On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

Abstract

We consider the problem

$$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$

where $\Omega$ is a bounded domain in ${\mathbb {R}}^2$ with smooth boundary, the exponent p is greater than 1, $\epsilon >0$ is a small parameter, V is a uniformly positive, smooth potential on $\bar{\Omega }$, and $\nu$ denotes the outward unit normal of $\partial \Omega$. Let $\Gamma$ be a curve intersecting orthogonally $\partial \Omega$ at exactly two points and dividing $\Omega$ into two parts. Moreover, $\Gamma$ satisfies stationary and non-degeneracy conditions with respect to the functional $\int _{\Gamma }V^{\sigma }$, where $\sigma =\frac{p+1}{p-1}-\frac{1}{2}$. We prove the existence of a solution $u_\epsilon$ concentrating along the whole of $\Gamma$, exponentially small in $\epsilon$ at any fixed distance from it, provided that $\epsilon$ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004).

Appendices

Appendices

Proofs of Lemmas 2.1 and 2.2

Proof of Lemma 2.1

The curves can be expressed in the following forms

$$\begin{aligned} {{\mathcal {C}}}_i:&\ {{\mathbb {H}}}\big ({{\tilde{t}}},{{\tilde{\varphi }}}_i({{\tilde{t}}})\big ) \,=\,\gamma \big ({{\tilde{\varphi }}}_i({{\tilde{t}}})\big )+{{\tilde{t}}}\,n\big ({{\tilde{\varphi }}}_i({{\tilde{t}}})\big ),\,\,\,\,i=1,2, \end{aligned}$$

(A.1)

$$\begin{aligned} \Gamma :&\ {{\mathbb {H}}}(0,\tilde{\theta })\,=\,\gamma (\tilde{\theta }). \end{aligned}$$

(A.2)

It follows that the tangent vectors of ${{\mathcal {C}}}_1$ at $P_1$ can be written as

$$\begin{aligned} \frac{{{\mathrm {d}}} {{\mathcal {C}}}_1}{{{\mathrm {d}}}{{\tilde{t}}}}\Big |_{{{\tilde{t}}}=0} \,=\,\frac{\partial \gamma }{\partial \tilde{\theta }}\Big |_{{{\tilde{\theta }}}\,=\,{{\tilde{\varphi }}_1(0)}} \cdot \frac{{{\mathrm {d}}} {{\tilde{\varphi }}_1}}{{{\mathrm {d}}} {{\tilde{t}}}}\Big |_{{{\tilde{t}}}=0}\,+\,n\big ({{\tilde{\varphi }}}_1(0)\big ), \end{aligned}$$

and the tangent vector of $\Gamma$ at $P_1$ is

$$\begin{aligned} \frac{\partial \gamma }{\partial \tilde{\theta }}\Big |_{\tilde{\theta }\,=\,{{\tilde{\varphi }}}_1(0)}. \end{aligned}$$

According to the condition: $\Gamma \bot \partial \Omega$, we have that

$$\begin{aligned} \left< \frac{{{\mathrm {d}}}{{\mathcal {C}}}_1}{{{\mathrm {d}}}{{\tilde{t}}}}\Big |_{{{\tilde{t}}}=0},\,\, \frac{\partial \gamma }{\partial \tilde{\theta }}\Big |_{\tilde{\theta }={{\tilde{\varphi }}}_1(0)}\right>=0. \end{aligned}$$

Combining with

$$\begin{aligned} \left<n({{\tilde{\varphi }}}_1(0)),\,\,\frac{\partial \gamma }{\partial \tilde{\theta }}\Big |_{\tilde{\theta }={{\tilde{\varphi }}}_1(0)}\right>=0, \end{aligned}$$

we have $\tilde{\varphi }_1'(0)=0.$ Similarly, we can show ${{\tilde{\varphi }}}_2'(0)=0$.

In the coordinate system $(y_1,y_2)$, $\gamma (\tilde{\theta })$ and $n(\tilde{\theta })$ can be expressed as follows

$$\begin{aligned} \gamma (\tilde{\theta })=\big (\gamma _1(\tilde{\theta }), \gamma _2(\tilde{\theta })\big ), \qquad n(\tilde{\theta })=\big (n_1(\tilde{\theta }), n_2(\tilde{\theta })\big ). \end{aligned}$$

The relations

$$\begin{aligned} |\gamma _1'(\tilde{\theta })|^2\,+\,|\gamma _2'(\tilde{\theta })|^2=1, \qquad |n_1(\tilde{\theta })|^2+|n_2(\tilde{\theta })|^2=1, \qquad \gamma _1'(\tilde{\theta })n_1(\tilde{\theta })+ \gamma _2'(\tilde{\theta })n_2(\tilde{\theta })=0, \end{aligned}$$

will give that

$$\begin{aligned} \gamma _1'(\tilde{\theta })n_2(\tilde{\theta })-\gamma _2'(\tilde{\theta })n_1(\tilde{\theta })\,=\,1, \end{aligned}$$

provided that the sign is taken as $'+'$ by suitable choice of the natural parameter of $\Gamma$. Then, the curve ${{\mathcal {C}}}_1$ can be expressed in the following form

$$\begin{aligned} \begin{aligned} {{\mathcal {C}}}_1: {{\mathbb {H}}}\big ({{\tilde{t}}},{{\tilde{\varphi }}}_1({{\tilde{t}}})\big ) \,&=\Big (\gamma _1\big ({{\tilde{\varphi }}}_1({{\tilde{t}}})\big )+\tilde{t}n_1\big ({{\tilde{\varphi }}}_1({{\tilde{t}}})\big ),\,\, \gamma _2\big ({{\tilde{\varphi }}}_1({{\tilde{t}}})\big )+\tilde{t}n_2\big ({{\tilde{\varphi }}}_1({{\tilde{t}}})\big )\Big ) \\&\equiv \,\big ( y_1(\tilde{t})\,,\,y_2(\tilde{t})\big ). \end{aligned} \end{aligned}$$

The calculations

$$\begin{aligned} y_i'({{\tilde{t}}})= & {} \,\gamma _i'({\tilde{\varphi }}_1)\cdot \frac{{{\mathrm {d}}}{{\tilde{\varphi }}_1}}{{{\mathrm {d}}} {{\tilde{t}}}}+n_i({\tilde{\varphi }}_1)+\tilde{t}\cdot n_i'({\tilde{\varphi }}_1)\cdot \frac{{{\mathrm {d}}} {{\tilde{\varphi }}_1}}{{{\mathrm {d}}} {{\tilde{t}}}},\qquad i=1,2\\ y_i''({{\tilde{t}}})= & {} \,\gamma _i''({\tilde{\varphi }}_1)\cdot \Big (\frac{{{\mathrm {d}}} {{\tilde{\varphi }}_1}}{{{\mathrm {d}}} {{\tilde{t}}}}\Big )^2+\gamma _i'({\tilde{\varphi }}_1)\cdot \frac{{{\mathrm {d}}}^2 {{\tilde{\varphi }}_1}}{{{\mathrm {d}}} {{\tilde{t}}}^2}+2\,n_i'({\tilde{\varphi }}_1)\cdot \frac{{{\mathrm {d}}} {{\tilde{\varphi }}_1}}{{{\mathrm {d}}} {{\tilde{t}}}} \\&+\tilde{t}\,n_i''({\tilde{\varphi }}_1)\cdot \Big (\frac{{{\mathrm {d}}} {{\tilde{\varphi }}_1}}{{{\mathrm {d}}} {{\tilde{t}}}}\Big )^2+\tilde{t}\,n_i'({\tilde{\varphi }}_1)\cdot \frac{{{\mathrm {d}}}^2 {{\tilde{\varphi }}_1}}{{{\mathrm {d}}} {{\tilde{t}}}^2},\qquad i=1,2 \end{aligned}$$

imply that

$$\begin{aligned} (y_1'(t))^2+(y_2'(t))^2\Big |_{\tilde{t}=0}\,=\,|n(0)|^2\,=\,1, \end{aligned}$$

and

$$\begin{aligned} y_1'(\tilde{t})\,y_2''(\tilde{t})-y_1''(\tilde{t})\,y_2'(\tilde{t})\Big |_{\tilde{t}=0}\, =\,\big (n_1(0)\,\gamma _2'(0)\,-\,n_2(0)\,\gamma _1'(0)\big )\,{\tilde{\varphi }}_1''(0)\,=\,{\tilde{\varphi }}_1''(0). \end{aligned}$$

Therefore, the signed curvature of the curve $\mathcal {C}_1$ at the point $P_1$ is

$$\begin{aligned} k_1=\frac{\, y_1'(0)y_2''(0)-y_1''(0)y_2'(0)\,}{\big ((y_1'(0))^2+(y_2'(0))^2\big )^{\frac{3}{2}}} =\tilde{\varphi }_1''(0). \end{aligned}$$

Similarly, we can show $k_2={{\tilde{\varphi }}}_2''(0)$. $\square$

Proof of Lemma 2.2

Let us begin by considering the first-order derivative

$$\begin{aligned} \frac{{\partial } F}{{\partial } t}(0,\theta )&\,=\,\Big [\gamma {\,'}(\Theta )\cdot \Theta _t \,+\,n(\Theta )\,+\,tn'(\Theta )\cdot {\Theta }_t\Big ]\Big |_{(0,\theta )} \\&\,=\,\gamma {\,'}(\Theta )\cdot 0\,+\,n(\Theta (0,\theta ))\,=\,n(\theta ). \end{aligned}$$

So as t is small enough, $\frac{{\partial } F}{{\partial } t}\ne 0$. Consider the second order derivative

$$\begin{aligned} \begin{aligned} q_1(\theta )&\equiv \frac{{\partial }^2F}{{\partial } t^2}(0,\theta ) \\&=\,\left[ \gamma ''(\Theta )\cdot (\Theta _t)^2\,+\,\gamma '(\Theta )\cdot \Theta _{tt}\,+\,2n'(\Theta )\cdot \Theta _{t}\,\right. \\&\quad \left. +\,tn''(\Theta )\cdot (\Theta _t)^2 \,+\,tn'(\Theta )\cdot \Theta _{tt}\right] \Big |_{(0,\theta )} \\&=\,\gamma '(\theta )\cdot \Theta _{tt}(0,\theta )\;\;{ \bot n(\gamma (\theta ))}. \end{aligned} \end{aligned}$$

The third-order derivative

$$\begin{aligned} \begin{aligned} q_2(\theta )&\equiv \frac{{\partial }^3F}{{\partial } t^3}(0,\theta )\\&=\Big [ \gamma {'''}(\Theta )\cdot (\Theta _t)^3 +3\gamma {''}(\Theta )\cdot \Theta _t\cdot \Theta _{tt} +\gamma {\,'}(\Theta )\cdot \Theta _{ttt} +3n{''}(\Theta )\cdot (\Theta _t)^2\\&\quad +3n{'}(\Theta )\cdot \Theta _{tt} +tn{'''}(\theta )\cdot (\Theta _t)^3+3tn{''}\cdot \Theta _t\cdot \Theta _{tt}+tn'(\theta )\cdot \Theta _{ttt} \Big ]\Big |_{(0,\theta )}\\&=\gamma {\,'}(\theta )\cdot \Theta _{ttt}(0,\theta )+3n{'}(\theta )\cdot \Theta _{tt}(0,\theta ) \qquad \bot n(\theta ). \end{aligned} \end{aligned}$$

This finishes the proof of the lemma. $\square$

As a conclusion, as t is small enough, there holds the asymptotic behavior

$$\begin{aligned} F(t,\theta )&\,=\,&\gamma (\theta ) +tn(\theta )+\frac{t^2}{2}q_1(\theta )+\frac{t^3}{6}q_2(\theta )+O(t^4),\nonumber \\&\quad \forall \, \theta \in [0,1], \forall t\in (-\delta _0,\delta _0), \end{aligned}$$

(A.3)

where $\delta _0>0$ is a small constant. This gives us that

$$\begin{aligned} \frac{{\partial } F}{{\partial } t}(t,\theta )\,= & {} \,n(\theta )+t q_1(\theta )+\frac{t^2}{2}q_2(\theta )+O(t^3),\\ \frac{{\partial } F}{{\partial } \theta }(t,\theta )\,= & {} \,\gamma '(\theta )-k(\theta )t\gamma {\,'}(\theta )+\frac{t^2}{2}q_1'(\theta )+\frac{t^3}{6}q_2'(\theta )+O(t^4). \end{aligned}$$

These formulas will play an important role in the derivation of the local form of (1.1), which will be given in Appendix B.

Local forms of the differential operators in (1.1)

In this section, we are devoted to presenting the expressions of the differential operators $\Delta$ and $\partial /\partial \nu$ in (1.1). The formulas will be provided in the local forms in the modified Fermi coordinates given in Sect. 2.

As first step, here are the computations of the metric matrix:

$$\begin{aligned} \begin{aligned} g_{11}&\,=\,\left<\frac{{\partial } F}{{\partial } t},\, \frac{{\partial } F}{{\partial } t}\right> \,=\,1+t^2|q_1|^2+O(t^3), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} g_{12}&\,=\,\left<\frac{{\partial } F}{{\partial } t},\,\,\frac{{\partial } F}{{\partial } \theta }\right> \,=\,t<q_1, \gamma '>+\frac{t^2}{2}\big (<q_2,\gamma '>-k<q_1, \gamma '>\big )+O(t^3), \end{aligned} \end{aligned}$$

where we have used the fact

$$\begin{aligned}<q_1', n>-k<q_1, \gamma '> \,=\,<q_1', n>+<q_1, n'> \,=\,\frac{\partial }{\partial \theta }<q_1, n> \,=\,0. \end{aligned}$$

The last element is

$$\begin{aligned} \begin{aligned} g_{22}&\,=\,\left<\frac{{\partial } F}{{\partial } \theta },\,\,\frac{{\partial } F}{{\partial } \theta }\right> \,=\,1-2kt+t^2\left(<q_1', \gamma '>+k^2\right)+O(t^3). \end{aligned} \end{aligned}$$

So the determinant of the metric matrix is

$$\begin{aligned} \begin{aligned} g&\,=\,\text{ det }(g_{ij}) \,=\,1-2kt+t^2\left(<q_1', \gamma '>+k^2\right)+O(t^3), \end{aligned} \end{aligned}$$

where we have used $|q_1|^2-<q_1, \gamma '>^2=0$ due to the expression of $q_1$ in Lemma 2.2.

Now, recall the definition of Laplace-Beltrami operator in the form

$$\begin{aligned} \begin{aligned} \triangle _{t,\theta }u&=g^{11}\frac{{\partial }^2 u}{{\partial } t^2}+2g^{12}\frac{{\partial }^2 u}{{\partial } t{\partial } \theta }+g^{22}\frac{{\partial }^2 u}{{\partial } \theta ^2}\\&+(\sqrt{g})^{ -1}\left[ \frac{{\partial }}{{\partial } t}(\sqrt{g}g^{11})+\frac{{\partial }}{{\partial } \theta }(\sqrt{g}g^{21})\right] \frac{{\partial } u}{{\partial } t} \\&\quad +(\sqrt{g})^{ -1}\left[ \frac{{\partial }}{{\partial } t}(\sqrt{g}g^{12})+\frac{{\partial }}{{\partial } \theta }(\sqrt{g}g^{22})\right] \frac{{\partial } u}{{\partial } \theta }. \end{aligned} \end{aligned}$$

(B.1)

Denote

$$\begin{aligned} \varpi (\theta )\,\equiv \,<q_1(\theta ), \gamma '(\theta )>\,=\,\Theta _{tt}(0, \theta ), \end{aligned}$$

(B.2)

where we have used the expression of $q_1$ in Lemma 2.2. It can be checked that the term $\triangle _y u$ in (1.1) has the following form in the modified Fermi coordinate system

$$\begin{aligned} \triangle _y u&=\left[1+O(t^3)\right]u_{tt} -2\left[\,\varpi \,t+\frac{t^2}{2}(3k\varpi +<q_2,\gamma '>)+O(t^3)\right]u_{t\theta } \nonumber \\&\quad \,\,+\left[1+2kt+t^2\big (3k^2-<q_1',\gamma '>+|q_1|^2\big )+O(t^3)\right]u_{\theta \theta } \nonumber \\&\quad \,\,+\left[-k-k^2t+O(t^2)\right]u_t +\left[-\varpi +tk'-t\big (2ka_0+<q_2,\gamma '>\big )+O(t^2)\right]u_\theta \nonumber \\&\equiv \left(1+a_2t^3\right)u_{tt} \,+\,\big (-2\,\varpi \,t+a_3t^2\big )u_{t\theta } \,+\,\big (1+2kt+a_1t^2\big )u_{\theta \theta } \nonumber \\&\quad \,\,+\big (-k-k^2t+a_4t^2\big )u_t \,+\,\big (-\varpi +a_5t\big )u_\theta , \end{aligned}$$

(B.3)

where $a_1$, $\ldots$, $a_5$ are smooth bounded functions. The terms in $\triangle _y u$ will be rearranged in the form

$$\begin{aligned} \triangle _y u\,\,=\,\,u_{tt}+u_{\theta \theta }+\bar{B}_1(u)+\bar{B}_0(u), \end{aligned}$$

(B.4)

where

$$\begin{aligned} \bar{B}_1(u)\,=\,-(k+k^2t)u_{t} -2\,\varpi \,tu_{t\theta } -\varpi u_\theta , \end{aligned}$$

(B.5)

and

$$\begin{aligned} \bar{B}_0(u) \,=\,\,2ktu_{\theta \theta }+a_1t^2u_{\theta \theta }+a_2t^3u_{tt} +a_3t^2u_{t\theta }+a_4t^2u_t +a_5tu_\theta . \end{aligned}$$

(B.6)

We finally show the expression of $\nu$, the outward unit normal of $\partial \Omega$ near $P_1, P_2$, i.e., when $\theta =0,1$ in the modified Fermi coordinates. This will provide the local expression of $\partial u/\partial \nu$ in (1.1). Suppose

$$\begin{aligned} \nu \,=\,\sigma _1\frac{{\partial } F}{{\partial } t}+\sigma _2\frac{{\partial } F}{{\partial } \theta }. \end{aligned}$$

Since ${{\partial } F}/{{\partial } t}\in T({\partial } \Omega )$, we have $<{{\partial } F}/{{\partial } t},\nu >=0$. Hence

$$\begin{aligned} \sigma _2\ne 0,\quad \frac{{\partial } F}{{\partial } \theta }\ne 0,\quad \mathrm{and}\quad \sigma _1g_{11}+\sigma _2g_{12}\,=\,0. \end{aligned}$$

On the other hand, $<\nu ,\nu >=1$, that is

$$\begin{aligned} \left<\sigma _1\frac{{\partial } F}{{\partial } t}+\sigma _2\frac{{\partial } F}{{\partial } \theta },\, \sigma _1\frac{{\partial } F}{{\partial } t}+\sigma _2\frac{{\partial } F}{{\partial } \theta }\right>\,=\,1, \end{aligned}$$

which implies that

$$\begin{aligned} \sigma _1^2g_{11}+\sigma _2^2g_{22}+2\sigma _1\sigma _2g_{12}\,=\,1. \end{aligned}$$

Combining above two equations, one can get

$$\begin{aligned} \sigma _1\,=\,\pm \frac{g^{12}}{\sqrt{g^{22}}}, \qquad \sigma _2\,=\,\pm \sqrt{g^{22}}. \end{aligned}$$

By choosing the ’+’, it is easy to check that

$$\begin{aligned} \begin{aligned} \sigma _2 \,=\,&\,1+kt+k^2t^2-\frac{1}{2}(k_2-k_1)t^2 +\frac{1}{2}\big [(k_2-k_1)\theta +k_1\big ]^2t^2+O(t^3), \end{aligned} \end{aligned}$$

and then

$$\begin{aligned} \begin{aligned} \sigma _1 \,=&\,-\,\big [(k_2-k_1)\theta +k_1\big ]t \,+\,k\big [(k_2-k_1)\theta +k_1\big ]t^2 \\&-\frac{1}{2}\left[(\tilde{\varphi }_2'''(0)-\tilde{\varphi }_1'''(0))\theta +\tilde{\varphi }_1'''(0)\right]t^2 \,+\,O(t^3). \end{aligned} \end{aligned}$$

In the modified Fermi coordinates $(t, \theta )$ in (2.4), the normal derivative $\partial u/\partial \nu$ has a local form as follows

$$\begin{aligned} \sigma _1\frac{\partial u}{\partial t} + \sigma _2\frac{\partial u}{\partial \theta }. \end{aligned}$$

More precisely, for $\theta \,=\,0$, it is

$$\begin{aligned} k_1tu_t \,+\, b_1 t^2u_t \,-\,u_{\theta } \,-\,k(0)tu_{\theta } \,+\,b_2t^2u_{\theta } \,+\,{\bar{D}}_0^0(u), \end{aligned}$$

(B.7)

where

$$\begin{aligned} {\bar{D}}_0^0(u)\,=\,\sigma _3(t)\,u_t\,+\,\sigma _4(t)u_\theta , \end{aligned}$$

(B.8)

and the constants $b_1$ and $b_2$ are given by

$$\begin{aligned} b_1\,=\,\frac{1}{2}\tilde{\varphi }_1'''(0)-k(0)k_1, \qquad b_2\,=\,\frac{1}{2}(k_2-k_1)-k^2(0)-\frac{1}{2}k_1^2. \end{aligned}$$

(B.9)

On the other hand, for $\theta =1$, it has the form

$$\begin{aligned} \begin{aligned}&k_2tu_t \,+\, b_3t^2u_t \,-\,u_{\theta } \,-\,k(1)tu_{\theta } \,+\,b_4 t^2u_{\theta } \,+\,{\bar{D}}_0^1(u), \end{aligned} \end{aligned}$$

(B.10)

with the notation

$$\begin{aligned} {\bar{D}}_0^1(u)&=\sigma _5(t)\,u_t\,+\,\sigma _6(t)u_\theta , \end{aligned}$$

(B.11)

$$\begin{aligned} b_3&=\frac{1}{2}\tilde{\varphi }_2'''(0)-k(1)k_2, \qquad b_4\,=\,\frac{1}{2}(k_2-k_1)-k^2(1)-\frac{1}{2}k_2^2. \end{aligned}$$

(B.12)

In the above, the functions $\sigma _3, \cdots , \sigma _6$ are smooth functions of t with the properties

$$\begin{aligned} |\sigma _i(t)|\le C|t|^3,\quad i=3, 4, 5, 6. \end{aligned}$$

The computations of (4.38)

The computation of (4.38) can be showed as follows. Since $S_6$, $S_8$, $M_{11}(x,z)$, $M_{21}(x,z)$, $M_{51}(x,z)$ are even functions in x, integration against $w_x$ therefore just vanish. This gives that

$$\begin{aligned} \begin{aligned} \text{ Left } \text{ hand } \text{ side } \text{ of } (4.38)&=\int _{\mathbb R}\Big [\epsilon ^2\,S_7\,+\,\epsilon ^2\,S_9\,+\,M_{31}(x,z)\,\\&\quad +\,M_{41}(x,z)\,+\,M_{61}(x,z)\,\Big ]\,w_x\,\mathrm{d}x \\&\equiv J_1\,+\,J_2\,+\,J_3\,+\,J_4\,+\,J_5. \end{aligned} \end{aligned}$$

Recalling the expression of $S_7$ in (4.1), direct computation leads to

$$\begin{aligned} J_{1}= & {} -\epsilon ^2\beta ^{-1}h{''}\int _{\mathbb R}w_x^2\,\mathrm{d}x -2\epsilon ^2\beta ^{-2}\beta 'h{'}\int _{\mathbb R}\big (w_x^2+x w_xw_{xx}\big )\,\mathrm{d}x \nonumber \\&-2\epsilon ^2\alpha ^{-1}\beta ^{-1}\alpha ' h'\int _{\mathbb R}w_x^2\,\mathrm{d}x-\epsilon ^2 h\Big (\beta ^{-1}k^2\int _{\mathbb R}w_x^2\,\mathrm{d}x+\beta ^{-3}V_{tt}\int _{\mathbb R}xww_x\,\mathrm{d}x\Big ) \nonumber \\= & {} -\epsilon ^2\varrho _1\beta ^{-1}h{''} \,-\, \epsilon ^2\varrho _1\beta ^{-1}\big (\beta ^{-1}\beta '+2\alpha ^{-1}\alpha '\big )h{'}\nonumber \\&-\epsilon ^2\varrho _1\beta ^{-1}\Big (k^2-\sigma \beta ^{-2}V_{tt}\Big )h, \end{aligned}$$

(C.1)

where we have used

$$\begin{aligned} -2\int _{\mathbb R}xww_x\,\mathrm{d}x\,=\int _{\mathbb R}w^2\,\mathrm{d}x \,=\,2\sigma \int _{\mathbb R}w_x^2\,\mathrm{d}x, \end{aligned}$$

(C.2)

and constant $\varrho _{1}$ defined in (5.9).

According to the definition of $S_9$, it follows that

$$\begin{aligned} \begin{aligned} J_{2}&=2\,\epsilon ^2\,\beta ^{-1}\,h'\,\varpi \,\int _{\mathbb R}\Big (xw_{xx}w_x+\frac{1}{2} w_x^2\Big )\,\mathrm{d}x-2\,\epsilon ^2\,\alpha ^{-1}\,\alpha '\,\beta ^{-1}\,h\,\varpi \,\int _{\mathbb R}w_x^2\,\mathrm{d}x \\&\quad -2\,\epsilon ^2\,\beta '\,\beta ^{-2}\,h\,\varpi \,\int _{\mathbb R}\big (w_x^2+xw_{xx}w_x\big )\,\mathrm{d}x \\&=-\epsilon ^2 \varrho _{1}\beta ^{-1}\Big [\,\varpi \,\beta '\,\beta ^{-1}+2\,\varpi \,\alpha ^{-1}\,\alpha '\Big ]\,h. \end{aligned} \end{aligned}$$

(C.3)

Since $w_1$ is an odd function and $w_2$ is a even function, we obtain

$$\begin{aligned} \begin{aligned} J_3 =\int _{\mathbb R}\,M_{31}(x,z)w_x\,\mathrm{d}x =&\,\epsilon ^2\beta ^{-1}\sigma ^{-1}k^2\,h\int _{\mathbb R}\Big [w_{2,x}w_x+\frac{1}{\sigma }xw_2w_x+w_1w_x\Big ]\,\mathrm{d}x. \end{aligned} \end{aligned}$$

(C.4)

By the definition of $M_{41}(x,z)$, we get

$$\begin{aligned} \begin{aligned} J_4&\,=\, -\epsilon ^2\, \varrho _{1}\,\beta ^{-1}\,\alpha _1(z)\,h' \,-\,\epsilon ^2\, \varrho _{1}\,\beta ^{-1}\,\alpha _2(z)\,h \,+\,\epsilon ^2\, \varrho _{1}\,\beta ^{-1}\,\big [G_1(z)+G_2(z)\big ], \end{aligned} \end{aligned}$$

(C.5)

where $\alpha _1(z)$ and $\alpha _2(z)$ are smooth functions defined in (4.42) and (4.43),

$$\begin{aligned} G_1(z)= & {} -\frac{k(\epsilon z)}{ \varrho _{1}}\xi (\epsilon z)\,\int _{\mathbb R} \Big [A\big (\mathfrak {a}(\epsilon z)\big )\,Z_x+\phi _{22,x}(x,z)\Big ]\,w_x\,\mathrm{d}x, \end{aligned}$$

(C.6)

$$\begin{aligned} G_2(z)= & {} -\frac{k(\epsilon z)}{\sigma \varrho _{1}}\xi (\epsilon z)\,\int _{\mathbb R} \,\Big [A\big (\mathfrak {a}(\epsilon z)\big )\,Z+\,\phi _{22}(x,z)\Big ]\,x\,w_x\,\mathrm{d}x. \end{aligned}$$

(C.7)

Since $w_1$ and $w_x$ are odd functions, while $w_2$, $\phi _{21}$ and $\phi _{22}$ are even functions, so we obtain that

$$\begin{aligned} \begin{aligned} J_5&=\epsilon ^2\,p(p-1)\int _{\mathbb R}\Big [a_{11}a_{12}\,h\,w_1\,w_2 +a_{11}w_1\,\xi (\epsilon z)(\phi _{21}+\phi _{22}) \Big ]\,w^{p-2}w_x\,\mathrm{d}x \\&=\epsilon ^2\,a_{11}\,a_{12}\,h\int _{\mathbb R}\,p(p-1)w^{p-2}\,w_1\,w_2\,w_x\,\mathrm{d}x \\&\quad +\epsilon ^2\,a_{11}\,\xi (\epsilon z)\int _{\mathbb R}p(p-1)w^{p-2}\,w_1\,(\phi _{21}+\phi _{22}) w_x\,\mathrm{d}x \\&\equiv \epsilon ^2\beta ^{-1}\sigma ^{-1}k^2\,h\int _{\mathbb R}\,p(p-1)w^{p-2}w_1\,w_2\,w_x\,\mathrm{d}x +\epsilon ^2 \varrho _{1}\,\beta ^{-1}G_3(z), \end{aligned} \end{aligned}$$

(C.8)

where

$$\begin{aligned} G_3(z)\,=\,a_{11}\,\beta \, \varrho _{1}^{-1}\,\xi (\epsilon z)\,p(p-1)\int _{\mathbb R}w^{p-2}\,w_1\Big [A\big (\mathfrak {a}(\epsilon z)\big )Z+\,\phi _{22}(x,z)\Big ]\, w_x\,\mathrm{d}x. \end{aligned}$$

(C.9)

By differentiating the Eq. (4.8) and using Eq. (4.7), we obtain

$$\begin{aligned} \int _{\mathbb R}p(p-1)w^{p-2}w_x w_1w_2\,\mathrm{d}x\,=\,-\int _{\mathbb R}w_xw_1\,\mathrm{d}x+\int _{\mathbb R}\Big (w_x+\frac{1}{\sigma }xw\Big )w_{2,x}\,\mathrm{d}x. \end{aligned}$$

(C.10)

Adding (C.4), (C.8) and using (C.10), we have

$$\begin{aligned} \begin{aligned} J_3+J_5&=\epsilon ^2\beta ^{-1}\sigma ^{-1}k^2h\int _{\mathbb R}\Big [p(p-1)w^{p-2}w_1w_2w_x+w_{2,x}w_x+\frac{1}{\sigma }xw_2w_x \\&\quad +w_1w_x\Big ]\,\mathrm{d}x+\epsilon ^2 \varrho _{1}\beta ^{-1}G_3(z) \\&=\epsilon ^2 \beta ^{-1}\sigma ^{-1}k^2h\int _{\mathbb R}\Big [2w_{2,x}w_x+\sigma ^{-1}x(w_2 w)_x\Big ]\,\mathrm{d}x +\epsilon ^2 \varrho _{1}\beta ^{-1}G_3(z) \\&=-\epsilon ^2 \varrho _{1}\beta ^{-1}\sigma ^{-1}k^2h \,+\,\epsilon ^2 \varrho _{1}\,\beta ^{-1}G_3(z), \end{aligned} \end{aligned}$$

(C.11)

where we have used (2.9) and the following integral identities

$$\begin{aligned} 2\int _{\mathbb R}w_{2,x}w_x\,\mathrm{d}x= & {} -\left( \frac{2}{p-1}+\frac{1}{2}\right) \int _{\mathbb R}w_x^2\,\mathrm{d}x, \qquad \\ \sigma ^{-1}\int _{\mathbb R}w_2w\,\mathrm{d}x= & {} \left( \frac{1}{2}-\frac{2}{p-1}\right) \int _{\mathbb R}w^2_x\,\mathrm{d}x. \end{aligned}$$

Consequently, we infer that

$$\begin{aligned} \begin{aligned} \text{ Left } \text{ hand } \text{ side } \text{ of } (4.38)&=-\epsilon ^2 \varrho _{1}\beta ^{-1}\Big [h'' +\big (\hbar _1(\epsilon z)+\alpha _1(z)\big )h'+(\hbar _2(\epsilon z)+\alpha _2( z))h\Big ] \\&\quad +\epsilon ^2 \varrho _{1}\beta ^{-1}\Big [G_1(z)\,+\,G_2(z)\,+\,G_3(z)\Big ], \end{aligned} \end{aligned}$$

where functions $\hbar _1(\theta )$, $\hbar _2(\theta )$, $\alpha _1(z)$ and $\alpha _2(z)$ are defined in (4.40), (4.41), (4.42) and (4.43).

The computations of (5.8) and (5.10)

First, we consider the estimate for the term $\int _{\mathbb R}{{\mathcal {E}}}w_x\,\mathrm{d}x$ in (5.8), where ${{\mathcal {E}}}$ is defined in (4.46) and $w_x$ is an odd function in x. Integration against all even terms in x, i.e., ${{\mathcal {E}}}_{11}$ and $S_4, M_{12}, M_{22}$ in ${{\mathcal {E}}}_{12}$, therefore just vanish. Note that

$$\begin{aligned} \int _{\mathbb R}\,{{\mathcal {E}}}\,w_x\,\mathrm{d}x&=\int _{\mathbb R}\epsilon ^2\,S_3\,w_x\,\mathrm{d}x +\int _{\mathbb R}\epsilon ^2\,S_5\,w_x\,\mathrm{d}x+\int _{\mathbb R}B_2(w)\,w_x\,\mathrm{d}x\nonumber \\&\quad +\int _{\mathbb R}\beta ^{-2}\epsilon \,\phi _{1,zz}\,w_x\,\mathrm{d}x+\int _{\mathbb R}\beta ^{-2}\epsilon ^2\,\phi _{4,zz}\,w_x\,\mathrm{d}x \nonumber \\&\quad +\int _{\mathbb R}M_{32}(x,z)\,w_x\,\mathrm{d}x+\int _{\mathbb R}M_{42}(x,z)\,w_x\,\mathrm{d}x \\&\quad +\int _{\mathbb R}M_{52}(x,z)\,w_x\,\mathrm{d}x+\int _{\mathbb R}M_{62}(x,z)\,w_x\,\mathrm{d}x\nonumber \\&\quad +\int _{\mathbb R}\big [B_3(\epsilon ^2 \phi _3)+B_3(\epsilon ^2 \phi _4)\big ]\,w_x\,\mathrm{d}x \nonumber \\&\equiv \text {I}_1\,+\,\text {I}_2\,+\,\text {I}_3\,+\,\text {I}_4\,+\,\text {I}_5 \,+\,\text {I}_6\,+\,\text {I}_7\,+\,\text {I}_8\,+\,\text {I}_9\,+\,\text {I}_{10}. \nonumber \end{aligned}$$

(D.1)

These terms will be estimated as follows.

By repeating the same computation used in (C.1) and (C.3), we get

$$\begin{aligned} \begin{aligned} \text {I}_{1} =&-\epsilon ^2\, \varrho _{1}\,\beta ^{-1}\,f{''} \,-\, \epsilon ^2\, \varrho _{1}\,\beta ^{-1}\,(\beta ^{-1}\,\beta '+2\,\alpha ^{-1}\,\alpha ')\,f{'}\\&-\epsilon ^2\, \varrho _{1}\,\beta ^{-1}\Big (k^2-\sigma \,\beta ^{-2}\,V_{tt}\Big )\,f, \end{aligned} \end{aligned}$$

(D.2)

and also

$$\begin{aligned} \text {I}_{2}\,=\,\int _{\mathbb R}\,\epsilon ^2\,S_5\,w_x\,\mathrm{d}x\,=\,-\epsilon ^2\, \varrho _{1}\,\beta ^{-1}\,\big [\,\varpi \,\beta '\beta ^{-1}+2\,\varpi \,\alpha ^{-1}\,\alpha '\big ]f. \end{aligned}$$

(D.3)

Recall the expression of $B_2(w)$ in (3.29) and $\epsilon \phi _1(x,z)$, it is easy to check that

$$\begin{aligned} \text {I}_{3}+\text {I}_{4}\,=\,\int _{\mathbb R}B_2(w)\,w_x\,\mathrm{d}x \,+\int _{\mathbb R}\beta ^{-2}\,\epsilon \phi _{1,zz}\,w_x\,\mathrm{d}x\,=\,\epsilon ^3 \,{{\mathbf {b}}}_{1\epsilon }\,f{''} +\epsilon ^3\,{{\mathbf {b}}}_{2\epsilon }. \end{aligned}$$

(D.4)

Since $\epsilon ^2 \phi _4(x,z)=\epsilon ^2 \phi _{41}\big (x, {\epsilon }z\big )+\epsilon ^2\phi _{42}(x,{\epsilon }z),$ it can be derived that

$$\begin{aligned} \text {I}_5\,=\,\int _{\mathbb R}\beta ^{-2}\,\epsilon ^2 \,\phi _{4,zz}\,w_x\,\mathrm{d}x\,=\,O(\epsilon ^4). \end{aligned}$$

(D.5)

Since $w_x$ is odd in x, we need only consider the odd terms in $M_{32}(x,z)$ and get

$$\begin{aligned} \begin{aligned} \text {I}_6 \,=\,&\epsilon ^2\,\beta ^{-1}\,\sigma ^{-1}\,k^2\,f\int _{\mathbb R}\Big [w_{2,x}\,w_x+\frac{1}{\sigma }\,x\,w_x\,w_2+w_1\,w_x\Big ]\,\mathrm{d}x+O(\epsilon ^3). \end{aligned} \end{aligned}$$

(D.6)

From the definition of $M_{42}(x,z)$, we can estimate the $I_7$ in (D.1) as the following

$$\begin{aligned} \text {I}_7&=-\,\frac{2\epsilon ^2}{\beta }\,f'\,\xi (\epsilon z)\int _{\mathbb R}\,\big [\epsilon \,A'(\mathfrak {a}(\epsilon z))\,\beta \, Z_x+\phi _{22,xz}\big ]\,w_x\,\mathrm{d}x\nonumber \\&\quad -\,\frac{2\epsilon ^2}{\beta }\,\varpi \,f\,\xi (\epsilon z)\int _{\mathbb R}\,\big [\epsilon \,A'(\mathfrak {a}(\epsilon z))\,\beta \, Z_x+\phi _{22,xz}\big ]\,w_x\,\mathrm{d}x+\epsilon ^3\, {{\mathbf {b}}}_{1\epsilon }\,f{''}+\epsilon ^3\,{{\mathbf {b}}}_{2\epsilon } \nonumber \\&\equiv \, \epsilon ^2\, \varrho _{1}\,\beta ^{-1}\,\alpha _1(z)\,f' \,+\,\epsilon ^2\, \varrho _{1}\,\beta ^{-1}\,\alpha _2(z)\,f+\epsilon ^3 \,{{\mathbf {b}}}_{1\epsilon }\,f{''}+\epsilon ^3\,{{\mathbf {b}}}_{2\epsilon }, \end{aligned}$$

(D.7)

where $\alpha _1(z)$ and $\alpha _2(z)$ are defined in (4.42)–(4.43).

From the definition of $M_{52}(x,z)$, we need only consider the odd terms and the higher order terms involving $e'$ and $e''$, so we get

$$\begin{aligned} \begin{aligned} \text {I}_8=&-\epsilon ^2\,k\,\beta ^{-1}\,e\int _{\mathbb R}\big [\,Z_{x}+\sigma ^{-1}\,x\,Z\big ]\,w_x\,\mathrm{d}x+2\,\epsilon ^4\, k\,\beta ^{-3}\,e{''}(\epsilon z)\int _{\mathbb R}x\,w_x\,Z(x)\,\mathrm{d}x \\&+\epsilon ^3\,\Big [{{\mathbf {b}}}_{1\epsilon }^1\,e'+ {{\mathbf {b}}}_{1\epsilon }\,f{''}+{{\mathbf {b}}}_{2\epsilon }\Big ] \\&\equiv \epsilon ^2 \varrho _{1}\beta ^{-1}\Big [\hbar _{31}({\epsilon }z)\,e+\epsilon ^2\hbar _4({\epsilon }z)\,e{''}\Big ] +\epsilon ^3\Big [{{\mathbf {b}}}_{1\epsilon }^1\,e'+{{\mathbf {b}}}_{1\epsilon }\,f{''}+{{\mathbf {b}}}_{2\epsilon }\Big ]. \end{aligned} \end{aligned}$$

(D.8)

It can easily be verified that

$$\begin{aligned} \text {I}_9= & {} \epsilon ^2\,\beta ^{-1}\,\sigma ^{-1}\,k^2\,f\,\int _{\mathbb R}p(p-1)\,w_x\,w^{p-2}\,w_1\,w_2\,\mathrm{d}x \nonumber \\&\quad +\,\epsilon ^2p(p-1)a_{11}\,e\int _{\mathbb R}w^{p-2}\,w_1\,Z\, w_x\,\mathrm{d}x+\epsilon ^3{{\mathbf {b}}}_{2\epsilon }. \\= & {} \epsilon ^2\,\beta ^{-1}\,\sigma ^{-1}\,k^2\,f\,\int _{\mathbb R}p(p-1)\,w_x\,w^{p-2}\,w_1\,w_2\,\mathrm{d}x+\epsilon ^2 \varrho _{1}\beta ^{-1}\,\hbar _{32}({\epsilon }z)\,e+\epsilon ^3{{\mathbf {b}}}_{2\epsilon }.\nonumber \end{aligned}$$

(D.9)

Adding (D.6) and (D.9), we get

$$\begin{aligned} \begin{aligned} \text {I}_6+ \text {I}_9&=\epsilon ^2\,\beta ^{-1}\,\sigma ^{-1}\,k^2\,f\,\int _{\mathbb R}\Big [p(p-1)w^{p-2}\,w_x\,w_1\,w_2+w_{2,x}\,w_x\\&\quad +\sigma ^{-1}\,x\,w_x\,w_2+w_1w_x\Big ]\,\mathrm{d}x \\&\quad +\epsilon ^2 \varrho _{1}\beta ^{-1}\,\hbar _{32}({\epsilon }z)\,e+\big [\epsilon ^3 {{\mathbf {b}}}_{1\epsilon }f{''}+\epsilon ^3{{\mathbf {b}}}_{2\epsilon }\Big ] \\&=-\epsilon ^2 \varrho _{1}\beta ^{-1}\sigma ^{-1}k^2f\,+\epsilon ^2 \varrho _{1}\beta ^{-1}\,\hbar _{32}({\epsilon }z)\,e+\,\epsilon ^3\big [{{\mathbf {b}}}_{1\epsilon }f{''}+{{\mathbf {b}}}_{2\epsilon }\big ]. \end{aligned} \end{aligned}$$

(D.10)

According to the fact that the terms in $B_3(\epsilon ^2 \phi _3)$ and $B_3(\epsilon ^2 \phi _4)$ are of order $O(\epsilon ^3)$, it follows that

$$\begin{aligned} \text {I}_{10}=\int _{\mathbb R}\Big [B_3(\epsilon ^2 \phi _3)+B_3(\epsilon ^2 \phi _4)\Big ]\,w_x\,\mathrm{d}x\,=\,\epsilon ^3\, {{\mathbf {b}}}_{1\epsilon }\,f{''}+\epsilon ^3\,{{\mathbf {b}}}_{2\epsilon }. \end{aligned}$$

(D.11)

As a conclusion, adding up all estimates together will give (5.8).

Second, we are in a position to provide the concise estimate for the integral $\int _{\mathbb R}{{\mathcal {E}}}\,Z\,\mathrm{d}x$ in (5.10). Using the decomposition of ${{\mathcal {E}}}$, we obtain

$$\begin{aligned} \int _{\mathbb R}{{\mathcal {E}}}\,Z\,\mathrm{d}x \,=\,\int _{\mathbb R}{{\mathcal {E}}}_{11}\,Z\,\mathrm{d}x +\int _{\mathbb R}{{\mathcal {E}}}_{12}\,Z\,\mathrm{d}x, \end{aligned}$$

where

$$\begin{aligned} \int _{\mathbb R}{{\mathcal {E}}}_{11}\,Z\,\mathrm{d}x \,=\,\epsilon \,\big [\epsilon ^2\,\beta ^{-2}\,e{''}+\lambda _0\,e\,\big ]\int _{\mathbb R}\,Z^2\,\mathrm{d}x \,=\,\epsilon ^3\,\beta ^{-2}\,e{''}+\epsilon \lambda _0\,e, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \int _{\mathbb R}{{\mathcal {E}}}_{12}\,Z\,\mathrm{d}x&=\int _{\mathbb R}\epsilon ^2\,S_4\,Z\,\mathrm{d}x+\int _{\mathbb R}B_2(w)\,Z\,\mathrm{d}x+\int _{\mathbb R}\beta ^{-2}\,\epsilon \,\phi _{1,zz}\,Z\,\mathrm{d}x \\&\quad +\int _{\mathbb R}\beta ^{-2}\,\epsilon ^2\,\phi _{4,zz}\,Z\,\mathrm{d}x+\int _{\mathbb R}M_{12}(x,z)\,Z\,\mathrm{d}x+\int _{\mathbb R}M_{22}(x,z)\,Z\,\mathrm{d}x \\&\quad +\int _{\mathbb R}M_{32}(x,z)\,Z\,\mathrm{d}x+\int _{\mathbb R}M_{42}(x,z)\,Z\,\mathrm{d}x+\int _{\mathbb R}M_{52}(x,z)\,Z\,\mathrm{d}x \\&\quad +\int _{\mathbb R}M_{62}(x,z)\,Z\,\mathrm{d}x+\int _{\mathbb R}B_3(\epsilon ^2 \phi _3)\,Z\,\mathrm{d}x+\int _{\mathbb R}B_3(\epsilon ^2 \phi _4)\,Z\,\mathrm{d}x \\&\equiv {\text {II}}_1\,+\,{\text {II}}_2\,+\,{\text {II}}_3\,+\,{\text {II}}_4\,+\,{\text {II}}_5\, +\,{\text {II}}_6\,+\,{\text {II}}_7\,+\,{\text {II}}_8\,\\&\quad +\,{\text {II}}_9 +{\text {II}}_{10}+{\text {II}}_{11}+{\text {II}}_{12}. \end{aligned} \end{aligned}$$

(D.12)

According to the expression of $S_4$ in (4.1) and the constraint of f in (3.23), it follows that

$$\begin{aligned} \begin{aligned} {\text {II}}_1&=\epsilon ^2\int _{\mathbb R}\big [ (f'^2+2f'h')w_{xx}+2\,\varpi \,(ff'+f'h)w_{xx}-\frac{1}{2\beta ^{2}}\,V_{tt}\,f^2\,w\big ]Z\,\mathrm{d}x \\&=\epsilon ^2\,\varrho _{2}\,f'\,h'+\epsilon ^2\,\varrho _{2}\,\varpi \,f'h+O(\epsilon ^3), \end{aligned} \end{aligned}$$

(D.13)

where constant $\varrho _{2}$ defined in (5.11).

The estimate of ${\text {II}}_7$ can be proved by the same way as employed in the above estimate.

$$\begin{aligned} \begin{aligned} {\text {II}}_7&=\int _{\mathbb R}M_{32}(x,z)\,Z\,\mathrm{d}x =\int _{\mathbb R}\Big (\epsilon ^2\,k^2\,\sigma ^{-2}\,f^2\,w_2+\epsilon ^3\,a_7({\epsilon }s, {\epsilon }z)\Big )\,Z\,\mathrm{d}x \\&=\,\epsilon ^3\,{{\mathbf {b}}}_{1\epsilon }\,f{''}+\epsilon ^3\,{{\mathbf {b}}}_{2\epsilon }. \end{aligned} \end{aligned}$$

(D.14)

Note that $B_2(w)=O(\epsilon ^3)$, it is easy to check that

$$\begin{aligned} {\text {II}}_{2}\,=\,\int _{\mathbb R}B_2(w)Z\,\mathrm{d}x \,=\,\epsilon ^3\,{{\mathbf {b}}}_{1\epsilon }\,f{''}+\epsilon ^3\,{{\mathbf {b}}}_{2\epsilon }. \end{aligned}$$

(D.15)

Since

$$\begin{aligned} \epsilon \phi _1(x, z)=\epsilon \,a_{11}({\epsilon }z)\,w_1(x)+\epsilon \,a_{12}({\epsilon }z)\,(f({\epsilon }z)+h({\epsilon }z))\,w_2(x), \end{aligned}$$

so we obtain

$$\begin{aligned} \begin{aligned} {\text {II}}_{3}&\,=\,\beta ^{-2}\int _{\mathbb R}\epsilon \phi _{1,zz}Z\,\mathrm{d}x \,=\,\epsilon ^3 \,\beta ^{-2}\,a_{12}(\epsilon z)\,f{''}\int _{\mathbb R}w_2(x)\,Z(x)\,\mathrm{d}x+\epsilon ^3 \,{{\mathbf {b}}}_{2\epsilon } \\&\,=\,\epsilon ^3\,{{\mathbf {b}}}_{1\epsilon }\,f''+\epsilon ^3 \,{{\mathbf {b}}}_{2\epsilon }. \end{aligned} \end{aligned}$$

(D.16)

Since $\epsilon ^2 \phi _4(x,z)=\epsilon ^2\, \phi _{41}(x,{\epsilon }z)+\epsilon ^2\,\phi _{42}(x,{\epsilon }z),$ it is easy to prove that

$$\begin{aligned} {\text {II}}_{4}\,=\,\beta ^{-2}\,\int _{\mathbb R}\epsilon ^2\phi _{4,zz}\,Z\,\mathrm{d}x\,=\,O(\epsilon ^4). \end{aligned}$$

(D.17)

According to the expression of $M_{12}(x,z)$ and $M_{22}(x,z)$, we know that the terms in $M_{12}(x,z)$ and $M_{22}(x,z)$ are of order $O(\epsilon ^3)$. Hence, it is easy to obtain that

$$\begin{aligned} {\text {II}}_{5}=\int _{\mathbb R}M_{12}(x,z)\,Z\,\mathrm{d}x=O(\epsilon ^3), \qquad {\text {II}}_{6}&=\int _{\mathbb R}M_{22}(x,z)\,Z\,\mathrm{d}x=O(\epsilon ^3). \end{aligned}$$

(D.18)

What’s more, we can compute that

$$\begin{aligned} {\text {II}}_{8}= & {} -\,2\epsilon ^2\,\beta ^{-1}\big (f'+\varpi \,f\big )\,\xi (\epsilon z)\,\int _{\mathbb R}\,\Big [\epsilon \,A'\big (\mathfrak {a}(\epsilon z)\big )\,\beta \, Z_x+\phi _{22,xz}\Big ]\,Z\,\mathrm{d}x+O(\epsilon ^3) \nonumber \\= & {} O(\epsilon ^3). \end{aligned}$$

(D.19)

We need only to compute those parts in $M_{52}(x,z)$ which are even in x. It is easy to check that

$$\begin{aligned} {\text {II}}_{9}=\int _{\mathbb R}M_{52}(x,z)\,Z\,\mathrm{d}x=-\epsilon ^2\beta ^{-1}\,k\,e\int _{\mathbb R}\big (Z_{x}+\sigma ^{-1}x\,Z\big )\,Z\,\mathrm{d}x+O(\epsilon ^3)=O(\epsilon ^3). \end{aligned}$$

(D.20)

Additionally, we also need to consider some higher order terms in ${\text {II}}_{9}$. The ones involving first derivative of e are

$$\begin{aligned} \begin{aligned}&2\epsilon ^3\,e'\,\Big (\frac{\beta '}{\beta ^{3}}-\frac{\varpi }{\beta ^{2}}\,\,\Big )\int _{\mathbb R}xZZ_x\,\mathrm{d}x+\epsilon ^3\,e'\,\Big (\frac{2\alpha '}{\alpha \beta ^{2}}-\frac{\varpi }{\beta ^{2}}\,\Big )\int _{\mathbb R}Z^2\,\mathrm{d}x \\&\quad =\epsilon ^3\,\Big (\frac{2\alpha '}{\alpha \beta ^2}-\frac{\beta '}{\beta ^3}\Big )e' \,\equiv \, \epsilon ^3\,\hbar _5({\epsilon }z)\,e', \end{aligned} \end{aligned}$$

(D.21)

where $\hbar _5({\epsilon }z)$ defined in (5.11). Moreover, the ones involving second derivative of e in ${\text {II}}_{9}$ are

$$\begin{aligned} \epsilon ^3\Big [\epsilon \, f\,\beta ^{-2}\,\hbar _6(\epsilon z)+O(\epsilon ^2)\Big ]\,e{''}(\epsilon z) \end{aligned}$$

with $O(\epsilon ^2)$ uniform in $\epsilon$.

In the terms of ${\text {II}}_{10}$ and ${\text {II}}_{12}$, we need only to consider those parts which are even in x. It can be derived from (4.35) that the even (in x) terms in ${\text {II}}_{10}$ is of order $o(\epsilon ^3)$. Moreover, the terms in $B_3(\epsilon ^2 \phi _3)$ and $B_3(\epsilon ^2 \phi _4)$ are of order $O(\epsilon ^3)$. Consequently, we deduce that

$$\begin{aligned} {\text {II}}_{10}+{\text {II}}_{11}+{\text {II}}_{12}= & {} \int _{\mathbb R}M_{62}(x,z)\,Z\,\mathrm{d}x+\int _{\mathbb R}B_3(\epsilon ^2 \phi _3)\,Z\,\mathrm{d}x+\int _{\mathbb R}B_3(\epsilon ^2 \phi _4)\,Z\,\mathrm{d}x\nonumber \\= & {} O(\epsilon ^3). \end{aligned}$$

(D.22)

This finish the computation of the integral in (5.10).