Existence of standing waves for quasi-linear Schrödinger equations on T n

where U is a complex-valued functions of (t, x), Tn is a n-dimensional at torus with n ≥ 3,4 is the LaplaceBeltrami operator, q ≥ 1, the terms a(s) and f (x, s) satisfy gauge invariant, i.e. a(eiφs) = eiφa(s) for almost every x ∈ Tn, all φ ∈ R and s ≥ 0. The problem in this general setting arises in various elds of mathematical physics, such as the superuid lm equation of uid 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 at-torus Tn. A standing wave is a solution of the form


Introduction and Main Results
This paper considers the quasi-linear Schrödinger equation where U is a complex-valued functions of (t, x), T n is a n-dimensional at torus with n ≥ , is the Laplace-Beltrami operator, q ≥ , the terms a(s) and f (x, s) satisfy gauge invariant, i.e. a(e iφ s) = e iφ a(s) for almost every x ∈ T n , all φ ∈ R and s ≥ . The problem in this general setting arises in various elds of mathematical physics, such as the superuid lm equation of uid 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 R n . 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 at-torus T n .
A standing wave is a solution of the form We make assumptions on nonlinear terms f , which include the standard tame estimates and Taylor tame estimates. f ∈ C ∞ (T n × R, R), f ( , ) = , ∂u f (x, ) = · · · = (∂ p− u )f (x, ) = , ∂ p u f (x, ) ≠ , ≤ p ≤ k, k ≥ and where s > s > , p > , ∀u, u ∈ Hs such that u s ≤ and u s ≤ . In particular, for s = s, In fact, when p = , assumption (1.3) and (1.4) are natural for f ∈ C ∞ (T n × R, R), which are tame estimates and Taylor tame estimates, respectively. Rescaling in (1.2) amplitude u(x) → δu(x), δ > , we solve the following problem 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 ≥ , the function a(x) is smooth and f (x) is ( π) -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 ≥ ).
We will divide into two cases to discuss the existence of solutions for (1.5). The rst case is a(x) = ax, where a ≠ is a constant, then the "small divisor" phenomenon appears. The second case is a(·) ∈ C ∞ (R). The second case is simpler than the rst case, and we can use the Nash-Moser iteration scheme constructed in the rst case to solve it. In what follows, we deal with the rst case, i.e. a((− ) q q u) = (− ) q a q u. Thus we can rewrite (1.5) as Assume that a is an irrational number and diophantine, i.e. there are constants γ > , τ > , such that (1.7) Then there exist γ > and τ > such that the rst order Melnikov nonreonance condition where ω j ; = |j| and j ∈ Z n . Our main results are based on the Nash-Moser iterative scheme, which is rstly 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 at torus.
We de ne the Sobolev scale of Hilbert spaces For the second case, we consider equation (1.5) and assume that a ∈ C ∞ (R), a( ) = , and where s > s > , < p ≤ k, ∀u, u ∈ Hs such that u s ≤ and u s ≤ . In particular, for s = s, For the second case, we have 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.

Analysis of the Linearized operator
This section is devoted to prove the invertible of linearized operator (2.1) We de ne the nite dimensional subspace of Hs as where A is a nite and symmetric subset of Z n+ and e j (x) = e ij·x . For ∀h = j∈Z n h j e j ∈ Hs, We denote which is a L -orthogonal projector on H A .
. Then the operator (2.1) can be de ned on We write the linearized operator in (2.2) by the block matrix In the L -orthonormal basis (e j ) j∈Ω N of H Ω N , D is represented a diagonal matrix with eigenvalues whereas T is represented by the self-adjoint Toepliz matrix (b j−j ) j,j ∈Ω N , the b j is the Fourier coe cients of the function b(x). Now we give the main result in this section.
Then the linearized operator L (N) (δ, u) is invertible and ∀ s > s >σ > , the linearized operator L (N) a satis es For xing ς > , we de ne 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 satis es (2.5). There exists ς (γ) such that for ς ∈ ( , ς (γ)] and a partition of the singular sites S which can be partitioned in pairwise disjoint clusters Ωα as We de ne the polynomially localized block matrices ). If s > s, then these holds A s ⊂ As.
The next lemma (see [4]) shows the algebra property of As and interpolation inequality. (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.
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 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 is totally convergent in | · |s with |L − R |s ≤ ς − , by taking ες − |T|s ≤ c(s ) small enough. Using (2.10) and (2.14), we have that ∀m ∈ N, which together with (2.18) implies that for ες − |T|s < c(s ) small enough, (2.16) holds.
By non-resonance condition (1.8) and sup x> (x y e −x ) = (ye − ) y , ∀y ≥ , we derive Then by (2.19), for any h ∈ H R , Thus using interpolation (2.12) and (2.16), we derive that for s < s < s , This completes the proof.
To show D is invertible, we only need to prove that Lα is invertible, ∀α ∈ l N .

Lemma 2.5. ∀α ∈ l N , Lα is invertible and L
The proof process of above Lemma is similar with Lemma 6.6 in [4], so we omit it.
The following result is taken from [4], so we omit the proof.

Nash-Moser-type iteration scheme
We de ne the nite dimensional subspaces Then we have the orthogonal splitting where i denotes the "i"th iterative step and ∀s ≤ k. For a given suitable N > , we take N i ≤ N i+ and and where La := − − µ + εa q .
The linearized operator of (3.2) has the following form where D denotes the Frechet derivative. By (3.2), we de ne Next we construct the rst step approximation.
Proof. Assume that the th step approximation solution u satis es Then the target is to get the th step approximation solution. Denote By (3.4), we have Then taking 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 de neẼ := −εΠ (N ) f (x, u ). (3.10)

Lemma 3.2.
Assume that a is diophantine. Then for any < α < σ, the following estimates hold: where C(α) is de ned in (3.12).
From the de nition of u in (3.5), by (2.7), (3.1) and (3.10), we derive By assumption (1.4) and the de nition of E , we have This completes the proof.
For i ∈ N and < σ <σ < σ < k − , set By (3.14)- (3.15), it follows that De ne P (u ) := u + u , for u ∈ H (N ) σ , In fact, to obtain the "i th" approximation solution u i ∈ H (N i ) σ i of system (3.4), we need to solve following equations Then, we get the 'i th' step approximation u i ∈ H (N i ) σ i : where As done in Lemma 3.2, it is easy to get that Hence, we only need to estimate R i− to prove the convergence of algorithm. In the following, a su cient 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).  (3.19) where c(ε, ς) is a constant depending on ε and ς. Note that N i = N i , ∀i ∈ N. By (3.17)- (3.19) and assumption (1.4), we have Hence, choosing small ε > such that For any xed p > , we derive lim But we will choose the initial step u = in this paper, which combining with (3.22) leads to u i σ i = , ∀i ∈ N. This contradicts with assumption ες − u i− p σ i− > . Hence, the case is not possible. Write h = u −ũ. Our target is to prove h = . By (3.4), we have . (3.23) Note that N i = N i , ∀i ∈ N. Thus, by (2.7) and (3.23), we have ≤ ( p c p (ε, ς, τ, s,s, γ , γ)N (τ+κ )p h σ ) p i .
Remark 3.1. The dependence upon the parameter, as is well known, is more delicated since it involves in the small divisors of ω j : it is, however, standard to check that this dependence is C on a bounded set of Diophantine numbers, for more details, see, for example, [3,4].