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Advances in Nonlinear Analysis

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Existence of standing waves for quasi-linear Schrödinger equations on Tn

Xin Zhao / Weiping Yan
Published Online: 2019-08-28 | DOI: https://doi.org/10.1515/anona-2020-0038


This paper is devoted to the study of the existence of standing waves for a class of quasi-linear Schrödinger equations on Tn with dimension n ≥ 3. By construction of a suitable Nash-Moser-type iteration scheme, we overcome the clusters of “small divisor” problem, then the existence of standing waves for quasi-linear Schrödinger equations is established.

Keywords: Schrödinger equation; Small divisors; Periodic solution

MSC 2010: 35J60; 35R03; 35B10

1 Introduction and Main Results

This paper considers the quasi-linear Schrödinger equation


where U is a complex-valued functions of (t, x), Tn is a n-dimensional flat torus with n ≥ 3, △ is the Laplace-Beltrami operator, q ≥ 1, the terms a(s) and f(x, s) satisfy gauge invariant, i.e. a(es) = ea(s) for almost every xTn, all φR and s ≥ 0.

The problem in this general setting arises in various fields of mathematical physics, such as the superfluid film equation of fluid mechanics [16] and ferromagnets and magnons [2, 15]. Lange etc [17] has obtained the local existence and uniqueness of smooth solution for a class of quasilinear Schrödinger equation. Poppenberg [22] used the Nash-Moser implicit function theorem to overcome “the loss of derivatives” introduced by the nonlinearity. Kenig etc [14] studied the Cauchy problem of a more general class of quasilinear Schrödinger equation. Bahrouni-Ounaies-Rǎdulescu [1] studied compactly supported solutions of Schrödinger equations with small perturbation. Zhang-Zhang-Xiang [27] obtained the existence of ground states for fractional Schrödinger equations involving a critical nonlinearity. Xue-Tang [25] showed that the existence of a bound state solution for quasilinear Schrödinger equations. One can see [8, 9, 12, 21] for more results on the existence of solution for elliptic equations in Rn. To our knowledge, there is no result on the existence of standing waves for a class quasilinear Schrödinger equation with higher derivatives in higher dimension flat-torus Tn.

A standing wave is a solution of the form


and for solutions of this form, quasi-linear Schrödinger equation (1.1) is reduced into a quasi-linear elliptic equation involving the parameter μ


We make assumptions on nonlinear terms f, which include the standard tame estimates and Taylor tame estimates. fC(Tn × R, R), f(0, 0) = 0, u f(x, 0) = ⋯ = (up1)f(x,0)=0,upf(x,0)0,1pk,k2 and



where s > s0 > 0, p > 1, ∀ u, u′ ∈ Hs such that ∥us0 ≤ 1 and ∥u′∥s0 ≤ 1. In particular, for s0 = s,


In fact, when p = 2, assumption (1.3) and (1.4) are natural for fC(Tn × R, R), which are tame estimates and Taylor tame estimates, respectively.

Rescaling in (1.2) amplitude u(x) ↦ δ u(x), δ > 0, we solve the following problem


where a(s) := asq, f(δ, u) := b(x)sp + O(δ), 1 ≤ pk and ε = δp−1.

The problem of solving nonlinear elliptic equations with a singular perturbation was inspired by the work of Rabinowitz [24]. By employing the Nash-Moser iteration process, he proved that the elliptic singular perturbation problem has a uniqueness spatial periodic solution. For more related work, we refer to [13, 20]. Han-Hong-Lin [10] partially extended the work of Rabinowitz [24], they considered the following singular perturbation problem


where q ≥ 4, the function a(x) is smooth and f(x) is (2π)2-periodic. Under some assumptions on a(x) and f(x), they employed the Nash-Moser iteration process to prove that above singular problem had spatial periodic solutions. But they only dealed with small divisors problem in one dimensional case. Beacuse there is the “clusters of small divisors” problem in higher dimensional case. The aim of the present paper is to focus on the solution of the small divisors problem in presence of large clusters and with smooth nonlinearities for singular perturbation elliptic problem (1.5) in higher dimensional case (n ≥ 3).

We will divide into two cases to discuss the existence of solutions for (1.5). The first case is a(x) = ax, where a ≠ 0 is a constant, then the “small divisor” phenomenon appears. The second case is a(⋅) ∈ C(R). The second case is simpler than the first case, and we can use the Nash-Moser iteration scheme constructed in the first case to solve it. In what follows, we deal with the first case, i.e. a((−1)qqu) = (−1)q aqu. Thus we can rewrite (1.5) as


Assume that a is an irrational number and diophantine, i.e. there are constants y0 > 0, τ0 > 1, such that


Then there exist y > 0 and τ > 1 such that the first order Melnikov nonreonance condition


where ωj2;=|j|2 and jZn.

Our main results are based on the Nash-Moser iterative scheme, which is firstly introduced by Nash [19] and Moser [18]. One can also see [11] for more details. Berti and Procesi [4] developed suitable linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces, and via the technique and the Nash-Moser implicit function theorem, they found a family of time-periodic solutions of nonlinear Schrödinger equations and wave equations. Inspired by the work of [4, 5, 26], we will construct a suitable Nash-Moser iteration scheme to study the elliptic-type singular perturbation problems (1.2) in higher dimensional flat torus.

We define the Sobolev scale of Hilbert spaces


for some s>n+12. There holds ∥uvs ≤ ∥usvs.

For the case a(x) = ax in (1.2), we have the following result.

Theorem 1.1

Assume that a > 0 is diophantine. For δ0 > 0, s0, kN and fC satisfying (1.3)-(1.4), Then there exists a positive measure Cantor set 𝓒 ⊂ [0, δ0] such that, ∀ a ∈ 𝓒, U(t, x) = eiμt uδ(x, ε) is a unique standing wave solution of (1.1). Furthermore, there exists a curve


For the second case, we consider equation (1.5) and assume that aC(R), a(0) = 0, and



where s > s0 > 0, 1 < pk, ∀ u, u′ ∈ Hs such that ∥us0 ≤ 1 and ∥u′∥s0 ≤ 1. In particular, for s0 = s,


For the second case, we have

Theorem 1.2

There exist s0 and kN such thatf, aC satisfying (1.3)-(1.4) and (1.9)-(1.10), respectively. Then equation (1.1) admits a unique standing wave solution U(t, x) = eiμtu(x) with u(x) ∈ Hs0.

The proof of Theorem 1.2 is similar to the proof of Theorem 1.1, hence we omit it.

The structure of the paper is as follows: In next section, we show that the linearized equation of (1.6) is solvable by means of proving the invertible of its linearized operator. Section 3 gives the proof of Theorem 1.1 by construction of a suitable Nash-Moser iteration scheme.

2 Analysis of the Linearized operator

This section is devoted to prove the invertible of linearized operator


We define the finite dimensional subspace of Hs as


where A is a finite and symmetric subset of Zn+1 and ej(x) = eijx.

For ∀ h = ∑jZn hj ejHs, We denote


which is a L2-orthogonal projector on HA.

Let A = ΩN := {jZn||j| ≤ N} and b(x) := −(uf)(δ, u). Then the operator (2.1) can be defined on HΩN := H(N), i.e.


where La := − △ − m − (− 1)q ε aq.

We write the linearized operator in (2.2) by the block matrix


In the L2-orthonormal basis (ej)jΩN of HΩN, D is represented a diagonal matrix with eigenvalues


whereas T is represented by the self-adjoint Toepliz matrix (bjj)j,j′∈ΩN, the bj is the Fourier coefficients of the function b(x).

Now we give the main result in this section.

Proposition 2.1

Assume that


anduσ̄ ≤ 1, ∀ 1 ≤ rN, ∀ κ ≥ 1,


Then the linearized operator L(N)(δ, u) is invertible ands2 > s1 > σ̄ > 0, the linearized operator La(N) satisfies


where C(s2s1) = c(s2s1)τ, c denotes a constant.

For fixing ς > 0, we define the regular sites R and the singular sites S as


The following result shows the separation of singular sites, and the proof can be found in the paper [3, 4], so we omit it.

Lemma 2.1

Assume that a is diophantine and a satisfies (2.5). There exists ς0(y) such that for ς ∈ (0, ς0(y)] and a partition of the singular sites S which can be partitioned in pairwise disjoint clusters Ωα as



  • (dyadic) ∀α, Mα ≤ 2mα, where Mα := maxjΩα|j|, mα := maxjΩα|j|.

  • (separation) ∃ λ, c > 0 such that d(Ωα, Ωβ) ≥ c(Mα + Mβ)λ, ∀αβ, where d(Ωα, Ωβ) := maxjΩα,j′∈Ωβ|jj′|.

We define the polynomially localized block matrices


where Ajj0:=supuH(N),u0=1Ajju0 is the L2-operator norm in 𝓛(H(N), H(N)). If s′ > s, then these holds 𝓐s ⊂ 𝓐s.

The next lemma (see [4]) shows the algebra property of 𝓐s and interpolation inequality.

Lemma 2.2

There holds




By Lemma 2.2, we can get, ∀ mN,



Then by the same method as the proof process of Lemma 6.3 in [4], we can prove the following result. Here we omit the proof.

Lemma 2.3

Let s > s′. For a real bHs+s, the matrix T=(Tjj)j,jJN+ defined in (2.3) is self-adjoint and belongs to the algebra of polynomially localized matrices 𝓐s with


Moreover, for any s > s′,


Since the decomposition


we can represent the operator L(N) as the self-adjoint block matrix


where LRS=(LSR),LR=LR,LS=LS.

Thus the invertibility of L(N) can be expressed via the “resolvent-type” identity


where the “quasi-singular” matrix


The reason of 𝓛 ∈ 𝓐s(S) is that 𝓛 is the restriction to S of the polynomially localized matrix




Lemma 2.4

Assume that a is diophantine. For s0<s1<s2<k1,|LR1|s02ς1, the operator LR satisfies



where L~1=LR1DR,c(y,τ,s2) is a constant depending on y, τ, s2.


It follows from (2.3) and (2.8) that DR is a diagonal matrix and satisfies |DR1|sς1. By (2.10), we have that the Neumann series


is totally convergent in |⋅|s1 with |LR1|s02ς1, by taking ες−1|T|s0c(s0) small enough.

Using (2.10) and (2.14), we have that ∀ mN,


which together with (2.18) implies that for ες−1|T|s0 < c(s0) small enough, (2.16) holds.

By non-resonance condition (1.8) and supx>0(xy ex) = (ye−1)y, ∀ y ≥ 0, we derive


Then by (2.19), for any hHR,


Thus using interpolation (2.12) and (2.16), we derive that for s1 < s < s2,


This completes the proof. □

Next we analyse the quasi-singular matrix 𝓛. By (2.9), the singular sites restricted to JN+ are


Since the decomposition S := ⨁αlN α, where Hα := ⨁jΩα𝓝j, we represent 𝓛 as the block matrix 𝓛 = (Lαβ)α,βlN, where Lαβ:=ΠHαL|Hβ. So we can rewrite


where 𝓓 := diagαlN(𝓛α), Lα:=Lαα,J:=(Lαβ)αβ.

We define a diagonal matrix corresponding to the matrix 𝓓 as := diagαlN(α), where α = diagjΩα(Dj).

To show 𝓓 is invertible, we only need to prove that 𝓛α is invertible, ∀αlN.

Lemma 2.5

αlN, 𝓛α is invertible and Lα10Cy11Mακ.

The proof process of above Lemma is similar with Lemma 6.6 in [4], so we omit it.

Lemma 2.6

Assume that a is diophantine. We have


where c(ς, s1, y1) is a constant which depends on ς, s1 and y1.


Note that hα0mαs1hαs1 and Mα = 2mα. So for any h = ∑αlN hαHα, hαHα,


Using interpolation and (2.8), for 0 < s1 < s2, it follows from (2.20) that


This completes the proof. □

The following result is taken from [4], so we omit the proof.

Lemma 2.7

s ≥ 0, ∀ mN, there hold:



Lemma 2.8

Assume that a is diophantine. For 0 < s0 < s1 < s2 < s3 < k − 1, we have



The Neumann series


is totally convergent in operator norm ∥⋅∥s0 with L1s0cy11Nτ, by using (2.21).

By (2.22) and (2.24), we have


Using supx>0(xy ex) = (ye−1)y, ∀ y ≥ 0, for 0 < s1 < s2 < s3, it follows from Lemma 2.4 that


Thus by (2.25) and (2.26), we derive


where 0 < s1 < s2 < s3 and εy11 ς−1(1 + |T|s0) ≤ c(k) small enough. □

Now we are ready to prove Proposition 2.1. Let


where hSHS, hRHR. Then by the resolvent identity (2.15),


Next we estimate the right hand side of (2.28) one by one. Using (2.12), (2.17) and (2.23), for 0 < s1 < s2 < s3 < s4 < k − 1, we have




The terms LR1hRs1 and ∥𝓛−1hRs1 can be controlled by using (2.17) and (2.23). Thus by (2.28)-(2.31), for 0 < s < , we conclude


which together with Lemma 2.8 gives (2.7).

3 Nash-Moser-type iteration scheme

We define the finite dimensional subspaces



Then we have the orthogonal splitting


where i denotes the “i”th iterative step and ∀sk. For a given suitable N0 > 1, we take NiNi+1 and and Ni = N0i, ∀iN.

The orthogonal projectors onto Hs(Ni) and Hs(Ni) denote by Π(Ni) : HsHs(Ni) and Π(Ni)⊥ : HsHs(Ni), which satisfy the “smoothing” properties:






The linearized operator of (3.2) has the following form


where D denotes the Frechet derivative.

By (3.2), we define


Next we construct the first step approximation.

Lemma 3.1

Assume that a is diophantine. Then system (3.4) has the first step approximation u1Hs(N1)


and the error term is



Assume that the 0th step approximation solution u0 satisfies


Then the target is to get the 1th step approximation solution.



By (3.4), we have


Then taking




By (3.8), we denote


On the other hand, by (3.4) and (3.7), we can obtain


This completes the proof.□

In order to prove the convergence of the Nash-Moser iteration scheme, the following estimate is needed. For convenience, we define


Lemma 3.2

Assume that a is diophantine. Then for any 0 < α < σ, the following estimates hold:



where C(α) is defined in (3.12).




From the definition of u1 in (3.5), by (2.7), (3.1) and (3.10), we derive


By assumption (1.4) and the definition of E1, we have


This completes the proof.□

For iN and 0 < σ0 < σ̄ < σ < k – 1, set



By (3.14)-(3.15), it follows that





In fact, to obtain the “i th" approximation solution uiHσi(Ni) of system (3.4), we need to solve following equations


Then, we get the ‘i th‘ step approximation uiHσi(Ni):




As done in Lemma 3.2, it is easy to get that



Hence, we only need to estimate Ri–1 to prove the convergence of algorithm. In the following, a sufficient condition on the convergence of the Nash-Moser iteration scheme is proved. This proof is based on Lemma 3.2. It also shows the existence of solutions for (3.4).

Lemma 3.3

Assume that a is diophantine. Then for sufficiently small ε, equations (3.2) has a solution


where 𝓑1(0) := {u|∥us ≤ 1, ∀s > > 0}.


We divide into two cases. If ες1ui1σi1p<1, by (2.7), (3.16) and (3.18), we derive


where c(ε, ς) is a constant depending on ε and ς.

Note that Ni = N0i, ∀iN. By (3.17)-(3.19) and assumption (1.4), we have


Hence, choosing small ε > 0 such that


For any fixed p > 1, we derive


If ες1ui1σi1p1, by (2.7), (3.16) and (3.18), we derive


But we will choose the initial step u0 = 0 in this paper, which combining with (3.22) leads to ∥uiσi = 0, ∀iN. This contradicts with assumption ες1ui1σi1p>1. Hence, the case is not possible. (3.2) has a solution


where 𝓑1(0) := {u|∥us ≤ 1, ∀s > > 0}. This completes the proof.□

Next result gives the local uniqueness of solutions for equation (3.2).

Lemma 3.4

Assume that a is diophantine. Equation (3.2) has a unique solution uHσ̄B1(0) obtained in Lemma 3.3.


Let u, ũHσ̄B1(0) be two solutions of system (3.4), where


Write h = uũ. Our target is to prove h = 0. By (3.4), we have


which implies that


Note that Ni = N0i, ∀iN. Thus, by (2.7) and (3.23), we have


Choosing δ < 8p2 cp2(ε, ς, τ, s, , y1, y)N0(τ+κ0)p, we obtain


This completes the proof.□

Remark 3.1

The dependence upon the parameter, as is well known, is more delicated since it involves in the small divisors of ωj: it is, however, standard to check that this dependence is C1 on a bounded set of Diophantine numbers, for more details, see, for example, [3, 4].

By Lemma 3.1, for sufficient small δ0 > 0 and given r > 0, we define


Then for a given function δq(δ)Ur(N), the set Gy1,κ0(N) is equivalent to


Choosing κ and y1 such that


Next we have the measure estimate. The proof of it will be given in Appendix.

Lemma 3.5

(Measure estimates) Assume that a is diophantine, ε0y–1Mτ+2q is sufficient small and (3.24) holds. Then Gy1,κ0(M)(0) = 𝓖, and 𝓖 satisfies


Furthermore, for any r′ > 0, there exists δ′ := δ′(y1, r′) such that the measure estimate


holds, where N′ ≥ NM, u1Ur(N),u1Ur(N) withu2u1σ̄Ne, e denotes a constant depending on κ0 and n.


This proof follows essentially the scheme of [3, 4, 5]. Note that |j| ≤ r and the eigenvalue of the operator La(r) has the form ωj2+μεaωj2qO(ε) of the operator La(r). Here jZn. For sufficient small ε0y–1Mτ+2q, by (1.8), all the eigenvalues of La(r) has modulus ≥ y(4rτ)–1y1(4rκ)–1. Thus 𝓖r = [0, δ0) and the measure estimate (3.25) for 𝓖 is standard. To prove the measure estimate (3.26), we divide the process of proof into two cases. For the case N,NNε0:=(cy1ε01)1τ+2ϱ,Gy1,κ0(N)(u2)=Gy1,κ0(N)(u1)=G, by the same process of proof of (3.25), one can prove (3.26) holds. For other cases, it is sufficient to prove


For fixed δ1 and the decomposition [0, δ0] = ∪n≥1[δ02n, δ02–(n–1)], we consider the complementary sets in [δ12,δ1)


If rNε0, then Grc(u2)G=. So it is sufficient to prove that, if ∥u1u2σ̄Ne, ed + n + 3, then


Note that (La(r))10 is the inverse of the eigenvalue of smallest modulus and


The sufficient and necessary condition of an eigenvalues of La(r)(u2) in [–4y1rτCεNe, 4y1rτ + CεNe] is that there exists an eigenvalues of La(r)(u1) in [–4y1rτ, 4y1rτ]. Thus, it leads to


Next we claim that if ε is small enough and I is a compact interval in [–y1, y1] of length |I|, then


Due to the C1 map δL(r)(δ, u1) and the selfadjoint property of L(r)(δ, u1), we have the corresponding eigenvalue function λk(δ, u1) with 1 ≤ kr. Denote the eigenspace of L(r)(δ, u1) by Eδ,k associated to λk(δ, u1), then by δbs=(u2f)(x,u)sCy11 and qh02h02, for sufficient small 0 < εε0(y1), we have


Hence we have |λk1(I,u1)[δ12,δ1]|C|I|δ1(2q2). The claim holds.

Thus, we obtain


Furthermore, by (3.27), we have |Grc(u2)|Cy1rd+nτ+1δ1(2q2). Therefore, we obtain


where C, C′ and C″ denote constants. This completes the proof.□


The first author is supported by Huizhou University Professor Doctor Launch Project Grant, No. 20187B037. The second author is supported by NSFC No 11771359, and the Fundamental Research Funds for the Central Universities (Grant No. 20720190070, 20720180009 and 201709000061).


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About the article

Received: 2019-06-07

Accepted: 2019-06-14

Published Online: 2019-08-28

Published in Print: 2019-03-01

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 978–993, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0038.

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© 2020 Xin Zhao and Weiping Yan, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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