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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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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

Abstract

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

iUtUa(qU)=f(x,|U|)U,(t,x)R×Tn,(1.1)

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

U(t,x)=eiμtu(x),μ>0,

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

uμua(qu)=f(x,|u|)u.(1.2)

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

uf(x,u)usc(s)(usp1+usus0p1),(1.3)

f(x,u+u)f(x,u)Duf(x,u)usc(s)(usus0p1+us0usp1),(1.4)

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

f(x,u+u)f(x,u)Duf(x,u)usc(s)usp.

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

uμuεa(qu)=εf(δ,u),(1.5)

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

u+u+εa(qu)=f(x),xR2,

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

uμu(1)qεaqu=εf(δ,u).(1.6)

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

|man|y0|n|τ0,(m,n)Z2{(0,0)}.(1.7)

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

|ωj2μεaωj2q|y|j|τ,(1.8)

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

Hs:=Hs(Tn,C)={u(x)=jTneijxuj,uj=uj|us2:=jTne2|j|s|uj|2<+}

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

u(x,ε)C1([0,δ0];Hs0)withu(δ)s0=O(δ).

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

ua(u)usc(s)(usp1+usus0p1),(1.9)

a(u+u)a(u)Dua(u)usc(s)(usus0p1+us0usp1),(1.10)

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

a(u+u)a(u)Dua(u)usc(s)usp.

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

L:=μ(1)qεaqεuf(δ,u).(2.1)

We define the finite dimensional subspace of Hs as

HA:=SpanjAej={kAhjej:hjC,hj=hj},

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

For ∀ h = ∑jZn hj ejHs, We denote

PAh=jAhjej,

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.

hL(N)[h]:=Lah+εPΩN(b(x)h),hH(N),(2.2)

where La := − △ − m − (− 1)q ε aq.

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

L(N)=D+εT,D:=La.(2.3)

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

Dj:=|j|2mεa|j|2q,(2.4)

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

|man|y1max(1,|m|32),0<y1<1,(m,n)Z2{(0,0)},(2.5)

anduσ̄ ≤ 1, ∀ 1 ≤ rN, ∀ κ ≥ 1,

(La(r)(δ,u))104rκy1.(2.6)

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

(L(N)(δ,u))1hs1C(s2s1)Nτ+κ01+ες1us2p3hs2,(2.7)

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

R:={jΩN||Dj|ς}andS:={jΩN||Dj|<ς}.(2.8)

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

S=αAΩα(2.9)

satisfying

  • (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

As:={A=(Ajj)j,jZn:|A|s2:=supjZnjZne2s|jj|Ajj02<},

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

|AB|sc(s)|A|s|B|s,A,BAs,s>s0>r+n+12,(2.10)

|AB|sc(s)(|A|s|B|s0+|A|s0|B|s),ss0,(2.11)

Ausc(s)(|A|sus0+|A|s0us),uHs,ss0.(2.12)

By Lemma 2.2, we can get, ∀ mN,

|Am|sc(s)m1|A|sm,(2.13)

|Am|sm(c(s)|A|s0)m1|A|s.(2.14)

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

|T|sK(s)bs+s.

Moreover, for any s > s′,

|T|sK(s)Nsbs.

Since the decomposition

H(N):=HRHS,

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

L(N)=LRLRSLSRLS,

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

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

(L(N))1=ILR1LRS0ILR100L1I0LSRLR1I,(2.15)

where the “quasi-singular” matrix

L:=LSLSRLR1LRSAs(S).

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

IS(LISLIRL~1IRLIS)ISAs,

where

L~1=I00LR.

Lemma 2.4

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

|L~R1|s1c(s1)(1+ες1|T|s1),(2.16)

LR1hs1c(y,τ,s2)(s2s1)τ(1+ες1|T|s2)hs2,(2.17)

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

Proof

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

L~R1=LR1DR=m0(ε)m(DR1TR)m(2.18)

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,

εm|(DR1TR)m|s1εmc(s)|(DR1TR)m|s1c(s)εmm(c(s)|DR1TR|s0)m1|DR1TR|s1c(s)εmς1(εc(s1)ς1|T|s0)m1|T|s1,

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

e2|j|(s2s1)|ωj2+μεaωj2q|2y1|j|τe2|j|(s2s1)c(y,τ)(s2s1)2τ.(2.19)

Then by (2.19), for any hHR,

LR1hs12=jRe2|j|s1LR1hjL22jRe2|j|s1|ωj2+μεaωj2q|2L~R1hjL22jRe2|j|(s2s1)|ωj2+μεaωj2q|2e2|j|s2L~R1hjL22c(y,τ)(s2s1)2τL~R1hs22.

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

LR1hs1c(y,τ)(s2s1)τL~R1hs2c(r,τ,s2)(s2s1)τ(|L~R1|s2hs+|L~R1|shs2)c(r,τ,s2)(s2s1)τ(1+ες1|T|s2)hs2.

This completes the proof. □

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

S=αlNΩα,wherelN:={αN|mαN}.

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

L=D+J,

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

D1D¯hs1c(ς,s1,y1)Nτhs2,

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

Proof

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

D1D¯hs12=αlNLα1L¯αhαs12αlNMα2s1Lα1L¯αhα02cy12αlNMα2(s1+τ)L¯αhα02cy12αlNMα2(s1+τ)mα2s1L¯αhαs12cy124s1αlNMα2τL¯αhαs12cy124s1N2ταlNL¯αhαs12=cy124s1N2τD¯hs12.(2.20)

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

D1D¯hs1cy112s1NτD¯hs1cy112s1Nτ(|D¯|s2hs1+|D¯|s1hs2)c(ς)y112s1+1Nτhs2.

This completes the proof. □

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

Lemma 2.7

s ≥ 0, ∀ mN, there hold:

c(s1)D1Js0<12,D1sc(s)y11Nτ,(2.21)

(D1J)mhs(εy1K(s))m(mNκ0|T|s|T|s0m1hs0+|T|s0mhs).(2.22)

Lemma 2.8

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

L1hs1c(ς,τ,s1,y1,y)Nτ+κ0(s3s2)τ(hs3+ε|T|s1hs2).(2.23)

Proof

The Neumann series

L1=(I+D1J)1D1=m0(1)m(D1J)mD1(2.24)

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

By (2.22) and (2.24), we have

L1hs1D1hs1+m1(D1J)mD1hs1D1hs1+D1hs1m1(εy11K(s)|T|s0)m+Nκ0K(s1)εy11|T|s1D1hs0m1m(K(s)εy11|T|s0)m1.(2.25)

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

D1hs12=D1D¯D¯1hs12c2(ς,s1,y1)N2τD¯1hs22=c2(ς,s1,y1)N2τjSe2|j|s2D¯1hjL22c2(ς,s1,y1)N2τjSe2|j|s2|ωj2+μεaωj2q|2hjL22c2(ς,s1,y1)N2τjSe2|j|(s3s2)|j|2e2|j|s3hjL22c2(ς,τ,s1,y1,y)N2τ(s3s2)2τhs32.(2.26)

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

L1hs1y11Nκ0K(s1)(D1hs1+ε|T|s1D1hs0)c(ς,τ,s1,y1,y)Nτ+κ0(s3s2)τ(hs3+ε|T|s1hs2),(2.27)

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

h=hR+hS,

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

(L(N))1hs1LR1hR+LR1LSRL1(hS+LRSLR1hR)s1+L1(hR+LRSLR1hR)s1LR1hRs1+LR1LSRL1hSs1+LR1LSRL1LRSLR1hRs1+L1hRs1+L1LRSLR1hRs1.(2.28)

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

LR1LSRL1hSs1c(y,τ,s2)(s2s1)τ(1+ες1|T|s2)LSRL1hSs2c(y,τ,s2)(s2s1)τ(1+ες1|T|s2)|T|s2L1hs2c(y,y1,ς,τ,s2)(s2s1)τ(s4s3)τNτ+κ0×(1+ες1|T|s2)|T|s2(hs3+ε|T|s2hs4),(2.29)

L1LRSLR1hRs1c(ς,τ,s1,y1,y)Nτ+κ0(s3s2)τ(LRSLR1hRs3+ε|T|s1LRSLR1hRs2)c(ς,τ,s1,s2,s3,y1,y)Nτ+κ0(s3s2)τ×(|T|s3LR1hRs3+ε|T|s1|T|s2LR1hRs2)c(ς,τ,s1,s2,s3,y1,y)Nτ+κ0(s3s2)τ(|T|s3(s4s3)τ(1+ες1|T|s4)hs4+ε|T|s1|T|s2(s3s2)τ(1+ες1|T|s3)hs3)c(ς,τ,s1,s2,s3,y1,y)Nτ+κ0(s3s2)τ|T|s3(1+ες1|T|s4)×((s4s3)τhs4+ε|T|s2(s3s2)τhs3),(2.30)

LR1LSRL1LRSLR1hRs1c(y,τ,s2)(s2s1)τ(1+ες1|T|s2)LSRL1LRSLR1hRs2c(y,τ,s2)(s2s1)τ(1+ες1|T|s2)|T|s2L1LRSLR1hRs2c(ς,τ,s1,s2,s3,y1,y)Nτ+κ0(s3s2)τ(s2s1)τ|T|s32×(1+ες1|T|s4)2((s4s3)τhs4+ε|T|s2(s3s2)τhs3).(2.31)

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

(L(N))1hsc(ς,τ,s,s~,y1,y)Nτ+κ0(1+ες1|T|s~)3(s~s)τhs~,

which together with Lemma 2.8 gives (2.7).

3 Nash-Moser-type iteration scheme

We define the finite dimensional subspaces

Hs(Ni)={uHs|u=|j|Niujeijx},

Hs(Ni)={uHs|u=|j|>Niujeijx}.

Then we have the orthogonal splitting

Hs=Hs(Ni)Hs(Ni),

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:

Π(Ni)us+deNidus,uXs,s,d0,Π(Ni)usNidus+d,uXs+d,s,d0.(3.1)

Consider

Lau=εf(x,u),(3.2)

where

La:=μ+εaq.

The linearized operator of (3.2) has the following form

La(Ni):=Π(Ni)(LaεDuf(δ,u))|Hs(Ni),(3.3)

where D denotes the Frechet derivative.

By (3.2), we define

J(u)=LauεΠ(Ni)f(x,u)=0.(3.4)

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)

u1=(La(N1))1E0Hs(N1),(3.5)

and the error term is

E1=R0=εΠ(N1)f(x,u0+u1)f(x,u0)Duf(x,u0)u1.(3.6)

Proof

Assume that the 0th step approximation solution u0 satisfies

f(x,u0)0.

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

Denote

E0=Lau0εΠ(N1)f(x,u0).(3.7)

By (3.4), we have

J(u0+u1)=La(u0+u1)εΠ(N1)f(x,u0+u1)=Lau0εΠ(N1)f(x,u0)+Lau1+εΠ(N1)Duf(x,u0)u1εΠ(N1)(f(x,u0+u1)f(x,u0)Duf(x,u0)u1)=E0+La(N1)u1+R0.(3.8)

Then taking

E0+La(N1)u1=0,

yields

u1=(La(N1))1E0Hs(N1).

By (3.8), we denote

E1:=R0=J1(u0+u1)=εΠ(N1)(f(x,u0+u1)f(x,u0)Duf(x,u0)u1).

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

E0=ε(IΠ(N0))Π(N1)f(x,u0).(3.9)

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

E~0:=εΠ(N1)f(x,u0).(3.10)

Lemma 3.2

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

u1σαC(α)(1+ες1u0σp)3E~0σ+τ+κ0,

E1σαCp(α)(1+ες1u0σp)3pE~0σ+τ+κ0p,(3.11)

where C(α) is defined in (3.12).

Proof

Denote

C(α)=c(ς,τ,s,s~,y1,y)ατ.(3.12)

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

u1σα=(La(N1))1E0σαC(α)N1τ+κ0(1+ες1u0σp)3E0σC(α)(1+ες1u0σp)3E~0σ+τ+κ0.(3.13)

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

E1σα=Π(N1)(f(x,u0+u1)f(x,u0)Duf(x,u0)u1)σαu1σαpCp(α)(1+ες1u0σp)3pE~0σ+τ+κ0p.

This completes the proof.□

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

σi:=σ¯+σσ¯2i,(3.14)

αi+1:=σiσi+1=σσ¯2i+1.(3.15)

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

σ0>σ1>>σi>σi+1>,foriN.

Define

P1(u0):=u0+u1,foru0Hσ0(N0),

Ei=J(k=0iuk)=J(P1i(u0)),

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

J(k=0iuk)=La(k=0i1uk)εΠ(Ni)f(x,k=0i1uk)+LauiεΠ(Ni)Duf(x,k=0i1uk)uiεΠ(Ni)f(x,k=0iuk)f(x,k=0i1uk)Duf(x,k=0i1uk)ui.

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

ui=(La(Ni))1Ei1,(3.16)

where

Ei1=La(k=0i1uk)εΠ(Ni)f(x,k=0i1uk)=ε(IΠ(Ni1))Π(Ni)f(x,k=0i1uk).

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

Ei:=Ri1=εΠ(Ni)(f(x,k=0i1uk)f(x,k=0iuk)Duf(x,k=0i1uk)ui),(3.17)

E~i=εΠ(Ni)f(x,k=0i1uk).(3.18)

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

u=k=0ukHσ¯B1(0),

where 𝓑1(0) := {u|∥us ≤ 1, ∀s > > 0}.

Proof

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

uiσi=(La(Ni))1Ei1σiC(αi)Niτ+κ0(1+ες1ui1σi1p)3Ei1σi1C(αi)(1+ες1ui1σi1p)3E~i1σi1+τ+κ02C(αi)E~i1σi1+τ+κ0,(3.19)

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

Eiσi=εΠ(Ni)(f(x,k=0iuk)f(x,k=0i1uk)Duf(x,k=0i1uk)ui)σiεc(s)uiσipεc(s)Ni(τ+κ0)pCp(αi)Ei1σi1p(εc(s))p+1Ni(τ+κ0)pNi1(τ+κ0)p2Cp(αi)Cp2(αi1)Ei2σi2p2(εc(s))k=1i1pk+1N0(τ+κ0)pi+2E0σ0pik=1iCpk(αi+1k)(εc(s))pi(ε,ς)(N0(τ+κ0)p2E0σ0)pik=1iCpk(αi+1k)(εc(s))pi(ε,ς)E~0σ0+(τ+κ0)p2pik=1iCpk(αi+1k)(8p2εc(s)cp2(τ,σ,σ~,y1,y)E~0σ0+(τ+κ0)p2)pi.(3.20)

Hence, choosing small ε > 0 such that

8p2εc(s)cp2(τ,σ,σ~,y1,y)E~0σ0+(τ+κ0)p2=8p2εc(s)cp2(τ,σ,σ~,y1,y)N0(τ+κ0)p2E~0σ0<1.

For any fixed p > 1, we derive

limiEiσi=0.(3.21)

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

uiσi=(La(Ni))1Ei1σiC(αi)Niτ+κ0(1+ες1ui1σi1p)3Ei1σi12ε3ς3C(αi)ui1σi13pE~i1σi1+τ+κ0(2ες1)3(p+1)C(αi)C3p(αi1)ui2σi2(3p)2E~i2σi2+τ+κ03pE~i1σi1+τ+κ0(2ες1)k=0i1(3p)ku0σ0(3p)ik=1iC(3p)k1(αi+1k)E~ikσik+τ+κ0(3p)k1.(3.22)

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

u:=k=0ukHσ¯B1(0),

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.

Proof

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

B1(0):={u|us<δ,forsomeδ<1,s>σ0}.

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

LahεΠ(Ni)Duf(x,u)hεΠ(Ni)(f(x,u)f(x,u~)Duf(x,u)h=0,

which implies that

h=ε(LaεΠ(Ni)Duf(x,u))1Π(Ni)(f(x,u)f(x,u~)Duf(x,u)h).(3.23)

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

hσi=ε(LaNi)1Π(Ni)(f(x,u)f(x,u~)Duf(x,u)h)σiC(αi)Niτ+κ0(1+ες1uσi1p)hσi1p2p+1Ni(τ+κ0)Ni1(τ+κ0)pC(αi)Cp(αi1)hσi2p2k=0i1pkN0(τ+κ0)(k=0i1pk)hσ0pik=1iCpk1(αi+1k)(8p2cp2(ε,ς,τ,s,s~,y1,y)N0(τ+κ0)phσ0)pi.

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

limihσ¯=0.

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

Yy1,κ0(N):={(δ,q)[0,δ0)×H(N)|qσ¯1,εδsatisfies(2.5)(2.6)},Ur(N):={uC1([0,δ0),H(N))|uσ¯1,δuσ¯r},Gy1,κ0(N):={δ[0,δ0)|(δ,u(δ))Yy1,κ0(N)anduUr(N)},Gr:={δ[0,δ0)|(La(r)(δ,q(δ)))14rκy1},G:={δ[0,δ0)|ω(δ)satisfies(2.5)}.

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

Gy1,κ0(N)=1rNGrG.

Choosing κ and y1 such that

κmax{τ,2+d+n+2q22q1(τ+2ϱ)},y1(0,y2],fory2y1.(3.24)

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

|(Gy1,κ0(M)(0))c[0,δ)|Cy1δ,δ(0,δ0].(3.25)

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

|(Gy1,κ0(N)(u2))c(Gy1,κ0(N)(u1))c[0,δ)|Cy1δN1,δ(0,δ](3.26)

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

Proof

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

|(Gy1,κ0(N)(u2))c(Gy1,κ0(N)(u1))c[δ12,δ1)|Cy1δN1,δ1[0,δ0].

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

(Gy1,κ0(N)(u2))c(Gy1,κ0(N)(u1))c=(Gy1,κ0(N)(u2))cGy1,κ0(N)(u1)[rN(Grc(u2)Gr(u1)G)][r>NGrc(u2)G].

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

Ω:=Nε<rN|Grc(u2)Gr(u1)|+r>max{N,Nε}|Grc(u2)|Cy1δ1N1.

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

La(r)(u2)La(r)(u1)0=O(εu2u1s0)=O(εNe).

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

Grc(u2)Gr(u1){δ[δ12,δ1]|atleataneigenvalueofLa(r)(δ,u1)withmodulusin[4y1rτ,4y1rτ+CεNe]}.

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

|{δ[δ12,δ]s.t.atleastaneigenvalueofL(r)(δ,u1)belongstoI}|Crd+n+1δ1(2q2)|I|.(3.27)

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

(δλk(δ,u1))maxhEδ,k,h0=1(δL(r))(δ,u1)h,h0maxhEδ,k,h0=1(2q1)δ2q2(qh,h)0+O(εy11)maxhEδ,k,h0=1(2q1)δ2q2qh02+O(εy11)maxhEδ,k,h0=1(2q1)δ2q2h02+O(εy11)(2q1)δ2q2+O(εy11)2(q1)δ12q2.

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

Thus, we obtain

|Grc(u2)Gr(u1)|Cεrd+n+1δ1(2q2)NeCδ1Nerd+n+1.

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

Ω=Nε<rN|Grc(u2)Gr(u1)|+r>max{N,Nε}|Grc(u2)|Cδ1(rNrd+n+1)Ne+Cy1δ1(2q2)r>max{N,Nε}rd+nτ+1Cδ1Nd+ne+2+y1δ1(2q2)(max{N,Nε})d+nτ+2Cy1δ1N1,

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

Acknowledgements

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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