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Advances in Nonlinear Analysis

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Homoclinics for strongly indefinite almost periodic second order Hamiltonian systems

Alexander Pankov
  • Corresponding author
  • Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA; and RUDN University, 6 Miklukho-Maklay St., Moscow 117198, Russia
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Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/anona-2017-0041

Abstract

Under certain assumptions, we prove the existence of homoclinic solutions for almost periodic second order Hamiltonian systems in the strongly indefinite case. The proof relies on a careful analysis of the energy functional restricted to the generalized Nehari manifold, and the existence and fine properties of special Palais–Smale sequences.

Keywords: Hamiltonian system; homoclinic solution; generalized Nehari manifold; Palais–Smale sequence

MSC 2010: 34C37; 58E05

1 Introduction

In this paper we employ a variational approach to study the existence of homoclinics for second order almost periodic Hamiltonian systems of the form

-u¨(t)+A(t)u(t)=Fu(t,u(t)),(1.1)

assuming that the linear part of (1.1) is indefinite.

In the special case of periodic system (1.1) with positive A(t), the first variational proof of the existence of homoclinics has been done by Rabinowitz [14]. Soon after, under similar assumptions, Coti Zelati and Rabinowitz [6] 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 [3], and Kryszewsky and Szulkin [8]. In [3] 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 [8] 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., [7]).

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 [10]. 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 [20]. In that paper a result on the existence of homoclinics is obtained in the case of system (1.1) with A(t)=I (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 un-un-10. 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 [20] are used in [15, 16] to study homoclinics in singular almost periodic Hamiltonian systems. In [1, 2, 5, 22, 23] the approach of [20] 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 [13]. 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 [20], 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 [12]. In turn, this requires a careful analysis of that manifold.

In the present paper we extend the result of [13] 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 [13] one of the main assumptions is inequality (3.5), which is used also in [12] and goes back to [10]. 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 [13] concerning Palais–Smale sequences with the special property can be transferred to our case almost automatically. Notice that, as in [13], 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 [20].

2 Preliminary results

First, let us introduce some basic spaces of functions on with values in d. 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 d, respectively.

By L2 we denote the space of square integrable functions. Endowed with the standard norm 2, L2 is a Hilbert space. The inner product in L2 is denoted by (,). The Sobolev space H1={uL2u˙L2} with the graph norm is a Hilbert space as well. We denote by L the space of all essentially bounded functions equipped with the standard norm .

The dual space to H1, with the norm *, is denoted by H-1. The symbol (,) stands for both the inner product

(u,v)=u(t)v(t)𝑑t

in L2 and for the duality pairing on H-1×H1. This does not lead to any confusion. It is well known that

H1L2H-1

continuously and densely. Moreover, H1 is continuously embedded into L. Actually, any H1-function is continuous and vanishes at infinity.

A locally integrable function u is Stepanov bounded if

uBS=supttt+1|u(τ)|dτ<.

Such functions form a Banach space denoted by BS. A Stepanov almost periodic function is a function uBS such that the set of shifts {Tτu}τ, where Tτu=u(+τ), is precompact in the space BS, i.e., for any sequence τk, there exists a subsequence τk such that the sequence Tτku converges in the space BS. The space S of all Stepanov almost periodic functions is a closed subspace of BS. For a function uS, the closure of {Tτu}τ in the space S is denoted by (u) 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 uh=limTτku(u) in the space S, then u=limT-τkuh in S.

Now let g:×dm be a Carathéodory function, i.e., g(t,u) is continuous in un for almost all t and Lebesgue measurable in t for all un. In the following, all functions of (t,u)×d are supposed to be Carathéodory functions. For any R>0, we set

gR=sup|u|R|g(,u)|BS.

A function g(t,u) is strictly Stepanov almost periodic in t [13] if gR< for all R>0, and for any sequence τk, there exists a subsequence τk and a function gh such that ghR< and

Tτkg-ghR0

for all R>0. Actually, this means that, being considered as a function of t with values in the Frechét space of all continuous maps nm, g is a Stepanov almost periodic function. All functions of the form gh constitute the envelop (g) of g. Any strictly Stepanov almost periodic function is Stepanov almost periodic in t uniformly with respect to u in compact subsets of n but not vise versa.

Now we discuss certain results from the spectral theory of operators of the form

LA=L0+A(t)=-d2dt2+A(t),

where the potential A is a real symmetric d×d-matrix. Under natural assumptions on the (matrix) potential, LA is a bounded below self-adjoint operator in L2, defined as the sum of quadratic forms

l0(u)=|u˙(t)|2dtanda(u)=(A(t)u(t))u(t)dt.

The form l0 is generated by the operator L0 and, hence, is a non-negative symmetric closed form on L2 with the form domain Q(l0)=H1. Then the operator LA extends to a bounded linear operator from H1 into H-1 still denoted by LA (see, e.g., [19, Section VIII.6]).

We start with the following simple lemma.

Lemma 2.1.

For every ε>0, there exists a constant Cε>0 such that

|φ(t)||ψ(t)|2dtφBS(εψ˙L22+CεψL22),

whenever φBS and ψH1.

Proof.

Any function ψH1 is continuous, and for every ε>0, there exists a constant Cε>0 such that

|ψ(τ)|2εnn+1|ψ˙(t)|2dt+Cεnn+1|ψ(t)|2dτ,τ[n,n+1]

for all n. Since

|φ(τ)||ψ(τ)|2dτ=n=-nn+1|φ(τ)||ψ(τ)|2dτ,

we obtain the required result. ∎

Proposition 2.2.

Suppose that ABS. Then the following hold:

  • (a)

    The operator of multiplication by A is L0 -form bounded with form bound 0 . In particular,

    Au*constu.

  • (b)

    Defined in terms of the sum of quadratic forms lA=l0+a , the operator LA is a self-adjoint bounded below operator in L2 with form domain Q(LA)=H1.

  • (c)

    The operator LA depends continuously on ABS with respect to the norm in the space of bounded linear operators from H1 into H-1 and in the sense of norm resolvent convergence.

Proof.

(a)  Lemma 2.1 implies that

|a(u)|ABS(εu˙L22+CεuL22),(2.1)

and the result follows.

(b)  This follow from part (a) and the KLMN theorem, [17, Theorem X.17].

(c)  Let LAn=L0+An and let ln be its quadratic form. Suppose that AnA in BS. Then

l(u)-ln(u)=a(u)-an(u)

for uH1, where an is the quadratic form generated by An. Hence,

supuH1,u0|l(u)-ln(u)|u2=supuH1,u0|a(u)-an(u)|u2A-AnBS0,

and the result follows from [19, Theorem VIII.25 (c)]. ∎

Remark 2.3.

It follows from the proof of Proposition 2.2 that if ABS, then the operator of multiplication by A is a bounded linear operator from H1 into H-1 and its norm is less than or equal to ABS. As a consequence, the operator LA extends to a bounded linear operator from H1 into H-1. This operator is still denoted by LA.

Proposition 2.4.

Suppose that ABS and uL2 is an eigenfunction of LA with an isolated eigenvalue λ of finite multiplicity. Then u decays at infinity exponentially fast, i.e., there exist constants α>0 and β>0 such that

|u(t)|+|u˙(t)|αexp(-β|t|),t.

Proof.

By inspection, the Combes–Thomas arguments (see, e.g., [21, Section C.3]) apply straightforwardly. Therefore, there exists β0>0 such that exp(β|t|)u(t)L2 for all β(0,β0).

To obtain a point-wise bound, without loss of generality, we may assume that LA is positive definite and, hence, λ>0. Fix any smooth function φ(t)0 such that φ(t)=|t|, and let χβ(t)=exp(βφ(t)) and uβ(t)=χβ(t)u(t). A straightforward calculation shows that uβL2 satisfies the equation

LA,βuβ=λuβ,

where the operator LA,β is defined by

LA,β=LA-2βφ˙(t)ddt-(βφ¨(t)+β2φ˙2(t)).

Since LA is positive definite, there exists a bounded inverse operator LA-1:H-1H1. It is easily seen that LA,β, as an operator from H1 into H-1, is a small perturbation of LA, provided β>0 is small enough. Hence, LA,β has a bounded inverse LA,β-1:H-1H1, whenever β>0 is sufficiently small. Since uβL2, we obtain that uβH1L. This implies the exponential decay of u.

From the above equation for uβ, it follows that u¨β is the sum of A(t)uβ and a function hL2 which automatically belongs to BS. Since uβ is bounded, we obtain that u¨βBS. Now the standard Sobolev embedding theorem for intervals of unit length (see, e.g., [4, Section 8.2]) shows that u˙βL, which implies that u˙ decays exponentially fast. The proof is complete. ∎

In what follows we denote by σ(LA) the spectrum of an operator LA. Assuming that 0σ(LA), we denote by P+ and P- the orthogonal projectors in L2 associated to the positive and negative parts of σ(LA), respectively. Since P+ and P- commute with LA and H1 is the form domain, it is easily seen that E±=P±H1H1. Furthermore, the spectral theorem for self-adjoint operators implies that P+ and P- are bounded linear operators in H1. Since these operators are projectors and P++P-=I, we conclude that E+ and E- are closed subspaces in H1 and H1=E+E-. Thus, every uH1 possesses the representation u=u++u-, where u+=P+u and u-=P-u. In particular, the standard norm in H1 is equivalent to the norm defined by (u+2+u-2)1/2. Notice that the quadratic form lA is positive on E+ and negative on E-.

Proposition 2.5.

Let ABS. If 0σ(LA), then there exists a constant κ>0 that depends on ABS and the distance between 0 and σ(LA), and such that

lA(u)κu2,uE+,(2.2)

and

lA(u)-κu2,uE-.(2.3)

Proof.

To prove inequality (2.2), let us denote by 2δ the distance between zero and σ(LA). Then

lA(u)2δuL22,uE+.

By inequality (2.1) with sufficiently small ε, there exist constants α(0,1) and β>0 that depend on ABS and such that

|a(u)|αl0(u)+βuL22=αu˙L22+βuL22,uH1.

Hence, for all uE+,

(1-α)u2lA(u)+CuL22,

with C=1+β-α. Expressing the right-hand side of this inequality in the form

C+δδ[δC+δ(lA(u)-δuL22)+δuL22]

and taking into account that δ/(C+δ)<1, we obtain that

(1-α)u2C+δδlA(u),uE+,

and the result follows.

The proof of (2.3) is similar. ∎

Under the assumptions of Proposition 2.5, we introduce a special inner product (,)A on H1, defined by

(u,v)A=(LAu+,v+)-(LAu-,v-),u,vH1,(2.4)

where LA is considered as an operator from H1 into H-1. By Proposition 2.5, (,)A is equivalent to the standard inner product in H1. An advantage of this inner product is that the operators P+ and P- are orthogonal projectors with respect to (,)A. Hence, the subspaces E+ and E- are orthogonal with respect to this inner product.

Now suppose that AS. The envelop (LA) of LA is the set of all operators of the form LAh, where Ah(A).

Proposition 2.6.

Suppose that AS. Then σ(LAh)=σ(LA) for all AhE(A).

Proof.

If Ah(A), then there exists a sequence τk such that TτkAAh in BS. It is easily seen that σ(LTτkA)=σ(LA). By Proposition 2.2 (c) and [19, Theorem VIII.23], σ(LA)σ(LAh). But T-τkAhA in BS. Hence, interchanging the role of A and Ah, we obtain the required result. ∎

Remark 2.7.

If AS and 0σ(LA)=σ(LAh), we denote by Eh+ and Eh- the positive and negative subspaces of the quadratic form lAh. By Proposition 2.6, the conclusion of Proposition 2.5 holds for lAh with the same constant κ.

3 Statement of the problem and variational formulation

We are looking for homoclinic solutions to second order Hamiltonian systems of the form

-u¨(t)+A(t)u(t)=χFu(t,u(t)),(3.1)

i.e., nonzero solutions that vanish at infinity. Here χ=±1. We make the following assumptions:

  • (i)

    The potential AS is a real symmetric d×d-matrix, the spectrum of the operator LA does not contain zero and the negative part of the spectrum is non-empty.

  • (ii)

    For almost all t, the function F(t,u) is twice continuously differentiable with respect to ud. The functions F(t,u), Fu(t,u) and Fuu(t,u) are strictly Stepanov almost periodic. For any ud, with |u|=r>0, the function F(t,u) is bounded below by a positive constant that depends on r only.

  • (iii)

    The nonlinear potential F satisfies F(,0)=0, Fu(,0)=0 and Fuu(,0)=0. Furthermore, for every R>0, there exists a constant μ(R)>0 such that

    |Fuu(t,u)-Fuu(t,v)|μ(R)|u-v|,t,|u|R.

  • (iv)

    The matrix Fuu(t,u) is positive definite for all u0 and almost all t. There exist constants θ(0,1) and q>2 such that for almost all t,

    0<Fuu(t,u)-1Fu(t,u)Fu(t,u)θFu(t,u)u,ud,u0,(3.2)

    and

    0<qF(t,u)Fu(t,u)u,ud,u0.(3.3)

  • (v)

    There exist r1>0 and r2>0 such that

    Fu(t,u)u|Fu(t,u)|2ν1>0,t,|u|r1,

    and

    Fu(t,u)u|Fu(t,u)|ν2>0,t,|u|r2.

Without loss of generality, we suppose that μ(R1)μ(R2) whenever R1R2.

Assumption (i) guarantees that the self-adjoint operator LA is well defined (see Section 2). By the mean value theorem, assumption (iii) implies that for almost all t,

|Fu(t,u)|μ(R)|u|2(3.4)

and

|F(t,u)|μ(R)|u|3,

whenever |u|R.

Inequality (3.3) is the standard Ambrosetti–Rabinowitz condition. In particular, it implies that for any ε>0, there exists a constant Cε>0 such that

F(t,u)-ε|u|2+Cε|u|q.

On the other hand, condition (3.2) is not standard. However, in the case when the nonlinear potential is radial, i.e., F(t,u)=F(t,|u|), inequality (3.2) is equivalent to

0<Fu(t,u)uθ(Fuu(t,u)u)u,ud,u0,(3.5)

which, in turn, implies (3.3). In the case d=1 the last condition is used in [13] (see also [12]).

In assumption (ii) it is sufficient to assume only that Fuu is strictly Stepanov almost periodic. Then the strict almost periodicity of F and Fu follows.

If d=1, then assumption (v) follows immediately from (3.4). In general, under assumptions (i)–(v), the inequalities in (v) hold for all r1>0 and r2>0, with ν1>0 and ν2>0 depending on r1 and r2, respectively. Therefore, we may assume that r1=r2=1.

Example.

The nonlinear potential F(t,u)=α(t)|u|p, where α(t) is a scalar function, satisfies assumptions (ii)–(iv), provided that αSL, essinfα>0, and p3.

Under our assumptions, the set of shifts {(TτA,TτF)}τ is precompact in the sense that for any sequence τk in , there exists a subsequence τk such that TτkAAh in BS, and TτkFFh, together with the first and second derivatives, with respect to the seminorms R for all R>0. The set of all such limits (Ah,Fh) 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

-u¨(t)+Ah(t)u(t)=χ(Fh)u(t,u(t)),h.(3.6)

Any equation in the envelop satisfies assumptions (i)–(iv), with the same function μ(R) and constants q, θ, ν1 and ν2.

Our main result is the following.

Theorem 3.1.

Under assumptions (i)(v), equation (3.1) has a nonzero solution uH1. Moreover, the solution u decays at infinity exponentially fast, i.e., there exist positive constants α and β such that

|u(t)|+|u˙(t)|αexp(-β|t|).(3.7)

The solution in Theorem 3.1 is a weak solution, i.e.,

(u˙(t)φ˙(t)+(A(t)u(t))φ(t))𝑑t=χFu(t,u(t))φ(t)𝑑t

for all smooth compactly supported functions φ.

Remark 3.2.

Theorem 3.1 applies to all equations (3.6) in the envelop of equation (3.1).

Associated to equation (3.1), we introduce the functional

J(u)=12(|u˙(t)|2+(A(t)u(t))u(t))𝑑t-χF(t,u(t))𝑑t=12lA(u)-χΦ(u).

Similarly, we introduce the functional Jh, h, associated to equation (3.6). Its non-quadratic part is denoted by Φh. The functionals Jh form the envelop of J.

Under the assumptions imposed above, the functional J is a well-defined C2,1-functional on the space H1. Its first and second derivatives are given by

(J(u),v)=(LAu,v)-χFu(t,u(t))v(t)𝑑t,u,vH1,

and

(J′′(u)v,w)=(LAv,w)-χ(Fuu(t,u(t))v(t))w(t)𝑑t,u,v,wH1.

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].

Proposition 3.3.

For any h1H, h2H and any R>0, we have that

|Jh1(u)-Jh2(u)|12Ah1-Ah2BS+(Fh1)uu-(Fh2)uuRu2

and

Jh1(u)-Jh2(u)*Ah1-Ah2BS+(Fh1)uu-(Fh2)uuRu,

provided that uH1 with uR.

Proposition 3.4.

If unu0 weakly in H1, then, uniformly with respect to hH,

Jh(un-u0)-Jh(un)+Jh(u0)0𝑎𝑛𝑑Jh(un-u0)-Jh(un)+Jh(u0)0

strongly in H-1.

Often it is convenient to use gradients of J instead of derivatives. These are defined by

(J(u),v)A=(J(u),v)and(2J(u)v,w)A=(J′′(u)v,w)

for all u,v,wH1, where (,)A is the inner product introduced in (2.4). Then J(u)H1, while 2J(u) is a linear bounded operator in H1.

The proof of Theorem 3.1 is given in the subsequent sections. Obviously, u=0 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 χ=+1. The other case is similar. We only need to replace the functional J by -J and interchange the role of the subspaces E+ and E- 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 𝒩 of the functional J consists of all nonzero uH1 such that

(J(u),u)=0and(J(u),v)=0for all vE-.

Equivalently, these equations can be written as (J(u),u)A=0 and P-J(u)=0, respectively.

For any wE-, we set

Ew={sw+v:s>0,vE-}andE¯w={sw+v:s,vE-}.

By the definition of 𝒩, if u is a critical point of J|Ew, then u𝒩. As consequence, 𝒩 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].

Lemma 4.1.

For every wE-, the functional J|Ew attains its positive global maximum. As a consequence, EwN for all wE-.

It is convenient to introduce the functional

I(u)=J(u)-12(J(u),u).

Obviously, J(u)=I(u) for all u𝒩. Now we prove the following technical result.

Lemma 4.2.

There exists a constant C>0 independent of uH1 such that

u2C(|(J(u),u)|+|(J(u),u-)|+μ(u)uu2)(4.1)

and

u2C(|(J(u),u)|+|(J(u),u-)|+(I1/2(u)+I(u))u)(4.2)

for all uH1.

Proof.

The identity

(J(u),u-)=lA(u-)-Fu(t,u)u-𝑑t

and Proposition 2.5 imply

κu-2-(J(u),u-)-Fu(t,u)u-𝑑t.(4.3)

Similarly, the identity

(J(u),u)=lA(u+)-Fu(t,u)u+𝑑t+(J(u),u-)

implies

κu+2(J(u),u)-(J(u),u-)+Fu(t,u)u+𝑑t.(4.4)

Adding inequalities (4.3) and (4.4), we obtain immediately that

u2C(|(J(u),u)|+|(J(u),u-)|+|Fu(t,u)||u+|dt+|Fu(t,u)||u-|dt).(4.5)

Now let R=u. Then, by inequality (3.4),

|Fu(t,u)||u±|dtμ(R)u±u22.

Hence,

u2C(|(J(u),u)|+|(J(u),u-)|+μ(R)u22(u++u-))C(|(J(u),u)|+|(J(u),u-)|+μ(R)uu2),

which proves (4.1).

Now we prove inequality (4.2). Given uH1, let

S1={t:|u(t)|1}andS2=S1.

We introduce the following integrals:

I1=S1|Fu(t,u)|2dtandI2=S2|Fu(t,u)|dt.

By inequality (3.3) and assumption (v),

I(u)ν(I1+I2),(4.6)

where ν depends on q, ν1 and ν2. Recall that we may take r1=r2=1 in assumption (v). Since

|Fu(t,u)||u±|dt(S1|Fu(u,t)|2dt)1/2(S1|u±|2dt)1/2+u±S2|Fu(u,t)|dt(I1(u)+I21/2(u))u±,

equations (4.5) and (4.6) yield (4.2). ∎

As an immediate consequence of Lemma 4.2, we obtain the following corollary.

Corollary 4.3.

There exists a constant ε0>0 such that uuε0, J(u)ε0 and

Fu(t,u)u𝑑t2ε0

for all uN.

Remark 4.4.

Obviously, Lemma 4.2 and Corollary 4.3 hold for all functionals Jh in the envelop of J with the same constants C and ε0.

Let E¯-=E-. Elements of this space are denoted by [h0,h], where h0 and hE-. We define the operator G:H1E¯- by

G(u)=[(J(u),u)A,P-J(u)],uH1.

It is not difficult to see that the operator G is a C1,1 map, and its derivative is given by the formula

G(u)v=[(2J(u)v,u)A+(J(u),v)A,P-2J(u)v]

for all u,vH1.

Lemma 4.5.

Let u0H1,

g0=(J(u0),u0)A𝑎𝑛𝑑g=P-J(u0)E-.

Then, for all h0R and hE-,

(G(u0)(h0u0+h),[h0,h])A2|g0|h02-κh2+32h02g+32gh2-h02(1-θ)Fu(t,u0)u0𝑑t,

where κ>0 and θ(0,1) are the constants from Proposition 2.2 and assumption (iv), respectively.

Proof.

Since, by the assumptions,

(LAu0,u0)=g0+Fu(t,u0)u0𝑑t

and

(LAu0,h)=(g,h)A+Fu(t,u0)h𝑑t,

a straightforward but a little bit tedious calculation yields the identity

(G(u0)(h0u0+h),[h0,h])A=2h02g0+(LAh,h)+3h0(g,h)A-(H(t)h02+2h0K(t)h+(M(t)h)h)𝑑t,

where

H(t)=(Fuu(t,u0)u0)u0-Fu(t,u0)u0,K(t)=Fuu(t,u0)u0-Fu(t,u0),M(t)=Fuu(t,u0).

Since, obviously,

|h0(g,h)A|12g(h02+h2)

and, by Proposition 2.2,

(LAh,h)-κh2,

it is enough to prove that

H(t)h02+2h0K(t)h(t)+(M(t)h(t))h(t)h02(1-θ)Fu(t,u0(t))u0(t).

This is trivial for all t such that u0(t)=0. Hence, we may assume that u0(t)0. Then the matrix M(t) is positive definite. Making use of the trivial identities

(M1/2h)(M1/2h)=(Mh)hand(M-1/2K)(M-1/2K)=(M-1K)K,

we obtain that

Hh02+2h0Kh+(Mh)h=(H-(M-1K)K))h02+(M1/2h+h0M-1/2K)(M1/2h+h0M-1/2K)(H-(M-1K)K))h02.

Making use of the following easily verified inequality:

H-(M-1K)K=Fu(t,u0)u0-Fuu(t,u0)-1Fu(t,u0)Fu(t,u0)

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 [13] to the present setting just by inspection of the proofs given there. In particular, the set 𝒩 is a non-empty closed C1,1-submanifold of H1. The tangent space to 𝒩 at a point u0𝒩 is Tu0=kerG(u0), while E¯u0 is a complementary subspace to Tu0. Furthermore, given R>0, there exist ρ>0 and C>0 such that for every u0𝒩, with u0R, there exists a C1,1-diffeomorphism from the ρ-neighborhood of 0 in Tu0 onto a neighborhood of u0 in 𝒩 such that the Lipschitz constant of its derivative is not greater than C.

Now we introduce the following quantities:

m=inf{J(u):u𝒩},mh=inf{Jh(u):u𝒩h},h,

where 𝒩h stands for the generalized Nehari manifold of the functional Jh. These numbers are strictly positive by Corollary 4.3 and Remark 4.4.

Proposition 4.6.

We have mh=m for all hH.

Proof.

See the proof of [13, Proposition 5.4]. ∎

Let us turn to Palais–Smale sequences. Recall that a sequence unH1 is a Palais–Smale sequence for the functional J at a level c if J(un)c and J(un)0 in H-1 (equivalently, J(un)0 in H1). A Palais–Smale sequence for J|𝒩 at a level c is a sequence un𝒩 such that J(un)c and τJ(un)0 in H1, where τ stands for the tangent component of the gradient.

As an immediate consequence of Lemma 4.2, we obtain the following corollary.

Corollary 4.7.

Let unH1 be a Palais–Smale sequence for the functional J at a level c. Then c0 and un is bounded in H1. Furthermore, un0 in H1 if and only if c=0.

As in [13, Propositions 6.2 and 6.3], we obtain the following result.

Proposition 4.8.

Every Palais–Smale sequence for J|N is a Palais–Smale sequence for J. Conversely, if unH1 is a Palais–Smale sequence for J at a level c>0, then there exists a Palais–Smale sequence u~nN for J|N such that un-u~n0 and, hence, cm.

Corollary 4.9.

If unH1 is a Palais–Smale sequence for J at a positive level, then lim infunε0, where ε0>0 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 J|𝒩.

Proposition 4.10.

For any ε>0, there exists a Palais–Smale sequence un for J|N, hence for J, at a level c[m,m+ε] such that

limun+1-un=0.

Also we need the following special version of concentration-compactness lemma (cf. [13, Lemma 6.1]).

Lemma 4.11.

Let unH1 be a Palais–Smale sequence for J at a level c[m,2m). Suppose that unu0 weakly in H1.

  • (a)

    If u00 , then unu0 strongly in H1 , and u0 is a critical point of the functional J with critical value c.

  • (b)

    If u0=0 , then there exists a sequence tn such that |tn| and a nontrivial critical point vh of Jh for some h at the level c with the property that along a subsequence, Ttnunvh and T-tnvh-un0 strongly in H1.

5 Proof of main result

We begin with the exponential decay of solutions.

Proposition 5.1.

For any solution uH1 of equation (3.1), there exist α>0 and β>0 such that u satisfies inequality (3.7).

Proof.

Given a solution uH1 of (3.1), let

B(t)=01Fuu(t,su(t))𝑑s.

This is a symmetric matrix and, by assumption (iii), B(t)L. Equation (3.1) implies that

-u¨(t)+[A(t)-B(t)]u(t)=0,

i.e., u is an eigenfunction of the operator LA-B, with the eigenvalue 0.

Since uH1, we have that u(t)0 as |t| and, hence, B(t)0 as |t| in the sense that esssup|t|r|B(t)|0 as r. Together with the embedding D(LA)H1 and by the local compactness of the embedding H1L2, this implies that the multiplication operator by B(t) is relatively compact with respect to LA and, hence, the essential spectra of LA and LA-B coincide (see, e.g., [18]). Therefore, 0 is an isolated eigenvalue of LA-B 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.

Proposition 5.2.

For every ε>0, the functional J possesses a critical value in the interval [m,m+ε].

Proof.

Obviously, we may assume that ε<m. Let un be the Palais–Smale sequence obtained in Proposition 4.10. If there is a weak limit point u00 of the sequence un, then, by Lemma 4.11 (a), u0 is a critical point of J with critical value c[m,m+ε].

Thus, assume that un0 weakly in H1. In this case, we use un to construct another Palais–Smale sequence that produces the required critical value. For any uH1, we set

r(t;u)=t(|u˙(s)|2+|u(s)|2)𝑑s.

Obviously, r(t;u) is a non-negative, continuous and non-increasing function, and r(t;u)0 as t+. Since, by construction, un𝒩, Corollary 4.3 implies that there exists tn (not necessarily unique) such that r(tn;un)=δ0=ε02/2, where ε0 is the constant from Corollary 4.3.

The key property of the sequence tn is the following:

limn|tn|=andlimn(tn-tn-1)=0.(5.1)

To prove this statement consider any subsequence tn. By Lemma 4.11 (b), there exist a subsequence un′′, a sequence τn′′, |τn′′|, and an nontrivial critical point vh of the functional Jh for some h such that Tτn′′un′′vh strongly in H1. As a consequence, Tτn′′un′′-1vh strongly in H1. This implies straightforwardly that

r(t;Tτn′′un′′)r(t;vh)andr(t;Tτn′′un′′-1)r(t;vh)

in L. By the uniqueness of the solution to the initial value problem for equation (3.6), |vh(t)|2+|vh(t)|2>0 and, hence, the function r(t;vh) is strictly decreasing. Therefore, there exists a unique th such that r(th;vh)=δ0. Now it is not difficult to see that

lim(tn′′-τn′′)=lim(tn′′-1-τn′′)=0,

which implies immediately (5.1).

Let vn=Ttnun. We prove that 0 is not a weak limit point of vn. Assume the contrary, i.e., along a subsequence, vn0 weakly in H1, and vn 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 τn, yet another functional Jh (h) and its nontrivial critical point vh such that lim|τn|=, and

TτnvnvhandT-τnvh-vn0

strongly in H1. This implies that

r(0;vn)-r(0;T-τnvh)=δ0-r(0;T-τnvh)0.

On the other hand, since |τn|, we obtain, by passing to a further subsequence, that either

r(0;T-τnvh)0orr(0;T-τnvh)vh22δ0.

This is a contradiction.

Suppose now that the sequence tn is unbounded above. It is well known (see, e.g., [9, 11]) that there exists a returning sequence ζn for the almost periodic functions A(t) and Fuu(t,u), i.e., TζkAA in BS and

TζkFuu-FuuR0for all R>0.

Since lim(tn-tn-1)=0, we obtain that there exists a subsequence tnk such that lim(ζk-tnk)=0. Along a subsequence, vnk converges weakly in H1 to a nonzero function v. By Proposition 3.3,

J(vnk)-Jxnk(vnk)*J(vnk)-Jζnk(vnk)*+Jζnk(vnk)-Jxnk(vnk)*0.

Since J is weakly continuous and un is the Palais–Smale sequence for the functional J, this implies that for any φH1,

(J(v),φ)=lim(J(vnk),φ)=lim(Jtnk(vnk),φ)=lim(J(unk),T-tnkφ)=0.

Thus, v is a critical point of J at a level in [m,m+ε]. The case when tn 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.

Theorem 6.1.

Let χ=1. Assume that the following hold:

  • (a’)

    AS and the operator LA is positive definite.

  • (b’)

    For almost all t , the function F(t,u) is C2 with respect to ud . The functions F(t,u), Fu(t,u) and Fuu(t,u) are strictly Stepanov almost periodic. For any ud , with |u|=r>0 , the function F(t,u) of t is bounded below by a positive constant that depends on u only.

  • (c’)

    F(t,0)=0, Fu(t,0)=0 and Fuu(t,0)=0.

  • (d’)

    The function F satisfies ( 3.3 ) with q>2.

Then equation (3.1) has a nontrivial solution uH1 that satisfies (3.7).

In this case the functional J possesses the mountain pass geometry. The arguments of [20] 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 χ=-1. Assume only that ABS, LA is positive definite, Fu(,u)BS and Fu(t,u)u>0 for all u0 and almost all t. Then, by multiplying (3.1) by u and integrating, we conclude that the equation has no nontrivial solution in H1.

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About the article

Received: 2017-02-25

Accepted: 2017-02-28

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.

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