In this paper we employ a variational approach to study the existence of homoclinics for second order almost periodic Hamiltonian systems of the form
assuming that the linear part of (1.1) is indefinite.
In the special case of periodic system (1.1) with positive , the first variational proof of the existence of homoclinics has been done by Rabinowitz . Soon after, under similar assumptions, Coti Zelati and Rabinowitz  have obtained a remarkable result on the existence of multi-bump homoclinic solutions. The case of periodic strongly indefinite systems has been considered by Arioli and Szulkin , and Kryszewsky and Szulkin . In  a homoclinic solution is obtained as the limit of subharmonics which, in turn, are shown to exist by means of the linking theorem. On the other hand, in  the existence of homoclinics is proven by means of the innovative generalized linking theorem obtained in that paper. Currently, the generalized linking theorem has received further development and numerous applications to other problems with built-in periodicity (see, e.g., ).
Typically, the generalized linking theorem as well as other critical point theorems without compactness produce a sequence of approximate critical points, say a Palais–Smale sequence at a certain level. Often this sequence is bounded, and one can employ concentration compactness arguments . As consequence, by translating the members of the sequence, we obtain a Palais–Smale sequence, with some compactness properties, for a limiting problem at infinity. In the periodic case, such a limiting problem is a translation of the original one, and we obtain a solution. However, in general, the collection of limiting problems is much bigger than the set of translated problem. Therefore, this approach breaks down, and one needs more control of the sequence of approximate critical points obtained at the very beginning. This is exactly what happens in the case of almost periodic problems.
The first application of variational methods to almost periodic problems has been done by Sera, Tarallo and Terracini . In that paper a result on the existence of homoclinics is obtained in the case of system (1.1) with (this assumption can be weakened) and an almost periodic nonlinearity of Ambrosetti–Rabinowitz type. First it was shown that at any level close to the mountain pass level of the energy functional, there exist a Palais–Smale sequence with the special property that . This result relies on the fact that the mountain pass minimax class is invariant under the negative gradient flow of the energy functional. Then the special property and returning sequences for almost periodic functions were used to extract a subsequence of the Palais–Smale sequence, constructed above, to extract a subsequence which can be converted, with the help of translations, into a Palais–Smale sequence for the original energy functional and such that it has a nonzero weak limit. Both these steps heavily depend on the positive definiteness of the linear part of the problem and are not transferable to strongly indefinite problems. Some ideas from  are used in [15, 16] to study homoclinics in singular almost periodic Hamiltonian systems. In [1, 2, 5, 22, 23] the approach of  has been extended in certain directions but still in the case when the energy functional possesses the mountain pass geometry.
To the best of our knowledge, the first variational result on homoclinic solutions for strongly indefinite almost periodic systems is obtained in . Motivated by certain problems on photonic crystals, in that paper we consider the nonlinear Schrödinger equation on the real line, i.e., a system of the form (1.1) with one-dimensional configuration space. As in , we use Palais–Smale sequences with the special property described above. To construct such sequences we employ the negative gradient flow of the restriction of the energy functional to the so-called generalized Nehari manifold introduced in . In turn, this requires a careful analysis of that manifold.
In the present paper we extend the result of  to Hamiltonian systems with the configuration space of arbitrary dimension. One of the key points is to find natural assumptions that guarantee nice properties of the generalized Nehari manifold. In  one of the main assumptions is inequality (3.5), which is used also in  and goes back to . In the vectorial case this assumption is no longer working, and its substitute, assumption (iv), is not obvious. It turns out that once nice properties of the generalized Nehari manifold are obtained, a substantial part of  concerning Palais–Smale sequences with the special property can be transferred to our case almost automatically. Notice that, as in , we use Stepanov almost periodicity, which is more general than Bohr almost periodicity. In general, Stepanov almost periodic functions are discontinuous. This is important because some applications lead to the so-called multi-component nonlinear Schrödinger equation, which is of the form (1.1) with higher dimensional configuration space. In this case, t stands for the spatial variable. Often in such applications the t-dependence in the equation is only piece-wise continuous.
The organization of the paper is as follows. Section 2 is devoted to preliminary results on the linear part of the problem. Some of the results of this section are well known in the literature but we provide them with their proofs. Section 3 contains the statement of the problem, its variational formulation, and the main result. In Section 4 we study the generalized Nehari manifold and Palais–Smale sequences. The proof of the main result is contained in Section 5. Finally, in Section 6 we discuss a minor extension of the result of .
2 Preliminary results
First, let us introduce some basic spaces of functions on with values in . In the notation of these spaces we drop the domain and target space because this does not lead to a confusion. We denote by and the standard Euclidean norm and the inner product in , respectively.
By we denote the space of square integrable functions. Endowed with the standard norm , is a Hilbert space. The inner product in is denoted by . The Sobolev space with the graph norm is a Hilbert space as well. We denote by the space of all essentially bounded functions equipped with the standard norm .
The dual space to , with the norm , is denoted by . The symbol stands for both the inner product
in and for the duality pairing on . This does not lead to any confusion. It is well known that
continuously and densely. Moreover, is continuously embedded into . Actually, any -function is continuous and vanishes at infinity.
A locally integrable function u is Stepanov bounded if
Such functions form a Banach space denoted by . A Stepanov almost periodic function is a function such that the set of shifts , where , is precompact in the space , i.e., for any sequence , there exists a subsequence such that the sequence converges in the space . The space S of all Stepanov almost periodic functions is a closed subspace of . For a function , the closure of in the space S is denoted by and is called the envelop of u. This is a compact subset of S. It is well known (see, e.g., [9, 11]) that if in the space S, then in S.
Now let be a Carathéodory function, i.e., is continuous in for almost all and Lebesgue measurable in for all . In the following, all functions of are supposed to be Carathéodory functions. For any , we set
A function is strictly Stepanov almost periodic in t  if for all , and for any sequence , there exists a subsequence and a function such that and
for all . Actually, this means that, being considered as a function of with values in the Frechét space of all continuous maps , g is a Stepanov almost periodic function. All functions of the form constitute the envelop of g. Any strictly Stepanov almost periodic function is Stepanov almost periodic in t uniformly with respect to u in compact subsets of but not vise versa.
Now we discuss certain results from the spectral theory of operators of the form
where the potential A is a real symmetric -matrix. Under natural assumptions on the (matrix) potential, is a bounded below self-adjoint operator in , defined as the sum of quadratic forms
The form is generated by the operator and, hence, is a non-negative symmetric closed form on with the form domain . Then the operator extends to a bounded linear operator from into still denoted by (see, e.g., [19, Section VIII.6]).
We start with the following simple lemma.
For every , there exists a constant such that
whenever and .
Any function is continuous, and for every , there exists a constant such that
for all . Since
we obtain the required result. ∎
Suppose that . Then the following hold:
The operator of multiplication by A is -form bounded with form bound 0 . In particular,
Defined in terms of the sum of quadratic forms , the operator is a self-adjoint bounded below operator in with form domain .
The operator depends continuously on with respect to the norm in the space of bounded linear operators from into and in the sense of norm resolvent convergence.
(a) Lemma 2.1 implies that
and the result follows.
(b) This follow from part (a) and the KLMN theorem, [17, Theorem X.17].
(c) Let and let be its quadratic form. Suppose that in . Then
for , where is the quadratic form generated by . Hence,
and the result follows from [19, Theorem VIII.25 (c)]. ∎
It follows from the proof of Proposition 2.2 that if , then the operator of multiplication by A is a bounded linear operator from into and its norm is less than or equal to . As a consequence, the operator extends to a bounded linear operator from into . This operator is still denoted by .
Suppose that and is an eigenfunction of with an isolated eigenvalue λ of finite multiplicity. Then u decays at infinity exponentially fast, i.e., there exist constants and such that
By inspection, the Combes–Thomas arguments (see, e.g., [21, Section C.3]) apply straightforwardly. Therefore, there exists such that for all .
To obtain a point-wise bound, without loss of generality, we may assume that is positive definite and, hence, . Fix any smooth function such that , and let and . A straightforward calculation shows that satisfies the equation
where the operator is defined by
Since is positive definite, there exists a bounded inverse operator . It is easily seen that , as an operator from into , is a small perturbation of , provided is small enough. Hence, has a bounded inverse , whenever is sufficiently small. Since , we obtain that . This implies the exponential decay of u.
From the above equation for , it follows that is the sum of and a function which automatically belongs to . Since is bounded, we obtain that . Now the standard Sobolev embedding theorem for intervals of unit length (see, e.g., [4, Section 8.2]) shows that , which implies that decays exponentially fast. The proof is complete. ∎
In what follows we denote by the spectrum of an operator . Assuming that , we denote by and the orthogonal projectors in associated to the positive and negative parts of , respectively. Since and commute with and is the form domain, it is easily seen that . Furthermore, the spectral theorem for self-adjoint operators implies that and are bounded linear operators in . Since these operators are projectors and , we conclude that and are closed subspaces in and . Thus, every possesses the representation , where and . In particular, the standard norm in is equivalent to the norm defined by . Notice that the quadratic form is positive on and negative on .
Let . If , then there exists a constant that depends on and the distance between 0 and , and such that
To prove inequality (2.2), let us denote by the distance between zero and . Then
By inequality (2.1) with sufficiently small ε, there exist constants and that depend on and such that
Hence, for all ,
with . Expressing the right-hand side of this inequality in the form
and taking into account that , we obtain that
and the result follows.
The proof of (2.3) is similar. ∎
Under the assumptions of Proposition 2.5, we introduce a special inner product on , defined by
where is considered as an operator from into . By Proposition 2.5, is equivalent to the standard inner product in . An advantage of this inner product is that the operators and are orthogonal projectors with respect to . Hence, the subspaces and are orthogonal with respect to this inner product.
Now suppose that . The envelop of is the set of all operators of the form , where .
Suppose that . Then for all .
If , then there exists a sequence such that in . It is easily seen that . By Proposition 2.2 (c) and [19, Theorem VIII.23], . But in . Hence, interchanging the role of A and , we obtain the required result. ∎
3 Statement of the problem and variational formulation
We are looking for homoclinic solutions to second order Hamiltonian systems of the form
i.e., nonzero solutions that vanish at infinity. Here . We make the following assumptions:
The potential is a real symmetric -matrix, the spectrum of the operator does not contain zero and the negative part of the spectrum is non-empty.
For almost all , the function is twice continuously differentiable with respect to . The functions , and are strictly Stepanov almost periodic. For any , with , the function is bounded below by a positive constant that depends on r only.
The nonlinear potential F satisfies , and . Furthermore, for every , there exists a constant such that
The matrix is positive definite for all and almost all . There exist constants and such that for almost all ,
There exist and such that
Without loss of generality, we suppose that whenever .
Assumption (i) guarantees that the self-adjoint operator is well defined (see Section 2). By the mean value theorem, assumption (iii) implies that for almost all ,
Inequality (3.3) is the standard Ambrosetti–Rabinowitz condition. In particular, it implies that for any , there exists a constant such that
In assumption (ii) it is sufficient to assume only that is strictly Stepanov almost periodic. Then the strict almost periodicity of F and follows.
If , then assumption (v) follows immediately from (3.4). In general, under assumptions (i)–(v), the inequalities in (v) hold for all and , with and depending on and , respectively. Therefore, we may assume that .
The nonlinear potential , where is a scalar function, satisfies assumptions (ii)–(iv), provided that , , and .
Under our assumptions, the set of shifts is precompact in the sense that for any sequence in , there exists a subsequence such that in , and , together with the first and second derivatives, with respect to the seminorms for all . The set of all such limits is denoted by and is a compact set in the topology just described. It is convenient to consider as a set parametrized by some index set . The envelope of equation (3.1) consists of all equations of the form
Any equation in the envelop satisfies assumptions (i)–(iv), with the same function and constants q, θ, and .
Our main result is the following.
Under assumptions (i)–(v), equation (3.1) has a nonzero solution . Moreover, the solution u decays at infinity exponentially fast, i.e., there exist positive constants α and β such that
The solution in Theorem 3.1 is a weak solution, i.e.,
for all smooth compactly supported functions φ.
Associated to equation (3.1), we introduce the functional
Similarly, we introduce the functional , , associated to equation (3.6). Its non-quadratic part is denoted by . The functionals form the envelop of J.
Under the assumptions imposed above, the functional J is a well-defined -functional on the space . Its first and second derivatives are given by
In particular, weak solutions of equation (3.1) are critical points of the functional J and vise versa.
The following statements are straightforward extensions of [13, Propositions 4.1 and 4.2].
For any , and any , we have that
provided that with .
If weakly in , then, uniformly with respect to ,
strongly in .
Often it is convenient to use gradients of J instead of derivatives. These are defined by
for all , where is the inner product introduced in (2.4). Then , while is a linear bounded operator in .
The proof of Theorem 3.1 is given in the subsequent sections. Obviously, is a trivial critical point of the functional J. We shall prove that J possesses a nontrivial critical point. In the course of the proof we consider equation (3.1) with . The other case is similar. We only need to replace the functional J by and interchange the role of the subspaces and introduced in Section 2.
In the subsequent sections we always suppose that assumptions (i)–(v) hold.
4 Generalized Nehari manifold and Palais–Smale sequences
The Generalized Nehari manifold of the functional J consists of all nonzero such that
Equivalently, these equations can be written as and , respectively.
For any , we set
By the definition of , if u is a critical point of , then . As consequence, contains all nontrivial critical points of J.
The proof of the following lemma is essentially the same as the proof of [13, Lemma 5.1].
For every , the functional attains its positive global maximum. As a consequence, for all .
It is convenient to introduce the functional
Obviously, for all . Now we prove the following technical result.
There exists a constant independent of such that
for all .
and Proposition 2.5 imply
Similarly, the identity
Now let . Then, by inequality (3.4),
which proves (4.1).
Now we prove inequality (4.2). Given , let
We introduce the following integrals:
By inequality (3.3) and assumption (v),
where ν depends on q, and . Recall that we may take in assumption (v). Since
As an immediate consequence of Lemma 4.2, we obtain the following corollary.
There exists a constant such that , and
for all .
Let . Elements of this space are denoted by , where and . We define the operator by
It is not difficult to see that the operator G is a map, and its derivative is given by the formula
for all .
Then, for all and ,
where and are the constants from Proposition 2.2 and assumption (iv), respectively.
Since, by the assumptions,
a straightforward but a little bit tedious calculation yields the identity
and, by Proposition 2.2,
it is enough to prove that
This is trivial for all such that . Hence, we may assume that . Then the matrix is positive definite. Making use of the trivial identities
we obtain that
Making use of the following easily verified inequality:
and inequality (3.2), we obtain the required result. ∎
The key Lemmas 4.1, 4.2 and 4.5 permit us to transfer some of auxiliary results of  to the present setting just by inspection of the proofs given there. In particular, the set is a non-empty closed -submanifold of . The tangent space to at a point is , while is a complementary subspace to . Furthermore, given , there exist and such that for every , with , there exists a -diffeomorphism from the ρ-neighborhood of 0 in onto a neighborhood of in such that the Lipschitz constant of its derivative is not greater than C.
Now we introduce the following quantities:
We have for all .
See the proof of [13, Proposition 5.4]. ∎
Let us turn to Palais–Smale sequences. Recall that a sequence is a Palais–Smale sequence for the functional J at a level c if and in (equivalently, in ). A Palais–Smale sequence for at a level c is a sequence such that and in , where stands for the tangent component of the gradient.
As an immediate consequence of Lemma 4.2, we obtain the following corollary.
Let be a Palais–Smale sequence for the functional J at a level c. Then and is bounded in . Furthermore, in if and only if .
As in [13, Propositions 6.2 and 6.3], we obtain the following result.
Every Palais–Smale sequence for is a Palais–Smale sequence for J. Conversely, if is a Palais–Smale sequence for J at a level , then there exists a Palais–Smale sequence for such that and, hence, .
If is a Palais–Smale sequence for J at a positive level, then , where is the constant from Corollary 4.3.
Inspecting the proof of [13, Proposition 6.4], we obtain the following statement on the existence of Palais–Smale sequences. Notice that the main ingredient of the proof is the negative gradient flow of the functional .
For any , there exists a Palais–Smale sequence for , hence for J, at a level such that
Also we need the following special version of concentration-compactness lemma (cf. [13, Lemma 6.1]).
Let be a Palais–Smale sequence for J at a level . Suppose that weakly in .
If , then strongly in , and is a critical point of the functional J with critical value c.
If , then there exists a sequence such that and a nontrivial critical point of for some at the level c with the property that along a subsequence, and strongly in .
5 Proof of main result
We begin with the exponential decay of solutions.
Given a solution of (3.1), let
This is a symmetric matrix and, by assumption (iii), . Equation (3.1) implies that
i.e., u is an eigenfunction of the operator , with the eigenvalue 0.
Since , we have that as and, hence, as in the sense that as . Together with the embedding and by the local compactness of the embedding , this implies that the multiplication operator by is relatively compact with respect to and, hence, the essential spectra of and coincide (see, e.g., ). Therefore, 0 is an isolated eigenvalue of with finite multiplicity. Due to Proposition 2.4, this completes the proof. ∎
The existence part of Theorem 3.1 is a consequence of the following proposition.
For every , the functional J possesses a critical value in the interval .
Obviously, we may assume that . Let be the Palais–Smale sequence obtained in Proposition 4.10. If there is a weak limit point of the sequence , then, by Lemma 4.11 (a), is a critical point of J with critical value .
Thus, assume that weakly in . In this case, we use to construct another Palais–Smale sequence that produces the required critical value. For any , we set
Obviously, is a non-negative, continuous and non-increasing function, and as . Since, by construction, , Corollary 4.3 implies that there exists (not necessarily unique) such that , where is the constant from Corollary 4.3.
The key property of the sequence is the following:
To prove this statement consider any subsequence . By Lemma 4.11 (b), there exist a subsequence , a sequence , , and an nontrivial critical point of the functional for some such that strongly in . As a consequence, strongly in . This implies straightforwardly that
in . By the uniqueness of the solution to the initial value problem for equation (3.6), and, hence, the function is strictly decreasing. Therefore, there exists a unique such that . Now it is not difficult to see that
which implies immediately (5.1).
Let . We prove that 0 is not a weak limit point of . Assume the contrary, i.e., along a subsequence, weakly in , and is a Palais–Smale sequence for some functional in the envelop. Lemma 4.4 (b) implies, along a further subsequence, that there exists a sequence , yet another functional () and its nontrivial critical point such that , and
strongly in . This implies that
On the other hand, since , we obtain, by passing to a further subsequence, that either
This is a contradiction.
Since , we obtain that there exists a subsequence such that . Along a subsequence, converges weakly in to a nonzero function v. By Proposition 3.3,
Since is weakly continuous and is the Palais–Smale sequence for the functional J, this implies that for any ,
Thus, v is a critical point of J at a level in . The case when is unbounded below is similar.
The proof is complete. ∎
6 Positive definite linear part
Let us consider the case when the linear part of equation (3.1) is positive definite.
Let . Assume that the following hold:
and the operator is positive definite.
For almost all , the function is with respect to . The functions , and are strictly Stepanov almost periodic. For any , with , the function of t is bounded below by a positive constant that depends on u only.
, and .
The function F satisfies ( 3.3 ) with .
In this case the functional J possesses the mountain pass geometry. The arguments of  can be easily adopted to our more general setting and show that Proposition 4.6 holds for the mountain pass level m. Then one can carry out the proof by making use of either the arguments of [20, Section 3], or our simpler approach given in Section 5.
Now consider the case . Assume only that , is positive definite, and for all and almost all . Then, by multiplying (3.1) by u and integrating, we conclude that the equation has no nontrivial solution in .
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About the article
Published Online: 2017-04-19
Funding Source: Ministry of Education and Science of the Russian Federation
Award identifier / Grant number: 02.a03.210008
The publication is partially supported by the Ministry of Education and Science of the Russian Federation, the agreement no. 02.a03.210008.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 372–385, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0041.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0