ∈ L p (R N ), From this and (4.1) we obtain Our main results in this sections are the following CCPs for W s,p(•,•) (R N ).
Theorem 4.1.Assume that (P 2 ) and ) ) For possible loss of mass at infinity, we have the following.
Theorem 4.2.Assume that (P 2 ) and Then ) Assume in addition that (E ∞ ) There exist lim |x|,|y|→∞ p(x, y) = p and lim |x|→∞ q(x) = q ∞ for p given by (P 2 ) and some Then The following example provides a nonconstant exponent p that fulfills the conditions in Theorems 4.1 and 4.2.

Auxiliary lemmas and proofs of the concentration-compactness principles.
The following auxiliary lemmas are useful to prove Theorems 4.1 and 4.2.
hold and let {u n } be as in Theorem 4.1.Then, we have hold and let {u n } be as in Theorem 4.1.Then, we have Let K > 4 be arbitrary and fixed and let ρ ∈ (0, ε 0 2K ).Clearly,

From this and the fact that |ψ
We first estimate J 1 (n, ρ).Decompose Hence, J 1 (n, ρ) ≤ Using the Hölder inequality we have From (4.20), (4.21) and the fact that u ∈ L p * s (R N ) (see (4.1)) we arrive at lim sup On the other hand, we have That is, Then, using (4.21) and the fact that u ∈ L p * s (R N ) again we obtain from (4.24) that lim sup In order to estimate J (3) 1 (n, ρ), we first note that Then, arguing as before we obtain lim sup we have That is, Invoking Proposition 2.3 again and using the boundedness of {u n } in L p * s (R N ), we deduce from the last inequality that Here and in the remainder of the proof C i (i ∈ N) is a positive constant independent of n, ρ and K.By changing variable x = x 0 + ρz we have Combining this with (4.28) and (4.29) we derive for all n ∈ N and all ρ ∈ (0, ε 0 2K ).Thus, lim sup Since K > 4 was chosen arbitrarily, the last inequality yields lim sup From (4.17), (4.27), and (4.31), we obtain (4.16) and the proof is complete.
Proof of Lemma 4.5.Let R > 2 and decompose First, we estimate I 1 (n, R).By rearranging we easily get Let t ∈ {p, p * }.We have We have Combining this with (4.34) gives On the other hand, we have Combining this and (4.34) yields and hence, lim Next, we estimate I 2 (n, R).Fix σ ∈ (0, 1/2) and decompose Thus, for x ∈ B R \ B σR we have From this and (4.34) we obtain Using (4.34) again, we have  We now prove the first concentration-compactness principle.
Proof of Theorem 4.1.Let v n = u n − u.Then, Invoking Theorem 3.2 we deduce from (4.44) that for any r ∈ C + (R N ) satisfying r(x) < p * s for all x ∈ R N due to Theorem 3.2.Hence, up to a subsequence we have v n (x) → 0 for a.e.x ∈ R N .(4.46) Using (4.6), (4.44), (4.46) and arguing as in [26], we have dy is bounded in L 1 (R N ).So up to a subsequence, we have for some nonnegative finite Radon measure µ on By (4.3), we have dy, and λ n := φv n s,p .Let ε > 0 be arbitrary and fixed.Then, there exists Then, invoking Proposition 3.1 again we deduce from (4.52) that By the symmetry of p we also have Thus, by the facts that supp(φ) ⊂ B R and λ p(x,y) n Before estimating the remaining integrals, we note that by (4.49) again, To estimate the last integral in the right-hand side of (4.54) we notice that for x ∈ B R , This together with (4.58) yields Using this, (4.56) and (4.58), we obtain from (4.54) that Letting Invoking Proposition 2.2, we easily obtain from the last estimate that . From (4.47), (4.50), (4.60) and the last inequality, we arrive at The fact that {x i } i∈I ⊂ C can be obtained by an argument similar to that of [26, Theorem 3.3] and we omit the proof.Next, we obtain the relation (4.9).Let i ∈ I and for ρ > 0, define ψ ρ as in Lemma 4.4 with x 0 replaced by x i .Thus 3) again, we have Taking the limit inferior as n → ∞ in the above inequality and using (4.6) we obtain Invoking Proposition 2.2, we have where q + i,ρ := max q(x).Thus, we obtain a lower bound of the left-hand side of (4.63) as follows: due to the continuity of q and the fact that x i ∈ C. To obtain an upper bound of the right-hand side of (4.63), we first prove that there exist ρ 0 ∈ (0, 1) and λ 0 ∈ (0, ∞) such that where λ n,ρ := ψ u n s,p .Indeed, by the continuity of q and the positiveness of ν i , we can choose ρ 0 ∈ (0, 1) such that From (4.62), (4.64) and (4.67), we infer Using (4.68), (4.69), the boundedness of {u n } in W s,p(•,•) (R N ) and invoking Proposition 3.1, we can easily show that there exists λ 0 ∈ (0, ∞) such that λ n,ρ < λ 0 for all n ∈ N and ρ ∈ (0, ρ 0 ).Thus, (4.66) has been proved.Next, let ε > 0 be arbitrary and fixed.We have dy dx.
Finally, to obtain (4.7) we note that for each φ ∈ C 0 (R N ), φ ≥ 0, the functional |x − y| N +sp(x,y) dy dx is convex and differentiable on W s,p(•,•) (R N ).From this and (4.5) we infer Thus, Extracting µ to its atoms, we get (4.7) and the proof is complete.
We conclude this section by proving Theorem 4.2.
Proof of Theorem 4.2.For each R > 0, define φ R as in Lemma 4.5.
In order to obtain (4.13), we decompose where U n is given by (4.71).By (4.11) and the fact that for all n ∈ N and R > 0, we obtain On the other hand, the fact that 1 in view of the Lebesgue dominated convergence theorem.From the last two equalities, we obtain lim From this and (4.72)-(4.74)we obtain (4.13).
(4.77) From (4.3) and (3.1), we have For R > R 1 , using (4.77) and Proposition 2.2 we have Next, we estimate the right-hand side of (4.78).To this end, denote σ n,R := φ R u n s,p for brevity.We will show that there exist Indeed, we first choose ε > 0 sufficiently small such that Then we can find for all R > R 2 .Finally, by (4.76), we can find for all R > R 2 .From (4.82) and (4.83) we get lim sup for all R > R 2 .This and (4.78) yield S q . By a similar argument to that obtained (4.66), invoking Lemma 4.5 and choosing R 2 larger if necessary, we can show that there exists σ ∈ (0, ∞) such that σ * ,R < σ for all R ∈ (R 2 , ∞).Thus, (4.80) has been proved.We now turn to estimate the right-hand side of (4.78).For each R > R 2 given, let Utilizing Proposition 3.1 and (4.51) again, we have This and the fact that 0 Taking limit superior as k → ∞ in the last inequality with noticing (4.80) and (4.84) we obtain y)  |x − y| N +sp(x,y) dy dx Now, taking the limit as R → ∞ in (4.85) with taking Lemma 4.5 and (4.73) into account, we deduce where σ * := lim inf R→∞ σ * ,R and hence, 0 < σ * < σ due to (4.80).From this, (4.78) and (4.79) we obtain Since ε was chosen arbitrarily in the last inequality, (4.14) follows.The proof of Theorem 4.2 is complete.

Application
5.1.The existence of solutions.In this section, we investigate the existence and multiplicity of solutions to the following problem where s, p, q satisfy (P 2 ), (Q 2 ) and (E ∞ ) with p + < q − , the operator L is defined as in (1.2), λ is a real parameter, and the nonlinear term f satisfies the following assumptions.(F1) f : R N × R → R is a Carathéodory function such that f is odd with respect to the second variable.
(F2) There exist functions r j , a j with a j (x)|u| r j (x)−1 for a.e.x ∈ R N and all u ∈ R.
Our main existence result is stated as follows.

5.2.
Proof of Theorem 5.1.In order to prove Theorem 5.1, we will make use of the following abstract result for symmetric C 1 functionals, which is a variant of Theorem 2.19 in [2] (see also [3,Theorem 10.20]).

Lemma 5.2 ( [2]
).Let E = V ⊕ X, where E is a real Banach space and V is finite dimensional.Suppose that J ∈ C 1 (E, R) is an even functional satisfying J(0) = 0 and (J 1) there exist constants ρ, β > 0 such that J(u) ≥ β for all u ∈ ∂B ρ ∩ X; (J 2) there exists a subspace E of E with dim V < dim E < ∞ and {u ∈ E : J(u) ≥ 0} is bounded in E; (J 3) for β and E respectively given in (J 1) and (J 2), J satisfies the (PS) c condition for any c ∈ [0, L] with L := sup u∈ E J(u).
Then J possesses at least dim E − dim V pairs of nontrivial critical points.
Proof.The proof is similar to that of [3,Theorem 10.20] for which we take E m in the proof of [3,Lemma 10.19] as E m = span{e 1 , • • • , e m }, where {e k } dim E k=1 is a basis of E.
To determine solutions to problem (5.1), we will apply Lemma 5.2 for E := W s,p(•,•) (R N ) endowed with the norm • := • s,p and J = J λ , where J λ : E → R is the energy functional associated with problem (5.1) defined as It is clear that under the assumptions (F1) − (F2), J λ is of class C 1 (E, R) and its Fréchet derivative J ′ λ : E → E * is given by Here, E * and •, • denote the dual space of E and the duality pairing between E and E * , respectively.Clearly, J λ is even in E, J λ (0) = 0, and each critical point of J λ is a solution to problem (5.1).The next lemma will be utilized for verifying (J 3).
Lemma 5.3.For any given λ > 0, J λ satisfies the (PS) c condition provided where h(x) := p q(x)−p for x ∈ R N and S q is defined as in (4.3).
Proof.Let {u n } be a (PS) c sequence for J λ with c satisfying (5.4).We first claim that {u n } is bounded in E. Indeed, by (F4) and invoking Proposition 3.1 we have that for n large, That is, Here and in the remaining proof, C i (i ∈ N) denotes a positive constant independent of n.
On the other hand, by the relation between modular and norm (see Proposition 3.1) and (F2) we have that for n large, Then, by the Young inequality we easily get From (5.5) and (5.6) we obtain This implies the boundedness of {u n } since p − > 1 and hence, in view of Proposition 3.1.Then, invoking Theorems 3.2, 4.1 and 4.2, up to a subsequence, we have u n ⇀ u in E, (5.9) ) |x−y| N+sp(x,y) dy for n ∈ N and x ∈ R N .Moreover, we have (5.15) We will show that I = ∅ and ν ∞ = 0.For this purpose we invoke (F4) to estimate Combining this with (5.14) gives (5.16) We now suppose on the contrary that I = ∅.Let i ∈ I and for ρ > 0, define ψ ρ as in Lemma 4.4 with x 0 replaced by x i .For an arbitrary and fixed ρ, it is not difficult to see that {u n ψ ρ } is a bounded sequence in E. Hence, This yields where Note that the boundedness of {u n } in E implies the boundedness of {u n } in L q(•) (R N ) due to Theorem 3.3.Hence, from (F2) and invoking Propositions 2.2 and 2.3 we have , ∀n ∈ N. (5.18) Here and in the remaining proof, C i (i ∈ N) denotes a positive constant independent of n and ρ.From (5.18), we obtain lim sup In order to estimate I 2 (n, ψ ρ ), let δ > 0 be arbitrary and fixed.By (5.7) and the Young inequality we have Now, by taking limit superior in (5.17) as ρ → 0 + with taking (5.19) and (5.21) into account, we obtain µ i = λν i .Plugging this into (5.12)we get
Consequently, we have ´RN |u n | q(x)−2 u n (u n − u) dx → 0 by invoking Proposition 2.3 and the boundedness of {u n } in L q(•) (R N ).Also, we easily obtain ´RN f (x, u n )(u n − u) dx → 0 by using (F2), (5.9), Proposition 2. We now conclude this section by proving Theorem 5.1.
Proof of Theorem 5.1.We will show that conditions (J 1) − (J 3) of Lemma 5.2 are fulfilled with E := W s,p(•,•) (R N ) and J = J λ .In order to verify (J 1), let {e n } ∞ n=1 be a Schauder basis of E and let {e * n } ∞ n=1 ⊂ E * be such that for each n ∈ N, (5.29) Then, E = V k ⊕ X k , ∀k ∈ N.
Extend ϕ k (x) to R N by putting ϕ k (x) = 0 for x ∈ R N \ B ε (x 0 ).Clearly, {ϕ k } ⊂ E. Define Let k ∈ N be arbitrary and fixed.We claim that there exists R k > 0 independent of λ such that J λ (u) ≤ 0, ∀u ∈ E k \ B R k .