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Ali Mai, Zhan Zhou, "Ground State Solutions for the Periodic Discrete Nonlinear Schrödinger Equations with Superlinear Nonlinearities", Abstract and Applied Analysis, vol. 2013, Article ID 317139, 11 pages, 2013. https://doi.org/10.1155/2013/317139
Ground State Solutions for the Periodic Discrete Nonlinear Schrödinger Equations with Superlinear Nonlinearities
We consider the periodic discrete nonlinear Schrödinger equations with the temporal frequency belonging to a spectral gap. By using the generalized Nehari manifold approach developed by Szulkin and Weth, we prove the existence of ground state solutions of the equations. We obtain infinitely many geometrically distinct solutions of the equations when specially the nonlinearity is odd. The classical Ambrosetti-Rabinowitz superlinear condition is improved.
The following discrete nonlinear Schrödinger equation (DLNS): where and is the discrete Laplacian operator, appears in many physical problems, like polarons, energy transfer in biological materials, nonlinear optics, and so forth (see ). The parameter characterizes the focusing properties of the equation: if , the equation is self-focusing, while corresponds to the defocusing equation. The given sequences and are assumed to be -periodic in , that is, and . Moreover, is a positive sequence. Here, is a positive integer. We assume that and the nonlinearity is gauge invariant, that is, We are interested in the existence of solitons of (1), that is, solutions which are spatially localized time-periodic and decay to zero at infinity. Thus, has the form where is a real-valued sequence and is the temporal frequency. Then, (1) becomes holds. Naturally, if we look for solitons of (1), we just need to get the solutions of (5) satisfying (6).
Actually, we consider a more general equation: with the same boundary condition (6). Here, is a second-order difference operator where and are real-valued -periodic sequences. When and , we obtain (5).
As it is well known, the operator is a bounded and self-adjoint operator in . The spectrum is a union of a finite number of closed intervals, and the complement consists of a finite number of open intervals called spectral gaps. Two of them are semi-infinite (see ). If , then finite gaps do not exist. However, in general, finite gaps exist, and the most interesting case in (7) is when the frequency belongs to a finite spectral gap. Let us fix any spectral gap and denote it by .
DNLS equation is one of the most important inherently discrete models. DNLS equation plays a crucial role in the modeling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology (see [1, 3–6] and references therein). In the past decade, solitons of the periodic DNLS have become a hot topic. The existence of solitons for the periodic DNLS equations with superlinear nonlinearity [7–10] and with saturable nonlinearity [11–13] has been studied, respectively. If is below or above the spectrum of the difference operator , solitons were shown by using the Nehari manifold approach and a discrete version of the concentration compactness principle in . If is a lower edge of a finite spectral gap, the existence of solitons was obtained by using variant generalized weak linking theorem in . If lies in a finite spectral gap, the existence of solitons was proved by using periodic approximations in combination with the linking theorem in  and the generalized Nehari manifold approach in , respectively. The results were extended by Chen and Ma in . In this paper, we employ the generalized Nehari manifold approach instead of periodic approximation technique to obtain the existence of a kind of special solitons of (7), which called ground state solutions, that is, nontrivial solutions with least possible energy in . We should emphasize that the results are obtained under more general super nonlinearity than the classical Ambrosetti-Rabinowitz superlinear condition [8, 9, 15].
This paper is organized as follows. In Section 2, we first establish the variational framework associated with (7) and transfer the problem on the existence of solutions in of (7) into that on the existence of critical points of the corresponding functional. We then present the main results of this paper and compare them with existing ones. Section 3 is devoted to the proofs of the main results.
2. Preliminaries and Main Results
The following are the basic hypotheses to establish the main results of this paper:),() and , and there exist and such that (),(), where is the primitive function of , that is, () is strictly increasing on and .
To state our results, we introduce some notations. Let
Consider the functional defined on by where is the inner product in and is the corresponding norm in . The hypotheses on imply that the functional and (7) is easily recognized as the corresponding Euler-Lagrange equation for . Thus, to find nontrivial solutions of (7), we need only to look for nonzero critical points of in .
For the derivative of , we have the following formula:
By , we have . So, corresponds to the spectral decomposition of with respect to the positive and negative parts of the spectrum, and For any , letting with and with , we can define an equivalent inner product and the corresponding norm on by respectively. So, can be rewritten as
We define for , the subspace and the convex subset of , where, as usual, . Let
In this paper, we also consider the multiplicity of solutions of (7).
For each , let which defines a -action on . By the periodicity of the coefficients, we know that both and are -invariants. Therefore, if is a critical point of , so is . Two critical points of are said to be geometrically distinct if for all .
Now, we are ready to state the main results.
Theorem 1. Suppose that conditions are satisfied. Then, one has the following conclusions.(1)If either and or and , then (7) has at least a nontrivial ground state solution.(2)If either and or and , then (7) has no nontrivial solution.
Theorem 2. Suppose that conditions are satisfied and is odd in . If either and or and , then (7) has infinitely many pairs of geometrically distinct solutions.
In what follows, we always assume that . The other case can be reduced to by switching to and to .
Remark 3. In , the author considered (7) with defined by which obviously satisfies ; the author also discussed the case where satisfies the Ambrosetti-Rabinowitz condition; that is, there exists such that Clearly, (23) implies that for . So, it is a stronger condition than .
Remark 4. In , the author assumed that satisfies the following condition: there exists such that Obviously, (24) implies (23) with , so it is a stronger condition than the Ambrosetti-Rabinowitz condition. In our paper, the nonlinearities satisfy more general superlinear assumptions instead of (24) which also implies . However, we do not assume that is differentiable and satisfies (24), is not a manifold of , and the minimizers on may not be critical points of . Hence, the method of  does not apply any more. Nevertheless, is still a topological manifold, naturally homeomorphic to the unit sphere in (see in detail in Section 3). We use the generalized Nehari manifold approach developed by Szulkin and Weth which is based on reducing the strongly indefinite variational problem to a definite one and prove that the minimizers of on are indeed critical points of .
Remark 5. In , it is shown that (7) has at least a nontrivial solution if satisfies , , , and the following conditions: for any and if , as , and there exist and such that if , where is a positive constant,
In our paper, we use (9) and instead of and .
3. Proofs of Main Results
We assume that and are satisfied from now on.
Lemma 6. and for all .
Proof. By and , it is easy to get that Set . It follows from that So, for all .
To continue the discussion, we need the following proposition.
Lemma 8. If , then Hence, is the unique global maximum of .
Proof. We rewrite by
Since , we have
Together with Proposition 7, we know that
The proof is complete.
(a) There exists such that , where .
(b) for every .
Proof. (a) By and , it is easy to show that for any , there exists such that
is equivalent to the norm on and for with . Hence, for any and , we have
which implies for some (small enough), where .
The first inequality is a consequence of Lemma 8 since for every , there is such that .
(b) For , by (25), we have Hence, .
Lemma 10. Let be a compact subset. Then, there exists such that on for every , where denotes the open ball with radius and center .
Proof. Suppose by contradiction that there exist and , , such that for all and as . Without loss of generality, we may assume that for . Then, there exists a subsequence, still denoted by the same notation, such that . Set . Then, By (25), we have Consequently, we know that and . Passing to a subsequence if necessary, we assume that , , , and for every . Hence, and . It follows that for with , , as . Then, by , we have which contradicts with (35).
Lemma 11. For each , the set consists of precisely one point which is the unique global maximum of .
Proof. By Lemma 8, it suffices to show that . Since , we may assume that . By Lemma 10, there exists such that on provided that is large enough. By Lemma 9 (a), for small . Moreover, on . Hence, .
Let in . Then, as for all after passing to a subsequence if necessary. Hence, . Let . Then, that is, is a weakly lower semicontinuous. From the weak lower semi-continuity of the norm, it is easy to see that is weakly upper semicontinuous on . Therefore, for some . By the proof of Lemma 10, is a critical point of . It follows that for all and hence . To summarize, .
According to Lemma 11, for each , we may define the mapping , , where is the unique point of .
Lemma 12. is coercive on ; that is, as , .
Proof. Suppose, by contradiction, that there exists a sequence such that and for some . Let . Then, there exists a subsequence, still denoted by the same notation, such that and for every as .
First, we know that there exist and such that Indeed, if not, then in as . By Lemma 9(b), , which means that is bounded. For , Then, in all . By (32), for any , which implies that as .
Since for , Lemma 8 implies that as . This is a contradiction if .
Due to the periodicity of coefficients, both and are invariant under -translation. Making such shifts, we can assume that in (39). Moreover, passing to a subsequence if needed, we can assume that is independent of . Next, we may extract a subsequence, still denoted by , such that for all . In particular, for , inequality (39) shows that and hence .
Since as , it follows again from and Fatou’s lemma that a contradiction again. The proof is finished.
(a) The mapping is continuous.
(b) The mapping is a homeomorphism between and , and the inverse of is given by , where .
(c) The mapping is the Lipschitz continuous.
Proof. (a) Let be a sequence with . Since , without loss of generality, we may assume that for all . Then, . By Lemma 10, there exists such that
It follows from Lemma 12 that is bounded. Passing to a subsequence if needed, we may assume that
where by Lemma 9(b). Moreover, by Lemma 11,
Therefore, using the weak lower semicontinuity of the norm and (defined in Lemma 11), we get
which implies that all inequalities above must be equalities and . By Lemma 11, and hence .
(b) This is an immediate consequence of (a).
(c) For , by , we have
We will consider the functional and defined by
(c) is a Palais-Smale sequence for if and only if is a Palais-Smale sequence for .
(d) is a critical point of if and only if is a nontrivial critical point of . Moreover, the corresponding values of and coincide and .
(a) We put , so we have . Let . Choose such that for and put . We may write with . From the proof of Lemma 13, the function is continuous. Then, . By Lemma 11 and the mean value theorem, we have
with some . Similarly,
with some . Combining these inequalities and the continuity of function , we have
Hence, the Gâteaux derivative of is bounded linear in and continuous in . It follows that is of class (see ).
(b) It follows from (a) by noting that since .
(c) Let be a Palais-Smale sequence for , and let . Since for every , we have an orthogonal splitting ; using , we have because for all and is orthogonal to . Using again, we have Therefore, According to Lemma 9(b) and Lemma 12, . Hence, is a Palais-Smale sequence for if and only if is a Palais-Smale sequence for .
(d) By (57), if and only if . The other part is clear.
Proof of Theorem 1. (1) We know that by Lemma 9(a). If satisfies , then is a minimizer of and therefore a critical point of and also a critical point of by Lemma 14. We shall show that there exists a minimizer of . Let be a minimizing sequence for . By Ekeland’s variational principle, we may assume that and as . Then, and as by Lemma 14(c), where . By Lemma 12, is bounded, and hence has a weakly convergent subsequence.
First, we show that there exist and such that Indeed, if not, then in as . From the simple fact that for , we have in all . By (32), we know that which implies that as . Therefore, Then, as , contrary to Lemma 9(b).
From the periodicity of the coefficients, we know that and are both invariant under -translation. Making such shifts, we can assume that in (58). Moreover, passing to a subsequence, we can assume that is independent of .
Next, we may extract a subsequence, still denoted by , such that and for all . Particularly, for , inequality (58) shows that , so . Moreover, we have that is, is a nontrivial critical point of .
Finally, we show that . By Lemma 6 and Fatou’s lemma, we have Hence, . That is, is a nontrivial ground state solution of (7).
(2) If , by way of contradiction, we assume that (7) has a nontrivial solution . Then, is a nonzero critical point of in . Thus, . But by Lemma 6, This is a contradiction, so the conclusion holds.
This completes the proof of Theorem 1.
Now, we are ready to prove Theorem 2. From now on, we always assume that is odd in . We need some notations. For , denote
It is easy to see that for every by Lemma 12.
Proof of Theorem 2. It is easy to see that mappings are equivariant with respect to the -action by Lemma 13; hence, the orbits consisting of critical points of are in - correspondence with the orbits consisting of critical points of by Lemma 14(d). Next, we may choose a subset such that and consists of a unique representative of -orbits. So, we only need to prove that the set is infinite. By contradiction, we assume that
where denotes genus and . We consider the sequence of the Lusternik-Schnirelmann values of defined by
Now, we claim that
Firstly, we show that In fact, there exist , and such that for all and Let . Passing to a subsequence, , , and either for all or . In the first case, for all . In the second case, and therefore . By (70), or .
Next, we consider a pseudogradient vector field of ; that is, there exists a Lipschitz continuous map : and for all , Let be the corresponding -decreasing flow defined by where , and , are the maximal existence times of the trajectory in negative and positive direction. By the continuity property of the genus, there exists such that , where and . Following the deformation argument (Lemma A.3), we choose such that Then, for every , there exists such that . Hence, we may define the entrance time map which satisfies for every . Since is not a critical value of by (74), it is easy to see that is a continuous and even map. It follows that the map is odd and continuous. Then, , and consequently, So, . Therefore, . Moreover, the definition of and of implies that if and if . Since , . Therefore, there is an infinite sequence of pairs of geometrically distinct critical points of with , which contradicts with (66). Therefore, the set is infinite.
This completes the proof of Theorem 2.
Here, we give a proof of (74). We state the discrete property of the Palais-Smale sequences. It yields nice properties of the corresponding pseudogradient flow.
Lemma A.1. Let . If , are two Palais-Smale sequences for , then either as or , where depends on but not on the particular choice of the Palais-Smale sequences.
Proof. Set and . Then, are the bounded Palais-Smale sequences for . We fix in and consider the following two cases.
(i) as .
By a straightforward calculation and (32), for any , there exist , and such that for all ,