The Nehari manifold method for discrete fractional p-Laplacian equations

The aim of this paper is to investigate the multiplicity of homoclinic solutions for a discrete fractional difference equation. First, we give a variational framework to a discrete fractional p-Laplacian equation. Then two nontrivial and nonnegative homoclinic solutions are obtained by using the Nehari manifold method.


Introduction and main result
Denote by Z the set of whole integers and let T be a positive real number. Set -T u(j) = 1 T 2 u (j + 1)T -2u(jT) + u (j -1)T for u : Z → R. The well-known second order difference equation can be regarded as the discrete version of the Schrödinger type equation, which can be used to describe a planetary system or an electron in an electromagnetic field. Here potential function V : Z → [0, ∞) and f : Z × R → R. Particularly, homoclinic orbits play a very important role in studying the dynamics of discrete Schrödinger equations. In recent tears, second order difference equations and homoclinic orbits have been the research focus. The literature on such a field is very rich, we collect some papers; see, for example, [2, 7, 18-20, 22, 23, 30]. Especially, Agarwal, Perera and O'Regan in [2] first considered the existence of solutions for second order difference equations like (1.1) by using variational methods.
Recently, Ciaurri et al. in [10] considered the following discrete fractional Laplace equation: where (-T ) s is the so-called discrete fractional Laplacian given by Here s ∈ (0, 1), is the Gamma function and v(t, j) = e t T u(j) is the solution of the following problem: provided u ∈ L s . As showed in [10, Theorem 1.1], there exist positive constants c s ≤ C s such that In particular, [10] stated that the solutions of (1.2) converge to the solutions of following fractional Laplacian problem: Here (-) s is the fractional Laplacian defined for any x ∈ R as For further details about the fractional Laplacian and fractional Sobolev spaces, we refer to [12]. The numerical analysis of fractional difference equations is difficulty, since the discrete fractional Laplace operator is nonlocal and singular; see for example [1,17] and the references cited therein. In [35], Xiang and Zhang first studied the following discrete fractional Laplacian equation: where s ∈ (0, 1), V : Z → (0, ∞), f : Z × R → R is a continuous function with respect to the second variable and satisfies asymptotically linear growth at infinity. Under some suitable hypotheses, two solutions were obtained by using the mountain pass theorem and Ekeland's variational principle.
Recently, the study of fractional Laplacian and related problems has been received an increasing amount of attention. The fractional Laplacian appears in many fields, such as anomalous diffusion, quantum mechanics, finance, optimization and game theory; see [4,8,21,31] and the references therein. For the applications of fractional operators, we refer to [3, 5, 9, 11, 14-16, 24-29, 33, 34, 36, 37] and the references therein.
Motivated by above papers, we study the following nonlinear discrete fractional p-Laplacian equation: where s ∈ (0, 1), V : Z → (0, ∞), 1 < q < p, a ∈ p p-q , p < r < ∞, b ∈ ∞ and (d ) s p is defined as follows: for each j ∈ Z, Here the discrete kernel K s,p satisfies the requirement that there exist constants 0 < c s,p ≤ C s,p < ∞ such that Note that when p = 2 the discrete fractional p-Laplacian (d ) s p reduces to (-T ) s with T = 1. As usual, we say that a function u : Z → R is a homoclinic solution of Eq. (1.4) if u(k) → 0 as |k| → ∞.
In this paper we always assume that the function V satisfies

4) admits at least two nontrivial and nonnegative homoclinic solutions.
To the best of our knowledge, our paper is the first time of use of the Nehari manifold method to study the multiplicity of solutions for discrete fractional p-Laplacian equations. It is worth mentioning that the weight function a may change sign in this paper. But for the case that both a and b are sign changing functions, the existence of two solutions is still an open problem. The authors will consider the case in the further.
The paper is organized as follows. In Sect. 2, we present a variational framework to Eq. (1.4) and show some basic results. In Sect. 3, we give the definitions of Nehari manifold and fibering map. Moreover, some properties of the fibering map are given. In Sect. 4, using the Nehari manifold method, we obtain two distinct nontrivial and nonnegative homoclinic solutions of Eq. (1.4).

Variational setting and preliminaries
In this section, we first recall some basic definitions, which can be found in [13,19,35]. Then we introduce a variational framework to Eq. (1.4) and discuss its properties. For any 1 ≤ ν < ∞, we define ν as with the norm Then ( ν , · ν ) and ( ∞ , · ∞ ) are Banach spaces; see [13]. Clearly, Define W as Equip W with the norm Proof The proof is similar to [32]. Let u ∈ p . Then where 0 < C = (32 p-1 C s,p k =0 1 |k| 1+2s ) 1/p < ∞. Therefore, the proof is complete.

Lemma 2.2 The norm
The proof is similar to [32], for completeness, we give its details. Using assumption (V ) and Lemma 2.1, we have which leads to u = ( j∈Z V (j)|u(j)| p ) 1/p being an equivalent norm of W . Finally we show that (W , · W ) is complete. Let {v n } n be a Cauchy sequence in W . Observe that Then {v n } n is also a Cauchy sequence in p . By the completeness of p , there exists u ∈ p such that v n → u in p . Furthermore, Lemma 2.1 and assumption (V ) show that v n → u strongly in W as n → ∞.
In conclusion, the proof is complete.
Moreover, we have the following compactness result.
Proof The proof is similar to that in [19] and [35]. We first show that the result holds for the case ν = p. It follows from assumption (V ) that which shows that the embedding W → p is continuous. Next we prove that W → p is compact. Let {v n } n ⊂ W and assume that there exists D > 0 such that v n p W ≤ D for all n ∈ N. Now we show that {v n } n strongly converges to some function in p . Using the reflexivity of W , there exist a subsequence of {v n } n still denoted by {v n } n and function u ∈ W such that v n u in W . By assumption (V ), for any δ > 0 there exists j 0 ∈ N such that for all |j| > j 0 Set I = [-j 0 , j 0 ] and define Observe that the dimension of W I is finite. Then {v n } n is a bounded sequence in W I , due to which yields {v n } n is bounded in p I . Thus, up to a subsequence we may assume that v n → u on I. Hence there exists n 0 ∈ N such that for all n ≥ n 0 Then, for all n > n 0 , Thus, we deduce that v n → u in p . Now we consider the case ν > p. Note that for all u ∈ p . This inequality together with the result of the case ν = p leads to the proof.
To obtain some properties of energy functional associated with Eq. (1.4), we need the following result.

Lemma 2.4 Assume that U is a compact subset of W . Then for any
Proof The proof can be found in [32].
For each u ∈ W , we define the associated energy functional with Eq. (1.4) as

Lemma 2.5 If V satisfies (V ), then is well-defined, of class C 1 (W , R) and
for all u, v ∈ W .
Proof By Lemma 2.1, we know that is well-defined on W . Fix u, v ∈ W . We first prove that Fix 0 < t < t 0 . For j, m ∈ Z, by the mean value theorem, we can choose 0 < t j,m < t such that where y(j) = u(j) + t j,m v(j). Clearly, y ∈ W and y W ≤ 2C. Observe that |j|≤h |m|>2h where φ p (τ ) := |τ | p-2 τ for all τ ∈ R. Thus, (2.1) holds true. An analogous argument gives Thus, we get Thus, is Gâteaux differentiable in W . Finally, we prove that : W → W * is continuous.
To this aim, we assume that {u n } n is a sequence in W such that u n → u in W as n → ∞. By Lemma 2.4, for any ε > 0 there exists h ∈ N such that |j|>h |m|>h u n (j)u n (m) p K s,p (jm) 1/p < ε for all n ∈ N and |j|>h |m|>h In addition, there exists n 0 ∈ N such that |j|≤2h |m|≤2h for all n ≥ n 0 , where p = p p-1 . For any v ∈ W with v W ≤ 1, and for any n ≥ n 0 , by the Hölder inequality and a similar discussion to above, we deduce Similarly, one can show that This means that is continuous. Consequently, we prove that ∈ C 1 (W , R).

Lemma 2.6 Assume that V satisfies
Proof Using the same discussion as [19] and [35], one can prove the lemma.

Lemma 2.7
Assume that V satisfies (V ), 1 < q < p < r < ∞, a ∈ p p-q and 0 ≤ b ∈ ∞ . Then a critical point of I λ is a homoclinic solution of Eq. (1.4) for all λ > 0.
Proof Let u ∈ W be a critical point of I λ , that is, I λ (u) = 0. Then for all v ∈ W . For each k ∈ Z, we define γ k as Clearly, γ k ∈ W . Choosing v = γ k in (2.6), we obtain which means that u is a solution of (1.4). Obviously, u(k) → 0 as |k| → ∞, this means that u is a homoclinic solution of (1.4).

Nehari manifold and Fibering map analysis
In this section, we give some definitions and properties of Nehari manifold. Some ideas are inspired from [6] and [32]. In present section, we always assume V satisfies (V ), a ∈ p p-q Define the Nehari manifold as follows: For each u ∈ W , we define the fibering map λ,u : (0, ∞) → R as for all t > 0. Then a simple calculation yields In particular, if u ∈ N λ , then λ,u (1) = 0 and λ,u (1) = (pq)λ j∈Z a(j) u(j) q + (pr) j∈Z b(j) u(j) r .
Since 1 may be a minimum point, maximum point, or saddle point of λ,u , we divide N λ into three subsets N + λ , Nλ and N 0 λ , which are defined respectively as which leads to t λ,u (t) = 0. Therefore, we can prove that tu ∈ N λ if and only if λ,u (t) = 0.

Lemma 3.2
If u is a local minimizer of I λ on N λ and u / ∈ N 0 λ , then I λ (u) = 0.
Proof The proof is similar to that in [6]; see also [32]. For completeness, we give its proof. Assume that u is a local minimizer of I λ on N λ . By Lagrange multipliers, there exists μ ∈ R such that I λ (u) = μJ (u), where J(u) is given by Since u ∈ N λ , we deduce I λ (u), u = 0. Hence, μ J (u), u = 0. It follows from u / ∈ N 0 λ that J (u), u = λ,u (1) = 0.
Consequently, μ = 0. Furthermore, we obtain I λ (u) = 0. Proof Using a similar discussion to Theorem 4.1, one can show that I λ possesses a minimizer u 1 on Nλ . Moreover, Lemma 3.6 shows that u 1 is nontrivial. Furthermore, one can use a similar discussion to Theorem 4.1 to prove that |u 1 | is a minimizer of I λ on Nλ . Therefore, the proof is complete.
Proof of Theorem 1.1 Gathering Theorem 4.1 with Theorem 4.2, we see that I λ has two nonnegative and nonnegative local minimizers. Then it follows from Lemma 3.2 that I λ has two critical points on W , which are two nontrivial and nonnegative local least energy solutions of problem (1.4).