Uniqueness and symmetry of ground states for higher-order equations

Abstract

We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schrödinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for higher-order approximations to the pseudo-relativistic ground state. Our proof adapts the strategy of Lenzmann (Anal PDE 2:1–27, 2009) using local uniqueness near the limit of ground states in a variational problem. However, in order to bypass difficulties from lack of symmetrization tools for higher-order differential operators, we employ the contraction mapping argument in our earlier work (Choi et al. 2017. arXiv:1705.09068) to construct radially symmetric real-valued solutions, as well as improving local uniqueness near the limit.

1 Introduction

Higher-order elliptic equations, whose higher-order differentials have small coefficients, arise in various physical contexts. For instance, in nonlinear optics, the envelope dynamics of wave trains in a weakly nonlinear medium is given by the equation

$$\begin{aligned} i\epsilon ^2\partial _t\psi =\omega (\epsilon \partial ) \psi -\epsilon ^2|\psi |^2\psi , \end{aligned}$$

where $\varepsilon >0$, $\psi =\psi (t,x):{\mathbb {R}}\times {\mathbb {R}}^d\rightarrow \mathbb {C}$ and $\omega (\partial )$ denotes the Fourier multiplier operator with a symbol $a=a(\xi ):{\mathbb {R}}^d\rightarrow {\mathbb {R}}$. Looking for a stationary solution, inserting the ansatz $\psi (t,x)=e^{i\mu t}u(x)$ with $\mu >0$, we obtain the time-independent equation

$$\begin{aligned} \omega (\epsilon \partial )u+\epsilon ^2\mu u=\epsilon ^2|u|^2u. \end{aligned}$$

When high-frequency dispersion is negligible and the medium is isotropic,¹ the above equation can be approximated by the second-order equation

$$\begin{aligned} -\Delta u+\mu u=|u|^2u \end{aligned}$$

(see [15]). However, if high frequency dispersion is weak but not negligible, one should consider a higher-order equation whose differential operator is a Taylor polynomial of $\omega (\epsilon \partial )$. Here, higher-order terms have small coefficients.

In astrophysics, the mean-field limit of stationary boson stars is described by the pseudo-relativistic nonlinear Hartree equation

$$\begin{aligned} \left( \sqrt{-c^2\Delta +m^2c^4}-mc^2\right) u+\mu u=\left( |x|^{-1}*|u|^2\right) {\quad \text {in } {\mathbb {R}}^3,} \end{aligned}$$

(1.1)

where $u=u(x):{\mathbb {R}}^3\rightarrow \mathbb {C}$, $m>0$ is the particle mass and $c>0$ stands for the speed of light. In applications, taking the formal Taylor polynomial of the pseudo-relativistic operator

$$\begin{aligned} \left( \sqrt{-c^2\Delta +m^2c^4}-mc^2\right) =mc^2\left( \sqrt{1-\frac{\Delta }{m^2c^2}}-1\right) =\frac{1}{2m}(-\Delta )-\frac{1}{8m^3c^2}(-\Delta )^2+\cdots , \end{aligned}$$

(1.2)

the higher-order model

$$\begin{aligned} \left( \sum _{j=1}^J\frac{(-1)^{j-1}\alpha _j}{m^{2j-1}c^{2j-2}}(-\Delta )^j\right) u+\mu u=\left( |x|^{-1}*|u|^2\right) u {\quad \text {in } {\mathbb {R}}^3,} \end{aligned}$$

(1.3)

where $\alpha _j=\frac{(2j-2)!}{j!(j-1)!2^{2j-1}}$, is employed to avoid possible complication from having a non-local operator (see [4, 5] and the references therein).

Moreover, given a previously known second-order model

$$\begin{aligned} -\Delta u+\mu u=f(|u|^2)u, \end{aligned}$$

a higher-order equation is sometimes introduced as a refinement taking additional physical effects in account. In this case, it is natural to put small coefficients on higher-order differentials, like

$$\begin{aligned} -\Delta u+\epsilon \Delta ^2u+\mu u=f(|u|^2)u, \end{aligned}$$

for consistency with the second-order model.

The purpose of this paper is to provide a general strategy to prove uniqueness and radial symmetry of ground states for a certain class of higher-order elliptic equations including the above examples.

Before proceeding, it should be pointed out that proving uniqueness and symmetry of ground states for higher-order equations is in general quite challenging. That is because some of useful tools, such as the diamagnetic inequality, the Pólya–Szegö inequality, the moving plane method and the shooting game argument, might not be available. Recall that for second-order equations, the standard variational approach employs the diamagnetic inequality $\Vert \nabla (|u|)\Vert _{L^2}\le \Vert \nabla u\Vert _{L^2}$ in the first step in order to obtain a non-negative ground state from a hypothetical possibly sign-changing ground state, and then symmetrization tools are applied to prove symmetry and uniqueness. When the symbol of a pseudo-differential operator is a Bernstein function, e.g., the pseudo-relativistic operator (1.2), the diamagnetic inequality as well as symmetrization tools can be recovered by a beautiful argument in [6, 13] involving the Bernstein’s theorem. However, this method does not work for higher-order operators.

In fact, some of analytic tools have been developed for higher-order operators e.g., for polyharmonic operators, and there might be a way to apply them for uniqueness and symmetry. For a comprehensive overview, we refer to the book by Gazzola et al. [7]. Nevertheless, they cannot be directly applied to the above examples. Even worse, the desired diamagnetic inequality does not seem to hold for higher-order differential operators, because even if u is smooth, its second derivative $\nabla _{x_j}^2(|u|)$ could be very singular near the set $\{x: u(x)=0\}$. In this paper, we go around the lack of the analytic tools rather than making an effort to build them up, by taking the advantage of higher-order differentials having small coefficients.

From now on, for concreteness of exposition, we restrict ourselves to the higher-order nonlinear Schrödinger equation (NLS)

$$\begin{aligned} P_\epsilon u+ u=|u|^{2k}u {\quad \text {in } {\mathbb {R}}^d}, \end{aligned}$$

(1.4)

where $k\in {\mathbb {N}}$ and $u=u(x):{\mathbb {R}}^d\rightarrow \mathbb {C}$, and the three-dimensional higher-order nonlinear Hartree equation (NLH)

$$\begin{aligned} P_\epsilon u+u=\left( |x|^{-1}*|u|^2\right) u {\quad \text {in } {\mathbb {R}}^3}, \end{aligned}$$

(1.5)

where $u=u(x):{\mathbb {R}}^3\rightarrow \mathbb {C}$. For an even integer J and $\epsilon \ge 0$ (including zero), the higher-order differential operator $P_\epsilon$ is defined by

$$\begin{aligned} P_\epsilon =P_\epsilon ^J=:-\Delta +\sum _{|\alpha |=3}^Jc_\alpha \epsilon ^{|\alpha |-2} (i\nabla )^\alpha , \end{aligned}$$

where $\alpha =(\alpha _1,\ldots ,\alpha _d)\in (\mathbb {Z}_{\ge 0})^d$ denotes a multi-index and $(i\nabla )^\alpha =i^{|\alpha |}\nabla _{x_1}^{\alpha _1}\ldots \nabla _{x_d}^{\alpha _d}$. We assume that the family of operators $\{P_{\epsilon }\}_{0\le \epsilon \le 1}$ is uniformly elliptic in the sense that there exists $\gamma >0$, independent of $\epsilon \ge 0$, such that

$$\begin{aligned} 1+P_\epsilon \ge \gamma (1-\Delta ). \end{aligned}$$

(1.6)

For NLS (1.4), we further assume that $1\le d\le 3$ and

$$\begin{aligned} \left\{ \begin{aligned}&k\in {\mathbb {N}}&\text {if }d=1,2,\\&k=1&\text {if }d=3 \end{aligned}\right. \end{aligned}$$

(1.7)

so that the odd-power nonlinearity is $H^1$-subcritical. We remark that as $\epsilon \rightarrow 0$, the higher-order NLS (1.4) formally converges to the standard second-order NLS

$$\begin{aligned} -\Delta u+ u=|u|^{2k}u, \end{aligned}$$

(1.8)

while the higher-order NLH (1.5) converges to the second-order NLH

$$\begin{aligned} -\Delta u+ u=\left( |x|^{-1}*|u|^2\right) u. \end{aligned}$$

(1.9)

A solution to the higher-order NLS (1.4) (resp., the higher-order NLH (1.5)) is called a ground state if it is a minimizer for the action functional

$$\begin{aligned} I_\epsilon (u):=\left\{ \begin{aligned}&\frac{1}{2}\int _{{\mathbb {R}}^d}(P_\epsilon +1) u\bar{u}\,dx-\frac{1}{2k+2}\int _{{\mathbb {R}}^d}|u|^{2k+2}dx\quad \text {(for NLS }(1.4))\\&\frac{1}{2}\int _{{\mathbb {R}}^3}(P_\epsilon +1) u\bar{u}\,dx-\frac{1}{4}\int _{{\mathbb {R}}^3}\left( |x|^{-1}*|u|^2\right) |u|^2dx\quad \text {(for NLH }(1.5)). \end{aligned}\right. \end{aligned}$$

restricted to the constraint

$$\begin{aligned} \langle I_\epsilon '(u),u\rangle _{L^2}=0\quad \text {and}\quad u\ne 0, \end{aligned}$$

where $I_\epsilon '$ is the Frechét derivative of $I_\epsilon$. When $\epsilon =0$, it is known that the second-order NLS (1.8) (resp., the second-order NLH (1.9)) has a smooth radially symmetric positive ground state, denoted by $Q_0$, and it is unique up to translation and phase shift² (see [1, 8, 10] for NLS, and [12] for NLH). Moreover, the ground state $Q_0$ is non-degenerate (see [10] for NLS and [11] for NLH). Indeed, linearizing the equation near the ground state $Q_0$, we obtain the linearized operator with the identification , where the linear operators ${\mathcal {L}}_0^\pm : H^2({\mathbb {R}}^d;{\mathbb {R}})\rightarrow L^2({\mathbb {R}}^d;{\mathbb {R}})$ are defined by

$$\begin{aligned} {\mathcal {L}}_0^+h:=\left\{ \begin{aligned}&-\Delta h+h-(2k+1)Q_0^{2k}h&\quad \text {(for NLS }(1.8))\\&-\Delta h+h-2\left( |x|^{-1}*(Q_0h)\right) Q_0-\left( |x|^{-1}*Q_0^2\right) h&\quad \text {(for NLH }(1.9)) \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned} {\mathcal {L}}_0^-h:=\left\{ \begin{aligned}&-\Delta h+h-Q_0^{2k}h&\quad \text {(for NLS }(1.8))\\&-\Delta h+h-\left( |x|^{-1}*Q_0^2\right) h&\quad \text {(for NLH } (1.9)), \end{aligned}\right. \end{aligned}$$

By non-degeneracy, we mean that the kernels of ${\mathcal {L}}_0^\pm$ are explicitly given by

$$\begin{aligned} \left\{ \begin{aligned} \text {Ker}{\mathcal {L}}_0^+&={\text {span}}\left\{ \partial _{x_1}Q_0,\ldots , \partial _{x_d}Q_0\right\} ,\\ \text {Ker}{\mathcal {L}}_0^-&=\text {span}\left\{ Q_0\right\} . \end{aligned}\right. \end{aligned}$$

When $\epsilon >0$, by standard variational arguments, one can show that the higher-order NLS (1.4) (resp., the higher-order NLH (1.5)) possesses a ground state $Q_\epsilon ^J$ and that it converges to $Q_0$ as $\epsilon \rightarrow 0$ (see Proposition 2.1).

Our main theorem establishes uniqueness and radial symmetry of ground states for the higher-order equations (1.4) and (1.5). We recall we only care the case $1 \le d \le 3$ and

$$\begin{aligned} \left\{ \begin{aligned}&k\in {\mathbb {N}}&\text {if }d=1,2,\\&k=1&\text {if }d=3 \end{aligned}\right. \end{aligned}$$

for NLS, and $d = 3$ for NLH.

Theorem 1.1

(Uniqueness and symmetry) Suppose that (1.6), as well as (1.7) for NLS and $d = 3$ for NLH), holds. Then, there exists $\epsilon _0>0$ such that for each $0<\epsilon \le \epsilon _0$, there exists a smooth radially symmetric and real-valued ground state $Q_\epsilon ^J$ for the higher-order NLS (1.4) (resp., the higher-order NLH (1.5)), and it is unique up to translation and phase shift. Moreover, the ground state $Q_\epsilon ^J$ is non-degenerate in the sense of Proposition 3.2 below.

Remark 1.2

An alternative concept of ground states to the higher order equations (1.4) or (1.5) can be given by a minimizer of the physical energy $E_\epsilon (u)$ given by $I_\epsilon (u) -\frac{1}{2}\Vert u\Vert ^2_{L^2}$ subject to the fixed mass $\Vert u\Vert ^2_{L^2} = N$, provided the minimizer exists. In second order case $\epsilon = 0$, a simple scaling argument says that this concept of ground states to (1.4) or (1.5) coincides with our one in Theorem 1.1. It is worth mentioning here that it is not known yet whether this also happens or not for the higher order case $\epsilon > 0$.

For the proof, we follow the roadmap in the important work by Lenzmann [11], where uniqueness of ground states for the pseudo-relativistic NLH (1.1) is established. The robust approach of Lenzmann [11] can be summarized in two steps.

• Step 1 Construct a ground state for the pseudo-relativistic NLH (1.1), and prove its convergence to the ground state $Q_0$ for the second-order NLH (1.9) as $c\rightarrow \infty$ up to translation and phase shift. Here, by construction (involving variational techniques), a pseudo-relativistic ground state must be positive and radially symmetric.

• Step 2 Prove uniqueness of a radially symmetric real-valued solution to the pseudo-relativistic NLH near the ground state $Q_0$. The proof of this local uniqueness relies on the non-degeneracy of the ground state $Q_0$, which is indeed one of the main contributions of Lenzmann [11].

• Conclusion If $c\ge 1$ is large enough, then a pseudo-relativistic ground state is close to $Q_0$, so it is unique up to translation and phase shift.

As for higher-order equations, however, we cannot make use of radial symmetry of a ground state for the proof of uniqueness, but we have to prove it instead, since we do not have symmetrization tools at hand. In order to overcome these obstacles, we employ several new ingredients, including the contraction mapping argument in our earlier work [3] and the improved local uniqueness near the ground state $Q_0$. Our proof can be summarized as follows.

• Step 1 Construct a ground state $Q_\epsilon ^J$ for the higher-order equation (1.4) (resp., (1.5)), and prove its convergence to the ground state $Q_0$ for the second-order equation (1.8) (resp., (1.9)) as $\epsilon \rightarrow 0$ up to translation and phase shift. We remark that contrary to Step 1 in [11], due to lack of symmetrization tools, it is not known that $Q_\epsilon ^J$ is radially symmetric and real-valued.

• Step 2 Construct a radially symmetric real-valued solution $u_\epsilon$ for the higher-order equation converging to the ground state $Q_0$ by the contraction mapping argument. By construction, the solution $u_\epsilon$ does not have any variational character as a ground state.

• Step 3 Prove uniqueness (up to translation and phase shift) for the higher-order equation near the ground state $Q_0$ without assuming that solutions are radially symmetric or real-valued.

• Conclusion If $\epsilon >0$ is small enough, then two solutions $Q_\epsilon ^J$ and $u_\epsilon$ are close to the ground state $Q_0$. Thus, identifying them by uniqueness, we conclude that $Q_\epsilon ^J$ is a unique radially symmetric real-valued ground state.

For the proof of local uniqueness in Step 2, we assume that there is a solution $\tilde{u}_\epsilon$, and then modify it by translation and phase shift to be perpendicular to the kernel of the linearized operator around $u_\epsilon$. Then, we prove that the modified $\tilde{u}_\epsilon$ is indeed $u_\epsilon$ itself. This argument, choosing the best modulation parameters, seems quite natural in the context of orbital stability [16]. However, to the best of authors’ knowledge, such a local uniqueness and its proof seem new.

Next, we consider the pseudo-relativistic NLH (1.1) and the higher-order NLH (1.3) in the non-relativistic regime ($c\gg 1$). In this case, it is shown in Lenzmann [11] that the pseudo-relativistic NLH (1.1) has a radially symmetric positive ground state $Q_c$, which is unique up to translation and phase shift. On the other hand, by the main theorem of this paper, the higher-order NLH (1.3) also has a radially symmetric and real-valued ground state $Q_c^J$, and it is unique up to translation and phase shift. Then, the contraction mapping argument in [3] can be applied to compare two ground states. As a consequence, we obtain the following error estimates for the higher-order approximations to the pseudo-relativistic ground state.

Theorem 1.3

(Higher-order approximations to a pseudo-relativistic ground state) Let J be an odd number, and $c>0$ be a sufficiently large number. We denote by $Q_c$ (resp., $Q_c^J$) the unique radially symmetric, real-valued ground state for the pseudo-relativistic NLH (1.1) (resp., the higher-order NLH (1.3)). Then,

$$\begin{aligned} \Vert Q_c^J-Q_c\Vert _{H^1}\lesssim \frac{1}{c^{2J}}. \end{aligned}$$

Remark 1.4

The higher-order Schrödinger operator in (1.3) is introduced as a higher-order approximation to the pseudo-relativistic operator $\sqrt{-c^2\Delta +m^2c^4}-mc^2$, provided that high-frequencies are not dominant. In [4, 5], the error estimates for the higher-order approximation to the linear evolution has been discussed. Theorem 1.3 first provides a precise error estimate for the higher-order approximation to the pseudo-relativistic ground state, which the simplest nonlinear object.

1.1 Notations

We denote the potential energy functional by

$$\begin{aligned} {\mathcal {N}}(u):=\left\{ \begin{aligned}&\frac{1}{2k+2}\int _{{\mathbb {R}}^d}|u|^{2k+2}dx&\text {(for NLS)}\\&\frac{1}{4}\int _{{\mathbb {R}}^3}\left( |x|^{-1}*|u|^2\right) |u|^2dx&\text {(for NLH)}. \end{aligned}\right. \end{aligned}$$

Then, the nonlinearity of the equation is given as its Frechét derivative

$$\begin{aligned} {\mathcal {N}}'(u):=\left\{ \begin{aligned}&|u|^{2k}u&\text {(for NLS)}\\&\left( |x|^{-1}*|u|^2\right) u&\text {(for NLH)}. \end{aligned}\right. \end{aligned}$$

We denote by $H_{P_\epsilon }^1=H_{P_\epsilon }^1({\mathbb {R}}^3;\mathbb {C})$ the Hilbert space equipped with the inner product

$$\begin{aligned} \langle f,g\rangle _{H_{P_\epsilon }^1}:=\int _{{\mathbb {R}}^3} \left( P_\epsilon +1\right) f(x)\overline{g(x)}dx, \end{aligned}$$

1.2 Organization of the paper

In Sect. 2, we prove existence of ground states $Q_\epsilon ^J$’s for the higher-order NLS (resp., the higher-order NLH) and their convergence to the ground state $Q_0$ for the second-order equation. In Sect. 3, we provide the non-degeneracy estimates, which are the key analytic tools in this paper. Using them, in Section 4, we construct radially symmetric real-valued solutions $u_\epsilon$’s converging to the ground state $Q_0$ for the second-order equation. In Sect. 5, we establish local uniqueness for higher-order equations near the ground state $Q_0$, and then identifying $Q_\epsilon ^J$ and $u_\epsilon$, we prove the main theorem (Theorem 1.1). Finally, in Sect. 6, we prove the error estimates for the higher-order approximation to the pseudo-relativistic ground state (Theorem 1.3).

2 Construction of ground states, and their limit

By the standard variational method, we construct ground states for higher-order equations (Proposition 2.1), and show their convergence to the ground state for the second-order equation (Proposition 2.3). In addition, we prove that in general, convergence to the ground state for the second-order equation in a low regularity norm can be upgraded to that in high regularity norms (Proposition 2.5).

Proposition 2.1

(Existence of a ground state) Suppose that (1.6) (as well as (1.7) for NLS) holds. Then, for any $\epsilon >0$, the higher-order equation (1.4) (resp., (1.5)) possesses a ground state $Q_\epsilon ^J\in H_{P_\epsilon }^1$.

Throughout this section, we denote the order of nonlinearity by

$$\begin{aligned} p=\left\{ \begin{aligned}&2k+1&\text {for NLS }(1.4)\\&3&\text {for NLH }(1.5). \end{aligned}\right. \end{aligned}$$

We observe that by algebra,

$$\begin{aligned} \langle {\mathcal {N}}'(u),u\rangle _{L^2}=(p+1){\mathcal {N}}(u). \end{aligned}$$

Hence, if u is admissible for the variational problem (2.3), equivalently

$$\begin{aligned} 0=\langle I_\epsilon '(u),u\rangle _{L^2}=\Vert u\Vert _{H_{P_\epsilon }^1}^2-(p+1){\mathcal {N}}(u)\quad \left( \Leftrightarrow \Vert u\Vert _{H_{P_\epsilon }^1}^2=(p+1){\mathcal {N}}(u)\right) , \end{aligned}$$

(2.1)

then the action functional can be written as

$$\begin{aligned} I_\epsilon (u)=\frac{1}{2}\Vert u\Vert _{H_{P_\epsilon }^1}^2-{\mathcal {N}}(u)=\frac{p-1}{2(p+1)}\Vert u\Vert _{H_{P_\epsilon }^1}^2=\frac{p-1}{2}{\mathcal {N}}(u). \end{aligned}$$

(2.2)

For each $\epsilon \ge 0$ (including $\epsilon =0$), the ground state energy level is defined by

$$\begin{aligned} {\mathcal {C}}_\epsilon := \inf \Big \{ I_\epsilon (u) ~|~ u \in H^1_{P_\epsilon }{\setminus }\{0\}\text { and }\langle I_\epsilon '(u),u \rangle _{L^2} = 0\Big \}. \end{aligned}$$

(2.3)

The following lemma is useful to prove the proposition.

Lemma 2.2

Suppose that (1.6) (as well as (1.7) for NLS) holds. Then, ${\mathcal {C}}_\epsilon$ is strictly positive, and

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0}{\mathcal {C}}_\epsilon \le {\mathcal {C}}_0. \end{aligned}$$

(2.4)

Proof

For NLS (1.4), by the Sobolev inequality with (1.6), we have

$$\begin{aligned} {\mathcal {N}}(u)=\frac{1}{2k+2}\Vert u\Vert _{L^{2k+2}}^{2k+2}\le C' \Vert u\Vert _{H_{P_\epsilon }^1}^{2k+2}, \end{aligned}$$

(2.5)

while for NLH (1.5), by the Hardy–Littlewood–Sobolev inequality and the Sobolev inequality with (1.6),

$$\begin{aligned} {\mathcal {N}}(u)\le \frac{1}{4}\left\| \frac{1}{|x|}*|u|^2\right\| _{L^6}\left\| |u|^2\right\| _{L^{6/5}}\le C \left\| |u|^2\right\| _{L^{6/5}}^2=C \Vert u\Vert _{L^{12/5}}^4\le C' \Vert u\Vert _{H_{P_\epsilon }^1}^4. \end{aligned}$$

(2.6)

Then, inserting the above inequality to the constraint (2.1), we get

$$\begin{aligned} 0\ge \Vert u\Vert _{H_{P_\epsilon }^1}^2\left( 1-(p+1)C' \Vert u\Vert _{H_{P_\epsilon }^1}^{p-2}\right) \quad \left( \Leftrightarrow \Vert u\Vert _{H_{P_\epsilon }^1}^2\ge ((p+1)C')^{-\frac{2}{p-2}} \right) \end{aligned}$$

Thus, by (2.2), $I(u_\epsilon )=\frac{p-1}{2(p+1)}\Vert u\Vert _{H_{P_\epsilon }^1}^2\ge \frac{p-1}{2(p+1)}((p+1)C')^{-\frac{2}{p-2}}$. Taking the infimum, we prove the lower bound on ${\mathcal {C}}_\epsilon$.

To show (2.4), we observe that the ground state $Q_0$ for the limit equation (1.8) (resp., (1.9)) is almost admissible for the variational problem (2.3) for sufficiently small $\epsilon >0$, because

$$\begin{aligned} \langle I_\epsilon '(Q_0),Q_0\rangle _{L^2}&=\langle (P_\epsilon +1)Q_0-{\mathcal {N}}'(Q_0), Q_0\rangle _{L^2}\\&=\langle (-\Delta +1)Q_0-{\mathcal {N}}'(Q_0), Q_0\rangle _{L^2}+\langle (P_\epsilon -(-\Delta ))Q_0, Q_0\rangle _{L^2}\\&=0+o_\epsilon (1)=o_\epsilon (1), \end{aligned}$$

where in the third identity, we used that $Q_0\in H^\ell$ for all $\ell \in {\mathbb {N}}$. Hence, for each $\epsilon >0$, there exists $t_\epsilon =1+o_\epsilon (1)$ such that $t_\epsilon Q_0$ is admissible. Then, it follows from the definition of the level set ${\mathcal {C}}_\epsilon$ and (2.2) that

$$\begin{aligned} {\mathcal {C}}_\epsilon \le I_\epsilon (t_\epsilon Q_0)=\frac{p-1}{2}{\mathcal {N}}(t_\epsilon Q_0)=t_\epsilon ^{p+1}\cdot \frac{p-1}{2}{\mathcal {N}}(Q_0)=(1+o_\epsilon (1))\cdot I_0(Q_0). \end{aligned}$$

Thus, taking $\limsup _{\epsilon \rightarrow 0}$, we prove (2.4). $\square$

Proof of Proposition 2.1

Let $\{u_n\}_{n=1}^\infty \subset H^1_{P_\epsilon }$ be a minimizing sequence for $I_\varepsilon (u)$ subject to the constraint $\langle I_\epsilon '(u),u \rangle _{L^2} = 0$ with $u \ne 0$, which is, by (2.2), bounded in $H_{P_\epsilon }^1$. We consider the Levy concentration function of $u_n$ (see [14])

$$\begin{aligned} M_n(r) := \sup _{x\in {\mathbb {R}}^d}\int _{B_r(x)}|u_n|^2\,dx, \end{aligned}$$

where $B_r(x)$ denotes the Euclidean ball of the radius r centered at x.

Suppose that there exists some $r > 0$ such that $M_n(r) \rightarrow 0$ as $n \rightarrow \infty$. It is shown in [14] that $\{u_n\}_{n=1}^\infty$ converges to zero in $L^p({\mathbb {R}}^d)$ for every $2< p < 2^*$, where $\frac{1}{2^*}=\max \{\frac{d-2}{2d},0\}$. Thus, by (2.2) and (2.5) (resp., (2.6)), it follows that $I_\epsilon (u_n)=\frac{p-1}{2}{\mathcal {N}}(u_n)\rightarrow 0$ as $n \rightarrow \infty$, but this contradicts to that ${\mathcal {C}}_\epsilon >0$ (see Lemma 2.2).

Now, passing to a subsequence, we assume that

$$\begin{aligned} M_0 := \lim _{n\rightarrow \infty } M(1) > 0. \end{aligned}$$

Then, there exists a sequence $\{x_n\}_{n=1}^\infty \subset {\mathbb {R}}^d$ such that for sufficiently large n,

$$\begin{aligned} \int _{B_1(x_n)}|u_n|^2\,dx > \frac{M_0}{2}. \end{aligned}$$

(2.7)

Translating the sequence, we introduce another minimizing sequence $\{v_n\}_{n=1}^\infty$ given by $v_n(x) = u_n(x+x_n)$, which is bounded in $H^1_{P_\epsilon }$. Let $v_0$ be the weak subsequential limit of $\{v_n\}_{n=1}^\infty$ in $H^1_{P_\epsilon }$ as $n\rightarrow \infty$. Note that $v_0\ne 0$, since $\{v_n\}_{n=1}^\infty$ is locally compact in $L^2({\mathbb {R}}^d)$ and it satisfies (2.7).

We claim that $v_0$ is admissible for the minimization problem (2.3), i.e.,

$$\begin{aligned} \langle I_\epsilon '(v_0), v_0 \rangle _{L^2} = 0. \end{aligned}$$

In order to prove the claim by contradiction, we assume that

$$\begin{aligned} \delta := \langle I_\epsilon '(v_0), v_0 \rangle _{L^2} > 0, \end{aligned}$$

and then applying the well-known Brezis-Lieb lemma, we decompose $v_n = v_0 +w_n$ such that

$$\begin{aligned} \Vert v_n\Vert _{H^1_{P_\epsilon }}^2&= \Vert v_0\Vert _{H^1_{P_\epsilon }}^2+\Vert w_n\Vert _{H^1_{P_\epsilon }}^2 +o_n(1), \nonumber \\ {\mathcal {N}}(v_n)&= {\mathcal {N}}(v_0) +{\mathcal {N}}(w_n)+o_n(1), \end{aligned}$$

(2.8)

and consequently,

$$\begin{aligned} \langle I_\epsilon '(w_n), w_n \rangle _{L^2} =\langle I_\epsilon '(v_n), v_n \rangle _{L^2} -\langle I_\epsilon '(v_0), v_0 \rangle _{L^2} +o_n(1)= -\delta +o_n(1). \end{aligned}$$

We observe that $f_n(t) := \langle I_\epsilon '(tw_n), tw_n \rangle _{L^2}$ is a polynomial of the form $a_nt^2 -b_nt^{p+1}$ with $a_n, b_n > 0$, and that $f_n(1)\le -\frac{\delta }{2}$ for large n. Hence, there exist a small $\eta \in (0, 1)$ and a sequence $\{t_n\}_{n=1}^\infty$, with $0< t_n \le 1-\eta$, such that $\{t_nw_n\}_{n=1}^\infty$ is admissible, i.e., $\langle I_\epsilon '(t_nw_n), t_nw_n \rangle _{L^2} = 0$. Then, by (2.2) and (2.8), we prove that

$$\begin{aligned} I_\epsilon (t_nw_n)&= \frac{p-1}{2(p+1)}\Vert t_nw_n\Vert _{H^1_{P_\epsilon }}^2= t_n^2\cdot \frac{p-1}{2(p+1)}\Vert w_n\Vert _{H^1_{P_\epsilon }}^2\\&\le (1-\eta )^2 \cdot \frac{p-1}{2(p+1)}\Vert v_n\Vert _{H^1_{P_\epsilon }}^2+o_n(1)\\&=(1-\eta )^2 I_\epsilon (v_n)+o_n(1)=(1-\eta )^2{\mathcal {C}}_\epsilon +o_n(1). \end{aligned}$$

However, this contradicts to minimality of ${\mathcal {C}}_\epsilon$. If $\delta := \langle I_\epsilon '(v_0), v_0 \rangle _{L^2} < 0$, repeating the same argument but switching the role of $v_0$ with $w_n$, we can again deduce a contradiction. Therefore, the claim is proved.

Finally, by the lower semi-continuity of the norm $\Vert \cdot \Vert _{H^1_{P_\epsilon }}$, we show that $v_0$ achieves the minimal energy,

$$\begin{aligned} I_\epsilon (v_0) = \frac{p-1}{2(p+1)}\Vert v_0\Vert _{H^1_{P_\epsilon }} \le \frac{p-1}{2(p+1)}\lim _{n\rightarrow \infty }\Vert v_n\Vert _{H^1_{P_\epsilon }} = \lim _{n\rightarrow \infty }I_\epsilon (v_n) = {\mathcal {C}}_\epsilon . \end{aligned}$$

This completes the proof by setting $Q_\epsilon ^J := v_0$. $\square$

Proposition 2.3

(Convergence of ground states) Suppose that (1.6) (as well as (1.7) for NLS) holds. Let $\{Q_\epsilon ^J\}_{\epsilon >0}$ be the family of ground states for the higher-order equation (1.4) (resp., (1.5)) given by Proposition 2.1. Then,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \Vert Q_\epsilon ^J-\tilde{Q}_0\Vert _{H_{P_\epsilon }^1}=0, \end{aligned}$$

where $\tilde{Q}_0$ is a ground state to the second-order equation (1.8) (resp., (1.9)).

Proof

By (1.6), (2.2) and Lemma 2.3, we see that $\{Q_\epsilon ^J\}_{\epsilon >0}$ is bounded in $H^1$,

$$\begin{aligned} \gamma \Vert Q_\epsilon ^J\Vert _{H^1}^2 \le \Vert Q_\epsilon ^J\Vert _{H^1_{P_\epsilon }}^2 = \frac{2(p+1)}{p-1}I_\epsilon (Q_\epsilon ^J) = \frac{2(p+1)}{p-1}{\mathcal {C}}_\epsilon =\frac{2(p+1)}{p-1}{\mathcal {C}}_0+o_\epsilon (1). \end{aligned}$$

Hence, $Q_\epsilon ^J$ weakly subsequentially converges to $\tilde{Q}_0$ in $H^1$ as $\epsilon \rightarrow 0$. As in the proof of Proposition 2.1, one can show that

$$\begin{aligned} \int _{B_1(0)}|Q_\epsilon ^J|^2\,dx \ge \frac{M_0}{2}, \end{aligned}$$

which implies $\tilde{Q}_0$ is nontrivial.

We claim that $\tilde{Q}_0$ is a smooth solution to (1.8) (resp., (1.9)). To show the claim, we recall that for any $\phi \in C_c^\infty$,

$$\begin{aligned} \langle (P_\epsilon +1) Q_\epsilon ^J- {\mathcal {N}}'(Q_\epsilon ^J), \phi \rangle _{L^2} = 0. \end{aligned}$$

However, by the weak convergence of $Q_\epsilon ^J$, up to a subsequence, we have

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0}\langle (P_\epsilon +1) Q_\epsilon ^J, \phi \rangle _{L^2} = \lim _{\varepsilon \rightarrow 0}\langle Q_\epsilon ^J, (P_\epsilon +1)\phi \rangle _{L^2}\\&\quad = \lim _{\varepsilon \rightarrow 0}\langle Q_\epsilon ^J, (-\Delta +1)\phi \rangle _{L^2}= \langle \tilde{Q}_0, (-\Delta +1)\phi \rangle _{L^2} \end{aligned}$$

and

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\langle {\mathcal {N}}'(Q_\epsilon ), \phi \rangle _{L^2} = \langle {\mathcal {N}}'(\tilde{Q}_0), \phi \rangle _{L^2}. \end{aligned}$$

Thus, sending $\epsilon \rightarrow 0$, we show that

$$\begin{aligned} \langle (-\Delta +1) \tilde{Q}_0- {\mathcal {N}}'(\tilde{Q}_0), \phi \rangle _{L^2} = 0. \end{aligned}$$

In other words, $\tilde{Q}_0$ is a weak solution to (1.8) (resp., (1.9)). Then, by the elliptic regularity (see [9], one can show that $\tilde{Q}_0 \in H^\ell$ for every $\ell \in {\mathbb {N}}$.

Next, using (2.2), we write

$$\begin{aligned} \begin{aligned} {\mathcal {C}}_0&\le I_0(\tilde{Q}_0) = \frac{p-1}{2(p+1)}\Vert \tilde{Q}_0\Vert _{H^1}^2\\&\le \frac{p-1}{2(p+1)}\left( \Vert \tilde{Q}_0\Vert _{H^1_{P_\epsilon }}^2 +\Vert Q_\epsilon ^J-\tilde{Q}_0\Vert _{H^1_{P_\epsilon }}^2\right) \\&= \frac{p-1}{2(p+1)}\Vert Q_\epsilon ^J\Vert _{H^1_{P_\epsilon }}^2-\frac{p-1}{p+1}\cdot \text {Re}\langle \tilde{Q}_0, Q_\epsilon ^J-\tilde{Q}_0\rangle _{H^1_{P_\epsilon }}\\&= I_\epsilon (Q_\epsilon ^J)-\frac{p-1}{p+1}\cdot \text {Re}\langle \tilde{Q}_0, Q_\epsilon ^J-\tilde{Q}_0\rangle _{H^1_{P_\epsilon }}. \end{aligned} \end{aligned}$$

(2.9)

However, since $\tilde{Q}_0$ is smooth and $Q_\epsilon ^J\rightharpoonup \tilde{Q}_0$ in $H^1$ as $\epsilon \rightarrow 0$, we have

$$\begin{aligned} \langle \tilde{Q}_0, Q_\epsilon ^J-\tilde{Q}_0\rangle _{H^1_{P_\epsilon }}=\langle \tilde{Q}_0, Q_\epsilon ^J-\tilde{Q}_0\rangle _{H^1}+\langle (P_\epsilon -(-\Delta ))\tilde{Q}_0, Q_\epsilon ^J-\tilde{Q}_0\rangle _{L^2}\rightarrow 0\ \end{aligned}$$

as $\epsilon \rightarrow 0$. Thus, by (2.2) again and Lemma 2.2, we get

$$\begin{aligned} {\mathcal {C}}_0\le {\mathcal {C}}_\epsilon +o_\epsilon (1)\le {\mathcal {C}}_0+o_\epsilon (1). \end{aligned}$$

Sending $\epsilon \rightarrow 0$ in (2.9), we conclude that $\tilde{Q}_0$ achieves the minimum value ${\mathcal {C}}_0$ of the action functional $I_0$ and that $\Vert Q_\epsilon ^J-\tilde{Q}_0\Vert _{H^1_{P_\epsilon }}\rightarrow 0$ as $\epsilon \rightarrow 0$. $\square$

Remark 2.4

Let $Q_0$ be the radially symmetric positive ground state for (1.8) (resp., (1.9)). By uniqueness of a ground state to the second-order equation, there exist $\theta \in {\mathbb {R}}$ and $x_0 \in {\mathbb {R}}^d$ such that $Q_0(x) = e^{i\theta }\tilde{Q}_0(x-x_0)$. Then, the modified profile $e^{i\theta }Q_\epsilon ^J(\cdot -x_0)$, which is also a ground state, converges to $Q_0$ as $\epsilon \rightarrow 0$.

Proposition 2.5

(Upgraded convergence) Suppose that (1.6) (as well as (1.7) for NLS) holds. For $\epsilon >0$, let $u_\epsilon \in H_{P_\epsilon }^1$ be a solution to the higher-order equation (1.4) (resp., (1.5)), which is not necessarily a ground state. Let $Q_0$ be the unique ground state to the second-order equation (1.8) (resp., (1.9)).

• (1) If $u_\epsilon \rightarrow Q_0$ in $H^1$ as $\epsilon \rightarrow 0$, then $u_\epsilon \rightarrow Q_0$ in $H^\ell$ as $\epsilon \rightarrow 0$ for all $\ell \in {\mathbb {N}}$.

• (2) As a consequence, for all $\ell \in {\mathbb {N}}$, $\Vert u_\epsilon \Vert _{H^\ell }$ is bounded uniformly in $0<\epsilon \le 1$.

Proof

We prove the proposition by induction. Let $r_\epsilon =u_\epsilon -Q_0$ be the difference between two solutions. Suppose that $\Vert r_\epsilon \Vert _{H^\ell }\rightarrow 0$ for some $\ell \in {\mathbb {N}}$. Then, by the equations, we have

$$\begin{aligned} (P_\epsilon +1)r_\epsilon&=(-\Delta -P_\epsilon )Q_0+(P_\epsilon +1)u_\epsilon -(-\Delta +1)Q_0\\&=(-\Delta -P_\epsilon )Q_0+{\mathcal {N}}'(u_\epsilon )-{\mathcal {N}}'(Q_0). \end{aligned}$$

Thus, it follows from (1.6) that

$$\begin{aligned} \gamma \Vert r_\epsilon \Vert _{H^{\ell +1}}&\le \Vert (P_\epsilon +1)r_\epsilon \Vert _{H^{\ell -1}}\\&\le \Vert (-\Delta -P_\epsilon )Q_0\Vert _{H^{\ell -1}}+\Vert {\mathcal {N}}'(Q_0+r_\epsilon )-{\mathcal {N}}'(Q_0)\Vert _{H^{\ell -1}}. \end{aligned}$$

For the first term on the right hand side, by smoothness of $Q_0$, $\Vert (-\Delta -P_\epsilon )Q_0\Vert _{H^{\ell -1}}\rightarrow 0$. For the second term, distributing derivatives and then applying Lemma A.2, one can show that

$$\begin{aligned} \Vert {\mathcal {N}}'(Q_0+r_\epsilon )-{\mathcal {N}}'(Q_0)\Vert _{H^{\ell -1}}\lesssim \Big \{\Vert Q_0\Vert _{H^\ell }+\Vert r_\epsilon \Vert _{H^\ell }\Big \}^{p-1}\Vert r_\epsilon \Vert _{H^\ell }\rightarrow 0, \end{aligned}$$

where $p=2k+1$ for (1.4) (resp., $p=3$ for (1.5)). Therefore, we conclude that $\Vert r_\epsilon \Vert _{H^{\ell +1}}\rightarrow 0$. $\square$

3 Non-degeneracy estimates

Let $\{u_\epsilon \}_{\epsilon >0}$ be a family of real-valued solutions to the higher-order equation such that $u_\epsilon \rightarrow Q_0$ in $H^1$ (as well as in $H^\ell$ for all $\ell \in {\mathbb {N}}$ by Proposition 2.5) as $\epsilon \rightarrow 0$, where $Q_0$ is the unique radially symmetric positive ground state for the second-order equation (1.8) (resp., (1.9)). For notational convenience, we denote $u_0:=Q_0.$ For $0\le \epsilon \le \epsilon _0$ (including 0), we consider the linear operators ${\mathcal {L}}_\epsilon ^\pm : H_{P_\epsilon }^1\rightarrow H_{P_\epsilon }^{-1}$, defined by

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned} {\mathcal {L}}_\epsilon ^+:&=P_\epsilon +1-{\mathcal {N}}_{u_\epsilon }^+\\ {\mathcal {L}}_\epsilon ^-:&=P_\epsilon +1-{\mathcal {N}}_{u_\epsilon }^-, \end{aligned}\right. \end{aligned} \end{aligned}$$

(3.1)

where

$$\begin{aligned} {\mathcal {N}}_{u}^+(g):=\left\{ \begin{aligned}&(2k+1)u^{2k}g&\text {(for NLS)}\\&2(|x|^{-1}*(u g))u+(|x|^{-1}*u^2)g&\text {(for NLH)} \end{aligned}\right. \end{aligned}$$

(3.2)

and

$$\begin{aligned} {\mathcal {N}}_{u}^-(g):=\left\{ \begin{aligned}&u^{2k}g&\text {(for NLS)}\\&(|x|^{-1}*u^2)g&\text {(for NLH)}. \end{aligned}\right. \end{aligned}$$

(3.3)

These linear operators naturally appear as the real and the imaginary parts of the linearized operator at the solution $u_\epsilon$. Factorizing out the differential operator $(1+P_\epsilon )$ in a symmetric form, we write

$$\begin{aligned} {\mathcal {L}}_\epsilon ^\pm =\sqrt{1+P_\epsilon }(\text {Id}-{\mathcal {A}}_\epsilon ^\pm ) \sqrt{1+P_\epsilon }, \end{aligned}$$

(3.4)

where

$$\begin{aligned} \left\{ \begin{aligned} {\mathcal {A}}_\epsilon ^+:&=\frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{u_\epsilon }^+\frac{1}{\sqrt{1+P_\epsilon }}\\ {\mathcal {A}}_\epsilon ^-:&=\frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{u_\epsilon }^-\frac{1}{\sqrt{1+P_\epsilon }}. \end{aligned}\right. \end{aligned}$$

In this section, we prove non-degeneracy of the solution $u_\epsilon$, and obtain uniform lower bounds for the linear operators $(\text {Id}-{\mathcal {A}}_\epsilon ^\pm )$, which are our main analytic tools.

To begin with, we consider the base case $\epsilon =0$. By the non-degeneracy of the ground state $Q_0$ and the relation (3.4) (see [10] for NLS and [11] for NLH), we have

$$\begin{aligned} \text {Ker}(\text {Id}-{\mathcal {A}}_0^+)=\text {span}\Big \{\partial _{x_1}\sqrt{1-\Delta }\,Q_0,\cdots ,\partial _{x_d}\sqrt{1-\Delta }\,Q_0\Big \} \end{aligned}$$

and

$$\begin{aligned} \text {Ker}(\text {Id}-{\mathcal {A}}_0^-)=\text {span}\Big \{\sqrt{1-\Delta }\,Q_0\Big \}. \end{aligned}$$

By the equation, the operator ${\mathcal {A}}_0^\pm$ sends an element of $\text {Ker}(\text {Id}-{\mathcal {A}}_0^\pm )$ to the same function, and thus $(\text {Id}-{\mathcal {A}}_0^\pm )$ maps $(\text {Ker}(\text {Id}-{\mathcal {A}}_0^\pm ))^\perp \subset L^2({\mathbb {R}}^d;{\mathbb {R}})$ to itself, where $A^\perp \subset H$ denotes the subspace orthogonal to A in the Hilbert space H. Moreover, the operator $(\text {Id}-{\mathcal {A}}_0^\pm )$ satisfies the following lower bounds.

Proposition 3.1

(Non-degeneracy estimates; base case) There exists $\beta _0>0$ such that

$$\begin{aligned} \Vert (\text {Id}-{\mathcal {A}}_0^\pm ) g\Vert _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\ge \beta _0\Vert g\Vert _{L^2({\mathbb {R}}^d;{\mathbb {R}})} \end{aligned}$$

for all $g\in (\text {Ker}(\text {Id}-{\mathcal {A}}_0^\pm ))^\perp \subset L^2({\mathbb {R}}^d;{\mathbb {R}})$.

Proof

We claim that both $\frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1-\Delta }}$ and $\frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^-\frac{1}{\sqrt{1-\Delta }}$ are compact on $L^2$. Indeed, for NLS, the integral kernel of ${\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1-\Delta }}$ (or ${\mathcal {N}}_{Q_0}^-\frac{1}{\sqrt{1-\Delta }}$, respectively) is given by

$$\begin{aligned} (2k+1)Q_0(x)^{2k}G_{-1}(x-y)\quad \Big (\text {or }Q_0(x)^{2k}G_{-1}(x-y),\text { respectively}\Big ), \end{aligned}$$

where $G_{-1}(x)=((1+|\xi |^2)^{-1})^\vee (x)$ is the Bessel potential. For NLH, it is given by

$$\begin{aligned}&2\int _{{\mathbb {R}}^3}\frac{Q_0(x)Q_0(z)}{|x-z|}G_{-1}(z-y)dz+(|x|^{-1}*Q_0^2)(x)G_{-1}(x-y)\\&\quad \Big (\text {or }(|x|^{-1}*Q_0^2)(x)G_{-1}(x-y), \text { respectively}\Big ). \end{aligned}$$

All of the above kernels are contained in $L^2({\mathbb {R}}_x^d\times {\mathbb {R}}_y^d)$, because $Q_0$ is smooth and rapidly decaying. Therefore, the associated operators are Hilbert–Schmidt (so, compact on $L^2({\mathbb {R}}^d)$). Since composition of a compact operator and a bounded operator is compact, this proves the claim. As a consequence, by the Fredholm alternative, the proposition is proved. $\square$

Next, we show that the non-degeneracy of the ground state $Q_0$ is stable along the family of solutions which converges to the ground state $Q_0$.

Proposition 3.2

(Stability of non-degeneracy) Let $\{u_\epsilon \}_{\epsilon >0}$ be a family of real-valued solutions to the higher-order equation (1.4) (resp., (1.5)) such that $u_\epsilon \rightarrow Q_0$ in $H^1$ as $\epsilon \rightarrow 0$. Then, there exists $\epsilon _0>0$ such that

$$\begin{aligned} \text {Ker}{\mathcal {L}}_\epsilon ^+=\text {span}\Big \{\partial _{x_1}u_\epsilon ,\ldots ,\partial _{x_d}u_\epsilon \Big \}\quad \text {and}\quad \text {Ker}{\mathcal {L}}_\epsilon ^-=\text {span}\big \{u_\epsilon \big \} \end{aligned}$$

for $0<\epsilon \le \epsilon _0$. Equivalently, we have

$$\begin{aligned} \text {Ker}(\text {Id}-{\mathcal {A}}_\epsilon ^+)=\text {span}\Big \{\partial _{x_1}\sqrt{1+P_\epsilon }\,u_\epsilon ,\ldots ,\partial _{x_d}\sqrt{1+P_\epsilon }\,u_\epsilon \Big \} \end{aligned}$$

(3.5)

and

$$\begin{aligned} \text {Ker}(\text {Id}-{\mathcal {A}}_\epsilon ^-)=\text {span}\Big \{\sqrt{1+P_\epsilon }\,u_\epsilon \Big \}. \end{aligned}$$

(3.6)

Proof

Following the argument in the proof of [11, Theorem 3], we prove (3.5) only, because (3.6) can be proved by the same way.

By the equation, it is easy to see that each $\partial _{x_j}\sqrt{1+P_\epsilon }u_\epsilon$ is contained in the kernel of $(\text {Id}-{\mathcal {A}}_\epsilon ^+)$. Therefore, it suffices to show that the dimension of $\text {Ker}(\text {Id}-{\mathcal {A}}_\epsilon ^+)$ is $\le d$. We recall that $\text {Ker}(\text {Id}-{\mathcal {A}}_\epsilon ^+)=\text {Im}({\mathcal {P}}_\epsilon )$, where ${\mathcal {P}}_\epsilon$ is the projection operator given by

$$\begin{aligned} {\mathcal {P}}_\epsilon :=\frac{1}{2\pi i}\oint _{|z|=c} (\text {Id}-{\mathcal {A}}_\epsilon ^+-z\text {Id})^{-1}dz \end{aligned}$$

for some sufficiently small $c>0$. We observe that by Lemma A.2,

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {A}}_\epsilon ^+-{\mathcal {A}}_0^+\Vert _{L^2\rightarrow L^2}&=\left\| \frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{u_\epsilon }^+\frac{1}{\sqrt{1+P_\epsilon }}-\frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1-\Delta }}\right\| _{L^2\rightarrow L^2}\\&\le \left\| \left( \frac{1}{\sqrt{1+P_\epsilon }}-\frac{1}{\sqrt{1-\Delta }}\right) {\mathcal {N}}_{u_\epsilon }^+\frac{1}{\sqrt{1+P_\epsilon }}\right\| _{L^2\rightarrow L^2}\\&\quad +\left\| \frac{1}{\sqrt{1-\Delta }}({\mathcal {N}}_{u_\epsilon }^+-{\mathcal {N}}_{Q_0}^+)\frac{1}{\sqrt{1+P_\epsilon }}\right\| _{L^2\rightarrow L^2}\\&\quad +\left\| \frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^+\left( \frac{1}{\sqrt{1+P_\epsilon }}-\frac{1}{\sqrt{1-\Delta }}\right) \right\| _{L^2\rightarrow L^2}\rightarrow 0, \end{aligned} \end{aligned}$$

(3.7)

and consequently, $\Vert {\mathcal {P}}_\epsilon -{\mathcal {P}}_0\Vert _{L^2\rightarrow L^2}\rightarrow 0$ as $\epsilon \rightarrow 0$. Suppose that $\text {Rank}({\mathcal {P}}_\epsilon )>\text {Rank}({\mathcal {P}}_0)$. Then, there exist $L^2$-orthonormal vectors $v_1,\ldots , v_{d+1}$ such that ${\mathcal {P}}_\epsilon v_j=v_j$. Hence, ${\mathcal {P}}_0v_1,\ldots , {\mathcal {P}}_0v_{d+1}$ are almost orthogonal, and they are linearly independent, which contradicts to the assumption. Therefore, we conclude that $\text {Rank}({\mathcal {P}}_\epsilon )\le \text {Rank}({\mathcal {P}}_0)=d$. $\square$

Using the non-degeneracy, we prove the inequality analogous to Proposition 3.1.

Proposition 3.3

(Non-degeneracy estimates; general case) Let $\{u_\epsilon \}_{\epsilon >0}$ be a family of real-valued solutions to the higher-order equation (1.4) (resp., (1.5)) such that $u_\epsilon \rightarrow Q_0$ in $H^1$ as $\epsilon \rightarrow 0$. Then, there exist $\epsilon _0>0$ and $\beta >0$ such that if $0<\epsilon \le \epsilon _0$, then

$$\begin{aligned} \Vert (\text {Id}-{\mathcal {A}}_\epsilon ^\pm ) g\Vert _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\ge \beta \Vert g\Vert _{L^2({\mathbb {R}}^d;{\mathbb {R}})} \end{aligned}$$

for all $g\in (\text {Ker}(\text {Id}-{\mathcal {A}}_\epsilon ^\pm ))^\perp \subset L^2({\mathbb {R}}^d;{\mathbb {R}})$, which is equivalent to

$$\begin{aligned} \Vert {\mathcal {L}}_\epsilon ^\pm g\Vert _{H_{P_\epsilon }^{-1}({\mathbb {R}}^d;{\mathbb {R}})}\ge \beta \Vert g\Vert _{H_{P_\epsilon }^1({\mathbb {R}}^d;{\mathbb {R}})} \end{aligned}$$

for all $g\in (\text {Ker}{\mathcal {L}}_\epsilon ^\pm )^\perp \subset H_{P_\epsilon }^1({\mathbb {R}}^d;{\mathbb {R}})$.

Proof

We show the proposition only for ${\mathcal {A}}_\epsilon ^+$, since the other inequality can be proved exactly by the same way.

Let $\beta =\frac{\beta _0}{4}>0$, where $\beta _0$ is given in Proposition 3.1. For $g\in L^2({\mathbb {R}}^d;{\mathbb {R}})$ and $\epsilon \ge 0$, we denote by $g_\epsilon ^\perp$ the orthogonal projection of g to $(\text {Ker}(\text {Id}-{\mathcal {A}}_\epsilon ^+))^\perp \subset L^2({\mathbb {R}}^d;{\mathbb {R}})$, precisely

$$\begin{aligned} g_\epsilon ^\perp :=g-\sum _{j=1}^d\langle g, e_{j;\epsilon }\rangle _{L^2}e_{j;\epsilon }, \end{aligned}$$

where $e_{j;\epsilon }:=\frac{\partial _{x_j}\sqrt{1+P_\epsilon }u_\epsilon }{\Vert \partial _{x_j}\sqrt{1+P_\epsilon }u_\epsilon \Vert _{L^2}^2}$. We fix $g\in (\text {Ker}(\text {Id}-{\mathcal {A}}_\epsilon ^+))^\perp$. Then, we decompose

$$\begin{aligned} (\text {Id}-{\mathcal {A}}_\epsilon ^+)g&=(\text {Id}-{\mathcal {A}}_0^+)g+({\mathcal {A}}_0^+-{\mathcal {A}}_\epsilon ^+)g\\&=(\text {Id}-{\mathcal {A}}_0^+)g_0^\perp +(\text {Id}-{\mathcal {A}}_0^+)(g-g_0^\perp )+({\mathcal {A}}_0^+-{\mathcal {A}}_\epsilon ^+)g. \end{aligned}$$

By the triangle inequalities and Proposition 3.1, we get

$$\begin{aligned} \begin{aligned} \Vert (\text {Id}-{\mathcal {A}}_\epsilon ^+)g\Vert _{L^2}&\ge \Vert (\text {Id}-{\mathcal {A}}_0^+)g_0^\perp \Vert _{L^2}-\Vert (\text {Id}-{\mathcal {A}}_0^+)(g-g_0^\perp )\Vert _{L^2}-\Vert ({\mathcal {A}}_0^+-{\mathcal {A}}_\epsilon ^+)g\Vert _{L^2}\\&\ge 4\beta \Vert g_0^\perp \Vert _{L^2}-\Vert g-g_0^\perp \Vert _{L^2}-\Vert {\mathcal {A}}_0^+(g-g_0^\perp )\Vert _{L^2}-\Vert ({\mathcal {A}}_0^+-{\mathcal {A}}_\epsilon ^+)g\Vert _{L^2}\\&\ge 4\beta \Vert g\Vert _{L^2}-(4\beta +1+\Vert {\mathcal {A}}_0^+\Vert _{L^2\rightarrow L^2})\Vert g_\epsilon ^\perp -g_0^\perp \Vert _{L^2}\quad \text {(by }g_\epsilon ^\perp =g\text {)}\\&\quad -\Vert {\mathcal {A}}_0^+-{\mathcal {A}}_\epsilon ^+\Vert _{L^2\rightarrow L^2}\Vert g\Vert _{L^2}. \end{aligned} \end{aligned}$$

(3.8)

On the other hand, we have

$$\begin{aligned} \Vert g_\epsilon ^\perp -g_0^\perp \Vert _{L^2}&\le \sum _{j=1}^d\left\| \langle g,e_{j;\epsilon }\rangle _{L^2}e_{j;\epsilon }-\langle g,e_{j;0}\rangle _{L^2}e_{j;0}\right\| _{L^2}\\&\le \sum _{j=1}^d|\langle g,e_{j;\epsilon }-e_{j;0}\rangle _{L^2}|+|\langle g,e_{j;0}\rangle _{L^2}|\left\| e_{j;\epsilon }-e_{j;0}\right\| _{L^2}\\&\le 2\Vert g\Vert _{L^2}\sum _{j=1}^d\left\| e_{j;\epsilon }-e_{j;0}\right\| _{L^2}\le o_\epsilon (1)\Vert g\Vert _{L^2}, \end{aligned}$$

because by Proposition 2.5,

$$\begin{aligned}&\left\| \partial _{x_j}\sqrt{1+P_\epsilon }u_\epsilon -\partial _{x_j}\sqrt{1-\Delta }u_0\right\| _{L^2}\\&\quad \le \left\| (\sqrt{1+P_\epsilon }-\sqrt{1-\Delta })\partial _{x_j}u_\epsilon \right\| _{L^2}+\left\| \partial _{x_j}\sqrt{1-\Delta }(u_\epsilon -u_0)\right\| _{L^2}\rightarrow 0 \end{aligned}$$

as $\epsilon \rightarrow 0$ and it implies $\Vert e_{j;\epsilon }-e_{j;0}\Vert _{L^2}\rightarrow 0$. Moreover, by (3.7), $\Vert {\mathcal {A}}_0^+-{\mathcal {A}}_\epsilon ^+\Vert _{L^2\rightarrow L^2}\rightarrow 0$ as $\epsilon \rightarrow 0$. Inserting these to (3.8), we prove the proposition. $\square$

By a little modification, we can also show the following inequality.

Lemma 3.4

There exists $\epsilon _0>0$ such that if $0<\epsilon \le \epsilon _0$, then

$$\begin{aligned} \text {Id}-\frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1+P_\epsilon }} \end{aligned}$$

is invertible on $L_{rad}^2({\mathbb {R}}^d;{\mathbb {R}})$. Moreover, its inverse is uniformly bounded,

$$\begin{aligned} \left\| \left( \text {Id}-\frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1+P_\epsilon }}\right) ^{-1}\right\| _{L_{rad}^2({\mathbb {R}}^d;{\mathbb {R}})\rightarrow L_{rad}^2({\mathbb {R}}^d;{\mathbb {R}})}\le \frac{2}{\beta _0}, \end{aligned}$$

where $\beta _0>0$ is given by Proposition 3.1.

Proof

By Proposition 3.1, the operator $(\text {Id}-\frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1-\Delta }})$ is invertible, because its kernel in $L_{rad}^2$ is empty ($\partial _{x_j}Q_0$’s are not radially symmetric). On the other hand, repeating the proof of (3.7), one can show that the difference

$$\begin{aligned}&\frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1+P_\epsilon }}-\frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1-\Delta }}\\&\quad =\left( \frac{1}{\sqrt{1+P_\epsilon }}-\frac{1}{\sqrt{1-\Delta }}\right) {\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1+P_\epsilon }}+\frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^+\left( \frac{1}{\sqrt{1+P_\epsilon }}-\frac{1}{\sqrt{1-\Delta }}\right) \end{aligned}$$

can be arbitrarily small in the operator norm on $L_{rad}^2({\mathbb {R}}^d;{\mathbb {R}})$, provided that $\epsilon >0$ is small enough. Therefore, we conclude that if $0<\epsilon \le \epsilon _0$, then $(\text {Id}-\frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1+P_\epsilon }})$ is invertible, and its inverse is uniformly bounded. $\square$

4 Construction of a solution by contraction

In this section, by the contraction mapping argument in [3] which relies on the non-degeneracy estimates in the previous section, we construct a radially symmetric real-valued solution $u_\epsilon$ to the higher-order equation, with small $\epsilon >0$, that converges to the ground state $Q_0$ as $\epsilon \rightarrow 0$, where $Q_0$ is the unique radially symmetric real-valued ground state for the second-order equation.

Proposition 4.1

(Construction of a solution by contraction) Suppose that (1.6) (as well as (1.7) for NLS) holds. Then, there exists $\epsilon _0>0$ such that a sequence of radially symmetric real-valued solutions $\{u_\epsilon \}_{0<\epsilon \le \epsilon _0}$ to the higher-order NLS (1.4) (resp., the higher-order NLH (1.5)) exists, with the convergence

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\Vert u_\epsilon -Q_0\Vert _{H_{P_\epsilon }^1}=0. \end{aligned}$$

Proof

Step 1. Reformulation of the equation Let $\epsilon >0$ be sufficiently small. Suppose that $u_\epsilon$ is a radially symmetric real-valued solution to the higher-order equation. Then, the difference

$$\begin{aligned} r_\epsilon :=u_\epsilon -Q_0 \end{aligned}$$

solves the equation

$$\begin{aligned} (P_\epsilon +1) r_\epsilon&=(P_\epsilon +1) u_\epsilon -(P_\epsilon +1) Q_0\\&=(-\Delta -P_\epsilon )Q_0+(P_\epsilon +1) u_\epsilon -(-\Delta +1)Q_0\\&=(-\Delta -P_\epsilon )Q_0+{\mathcal {N}}'(u_\epsilon )-{\mathcal {N}}'(Q_0)\\&=(-\Delta -P_\epsilon )Q_0+{\mathcal {N}}'(Q_0+r_\epsilon )-{\mathcal {N}}'(Q_0). \end{aligned}$$

Moving the linear terms with respect to $r_\epsilon$ on the right hand side to the left, we write

$$\begin{aligned} \left( P_\epsilon +1-{\mathcal {N}}_{Q_0}^+\right) r_\epsilon =(-\Delta -P_\epsilon )Q_0+{\mathcal {N}}'(Q_0+r_\epsilon )-{\mathcal {N}}'(Q_0)-{\mathcal {N}}_{Q_0}^+(r_\epsilon ) \end{aligned}$$

(see (3.2) for the definition of ${\mathcal {N}}_{Q_0}^+$). Then, inverting the operator

$$\begin{aligned} \left( P_\epsilon +1-{\mathcal {N}}_{Q_0}^+\right) =\sqrt{1+P_\epsilon }\left( \text {Id}-\frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1+P_\epsilon }}\right) \sqrt{1+P_\epsilon } \end{aligned}$$

by Lemma 3.4, we reformulate the higher-order equation as

$$\begin{aligned} r_\epsilon&=\left( P_\epsilon +1-{\mathcal {N}}_{Q_0}^+\right) ^{-1}\Big \{(-\Delta -P_\epsilon )Q_0+{\mathcal {N}}'(Q_0+r_\epsilon )-{\mathcal {N}}'(Q_0)-{\mathcal {N}}_{Q_0}^+(r_\epsilon )\Big \}\\&=:\Phi (r_\epsilon ). \end{aligned}$$

Step 2. Construction of a solution We set

$$\begin{aligned} \delta _\epsilon :=\frac{4}{\beta _0}\left\| (-\Delta -P_\epsilon )u_0\right\| _{H_{P_\epsilon }^{-1}}, \end{aligned}$$

where $\beta _0>0$ is the constant given in Lemma 3.4. Then, by (1.6), we have

$$\begin{aligned} \delta _\epsilon \le \frac{4}{\beta _0}\sum _{|\alpha |=3}^J|c_\alpha | \epsilon ^{|\alpha |-2}\Vert \nabla ^\alpha Q_0\Vert _{H_{P_\epsilon }^{-1}}\le \frac{4}{\beta _0\gamma }\sum _{|\alpha |=3}^J|c_\alpha | \epsilon ^{|\alpha |-2}\Vert \nabla ^\alpha Q_0\Vert _{H^{-2}}\rightarrow 0 \end{aligned}$$

as $\epsilon \rightarrow 0$. Let $\epsilon _0$ be a sufficiently small number to be chosen so that all of the following estimates hold. Suppose that $0<\epsilon \le \epsilon _0$ (and thus $\delta _\epsilon >0$ is also small enough). If $\Vert r\Vert _{H_{P_\epsilon }^1}, \Vert \tilde{r}\Vert _{H_{P_\epsilon }^1}\le \delta _\epsilon$, then by Lemma 3.4 and Lemma A.2,

$$\begin{aligned} \Vert \Phi (r)\Vert _{H_{P_\epsilon }^1}&=\Bigg \Vert \left( \text {Id}-\frac{1}{\sqrt{1+P_\epsilon }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1+P_\epsilon }}\right) ^{-1}\frac{1}{\sqrt{1+P_\epsilon }}\\&\quad \quad \Big \{(-\Delta -P_\epsilon )Q_0+{\mathcal {N}}'(Q_0+r)-{\mathcal {N}}'(Q_0)-{\mathcal {N}}_{Q_0}^+(r)\Big \}\Bigg \Vert _{L^2}\\&\le \frac{2}{\beta _0}\left\| (-\Delta -P_\epsilon )Q_0\right\| _{H_{P_\epsilon }^{-1}}+\frac{2}{\beta _0}\left\| {\mathcal {N}}'(Q_0+r)-{\mathcal {N}}'(Q_0)-{\mathcal {N}}_{Q_0}^+(r)\right\| _{H_{P_\epsilon }^{-1}}\\&\le \frac{2}{\beta _0}\left\| (-\Delta -P_\epsilon )u_0\right\| _{H_{P_\epsilon }^{-1}}+\frac{1}{2}\Vert r\Vert _{H_{P_\epsilon }^1}\le \delta _\epsilon \end{aligned}$$

and similarly,

$$\begin{aligned} \Vert \Phi (r)-\Phi (\tilde{r})\Vert _{H_{P_\epsilon }^1}&\le \frac{2}{\beta _0}\Big \Vert \left( {\mathcal {N}}'(Q_0+r)-{\mathcal {N}}'(Q_0)-{\mathcal {N}}_{Q_0}^+(r)\right) \\&\quad \quad \quad -\left( {\mathcal {N}}'(Q_0+\tilde{r})-{\mathcal {N}}'(Q_0)-{\mathcal {N}}_{Q_0}^+(\tilde{r})\right) \Big \Vert _{H_{P_\epsilon }^{-1}}\\&\le \frac{1}{2}\Vert r-\tilde{r}\Vert _{H_{P_\epsilon }^1}. \end{aligned}$$

Therefore, we conclude that $\Phi$ is contractive, and it has a unique fixed point, denoted by $r_\epsilon$, on the ball of radius $\delta _\epsilon$ centered at 0 in $H_{P_\epsilon }^1$. As a consequence, $u_\epsilon =Q_0+r_\epsilon$ solves the higher-order equation, and $\Vert u_\epsilon -Q_0\Vert _{H_{P_\epsilon }^1}=\Vert r_\epsilon \Vert _{H_{P_\epsilon }^1}\rightarrow 0$ as $\epsilon \rightarrow 0$. $\square$

5 Local uniqueness

The solution $u_\epsilon$, given by Proposition 4.1, is unique in a small ball of radially symmetric real-valued functions whose radius may depend on $\epsilon >0$. In this section, we upgrade this uniqueness to that in a small ball of all complex-valued functions whose radius is independent of $\epsilon >0$.

Proposition 5.1

(Uniqueness) Suppose that (1.6) (as well as (1.7) for NLS) holds. Then, there exist $\delta >0$ and $\epsilon _0>0$ such that if $0<\epsilon \le \epsilon _0$, then the solution $u_\epsilon$ to the higher-order equation (1.4) (resp., (1.5)), constructed in Proposition 4.1, is unique in a $\delta$-ball centered at $u_0$ in $H_{P_\epsilon }^1({\mathbb {R}}^d;\mathbb {C})$ up to translation and phase shift.

Proof

We prove the proposition only for the higher-order NLH, because the proof for the higher-order NLS follows similarly. Let $\epsilon _0>0$ be a small number given in Proposition 3.3, and let $\delta >0$ be sufficiently small numbers to be chosen later. Suppose that $0<\epsilon \le \epsilon _0$ and $\tilde{u}_\epsilon$ is another solution to the higher-order equation in a $\delta$-ball centered at $u_0$ in $H_{P_\epsilon }^1({\mathbb {R}}^3;\mathbb {C})$.

First, we aim to show that the imaginary part of $\tilde{u}_\epsilon$ is orthogonal to $u_\epsilon$ up to phase shift. To this end, we consider

$$\begin{aligned} \tilde{U}_\epsilon =\frac{\overline{\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}}}{|\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}|}\tilde{u}_\epsilon , \end{aligned}$$

which also solves the higher-order equation. Here, since $u_\epsilon$ and $\tilde{u}_\epsilon$ are assumed to be sufficiently close to $u_0$, the denominator $\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}\ne 0$. Note that $\tilde{u}_\epsilon \mapsto \tilde{U}_\epsilon$ is a natural action, because if $\tilde{u}_\epsilon$ is simply a rotated $u_\epsilon$ on the complex plane, then this action rotates it back to $u_\epsilon$. Moreover, we have

$$\begin{aligned} \left\| \tilde{U}_\epsilon -u_\epsilon \right\| _{H_{P_\epsilon }^1}^2&=\Vert \tilde{U}_\epsilon \Vert _{H_{P_\epsilon }^1}^2+\Vert u_\epsilon \Vert _{H_{P_\epsilon }^1}^2-2\text {Re}\langle \tilde{U}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}\\&=\Vert \tilde{u}_\epsilon \Vert _{H_{P_\epsilon }^1}^2+\Vert u_\epsilon \Vert _{H_{P_\epsilon }^1}^2-2|\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}|\\&\le \left| \Vert \tilde{u}_\epsilon \Vert _{H_{P_\epsilon }^1}^2+\Vert u_\epsilon \Vert _{H_{P_\epsilon }^1}^2-2\text {Re}\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}\right| =\Vert \tilde{u}_\epsilon -u_\epsilon \Vert _{H_{P_\epsilon }^1}^2\\&\le \left\{ \Vert \tilde{u}_\epsilon -u_0\Vert _{H_{P_\epsilon }^1}+\Vert u_\epsilon -u_0\Vert _{H_{P_\epsilon }^1}\right\} ^2\le 4\delta ^2 \end{aligned}$$

and

$$\begin{aligned} \langle \text {Im}(\tilde{U}_\epsilon ), u_\epsilon \rangle _{H_{P_\epsilon }^1}=\text {Im}\left\{ \frac{\overline{\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}}}{|\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}|}\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}\right\} =\text {Im}\Big \{|\langle \tilde{u}_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}|\Big \}=0. \end{aligned}$$

Therefore, replacing $\tilde{u}_\epsilon$ by $\tilde{U}_\epsilon$ and $\delta$ by $\frac{\delta }{2}$, we may assume that the imaginary part of $\tilde{u}_\epsilon$ is orthogonal to $u_\epsilon$ in $H_{P_\epsilon }^1$.

We denote the difference between two solutions by

$$\begin{aligned} r_\epsilon :=\tilde{u}_\epsilon -u_\epsilon =v_\epsilon +iw_\epsilon \quad \left( \Leftrightarrow \tilde{u}_\epsilon =(u_\epsilon +v_\epsilon )+iw_\epsilon \right) , \end{aligned}$$

where $v_\epsilon$ and $w_\epsilon$ are real-valued, and $\langle w_\epsilon , u_\epsilon \rangle _{H_{P_\epsilon }^1}=0$. When $u_\epsilon$ and $\tilde{u}_\epsilon$ are solutions to the higher-order NLH, then the difference $r_\epsilon$ satisfies

$$\begin{aligned} (P_\epsilon +1) r_\epsilon&=\left( |x|^{-1}*|\tilde{u}_\epsilon |^2\right) \tilde{u}_\epsilon -\left( |x|^{-1}*|u_\epsilon |^2\right) u_\epsilon \\&=\Big (|x|^{-1}*\left( u_\epsilon ^2+2u_\epsilon v_\epsilon +|r_\epsilon |^2\right) \Big )\left( (u_\epsilon +v_\epsilon )+iw_\epsilon \right) -\left( |x|^{-1}*|u_\epsilon |^2\right) u_\epsilon \\&=\Big (|x|^{-1}*\left( 2u_\epsilon v_\epsilon +|r_\epsilon |^2\right) \Big )u_\epsilon +\Big (|x|^{-1}*\left( u_\epsilon ^2+2u_\epsilon v_\epsilon +|r_\epsilon |^2\right) \Big )(v_\epsilon +iw_\epsilon ). \end{aligned}$$

Moving the linear terms on the right hand side to the left then using (3.4), the imaginary part of the equation (for $w_\epsilon$) can be written as

$$\begin{aligned} {\mathcal {L}}_\epsilon ^-w_\epsilon =\Big (|x|^{-1}*\left( 2u_\epsilon v_\epsilon +|r_\epsilon |^2\right) \Big )w_\epsilon . \end{aligned}$$

Then, by Proposition 3.3 and the nonlinear estimate (Lemma A.2), we prove that

$$\begin{aligned} \beta \Vert w_\epsilon \Vert _{H_{P_\epsilon }^1}&\le \Vert {\mathcal {L}}_\epsilon ^-w_\epsilon \Vert _{H_{P_\epsilon }^{-1}}=\left\| \Big (|x|^{-1}*\left( 2u_\epsilon v_\epsilon +|r_\epsilon |^2\right) \Big )w_\epsilon \right\| _{H_{P_\epsilon }^{-1}}\\&\le C \left( \Vert u_\epsilon \Vert _{H_{P_\epsilon }^1}+\Vert r_\epsilon \Vert _{H_{P_\epsilon }^1}\right) \Vert r_\epsilon \Vert _{H_{P_\epsilon }^1}\Vert w_\epsilon \Vert _{H_{P_\epsilon }^1}\\&\le C\left( \Vert u_0\Vert _{H_{P_\epsilon }^1}+2\delta \right) \delta \Vert w_\epsilon \Vert _{H_{P_\epsilon }^1}. \end{aligned}$$

Therefore, choosing small $\delta$ such that $C(\Vert u_0\Vert _{H_{P_\epsilon }^1}+2\delta )\delta <\beta$, we conclude that $w_\epsilon =0$.

By a suitable phase shift, we may assume that $\tilde{u}_\epsilon$ is real-valued. Furthermore, by translating $\tilde{u}_\epsilon$ so that $\Vert \tilde{u}_\epsilon (\cdot -a)-u_\epsilon \Vert _{H_{P_\epsilon }^1}=\Vert \tilde{u}_\epsilon -u_\epsilon (\cdot +a)\Vert _{H_{P_\epsilon }^1}$ is minimized, equivalently

$$\begin{aligned} \frac{\partial }{\partial _{x_j}}\Bigg |_{a=0}\Vert \tilde{u}_\epsilon -u_\epsilon (\cdot +a)\Vert _{H_{P_\epsilon }^1}^2=0, \end{aligned}$$

we may assume that $\tilde{u}_\epsilon$ is orthogonal to $\partial _{x_j}u_\epsilon$ in $H_{P_\epsilon }^1$ for all $j=1,2,3$. Now, we write the equation for the difference $r_\epsilon =\tilde{u}_\epsilon -u_\epsilon$,

$$\begin{aligned} (P_\epsilon +1) r_\epsilon&={\mathcal {N}}'(\tilde{u}_\epsilon )-{\mathcal {N}}'(u_\epsilon )={\mathcal {N}}'(u_\epsilon +r_\epsilon )-{\mathcal {N}}'(u_\epsilon )\\ \Rightarrow {\mathcal {L}}_\epsilon ^+r_\epsilon&={\mathcal {N}}'(u_\epsilon +r_\epsilon )-{\mathcal {N}}'(u_\epsilon )-{\mathcal {N}}_{u_\epsilon }^+(r_\epsilon ), \end{aligned}$$

where ${\mathcal {N}}_{u_\epsilon }^+$ is defined in (3.2). Since $r_\epsilon =\tilde{u}_\epsilon -u_\epsilon$ is orthogonal to $\partial _{x_j}u_\epsilon$ in $H_{P_\epsilon }^1$, by Proposition 3.3 and the nonlinear estimate (Lemma A.2) again, we obtain

$$\begin{aligned} \beta \Vert r_\epsilon \Vert _{H_{P_\epsilon }^1}&\le \Vert {\mathcal {L}}_\epsilon ^+r_\epsilon \Vert _{H_{P_\epsilon }^{-1}}=\left\| \left( |x|^{-1}*r_\epsilon ^2\right) u_\epsilon +\Big (|x|^{-1}*\left( 2u_\epsilon r_\epsilon +r_\epsilon ^2\right) \Big )r_\epsilon \right\| _{H_{P_\epsilon }^{-1}}\\&\le \tilde{C} \left( \Vert u_\epsilon \Vert _{H_{P_\epsilon }^1}+\Vert r_\epsilon \Vert _{H_{P_\epsilon }^1}\right) \Vert r_\epsilon \Vert _{H_{P_\epsilon }^1}^2\\&\le \tilde{C} \left( \Vert u_0\Vert _{H_{P_\epsilon }^1}+2\delta \right) \delta \Vert r_\epsilon \Vert _{H_{P_\epsilon }^1}. \end{aligned}$$

Choosing even smaller $\delta >0$ such that $\tilde{C}(\Vert u_0\Vert _{H_{P_\epsilon }^1}+2\delta )\delta <\beta$ if necessary, we prove that $r_\epsilon =0$. Therefore, we conclude that $\tilde{u}_\epsilon =u_\epsilon$ up to translation and phase shift. $\square$

Now, we are ready to prove the main theorem.

Proof of Theorem 1.1

By Proposition 2.3, if $\epsilon >0$ is small enough, then ground states $Q_\epsilon ^J$’s are close to the reference ground state $Q_0$ in $H_{P_\epsilon }^1$ (modifying the sequence by translation and phase shift if necessary). However, by uniqueness in Proposition 5.1, $Q_\epsilon ^J$ is identified with the radially symmetric real-valued solution $u_\epsilon$, constructed in Proposition 4.1. Moreover, by Proposition 3.2, it is non-degenerate. Therefore, we prove the main theorem. $\square$

6 Proof of Theorem 1.3

We proceed exactly as in the proof of Proposition 4.1. We denote by

$$\begin{aligned} r_c^J:=Q_c^J-Q_c \end{aligned}$$

the difference between two solutions. Then, it satisfies

$$\begin{aligned} (P_c^J+1) r_c^J&=(P_c^J+1)Q_c^J-(P_c^J+1)Q_c\\&=\left( P_c-P_c^J\right) Q_c+(P_c^J+1)Q_c^J-(P_c+1)Q_c\\&=\left( P_c-P_c^J\right) Q_c+{\mathcal {N}}'({Q_c^J})-{\mathcal {N}}'({Q_c})\\&=\left( P_c-P_c^J\right) Q_c+{\mathcal {N}}'({Q_c+r_c})'-{\mathcal {N}}'({Q_c}), \end{aligned}$$

where $P_c=\sum _{j=1}^J\frac{(-1)^{j-1}\alpha _j}{m^{2j-1}c^{2j-2}}$ and $\alpha _j=\frac{(2j-2)!}{j!(j-1)!2^{2j-1}}$. Moving the linear terms on the right hand side to the left as above, we write

$$\begin{aligned} {\mathcal {L}}_c^{J,+} r_c^J=\left( P_c-P_c^J\right) Q_c+{\mathcal {N}}'({Q_c+r_c})'-{\mathcal {N}}'({Q_c})-{\mathcal {N}}_{Q_c}^+(r_c), \end{aligned}$$

(6.1)

where

$$\begin{aligned} {\mathcal {L}}_c^{J,+}=P_c^J+1-{\mathcal {N}}_{Q_c}^+=\sqrt{1+P_c^J}\left( \text {Id}-\frac{1}{\sqrt{1+P_c^J}}{\mathcal {N}}_{Q_c}^+\frac{1}{\sqrt{1+P_c^J}}\right) \sqrt{1+P_c^J}. \end{aligned}$$

Repeating the proof of Lemma 3.4 together with $Q_c\rightarrow Q_0$ in $H^1$ as $c\rightarrow \infty$, one can show that

$$\begin{aligned}&\text {Id}-\frac{1}{\sqrt{1+P_c^J}}{\mathcal {N}}_{Q_c}^+\frac{1}{\sqrt{1+P_c^J}}\\&\quad =\left( \text {Id}-\frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1-\Delta }}\right) +\left( \frac{1}{\sqrt{1-\Delta }}{\mathcal {N}}_{Q_0}^+\frac{1}{\sqrt{1-\Delta }}-\frac{1}{\sqrt{1+P_c^J}}{\mathcal {N}}_{Q_c}^+\frac{1}{\sqrt{1+P_c^J}}\right) \end{aligned}$$

is invertible on $L_{rad}^2({\mathbb {R}}^d;{\mathbb {R}})$ and its inverse is uniformly bounded for sufficiently large $c\ge 1$. Hence, applying the trivial embedding $H_{P_c^J}^1\hookrightarrow H^1$ (from Lemma B.1) and its dual embedding, we obtain

$$\begin{aligned} \Vert r_c^J\Vert _{H^1}&\lesssim \Vert r_c^J\Vert _{H_{P_c^J}^1}=\left\| ({\mathcal {L}}_c^{J,+})^{-1}\Bigg \{\left( P_c-P_c^J\right) Q_c+{\mathcal {N}}'({Q_c+r_c})'-{\mathcal {N}}'({Q_c})-{\mathcal {N}}_{Q_c}^+(r_c)\Bigg \}\right\| _{H_{P_c^J}^1}\\&\lesssim \left\| \left( P_c-P_c^J\right) Q_c+{\mathcal {N}}'({Q_c+r_c})'-{\mathcal {N}}'({Q_c})-{\mathcal {N}}_{Q_c}^+(r_c)\right\| _{H_{P_c^J}^{-1}}\\&\lesssim \left\| \left( P_c-P_c^J\right) Q_c\right\| _{H^{-1}}+\left\| {\mathcal {N}}'({Q_c+r_c})'-{\mathcal {N}}'({Q_c})-{\mathcal {N}}_{Q_c}^+(r_c)\right\| _{H^{-1}}, \end{aligned}$$

where the implicit constants do not depend on $c\ge 1$. Therefore, it follows from the nonlinear estimates (Lemma A.2) that for sufficiently large $c\ge 1$,

$$\begin{aligned} \Vert r_c^J\Vert _{H^1}\lesssim \left\| \left( P_c-P_c^J\right) Q_c\right\| _{H^{-1}}\lesssim \frac{1}{c^{2J}}\Vert Q_c\Vert _{H^{2J+1}}, \end{aligned}$$

because by Taylor’s theorem,

$$\begin{aligned} \left| \left( \sqrt{c^2s+m^2c^4}-mc^2\right) -\sum _{j=1}^J\frac{(-1)^{j-1}\alpha _j}{m^{2j-1}c^{2j-2}}s^j\right| \lesssim \frac{s^{J+1}}{c^{2J}}. \end{aligned}$$

Finally, by the uniform bound on high Sobolev norm of $Q_c$ (see Proposition 2.5 or [2]), we conclude that $\Vert Q_c^J-Q_c\Vert _{H^1}=\Vert r_c^J\Vert _{H^1}\lesssim _J c^{-2J}$.

Appendices

Appendix A: Nonlinear estimates

We show the nonlinear estimates which are used in the contraction mapping argument.

Lemma A.1

(Nonlinear estimates) Let $u\in H^1$ be real-valued. For any $\eta >0$, there exists $\delta _0>0$, depending on $\Vert u\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}$ and $\eta$, such that if $0<\delta \le \delta _0$ and

$$\begin{aligned} \Vert r\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}, \Vert \tilde{r}\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}\le \delta , \end{aligned}$$

then

$$\begin{aligned} \left\| {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right\| _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\le \eta \Vert r\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})} \end{aligned}$$

and

$$\begin{aligned} \left\| \left( {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right) -\left( {\mathcal {N}}'(u+\tilde{r})-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(\tilde{r})\right) \right\| _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\le \eta \Vert r-\tilde{r}\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}. \end{aligned}$$

The above lemma follows from the multilinear estimates.

Lemma A.2

(Multilinear estimates) We have

$$\begin{aligned} \left\| \left( \frac{1}{|x|}*(\phi _1\phi _2)\right) \phi _3\right\| _{L^2({\mathbb {R}}^3;{\mathbb {R}})}\lesssim \prod _{j=1}^3\Vert \phi _j\Vert _{H^1({\mathbb {R}}^3;{\mathbb {R}})}. \end{aligned}$$

Moreover, if $d=1,2$ and $k\in {\mathbb {N}}$ or if $d=3$ and $k=1$, then

$$\begin{aligned} \left\| \prod _{j=1}^{2k+1}\phi _j\right\| _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\lesssim \prod _{j=1}^{2k+1}\Vert \phi _j\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}. \end{aligned}$$

Proof

By the Hölder, Young’s and Sobolev inequalities, we prove that

$$\begin{aligned} \left\| \left( \frac{1}{|x|}*(\phi _1\phi _2)\right) \phi _3\right\| _{L^2({\mathbb {R}}^3;{\mathbb {R}})}&\le \left\| \left( \frac{1}{|x|}*(\phi _1\phi _2)\right) \right\| _{L^9({\mathbb {R}}^3;{\mathbb {R}})}\Vert \phi _3\Vert _{L^{18/7}({\mathbb {R}}^3;{\mathbb {R}})}\\&\lesssim \Vert \phi _1\phi _2\Vert _{L^{9/7}({\mathbb {R}}^3;{\mathbb {R}})}\Vert \phi _3\Vert _{L^{18/7}({\mathbb {R}}^3;{\mathbb {R}})}\\&\lesssim \prod _{j=1}^3\Vert \phi _j\Vert _{L^{18/7}({\mathbb {R}}^3;{\mathbb {R}})}\lesssim \prod _{j=1}^3\Vert \phi _j\Vert _{H^1({\mathbb {R}}^3;{\mathbb {R}})} \end{aligned}$$

and similarly,

$$\begin{aligned} \left\| \prod _{j=1}^{2k+1}\phi _j\right\| _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\le \prod _{j=1}^{2k+1}\Vert \phi _j\Vert _{L^{2(2k+1)}({\mathbb {R}}^d;{\mathbb {R}})}\lesssim \prod _{j=1}^{2k+1}\Vert \phi _j\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}. \end{aligned}$$

Proof of Lemma A.1

Suppose that $\Vert r\Vert _{H^1},\Vert \tilde{r}\Vert _{H^1}\le \Vert u\Vert _{H^1}$. For the Hartree nonlinearity, by algebra, we write

$$\begin{aligned} {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)=\left( \frac{1}{|x|}*r^2\right) u+2\left( \frac{1}{|x|}*(ur)\right) r+\left( \frac{1}{|x|}*r^2\right) r \end{aligned}$$

and

$$\begin{aligned}&\left( {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right) -\left( {\mathcal {N}}'(u+\tilde{r})-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(\tilde{r})\right) \\&\quad =\left( \frac{1}{|x|}*\left( (r+\tilde{r})(r-\tilde{r})\right) \right) u+2\left( \frac{1}{|x|}*\left( u(r-\tilde{r})\right) \right) r+2\left( \frac{1}{|x|}*(u\tilde{r})\right) (r-\tilde{r})\\&\quad +\left( \frac{1}{|x|}*\left( (r+\tilde{r})(r-\tilde{r})\right) \right) r+\left( \frac{1}{|x|}*\tilde{r}^2\right) (r-\tilde{r}) \end{aligned}$$

Thus, by the multilinear estimate (Lemma A.2),

$$\begin{aligned} \left\| {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right\| _{L^2}\le C\Big (\Vert u\Vert _{H^1}+\Vert r\Vert _{H^1}\Big )\Vert r\Vert _{H^1}^2\le 2C\delta \Vert u\Vert _{H^1}\Vert r\Vert _{H^1} \end{aligned}$$

and

$$\begin{aligned}&\left\| \left( {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right) -\left( {\mathcal {N}}'(u+\tilde{r})-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(\tilde{r})\right) \right\| _{L^2}\\&\quad \le C\Big (\Vert u\Vert _{H^1}+\Vert r\Vert _{H^1}+\Vert \tilde{r}\Vert _{H^1}\Big )\Big (\Vert r\Vert _{H^1}+\Vert \tilde{r}\Vert _{H^1}\Big )\Vert r-\tilde{r}\Vert _{H^1}\\&\quad \le 6C\delta \Vert u\Vert _{H^1}\Vert r-\tilde{r}\Vert _{H^1}. \end{aligned}$$

Then, taking $\delta _0=\eta \min \{\frac{1}{6C\Vert u\Vert _{H^1}},\Vert u\Vert _{H^1}\}$, we prove the lemma for the Hartree nonlinearity.

Similarly, for the polynomial nonlinearity, by the multilinear estimate (Lemma A.2),

$$\begin{aligned} \left\| {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right\| _{L^2}&=\left\| \sum _{j=2}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} u^{2k+1-j} r^j\right\| _{L^2}\\&\le \sum _{j=2}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} \left\| u^{2k+1-j} r^j\right\| _{L^2}\\&\le C\sum _{j=2}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} \Vert u\Vert _{H^1}^{2k+1-j} \Vert r\Vert _{H^1}^j\\&\le C_k\delta \Vert u\Vert _{H^1}^{2k-1}\Vert r\Vert _{H^1} \end{aligned}$$

and

$$\begin{aligned}&\left\| \left( {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right) -\left( {\mathcal {N}}'(u+\tilde{r})-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(\tilde{r})\right) \right\| _{L^2}\\&\quad =\left\| \sum _{j=3}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} u^{2k+1-j} (r-\tilde{r})(r^{j-1}+r^{j-2}\tilde{r}+\cdots +\tilde{r}^{j-1})\right\| _{L^2}\\&\quad \le \sum _{j=3}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} \left\| u^{2k+1-j} (r-\tilde{r})(r^{j-1}+r^{j-2}\tilde{r}+\cdots +\tilde{r}^{j-1})\right\| _{L^2}\\&\quad \le C\sum _{j=3}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} \Vert u\Vert _{H^1}^{2k+1-j} \Vert r-\tilde{r}\Vert _{H^1}\left( \Vert r\Vert _{H^1}^{j-1}+\Vert r\Vert _{H^1}^{j-2}\Vert \tilde{r}\Vert _{H^1}+\cdots +\Vert \tilde{r}\Vert _{H^1}^{j-1}\right) \\&\quad \le C_k\delta \Vert u\Vert _{H^1}^{2k-1}\Vert r-\tilde{r}\Vert _{H^1} \end{aligned}$$

for some constant $C_k>0$. Then, taking $\delta _0=\eta \min \{\frac{1}{2C_k\Vert u\Vert _{H^1}^{2k-1}},\Vert u\Vert _{H^1}\}$, we complete the proof of the lemma for the polynomial nonlinearity. $\square$

Appendix B: Uniform lower bound for higher-order operators in (1.3)

Lemma B.1

(Uniform lower bound for higher-order operators in (1.3)) For any $\xi \in {\mathbb {R}}^3$, we have

$$\begin{aligned} \sum _{j=1}^{2k-1}\frac{(-1)^{j-1}\alpha _j}{m^{2j-1}c^{2j-2}}|\xi |^{2j}\ge \frac{|\xi |^2}{2m}, \end{aligned}$$

where $a_j=\frac{(2j-2)!}{j!(j-1)! 2^{2j-1}}$.

Proof

By change of variables $\frac{\xi }{m}\mapsto \xi$, it suffices to prove the lemma assuming $m=1$. The inequality is trivial when $k=1$. Suppose that $k\ge 2$. Splitting the positive and the negative terms and then applying the Cauchy–Schwarz inequality for the negative terms, we obtain

$$\begin{aligned} \sum _{j=1}^{2k-1}\frac{(-1)^{j-1}\alpha _j}{c^{2j-2}}|\xi |^{2j}&=\sum _{j=1}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}-\sum _{j=1}^{k-1}\frac{\alpha _{2j}}{c^{4j-2}}|\xi |^{4j}\\&\ge \sum _{j=1}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}\\&-\frac{1}{2}\sum _{j=1}^{k-1} \left\{ \frac{(\alpha _{2j})^2}{\alpha _{2j-1}\alpha _{2j+1}}\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}+\frac{\alpha _{2j+1}}{c^{4j}}|\xi |^{4j+2}\right\} . \end{aligned}$$

Since $\frac{(\alpha _{2j})^2}{\alpha _{2j-1}\alpha _{2j+1}}=\cdots =\frac{(4j-3)(2j+1)}{(4j-1)2j}\le 1$ for all $j\ge 1$, it is bounded below from

$$\begin{aligned}&\sum _{j=1}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}-\frac{1}{2}\sum _{j=1}^{k-1} \left\{ \frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}+\frac{\alpha _{2j+1}}{c^{4j}}|\xi |^{4j+2}\right\} \\&\quad = \sum _{j=1}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}-\frac{1}{2}\sum _{j=1}^{k-1}\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}-\frac{1}{2}\sum _{j=2}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}\\&\quad =\frac{1}{2}|\xi |^2+\frac{\alpha _{2k-1}}{2c^{4k-4}}|\xi |^{4k-2}\ge \frac{1}{2}|\xi |^2. \end{aligned}$$

$\square$