Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition

The purpose of this paper is to study $T$-periodic solutions to [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u)&\mbox{in} (0,T)^{N} (P) u(x+Te_{i})=u(x)&\mbox{for all} x \in \R^{N}, i=1, \dots, N where $s\in (0,1)$, $N>2s$, $T>0$, $m>0$ and $f(x,u)$ is a continuous function, $T$-periodic in $x$ and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator $(-\Delta_{x}+m^{2})^{s}$ can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder $\mathcal{S}_{T}=(0,T)^{N}\times (0,\infty)$. By using a variant of the Linking Theorem, we show that the extended problem in $\mathcal{S}_{T}$ admits a nontrivial solution $v(x,\xi)$ which is $T$-periodic in $x$. Moreover, by a procedure of limit as $m\rightarrow 0$, we also prove the existence of a nontrivial solution to (P) with $m=0$.


Introduction
The aim of this paper is to investigate the existence of periodic solutions to the problem for all x ∈ R N , i = 1, . . . , N , (1.1) where s ∈ (0, 1), N > 2s, m ≥ 0, (e i ) is the canonical basis in R N and the nonlinearity f : R N × R → R is a superlinear continuous function. The operator (−∆ x + m 2 ) s appearing in (1.1), is defined through the spectral decomposition, by using the powers of the eigenvalues of −∆ x + m 2 with periodic boundary conditions. Let u ∈ C ∞ T (R N ), that is u is infinitely differentiable in R N and T -periodic in each variable. Then u has a Fourier series expansion: are the Fourier coefficients of u. The operator (−∆ x + m 2 ) s is defined by setting can be extended by density to a quadratic form on the Hilbert space When m = 1 we set H s T = H s 1,T . We would remind that in the last decade a great attention has been devoted to the study of fractional Sobolev spaces and non-local operators. In fact such operators arise in several fields such as optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, minimal surfaces and water waves; see for instance [4], [5], [6], [8], [10], [13], [16], [21], [22], [23] and references therein. Now we state the assumptions on the nonlinear term f in (1.1). Along the paper, we will suppose that f : R N ×R → R verifies the following conditions: for any x ∈ R N and t ∈ R; (f 5) lim |t|→∞ F (x, t) |t| 2 = +∞ uniformly for any x ∈ R N . Here F (x, t) = t 0 f (x, τ )dτ ; (f 6) There exists γ ≥ 1 such that for any x ∈ R N G(x, θt) ≤ γ G(x, t) for any x ∈ R N , t ∈ R and θ ∈ [0, 1], where G(x, t) := f (x, t)t − 2F (x, t). Our first main result is the following Theorem 1. Let m > 0 and let us assume that f : R N +1 → R is a function satisfying the assumptions (f 1)−(f 6). Then there exists a solution u ∈ H s m,T to (1.1). In particular u ∈ C 0,α ([0, T ] N ) for some α ∈ (0, 1).
To study the problem (1.1), we will make use of a Caffarelli-Silvestre typeextension in the periodic framework (see [2,3]). This method, which has been originally introduced in [7] to investigate the fractional Laplacian in R N , is very useful because it allows us to reformulate the non-local problem (1.1) in terms of a local degenerate elliptic problem with a Neumann boundary condition in one dimension higher.
We will exploit this fact and we will look for solutions to is the conormal exterior derivative of v.
Since (1.2) has a variational nature, its solutions can be found as critical points of the functional defined on the space X s m,T , which is the closure of the set of smooth and T -periodic (in x) functions in R N +1 + with respect to the norm ||v|| X s m,T := Under the assumptions (f 1) − (f 6) we are able to prove that for any m > 0 fixed, J m has a Linking geometry. We recall that in [2,3] we proved that then we can obtain a nontrivial solution to (1.2) by applying the standard Linking Theorem [19,24,25]. This is due to the fact that (AR) guarantees the boundedness of Palais-Smale sequences for the functional associated with the problem under consideration. However, although (AR) is a quite natural condition when we have to deal with superlinear elliptic problems, it is somewhat restrictive. In fact, by a direct integration of (AR), we can deduce the existence of A, B > 0 such that which implies, being µ > 2, the condition (f 5). If we consider the function f (x, t) = t log(1 + |t|), then it is easy to prove that f verifies (f 5) but it does not verify (AR). This means that that there are functions which are superlinear at infinity, but do not satisfy (AR). For this reason, in several works concerning superlinear problems of the type where L is a second order elliptic operator and Ω ⊂ R N is a smooth bounded domain, some authors tried to drop the condition (AR); see for instance [11,14,17,18,20] and references therein.
In this paper, we claim to study the non-local counterpart of (1.3) with L = −∆+m 2 , in periodic setting, without assuming (AR). Our basic assumptions on the nonlinearity f are (f 1) − (f 6). We recall that the hypothesis (f 6) was introduced for the first time by Jeanjean in [14] to show the existence of a bounded Palais-Smale sequence for functionals having a Mountain Pass structure Here, in order to study the problem (1.2), we invoke a variant of the Linking Theorem proved by Li and Wang in [15], in which the Palais-Smale condition is replaced by the Cerami condition [9]; namely any Cerami sequence {v j } in X s m,T at the level α ∈ R, that is such that J m (v j ) → α and ||J ′ m (v j )|| (X s m,T ) * (1 + ||v j || X s m,T ) → 0 as j → ∞, admits a convergent subsequence. At this point, to get the existence of a weak solution to (1.1), it will be sufficient to show that every Cerami sequence is bounded and that it possesses a convergent subsequence. This step will constitute the heart of the proof of Theorem 1.
Finally, we will also consider the problem (1.1) when m = 0, that is (1.4) By passing to the limit in (1.2) as m → 0, we prove the existence of a nontrivial periodic solution to (1.4). A such procedure of limit is justified by the fact that we are able to estimate from below and from above, the critical levels α m of the functionals J m independently of m, provided that m is sufficiently small. Therefore our second result can be stated as follows Then the problem (1.4) admits a nontrivial solution u ∈ H s T . The paper is organized as follows: in Section 2 we give some notations and preliminaries results about the involved fractional Sobolev spaces; in Section 3 we consider the extension problem (1.2) which localizes the nonlocal problem (1.1); in Section 4 we investigate the existence of solutions to (1.2) via variational methods; finally we show that we can find a nontrivial solution to (1.4) by passing to the limit in (1.2) as m → 0.

Notations and Preliminaries
In this section we introduce the notations and we collect some facts which will be used along the paper. Let be the upper half-space in R N +1 . We denote by S T = (0, T ) N × (0, ∞) the half-cylinder in R N +1 + with basis ∂ 0 S T = (0, T ) N × {0} and lateral boundary ∂ L S T = ∂(0, T ) N × [0, +∞). With ||v|| L r (S T ) we always denote the norm of v ∈ L r (S T ) and with |u| L r (0,T ) N the L r (0, T ) N norm of u ∈ L r (0, T ) N . Let s ∈ (0, 1) and m > 0. Let A ⊂ R N be a domain.
We define L 2 (A×R + , ξ 1−2s ) as the set of measurable functions v on A×R + such that Let u ∈ C ∞ T (R N ). Then we know that We will also use the notation to denote the Gagliardo semi-norm of u. When m = 1, we set H s T = H s 1,T and | · | H s T = | · | H s 1,T . Finally we introduce the functional space X s m,T defined as the completion of If m = 1, we set X s T = X s 1,T and || · || X s T = || · || X s 1,T . In order to lighten the notation, it is convenient to omit the indices s and T (which are fixed) appearing in the Sobolev spaces and norms just defined. From now on we will write X m , X, H m , H, || · || Xm , || · || X , | · | H and | · | Hm . Now we recall that it is possible to define a trace operator from X m to H m : There exists a surjective linear operator Tr : X m → H m such that: (ii) Tr is bounded and In particular, equality holds in Finally we have the following embeddings:

Extension Method
As mentioned in the introduction, crucial to our results is that (−∆ x + m 2 ) s is a nonlocal operator which can be realized through a local problem in S T . This result can be stated more precisely as follows: where H * m is the dual of H m . We call v the extension of u.
It is known (see [12]) that θ(ξ) = 2 Γ(s) By using this fact we can deduce that ||v|| Xm = √ κ s |u| Hm . Now we take advantage of the previous result to reformulate nonlocal problems with periodic boundary conditions, in a local way. Let g ∈ H * m and consider the following two problems: Definition 2. We say that u ∈ H m is a weak solution to (3.4) and v is a weak solution to (3.5).
Remark 2. Later, with abuse of notation, we will denote by v(·, 0) the trace Tr(v) of a function v ∈ X m .

Periodic solutions in the cylinder S T
In this section we prove the existence of a nontrivial solution to (1.1). As explained in the previous section, we know that the study of (1.1) is equivalent to investigate the existence of weak solutions to For simplicity, we will assume that κ s = 1.
We introduce the following functional on X m By conditions (f 2)-(f 4) we know that for any ε > 0 there exists C ε > 0 such that and Then, by Theorem 4, follows that J m is well defined on X m and J m ∈ C 1 (X m , R). By using Theorem 3, we notice that the quadratic part of J m is nonnegative, that is (see Theorem 7 in [3]). As observed in [2,3], X m can be splitted as where dim Y m < ∞ and Z m is the orthogonal complement of Y m with respect to the inner product in X m . In order to find critical points of J m we will make use of a suitable version of the Linking Theorem due to Li and Wang [15]. Firstly we recall the following definitions.
Definition 3. Let (X, || · || X ) be a real Banach space with its dual space (X ′ , || · || X ′ ), J ∈ C 1 (X, R) and c ∈ R. We say that {v n } ⊂ X is a Cerami sequence for J at the level c if In particular, if J satisfies the (C) α condition, then α is a critical value of J.
Then, we are going to verify that J m satisfies the assumptions of the above Theorem 6. We begin proving a series of lemmas, which ensure us that J m possesses a Linking geometry.
Proof. Firstly we show that F (x, t) ≥ 0 for any x ∈ R N and t ∈ R. By (f 6) we Arguing similarly for the case t ≤ 0, eventually we obtain that F ≥ 0 in R N × R. As a consequence, recalling that ||v|| 2 Xm = m 2s |v(·, 0)| 2 L 2 (0,T ) N for any v ∈ Y m by (4.5), we can see that Proof. By using Lemma 3 in [3] we know that there is a constant C m > 0 such that for any v ∈ Z m . Then, taking into account (4.3), (4.7) and Theorem 4 we have Xm for any v ∈ Z m . Choosing ε ∈ (0, mC m ), we can find r m > 0 such that Lemma 3. There exists ρ m > r m and z ∈ Z m with ||z|| Xm = r m such that, denoted by We observe that w ∈ Z m (since T 0 sin(ωx)dx = 0) and by (4.9) follows that there exist C 1 , C 2 , C 3 > 0 (independent of m) such that Now, let z = r m w ||w|| Xm . It is clear that z ∈ Z m and ||z|| Xm = r m .
Moreover, by (4.10) we obtain (4.11) Take v = y + λz ∈ Y m ⊕ R + z. We recall that if y ∈ Y m then y(x, ξ) = d m θ(mξ), d m ∈ R and ||y|| 2 Xm = m 2s |y(·, 0)| 2 L 2 (0,T ) N . Then, by using (4.11), we have By using (f 2) and (f 5), we know that for any A > 0 there exists Taking into account (4.12) and (4.13), we get for any and Proof. We start proving that {v j } is bounded in X m . We proceed as in [17]. We argue by contradiction and assume that Then ||z j || Xm = 1 and by using Theorem 4 we can assume, up to a subsequence, that z j ⇀ z in X m z j (·, 0) → z(·, 0) in L q (0, T ) N for any q ∈ [2, 2 ♯ s ) (4.18) z j (·, 0) → z(·, 0) a.e in (0, T ) N and there exists h ∈ L p+1 (0, T ) N such that Now we distinguish two cases. Firstly we suppose that As in [14], we can choose Since ||v j || Xm → ∞ we can take r n = 2 √ n such that provided j is large enough. By (4.18) and the continuity of F , we can see as j → ∞ and n ∈ N. In particular, integrating (f 4) and exploiting (4.19) we get a.e. x ∈ (0, T ) N and n, j ∈ N. Then, taking into account (4.23), (4.24) and by using the Dominated Convergence Theorem we deduce that F (x, r n z j (x, 0)) → F (x, r n z(x, 0)) in L 1 (0, T ) N .
provided j is large enough and for any n ∈ N. As a consequence Since J m (0) = 0 and J m (v j ) → c we deduce that t j ∈ (0, 1). Thus, by (4.21) we have Taking into account (f 6), (4.14), (4.15) and (4.28) we get which contradicts (4.27). Secondly, we suppose that z ≡ 0. By (4.14), (4.16) and F ≥ 0 we can easily deduce that Xm dx as j → ∞. Putting together (4.31) and (4.32) we get a contradiction. Thus the sequence {v j } is bounded in X m . By Theorem 4 we can assume, up to a subsequence, that  Taking into account (f 2), (f 4), (4.33), (4.34) and the Dominated Convergence Theorem we get as j → ∞. By using (4.15) and the boundedness of {v j } j∈N in X m , we deduce that as j → ∞. By (4.33), (4.35) and (4.37) we have Moreover, by (4.15) and Taking into account (4.33), (4.34), (4.36) and (4.39) we get Since X m is a Hilbert space, we have ||v j − v|| 2 Xm = ||v j || 2 Xm + ||v|| 2 Xm − 2 v j , v Xm and using v j ⇀ v in X m and (4.41) we can conclude that v j → v in X m , as j → ∞.
Proof of Theorem 1. Taking into account Lemma 1 -Lemma 4 we can see that J m satisfies the assumptions of Theorem 6. Then, we deduce that for any fixed m > 0, there exists a function v m ∈ X m such that Hence v m is a nontrivial weak solution to (4.1), and by Theorem 9 in [3] follows that v m (·, 0) ∈ C 0,α ([0, T ] N ) for some α ∈ (0, 1).

Periodic solution in the case m = 0
In this last section, we show that it is possible to find a nontrivial weak solution to (1.4). In Section 4 we proved that for any m > 0 there exists v m ∈ X m such that where α m is defined as in (4.43). Now, we will prove that we can estimate from below and from above the critical levels of the functional J m independently of m, when m is sufficiently small. This allows us to pass to the limit in (4.1) as m → 0 and to prove the existence of a nontrivial solution to Let us assume that 0 < m < m 0 := ω 2s 2 . Firstly we prove that there exists K 1 > 0 independent of m, such that In order to achieve our aim, we will estimate the L q -norm of the trace of v, with q ∈ [2, 2 ♯ s ). Let v ∈ Z m and we denote by c k its Fourier coefficients. By c 0 = 0 and Theorem 3 follow that Now, fix 2 < q < 2 ♯ s and we denote by q ′ its conjugate exponent. Taking into account c 0 = 0, by using Hölder inequality and Theorem 3 we get because of 1 < 2N N +2s < q ′ < 2. As a consequence, by using the Theorem of Hausdorff-Young-Riesz (see pages 101-102 in [26]) follows that and taking q = p + 1 we have for some C ′′ := C ′′ (T, N, s, p) > 0 independent of m.
(5.6) By using (4.10) and 0 < m < m 0 we know that where w is the function defined in (4.8).
Recalling that 0 < m < m 0 , we can see that (4.12) in Section 4 can be replaced by . By using (4.13) and 0 < m < m 0 , we have for any Therefore, taking into account (5.1), (5.3) and (5.6) we deduce that Now, we show how to exploit this last information to pass to the limit in (1.2) as m → 0. Firstly we begin proving that for any δ > 0, holds the following inequality and applying the Hölder inequality we deduce Multiplying both members of (5.10) by ξ 1−2s we have Integrating (5.11) over (0, T ) N × (0, δ) we have Finally, we notice that (5.8) implies that Xm and by applying (4.13) with A = 1, we can deduce that Then, taking into account (5.12), (5.13), (5.14), (5.15), and Theorem 4, it is enough to prove lim sup m→0 ||v m || Xm < ∞ (5. 16) to deduce the existence of a subsequence, that for simplicity we will denote again with {v m }, and a function v such that v ∈ L 2 loc (S T , ξ 1−2s ) and ∇v ∈ L 2 (S T , , as m → 0. To show the validity of (5.16), we proceed as in the first part of the proof of Lemma 4 in which we demonstrated the boundedness of Cerami sequences. We assume by contradiction that, up to a subsequence, ||v m || Xm → ∞ as m → 0. Taking into account (5.20), (5.23), (5.24) and by using Theorem 4, we can deduce the existence of a subsequence, which we will denote again with {w m }, and a function w such that w ∈ L 2 loc (S T , ξ 1−2s ) and ∇w ∈ L 2 (S T , ξ 1−2s ); , as m → 0.