Ground state solutions for periodic Discrete nonlinear Schr¨odinger equations

: In this paper, we consider the following periodic discrete nonlinear Schr¨odinger equation


Introduction and main results
In this paper, we are interested in the following periodic discrete nonlinear Schrödinger (DNLS) equation in m dimensional lattices is the discrete Laplacian in m spatial dimension and {ε n } is a real-valued and T -periodic in n, i.e., for n = (n 1 , n 2 , ..., n m ) ∈ Z m , here T = (T 1 , T 2 , ..., T m ), T i is a positive integer, i = 1, 2, ..., m.Moreover, we assume the nonlinearity g n (u) is T -periodic in n and gauge invariant in u, i.e., g n (e iθ u) = e iθ g n (u), θ ∈ R.
Since breathers are spatially localized time-periodic solutions and decay to zero at infinity, we know ψ n has the form ψ n = u n e −iωt and lim |n|→∞ ψ n = 0.
Then (1.1) becomes to and lim where is the length of multi-index n, u n is a real-valued sequence, m is a positive integer and ω ∈ R is the temporal frequency.If u n 0, then u is called a nontrivial solitons.Usually, if (1.3) holds, we say that a solution u = {u n } of (1.2) is homoclinic to 0.
The DNLS equation has been widely investigated because it is one of the most important inherently discrete models, ranging from solid state and condensed matter physics to biology.For examples, it has a physical meaning as a quantum Hamiltonian for a conduction electron in a magnetic field in a particular case of so-called tight-binding model (Aubry-Andre-Azbel-Harper model) [2,4,15], and also has been studied widely in nonlinear optics [8], biomolecular chains [20], Bose-Einstein condensates [23,35] and the denaturation of the DNA double strand [19].Here, we would like to mention the survey paper of Hennig and Tsironis [16] on wave transmission properties in one dimensional nonlinear lattices for an overview and more references cited in, some latest advances in both theory and applications of chaotic breather formation [17,27], and some important works on discrete breathers [10][11][12].We refer to [1,3,9,13,14] and the references cited therein for more physical background.
First, we deal with the suplinear case.To our knowledge, it seems that the earlier works investigating the existence of ground state solutions for problem (1.2) with superquadratic potentials is [25,26].In [25], Mai and Zhou considered equation (1.2) with m = 1 and superquadratic nonlinearity, they required the condition (V 1 ) ω σ(L), the spectrum of L := −∆ + ε n and ω belonging to a finite gap (α, β); |s| is increasing(−∞, 0) ∪ (0, +∞) for all n ∈ Z m , and followed the same way as in [22] to obtain the ground state solutions.
Motivated by papers [26,33], we will study the existence of ground state solutions of problem (1.2) under more general assumptions on g n .More precisely, we make the following hypothesis: (g 4 ) for each n ∈ Z m , sg n (s) ≥ 0, and there exists a constant η 0 ∈ (0, 1) such that Under our assumptions (g 1 )-(g 2 ), the functional associated to (1.1) is of class C 1 on E := l 2 (Z m ) (see section 2), and the critical points of Φ are weak solutions of (1.2).Our main result is the following theorem.
Theorem 1.1.If (V 1 ), (g 1 ) − (g 4 ) hold, then (1.2) has a ground state soliton, i.e. a nontrivial soliton v such that Φ(v) = inf{Φ(u) | u 0, Φ (u) = 0}.Moreover, the solution u decays exponentially at infinity, that is, there exist constants C > 0 and γ > 0 such that Now we handle the asymptotically linear case.More precisely, we assume ), (g 1 ), (g 2 ), (g 4 ), (g 5 ) hold, then (1.2) has a ground state soliton, i.e. a nontrivial soliton v such that Φ(v) = inf{Φ(u) | u 0, Φ (u) = 0}.Moreover, the solution u decays exponentially at infinity, that is, there exist constants C > 0 and γ > 0 such that Remark 1.3.It is worth mentioning that we have removed condition (S 1 ) which has been commonly assumed to be satisfied in all existing works mentioned above.Next, it is easy to check that (S 3 ) implies (g 4 ).Moreover, let g n (s) = ϕ(s) for all n, where ϕ(−s) = −ϕ(s) for s ≥ 0, and it is not difficult to verify that g n (u) satisfies condition (g 4 ) with θ 0 = 1 10 , but does not satisfy condition(S 3 ).This shows that condition (g 4 ) is weaker than condition (S 3 ).We remark that assumptions (S 1 ) and (S 3 ) play essential roles in verifying the isolation of trivial critical point and boundedness of Cerami seuqences for the energy functional Φ in [26].Furthermore, it is not difficult to verify that (g 4 ) implies that H n (s We mention that (H) has made significant contributions to dealing with the asymptotically linear case, for example, see [6,31,37] along this direction.Therefore, our results would be applied to more general situations.To the best of our knowledge, there is no work devoted to handling with problem (1.2) under our weaker conditions.Hence our result is new and sharply extends and improves many existing ones in the literature, such as those in [6,25,26,29,31,37].
Before we close this section, let us outline the proof of our main results and explain the difficulties we will encounter.To prove the existence of ground state solutions, we will adopt a technique developed in [22] (see also [18].We set m = inf{I(u) : u ∈ K \ {0}} where K denotes the critical points set of Φ.If K \ {0} ∅, then one can find a minimizing Cerami sequence for Φ on K \ {0}.Next, the main task is to verify boundedness of this Cerami sequence.Finally, we must show that m is achieved at a nontrivial critical point.Fortunately, we can prove that all Cerami sequences of Φ are bounded (see Lemma 3.3 bellow).Then K \ {0} ∅ follows directly from the generalized linking theorem due to Li and Szulkin [21].Furthermore, we can show that the zero function 0 is an isolated critical point of Φ (see Lemma 2.5 in section 2) which plays a crucial role in verifying that m is achieved at a nontrivial critical point.
The remaining of this paper is organized as follows.In Section 2, we establish the variational framework associated with (1.2), and give some preliminary lemmas.In Section 3, we present the detailed proofs of our main results.

Variational setting and preliminary lemmas
First, we will establish the variational framework associated with (1.2).Let Then the following embedding between l p spaces holds, •) E is the inner product in E, and the corresponding norm in E is denoted by • E .Then the derivative of Φ has the following formula, . Therefore, E possesses the orthogonal decomposition Obviously, (Au, u) E is positive on E + and negative on E − , respectively.Moreover, we may define an equivalent norm • on E ± by u ± 2 = ±(Au ± , u ± ) E , ∀u ± ∈ E ± .Therefore, Φ can be rewritten as where ) and the derivative is given by To state the generalized linking theorem of Li and Szulkin [21], we introduce some notation.let R > r > 0 and let Then M is a submanifold of E − ⊕ R + z 0 with boundary ∂M.The generalized linking theorem is stated as follows: ) is bounded from below, weakly sequentially lower semicontinuous and Ψ is weakly sequentially continuous.Let Φ be a functional on E of the form (2.3).
then for some κ ≤ c ≤ sup M Φ, there is a sequence {u ( j) } ⊂ E such that Such a sequence is called a Cerami sequence on the level c, or a (C) c sequence.
Proof.From (g 2 ) and (g 4 ), it is easy to see that Thus, by Fatou lemma, we deduce that proving that Ψ is weakly sequentially lower semi-continuous.
To show that Ψ is weakly sequentially continuous, let u ( j) u in E. We have that u ( j) n → u n as j → ∞ for all n ∈ Z m , and there exists C > 0 such that u ( j) E ≤ C and u E ≤ C. By (g 1 ) and (g 2 ), there exists M > 0, such that |g n (s)| ≤ M|s| for |s| ≤ C and all n ∈ N.
Given ν ∈ E, for any ε > 0, there exists N ∈ N such that (2.5) For such N, by (g 1 ), for large j, we have Then using (2.5), (2.6) and Hölder's inequality in l 2 , for j large enough, one has < ε, which implies the weakly sequentially continuity of Ψ .
Next, we discuss the linking structure of the function Φ.In the following, for the asymptotically quadratic case we set θ = inf V n and for the superquadratic case we choose θ = 2(β−ω).Take a number µ satisfying β − ω < µ < θ. (2.7) Since σ(H) is absolutely continuous, the subspace Y 0 := P µ − P β−ω l 2 is infinite dimensional, where (P λ ) λ∈R denotes the spectrum family of H.By definition and (2.2), Lemma 2.3.The following statements hold true: (a) There exists r > 0 such that κ : Under assumption (g 1 ) and (g 2 ), there exists . So the inequality follows if we assume that r > 0 is sufficiently small.
(b) Arguing indirectly, suppose that, there exists u ( j) = s ( j) z 0 + u ( j) − with u ( j) → ∞ as j → ∞ such that Φ(u ( j) ) ≥ 0 for all j ∈ N. Setting v ( j) = u ( j) / u ( j) , we have v Passing to a subsequence, we may assume that s v in E, v ( j) − v − , and v ( j) n → v n for every n.It follows that v = sz 0 + v − 0. First we consider the super linear case.Recalling that v 0, there exists n 0 ∈ Z m such that v n 0 0 and |u ( j) Hence, it follows from (g 3 ) and Fatou's lemma that contradicting with (2.8).
Next we consider the asymptotically linear case.Noting that v + = sz 0 0, by (g 5 ) and (2.7), there Hence there exixts a finite set By (g 1 ), (g 2 ) and (g 5 ), one gets that |F n (t)| ≤ C|t| 2 and F n (t)/t 2 → 0 as |t| → ∞ uniformly in n.Noting that it follows from Lebesgue's dominated convergence theorem and the fact n∈A |v Therefore, by (2.8) and (2.9), we conclude According to Lemma 2.3 and proposition 2.1, we obtain the following result.
Let K := {u ∈ E : Φ (u) = 0} denote the set of all the critical points of Φ.We have the following result which plays an important role in showing the ground state solution is nontrivial.

Proof of main Theorems
In order to prove the boundedness of (C) c -sequence of Φ, we need the following important technical lemma.
Proof.Let {u ( j) } be a (C) c sequence of Φ, that is, {u ( j) } satisfying (2.3).Suppose {u ( j) } is unbounded.Then, passing to a subsequence, we assume that u ( j) → ∞.Setting h ( j) = u ( j) / u ( j) , we have h ( j) = 1.Then, up to a subsequence, there exists h ∈ E such that h ( j)  h in E. Let h ( j) ± be the orthogonal projection of h ( j) on E ± , respectively.Since G n (s) ≥ 0, for large j we have This and h ( j) + 2 + h ( j) − 2 = 1 imply that h ( j) + 2 ≥ 1 4 .Given any R > √ 8(|c| + 4), noting that h ( j) + is bounded in E, there exists δ > 0 and n j ∈ Z m such that and Set s ( j) = R u ( j) , by using (3.4), (3.5) and Lemma 3.2, for large j we have this yields a contradiction since R > √ 8(|c| + 4).For the super linear case, from the periodicity of the coefficients, we know Φ and Φ are both invariant under translation, i.e., v ( j) = v ( j) n = h ( j)  n+KT .Making such shifts, we can assume that 1 ≤ n j ≤ T − 1 in (3.3), moreover passing to a subsequence, we can assume that n j = n 0 is independent of j.Next we may extract a subsequence, still denoted by h ( j) , such that h ( j) + → h + and h ( j) + n → h + n for all n.Specially, for n = n 0 , inequality (3.3) shows that |h + n 0 | ≥ δ.Moreover, we have Consequently, we obtain as j → ∞, which is a contradiction.For the asymptotically linear case, let n ∈ Z m be such that h n 0, then u ( j) n = h ( j) n • u ( j) → ∞ as j → ∞.We denote l 0 the vector space of all finite sequences, i.e., sequences u = {u(n)} such that suppu = {n ∈ Z m : u(n) 0} is a finite set.It is well known that l 0 is a dense subspace of l p with 1 ≤ p < ∞.For any φ ∈ l 0 , we have From (2.4), we derive V n h n φ n .

Note that
This gives a contradiction since it is well known that the operator L − ω − V has no eigenvalue in E, where the operator V is defined as follows: V : E → E, (Vu) n = V n u n .
Thus u ( j) is bounded and so the lemma is proved.Now we are in a position to give the proof of our Theorems.
This contradicts with Lemma 2.5.Therefore, |u ( j) n j | ≥ δ.With the same argument above, after a suitable translation, a subsequence of {u ( j) } converges weakly to some v 0, a nontrivial critical of Φ.Then by (3.7) and Fatou's lemma, we have Hence v is a nontrivial critical point of Φ with Φ(v) = c, and Theorem 1.1 is proved.Moreover, one can follow the same way as in the proof of Theorem 6.1 in [28] to prove that the solution obtained above decays exponentially at infinity, that is, there exist constants C > 0 and γ > 0 such that |u n | ≤ Ce −γ|n| for n ∈ Z m , we omit it here.The proof is complete.
Proof of Theorem 1.2.The proof is similar to that of Theorem 1.1 and is omitted.

Conclusions
This study set out to prove the existence of ground state solitons of the discrete nonlinear Schrödinger equation in m dimensional lattices under weaker conditions on g n .In general, therefore, it seems that our results woulds be applied to more general situations.The present study sharply extends and improves many existing ones in the literature.We hope that this work will bring a new perspective for researchers.