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Volume 9, Issue 1

# 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.

MSC 2010: 35J60; 35R03; 35B10

## 1 Introduction and Main Results

This paper considers the quasi-linear Schrödinger equation

$iUt−△U−a(∇qU)=f(x,|U|)U,(t,x)∈R×Tn,$(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

$−△u−μu−a(∇qu)=f(x,|u|)u.$(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) = ⋯ = $\begin{array}{}\left({\mathrm{\partial }}_{u}^{p-1}\right)f\left(x,0\right)=0,{\mathrm{\partial }}_{u}^{p}f\left(x,0\right)\ne 0,1\le p\le k,k\ge 2\end{array}$ and

$∥∂uf(x,u′)u∥s≤c(s)(∥u∥sp−1+∥u′∥s∥u∥s0p−1),$(1.3)

$∥f(x,u+u′)−f(x,u′)−Duf(x,u′)u∥s≤c(s)(∥u′∥s∥u∥s0p−1+∥u∥s0∥u∥sp−1),$(1.4)

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

$∥f(x,u+u′)−f(x,u′)−Duf(x,u′)u∥s≤c(s)∥u∥sp.$

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

$−△u−μu−εa(∇qu)=εf(δ,u),$(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

$−△u+u+εa(∇qu)=f(x), x∈R2,$

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

$−△u−μu−(−1)qεa△qu=εf(δ,u).$(1.6)

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

$|m−an|≥y0|n|τ0, ∀(m,n)∈Z2∖{(0,0)}.$(1.7)

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

$|ωj2−μ−εaωj2q|≥y|j|τ,$(1.8)

where $\begin{array}{}{\omega }_{j}^{2};=|j{|}^{2}\end{array}$ 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

$Hs:=Hs(Tn,C)={u(x)=∑j∈Tneij⋅xuj, uj∗=u−j|∥u∥s2:=∑j∈Tne2|j|s|uj|2<+∞}$

for some $\begin{array}{}s>\frac{n+1}{2}.\end{array}$ 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

$u(x,ε)∈C1([0,δ0];Hs0) with ∥u(δ)∥s0=O(δ).$

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

$∥∂ua(u′)u∥s≤c(s)(∥u∥sp−1+∥u′∥s∥u∥s0p−1),$(1.9)

$∥a(u+u′)−a(u′)−Dua(u′)u∥s≤c(s)(∥u′∥s∥u∥s0p−1+∥u∥s0∥u∥sp−1),$(1.10)

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

$∥a(u+u′)−a(u)−Dua(u)u∥s≤c(s)∥u∥sp.$

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

$L:=−△−μ−(−1)qεa△q−ε∂uf(δ,u).$(2.1)

We define the finite dimensional subspace of Hs as

$HA:=Spanj∈Aej={∑k∈Ahjej:hj∈C,hj∗=h−j},$

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

For ∀ h = ∑jZn hj ejHs, We denote

$PAh=∑j∈Ahjej,$

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.

$h↦L(N)[h]:=Lah+εPΩN(b(x)h), ∀h∈H(N),$(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

$Dj:=|j|2−m−εa|j|2q,$(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

$|m−an|≥y1max(1,|m|32), 0(2.5)

anduσ̄ ≤ 1, ∀ 1 ≤ rN, ∀ κ ≥ 1,

$∥(La(r)(δ,u))−1∥0≤4rκy1.$(2.6)

Then the linearized operator L(N)(δ, u) is invertible ands2 > s1 > σ̄ > 0, the linearized operator $\begin{array}{}{L}_{a}^{\left(N\right)}\end{array}$ satisfies

$∥(L(N)(δ,u))−1h∥s1≤C(s2−s1)Nτ+κ01+ες−1∥u∥s2p3∥h∥s2,$(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

$R:={j∈ΩN||Dj|≥ς} and S:={j∈ΩN||Dj|<ς}.$(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

$S=⋃α∈AΩα$(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

$As:={A=(Ajj′)j,j′∈Zn:|A|s2:=supj∈Zn∑j′∈Zne2s|j−j′|∥Ajj′∥02<∞},$

where $\begin{array}{}\parallel {A}_{j}^{{j}^{\prime }}{\parallel }_{0}:=\underset{u\in {\mathbf{H}}^{\left(N\right)},\parallel u{\parallel }_{0}=1}{sup}\parallel {A}_{j}^{{j}^{\prime }}u{\parallel }_{0}\end{array}$ 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

$|AB|s≤c(s)|A|s|B|s, ∀A,B∈As, s>s0>r+n+12,$(2.10)

$|AB|s≤c(s)(|A|s|B|s0+|A|s0|B|s), s≥s0,$(2.11)

$∥Au∥s≤c(s)(|A|s∥u∥s0+|A|s0∥u∥s), ∀u∈Hs, s≥s0.$(2.12)

By Lemma 2.2, we can get, ∀ mN,

$|Am|s≤c(s)m−1|A|sm,$(2.13)

$|Am|s≤m(c(s)|A|s0)m−1|A|s.$(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 $\begin{array}{}T=\left({T}_{j}^{{j}^{\prime }}{\right)}_{j,{j}^{\prime }\in {J}_{N}^{+}}\end{array}$ defined in (2.3) is self-adjoint and belongs to the algebra of polynomially localized matrices 𝓐s with

$|T|s≤K(s)∥b∥s+s′.$

Moreover, for any s > s′,

$|T|s≤K′(s)Ns′∥b∥s.$

Since the decomposition

$H(N):=HR⊕HS,$

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

$L(N)=LRLRSLSRLS,$

where $\begin{array}{}{L}_{R}^{S}=\left({L}_{S}^{R}{\right)}^{†},{L}_{R}={L}_{R}^{†},{L}_{S}={L}_{S}^{†}.\end{array}$

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

$(L(N))−1=I−LR−1LRS0ILR−100L−1I0−LSRLR−1I,$(2.15)

where the “quasi-singular” matrix

$L:=LS−LSRLR−1LRS∈As(S).$

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

$IS(L−ISLIRL~−1IRLIS)IS∈As,$

where

$L~−1=I00LR.$

#### Lemma 2.4

Assume that a is diophantine. For $\begin{array}{}{s}_{0}<{s}_{1}<{s}_{2} the operator LR satisfies

$|L~R−1|s1≤c(s1)(1+ες−1|T|s1),$(2.16)

$∥LR−1h∥s1≤c(y,τ,s2)(s2−s1)−τ(1+ες−1|T|s2)∥h∥s2,$(2.17)

where $\begin{array}{}{\stackrel{~}{L}}^{-1}={L}_{R}^{-1}{D}_{R},c\left(y,\tau ,{s}_{2}\right)\end{array}$ 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 $\begin{array}{}|{D}_{R}^{-1}{|}_{s}\le {\varsigma }^{-1}.\end{array}$ By (2.10), we have that the Neumann series

$L~R−1=LR−1DR=∑m≥0(−ε)m(DR−1TR)m$(2.18)

is totally convergent in |⋅|s1 with $\begin{array}{}|{L}_{R}^{-1}{|}_{{s}_{0}}\le 2{\varsigma }^{-1},\end{array}$ by taking ες−1|T|s0c(s0) small enough.

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

$εm|(DR−1TR)m|s1≤εmc(s)|(DR−1TR)m|s1≤c(s)εmm(c(s)|DR−1TR|s0)m−1|DR−1TR|s1≤c′(s)εmς−1(εc(s1)ς−1|T|s0)m−1|T|s1,$

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

$e−2|j|(s2−s1)|ωj2+μ−εaωj2q|−2≤y−1|j|τe−2|j|(s2−s1)≤c(y,τ)(s2−s1)−2τ.$(2.19)

Then by (2.19), for any hHR,

$∥LR−1h∥s12=∑j∈Re2|j|s1∥LR−1hj∥L22≤∑j∈Re2|j|s1|ωj2+μ−εaωj2q|−2∥L~R−1hj∥L22≤∑j∈Re−2|j|(s2−s1)|ωj2+μ−εaωj2q|−2e2|j|s2∥L~R−1hj∥L22≤c(y,τ)(s2−s1)−2τ∥L~R−1h∥s22.$

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

$∥LR−1h∥s1≤c(y,τ)(s2−s1)−τ∥L~R−1h∥s2≤c(r,τ,s2)(s2−s1)τ(|L~R−1|s2∥h∥s+|L~R−1|s∥h∥s2)≤c(r,τ,s2)(s2−s1)τ(1+ες−1|T|s2)∥h∥s2.$

This completes the proof. □

Next we analyse the quasi-singular matrix 𝓛. By (2.9), the singular sites restricted to $\begin{array}{}{J}_{N}^{+}\end{array}$ are

$S=⋃α∈lNΩα, where lN:={α∈N|mα≤N}.$

Since the decomposition S := ⨁αlN α, where Hα := ⨁jΩα𝓝j, we represent 𝓛 as the block matrix 𝓛 = $\begin{array}{}\left({\mathcal{L}}_{\alpha }^{\beta }{\right)}_{\alpha ,\beta \in {l}_{N}},\end{array}$ where $\begin{array}{}{\mathcal{L}}_{\alpha }^{\beta }:={\mathit{\Pi }}_{{\mathbf{H}}_{\alpha }}\mathcal{L}{|}_{{\mathbf{H}}_{\beta }}.\end{array}$ So we can rewrite

$L=D+J,$

where 𝓓 := diagαlN(𝓛α), $\begin{array}{}{\mathcal{L}}_{\alpha }:={\mathcal{L}}_{\alpha }^{\alpha },\mathcal{J}:=\left({\mathcal{L}}_{\alpha }^{\beta }{\right)}_{\alpha \ne \beta }.\end{array}$

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 $\begin{array}{}\parallel {\mathcal{L}}_{\alpha }^{-1}{\parallel }_{0}\le C{y}_{1}^{-1}{M}_{\alpha }^{\kappa }.\end{array}$

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

$∥D−1D¯h∥s1≤c(ς,s1,y1)Nτ∥h∥s2,$

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

#### Proof

Note that $\begin{array}{}\parallel {h}_{\alpha }{\parallel }_{0}\le {m}_{\alpha }^{-{s}_{1}}\parallel {h}_{\alpha }{\parallel }_{{s}_{1}}\end{array}$ and Mα = 2mα. So for any h = ∑αlN hαHα, hαHα,

$∥D−1D¯h∥s12=∑α∈lN∥Lα−1L¯αhα∥s12≤∑α∈lNMα2s1∥Lα−1L¯αhα∥02≤cy1−2∑α∈lNMα2(s1+τ)∥L¯αhα∥02≤cy1−2∑α∈lNMα2(s1+τ)mα−2s1∥L¯αhα∥s12≤cy1−24s1∑α∈lNMα2τ∥L¯αhα∥s12≤cy1−24s1N2τ∑α∈lN∥L¯αhα∥s12=cy1−24s1N2τ∥D¯h∥s12.$(2.20)

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

$∥D−1D¯h∥s1≤cy1−12s1Nτ∥D¯h∥s1≤cy1−12s1Nτ(|D¯|s2∥h∥s1+|D¯|s1∥h∥s2)≤c(ς)y1−12s1+1Nτ∥h∥s2.$

This completes the proof. □

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

#### Lemma 2.7

s ≥ 0, ∀ mN, there hold:

$c(s1)∥D−1J∥s0<12, ∥D−1∥s≤c(s)y1−1Nτ,$(2.21)

$∥(D−1J)mh∥s≤(εy−1K(s))m(mNκ0|T|s|T|s0m−1∥h∥s0+|T|s0m∥h∥s).$(2.22)

#### Lemma 2.8

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

$∥L−1h∥s1≤c(ς,τ,s1,y1,y)Nτ+κ0(s3−s2)−τ(∥h∥s3+ε|T|s1∥h∥s2).$(2.23)

#### Proof

The Neumann series

$L−1=(I+D−1J)−1D−1=∑m≥0(−1)m(D−1J)mD−1$(2.24)

is totally convergent in operator norm ∥⋅∥s0 with $\begin{array}{}\parallel {\mathcal{L}}^{-1}{\parallel }_{{s}_{0}}\le c{y}_{1}^{-1}{N}^{\tau },\end{array}$ by using (2.21).

By (2.22) and (2.24), we have

$∥L−1h∥s1≤∥D−1h∥s1+∑m≥1∥(D−1J)mD−1h∥s1≤∥D−1h∥s1+∥D−1h∥s1∑m≥1(εy1−1K(s)|T|s0)m+Nκ0K(s1)εy1−1|T|s1∥D−1h∥s0∑m≥1m(K(s)εy1−1|T|s0)m−1.$(2.25)

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

$∥D−1h∥s12=∥D−1D¯D¯−1h∥s12≤c2(ς,s1,y1)N2τ∥D¯−1h∥s22=c2(ς,s1,y1)N2τ∑j∈Se2|j|s2∥D¯−1hj∥L22≤c2(ς,s1,y1)N2τ∑j∈Se2|j|s2|ωj2+μ−εaωj2q|−2∥hj∥L22≤c2(ς,s1,y1)N2τ∑j∈Se−2|j|(s3−s2)|j|−2e2|j|s3∥hj∥L22 ≤c2(ς,τ,s1,y1,y)N2τ(s3−s2)−2τ∥h∥s32.$(2.26)

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

$∥L−1h∥s1≤y1−1Nκ0K′(s1)(∥D−1h∥s1+ε|T|s1∥D−1h∥s0)≤c(ς,τ,s1,y1,y)Nτ+κ0(s3−s2)−τ(∥h∥s3+ε|T|s1∥h∥s2),$(2.27)

where 0 < s1 < s2 < s3 and $\begin{array}{}\epsilon {y}_{1}^{-1}\end{array}$ 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),

$∥(L(N))−1h∥s1≤∥LR−1hR+LR−1LSRL−1(hS+LRSLR−1hR)∥s1+∥L−1(hR+LRSLR−1hR)∥s1≤∥LR−1hR∥s1+∥LR−1LSRL−1hS∥s1+∥LR−1LSRL−1LRSLR−1hR∥s1+∥L−1hR∥s1+∥L−1LRSLR−1hR∥s1.$(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

$∥LR−1LSRL−1hS∥s1≤c(y,τ,s2)(s2−s1)−τ(1+ες−1|T|s2)∥LSRL−1hS∥s2≤c(y,τ,s2)(s2−s1)−τ(1+ες−1|T|s2)|T|s2∥L−1h∥s2≤c(y,y1,ς,τ,s2)(s2−s1)−τ(s4−s3)−τNτ+κ0×(1+ες−1|T|s2)|T|s2(∥h∥s3+ε|T|s2∥h∥s4),$(2.29)

$∥L−1LRSLR−1hR∥s1≤c(ς,τ,s1,y1,y)Nτ+κ0(s3−s2)−τ(∥LRSLR−1hR∥s3+ε|T|s1∥LRSLR−1hR∥s2)≤c(ς,τ,s1,s2,s3,y1,y)Nτ+κ0(s3−s2)−τ×(|T|s3∥LR−1hR∥s3+ε|T|s1|T|s2∥LR−1hR∥s2)≤c(ς,τ,s1,s2,s3,y1,y)Nτ+κ0(s3−s2)−τ(|T|s3(s4−s3)−τ(1+ες−1|T|s4)∥h∥s4+ε|T|s1|T|s2(s3−s2)−τ(1+ες−1|T|s3)∥h∥s3)≤c(ς,τ,s1,s2,s3,y1,y)Nτ+κ0(s3−s2)−τ|T|s3(1+ες−1|T|s4)×((s4−s3)−τ∥h∥s4+ε|T|s2(s3−s2)−τ∥h∥s3),$(2.30)

$∥LR−1LSRL−1LRSLR−1hR∥s1≤c(y,τ,s2)(s2−s1)−τ(1+ες−1|T|s2)∥LSRL−1LRSLR−1hR∥s2≤c(y,τ,s2)(s2−s1)−τ(1+ες−1|T|s2)|T|s2∥L−1LRSLR−1hR∥s2≤c(ς,τ,s1,s2,s3,y1,y)Nτ+κ0(s3−s2)−τ(s2−s1)−τ|T|s32×(1+ες−1|T|s4)2((s4−s3)−τ∥h∥s4+ε|T|s2(s3−s2)−τ∥h∥s3).$(2.31)

The terms $\begin{array}{}\parallel {L}_{R}^{-1}{h}_{R}{\parallel }_{{s}_{1}}\end{array}$ and ∥𝓛−1hRs1 can be controlled by using (2.17) and (2.23). Thus by (2.28)-(2.31), for 0 < s < , we conclude

$∥(L(N))−1h∥s≤c(ς,τ,s,s~,y1,y)Nτ+κ0(1+ες−1|T|s~)3(s~−s)−τ∥h∥s~,$

which together with Lemma 2.8 gives (2.7).

## 3 Nash-Moser-type iteration scheme

We define the finite dimensional subspaces

$Hs(Ni)={u∈Hs|u=∑|j|≤Niujeij⋅x},$

$Hs(Ni)⊥={u∈Hs|u=∑|j|>Niujeij⋅x}.$

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 = $\begin{array}{}{N}_{0}^{i}\end{array}$, ∀iN.

The orthogonal projectors onto $\begin{array}{}{\mathbf{H}}_{s}^{\left({N}_{i}\right)}\end{array}$ and $\begin{array}{}{\mathbf{H}}_{s}^{\left({N}_{i}\right)\perp }\end{array}$ denote by Π(Ni) : Hs$\begin{array}{}{\mathbf{H}}_{s}^{\left({N}_{i}\right)}\end{array}$ and Π(Ni)⊥ : Hs$\begin{array}{}{\mathbf{H}}_{s}^{\left({N}_{i}\right)\perp }\end{array}$, which satisfy the “smoothing” properties:

$∥Π(Ni)u∥s+d≤eNid∥u∥s, ∀u∈Xs, ∀s, d≥0,∥Π(Ni)⊥u∥s≤Ni−d∥u∥s+d, ∀u∈Xs+d, ∀s, d≥0.$(3.1)

Consider

$Lau=εf(x,u),$(3.2)

where

$La:=−△−μ+εa△q.$

The linearized operator of (3.2) has the following form

$La(Ni):=Π(Ni)(La−εDuf(δ,u))|Hs(Ni),$(3.3)

where D denotes the Frechet derivative.

By (3.2), we define

$J(u)=Lau−εΠ(Ni)f(x,u)=0.$(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 u1$\begin{array}{}{\mathbf{H}}_{s}^{\left({N}_{1}\right)}\end{array}$

$u1=−(La(N1))−1E0∈Hs(N1),$(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

$E0=Lau0−εΠ(N1)f(x,u0).$(3.7)

By (3.4), we have

$J(u0+u1)=La(u0+u1)−εΠ(N1)f(x,u0+u1)=Lau0−εΠ(N1)f(x,u0)+Lau1+εΠ(N1)Duf(x,u0)u1−εΠ(N1)(f(x,u0+u1)−f(x,u0)−Duf(x,u0)u1)=E0+La(N1)u1+R0.$(3.8)

Then taking

$E0+La(N1)u1=0,$

yields

$u1=−(La(N1))−1E0∈Hs(N1).$

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

$E0=−ε(I−Π(N0))Π(N1)f(x,u0).$(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:

$∥u1∥σ−α≤C(α)(1+ες−1∥u0∥σp)3∥E~0∥σ+τ+κ0,$

$∥E1∥σ−α≤Cp(α)(1+ες−1∥u0∥σp)3p∥E~0∥σ+τ+κ0p,$(3.11)

where C(α) is defined in (3.12).

#### Proof

Denote

$C(α)=c(ς,τ,s,s~,y1,y)α−τ.$(3.12)

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

$∥u1∥σ−α=∥−(La(N1))−1E0∥σ−α≤C(α)N1τ+κ0(1+ες−1∥u0∥σp)3∥E0∥σ≤C(α)(1+ες−1∥u0∥σp)3∥E~0∥σ+τ+κ0.$(3.13)

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

$∥E1∥σ−α=∥Π(N1)(f(x,u0+u1)−f(x,u0)−Duf(x,u0)u1)∥σ−α≤∥u1∥σ−αp≤Cp(α)(1+ες−1∥u0∥σp)3p∥E~0∥σ+τ+κ0p.$

This completes the proof.□

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

$σi:=σ¯+σ−σ¯2i,$(3.14)

$αi+1:=σi−σi+1=σ−σ¯2i+1.$(3.15)

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

$σ0>σ1>…>σi>σi+1>…, for i∈N.$

Define

$P1(u0):=u0+u1, for u0∈Hσ0(N0),$

$Ei=J(∑k=0iuk)=J(P1i(u0)),$

In fact, to obtain the “i th" approximation solution ui$\begin{array}{}{\mathbf{H}}_{{\sigma }_{i}}^{\left({N}_{i}\right)}\end{array}$ of system (3.4), we need to solve following equations

$J(∑k=0iuk)=La(∑k=0i−1uk)−εΠ(Ni)f(x,∑k=0i−1uk)+Laui−εΠ(Ni)Duf(x,∑k=0i−1uk)ui−εΠ(Ni)f(x,∑k=0iuk)−f(x,∑k=0i−1uk)−Duf(x,∑k=0i−1uk)ui.$

Then, we get the ‘i th‘ step approximation ui$\begin{array}{}{\mathbf{H}}_{{\sigma }_{i}}^{\left({N}_{i}\right)}\end{array}$:

$ui=−(La(Ni))−1Ei−1,$(3.16)

where

$Ei−1=La(∑k=0i−1uk)−εΠ(Ni)f(x,∑k=0i−1uk)=−ε(I−Π(Ni−1))Π(Ni)f(x,∑k=0i−1uk).$

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

$Ei:=Ri−1=−εΠ(Ni)(f(x,∑k=0i−1uk)−f(x,∑k=0iuk)−Duf(x,∑k=0i−1uk)ui),$(3.17)

$E~i=−εΠ(Ni)f(x,∑k=0i−1uk).$(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

$u∞=∑k=0∞uk∈Hσ¯∩B1(0),$

where 𝓑1(0) := {u|∥us ≤ 1, ∀s > > 0}.

#### Proof

We divide into two cases. If $\begin{array}{}\epsilon {\varsigma }^{-1}\parallel {u}_{i-1}{\parallel }_{{\sigma }_{i-1}}^{p}<1,\end{array}$ by (2.7), (3.16) and (3.18), we derive

$∥ui∥σi=∥−(La(Ni))−1Ei−1∥σi≤C(αi)Niτ+κ0(1+ες−1∥ui−1∥σi−1p)3∥Ei−1∥σi−1≤C(αi)(1+ες−1∥ui−1∥σi−1p)3∥E~i−1∥σi−1+τ+κ0≤2C(αi)∥E~i−1∥σi−1+τ+κ0,$(3.19)

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

Note that Ni = $\begin{array}{}{N}_{0}^{i}\end{array}$, ∀iN. By (3.17)-(3.19) and assumption (1.4), we have

$∥Ei∥σi=ε∥Π(Ni)(f(x,∑k=0iuk)−f(x,∑k=0i−1uk)−Duf(x,∑k=0i−1uk)ui)∥σi≤εc(s)∥ui∥σip≤εc(s)Ni(τ+κ0)pCp(αi)∥Ei−1∥σi−1p≤(εc(s))p+1Ni(τ+κ0)pNi−1(τ+κ0)p2Cp(αi)Cp2(αi−1)∥Ei−2∥σi−2p2≤⋯≤(εc(s))∑k=1i−1pk+1N0(τ+κ0)pi+2∥E0∥σ0pi∏k=1iCpk(αi+1−k)≤(εc(s))pi(ε,ς)(N0(τ+κ0)p2∥E0∥σ0)pi∏k=1iCpk(αi+1−k)≤(εc(s))pi(ε,ς)∥E~0∥σ0+(τ+κ0)p2pi∏k=1iCpk(αi+1−k)≤(8p2εc(s)cp2(τ,σ,σ~,y1,y)∥E~0∥σ0+(τ+κ0)p2)pi.$(3.20)

Hence, choosing small ε > 0 such that

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

For any fixed p > 1, we derive

$limi⟶∞∥Ei∥σi=0.$(3.21)

If $\begin{array}{}\epsilon {\varsigma }^{-1}\parallel {u}_{i-1}{\parallel }_{{\sigma }_{i-1}}^{p}\ge 1\end{array}$, by (2.7), (3.16) and (3.18), we derive

$∥ui∥σi=∥−(La(Ni))−1Ei−1∥σi≤C(αi)Niτ+κ0(1+ες−1∥ui−1∥σi−1p)3∥Ei−1∥σi−1≤2ε3ς−3C(αi)∥ui−1∥σi−13p∥E~i−1∥σi−1+τ+κ0≤(2ες−1)3(p+1)C(αi)C3p(αi−1)∥ui−2∥σi−2(3p)2∥E~i−2∥σi−2+τ+κ03p∥E~i−1∥σi−1+τ+κ0≤⋯≤(2ες−1)∑k=0i−1(3p)k∥u0∥σ0(3p)i∏k=1iC(3p)k−1(αi+1−k)∥E~i−k∥σi−k+τ+κ0(3p)k−1.$(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 $\begin{array}{}\epsilon {\varsigma }^{-1}\parallel {u}_{i-1}{\parallel }_{{\sigma }_{i-1}}^{p}>1.\end{array}$ Hence, the case is not possible. (3.2) has a solution

$u∞:=∑k=0∞uk∈Hσ¯∩B1(0),$

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

$B1(0):={u|∥u∥s<δ, for some δ<1, ∀s>σ0}.$

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

$Lah−εΠ(Ni)Duf(x,u)h−εΠ(Ni)(f(x,u)−f(x,u~)−Duf(x,u)h=0,$

which implies that

$h=ε(La−εΠ(Ni)Duf(x,u))−1Π(Ni)(f(x,u)−f(x,u~)−Duf(x,u)h).$(3.23)

Note that Ni = $\begin{array}{}{N}_{0}^{i}\end{array}$, ∀iN. Thus, by (2.7) and (3.23), we have

$∥h∥σi=ε∥(LaNi)−1Π(Ni)(f(x,u)−f(x,u~)−Duf(x,u)h)∥σi≤C(αi)Niτ+κ0(1+ες−1∥u∥σi−1p)∥h∥σi−1p≤2p+1Ni(τ+κ0)Ni−1(τ+κ0)pC(αi)Cp(αi−1)∥h∥σi−2p≤⋯≤2∑k=0i−1pkN0(τ+κ0)(∑k=0i−1pk)∥h∥σ0pi∏k=1iCpk−1(αi+1−k)≤(8p2cp2(ε,ς,τ,s,s~,y1,y)N0(τ+κ0)p∥h∥σ0)pi.$

Choosing δ < 8p2 cp2(ε, ς, τ, s, , y1, y)$\begin{array}{}{N}_{0}^{-\left(\tau +{\kappa }_{0}\right)p},\end{array}$ we obtain

$limi⟶∞∥h∥σ¯=0.$

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

$Yy1,κ0(N):={(δ,q′)∈[0,δ0)×H(N)|∥q′∥σ¯≤1,εδ satisfies (2.5)−(2.6)},Ur(N):={u∈C1([0,δ0),H(N))|∥u∥σ¯≤1, ∥∂δu∥σ¯≤r},Gy1,κ0(N):={δ∈[0,δ0)|(δ,u(δ))∈Yy1,κ0(N) and u∈Ur(N)},Gr:={δ∈[0,δ0)|∥(La(r)(δ,q′(δ)))−1∥≤4rκy1},G:={δ∈[0,δ0)|ω(δ) satisfies (2.5)}.$

Then for a given function $\begin{array}{}\delta ↦{q}^{\prime }\left(\delta \right)\in {U}_{r}^{\left(N\right)},\end{array}$ the set $\begin{array}{}{\mathcal{G}}_{{y}_{1},{\kappa }_{0}}^{\left(N\right)}\end{array}$ is equivalent to

$Gy1,κ0(N)=∩1≤r≤NGr∩G.$

Choosing κ and y1 such that

$κ≥max{τ,2+d+n+2q−22q−1(τ+2ϱ)}, y1∈(0,y2], for y2≤y1.$(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 $\begin{array}{}{\mathcal{G}}_{{y}_{1},{\kappa }_{0}}^{\left(M\right)}\end{array}$(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

$|(Gy1,κ0(N′)(u2))c∖(Gy1,κ0(N)(u1))c∩[0,δ)|≤Cy1δN−1, ∀δ∈(0,δ′]$(3.26)

holds, where N′ ≥ NM, $\begin{array}{}{u}_{1}\in {U}_{{r}^{\prime }}^{\left(N\right)},{u}_{1}\in {U}_{{r}^{\prime }}^{\left(N\right)}\end{array}$ 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 $\begin{array}{}{L}_{a}^{\left(r\right)}\end{array}$ has the form $\begin{array}{}{\omega }_{j}^{2}+\mu -\epsilon a{\omega }_{j}^{2q}-O\left(\epsilon \right)\end{array}$ of the operator $\begin{array}{}{L}_{a}^{\left(r\right)}\end{array}$. Here jZn. For sufficient small ε0y–1Mτ+2q, by (1.8), all the eigenvalues of $\begin{array}{}{L}_{a}^{\left(r\right)}\end{array}$ 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 $\begin{array}{}N,{N}^{\prime }\le {N}_{{\epsilon }_{0}}:=\left(c{y}_{1}{\epsilon }_{0}^{-1}{\right)}^{\frac{1}{\tau +2\varrho }},{\mathcal{G}}_{{y}_{1},{\kappa }_{0}}^{\left({N}^{\prime }\right)}\left({u}_{2}\right)={\mathcal{G}}_{{y}_{1},{\kappa }_{0}}^{\left(N\right)}\left({u}_{1}\right)=\mathcal{G},\end{array}$ by the same process of proof of (3.25), one can prove (3.26) holds. For other cases, it is sufficient to prove

$|(Gy1,κ0(N′)(u2))c∖(Gy1,κ0(N)(u1))c∩[δ12,δ1)|≤Cy1δN−1, ∀δ1∈[0,δ0].$

For fixed δ1 and the decomposition [0, δ0] = ∪n≥1[δ02n, δ02–(n–1)], we consider the complementary sets in $\begin{array}{}\left[\frac{{\delta }_{1}}{2},{\delta }_{1}\right)\end{array}$

$(Gy1,κ0(N′)(u2))c∖(Gy1,κ0(N)(u1))c=(Gy1,κ0(N′)(u2))c∩Gy1,κ0(N)(u1)⊂[∪r≤N(Grc(u2)∩Gr(u1)∩G)]∪[∪r>NGrc(u2)∩G].$

If rNε0, then $\begin{array}{}{\mathcal{G}}_{r}^{c}\left({u}_{2}\right)\cap \mathcal{G}=\mathrm{\varnothing }.\end{array}$ So it is sufficient to prove that, if ∥u1u2σ̄Ne, ed + n + 3, then

$Ω:=∑Nεmax{N,Nε}|Grc(u2)|≤C′y1δ1N−1.$

Note that $\begin{array}{}\parallel \left({L}_{a}^{\left(r\right)}{\right)}^{-1}{\parallel }_{0}\end{array}$ is the inverse of the eigenvalue of smallest modulus and

$∥La(r)(u2)−La(r)(u1)∥0=O(ε∥u2−u1∥s0)=O(εN−e).$

The sufficient and necessary condition of an eigenvalues of $\begin{array}{}{L}_{a}^{\left(r\right)}\end{array}$(u2) in [–4y1rτCεNe, 4y1rτ + CεNe] is that there exists an eigenvalues of $\begin{array}{}{L}_{a}^{\left(r\right)}\end{array}$(u1) in [–4y1rτ, 4y1rτ]. Thus, it leads to

$Grc(u2)∩Gr(u1)⊂{δ∈[δ12,δ1]|∃ at leat an eigenvalue of La(r)(δ,u1) with modulus in [4y1r−τ,4y1r−τ+CεN−e]}.$

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

$|{δ∈[δ12,δ] s.t. at least∃ an eigenvalue of L(r)(δ,u1) belongs to I}|≤Crd+n+1δ1−(2q−2)|I|.$(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 $\begin{array}{}\parallel {\mathrm{\partial }}_{\delta }b{\parallel }_{s}=\parallel \left({\mathrm{\partial }}_{u}^{2}f\right)\left(x,u\right){\parallel }_{s}\le C{y}_{1}^{-1}\end{array}$ and $\begin{array}{}\parallel {\mathrm{\nabla }}^{q}h{\parallel }_{0}^{2}\ge \parallel h{\parallel }_{0}^{2},\end{array}$ for sufficient small 0 < εε0(y1), we have

$(∂δλk(δ,u1))≤maxh∈Eδ,k,∥h∥0=1(∂δL(r))(δ,u1)h,h0≤maxh∈Eδ,k,∥h∥0=1(2q−1)δ2q−2(△qh,h)0+O(εy1−1)≤maxh∈Eδ,k,∥h∥0=1−(2q−1)δ2q−2∥∇qh∥02+O(εy1−1)≤maxh∈Eδ,k,∥h∥0=1−(2q−1)δ2q−2∥h∥02+O(εy1−1)≤−(2q−1)δ2q−2+O(εy1−1)≤−2(q′−1)δ12q−2.$

Hence we have $\begin{array}{}|{\lambda }_{k}^{-1}\left(I,{u}_{1}\right)\cap \left[\frac{{\delta }_{1}}{2},{\delta }_{1}\right]|\le C|I|{\delta }_{1}^{-\left(2q-2\right)}.\end{array}$ The claim holds.

Thus, we obtain

$|Grc(u2)∩Gr(u1)|≤Cεrd+n+1δ1−(2q−2)N−e≤Cδ1N−erd+n+1.$

Furthermore, by (3.27), we have $\begin{array}{}|{\mathcal{G}}_{r}^{c}\left({u}_{2}\right)|\le C{y}_{1}{r}^{d+n-\tau +1}{\delta }_{1}^{-\left(2q-2\right)}.\end{array}$ Therefore, we obtain

$Ω=∑Nεmax{N,Nε}|Grc(u2)|≤Cδ1(∑r≤Nrd+n+1)N−e+Cy1δ1−(2q−2)∑r>max{N,Nε}rd+n−τ+1≤C′δ1Nd+n−e+2+y1δ1−(2q−2)(max{N,Nε})d+n−τ+2≤C″y1δ1N−1,$

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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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,

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