# Ground state of the Gross-Pitaevskii equation with a harmonic potential in
the energy-critical case

Dmitry E. Pelinovsky  Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>and
Szymon Sobieszek  Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>
###### Abstract.

Ground state of the energy-critical Gross–Pitaevskii equation with a harmonic
potential can be constructed variationally. It exists in a finite interval of
the eigenvalue parameter. The supremum norm of the ground state vanishes at
one end of this interval and diverges to infinity at the other end. We explore
the shooting method in the limit of large norm to prove that the ground state
is pointwise close to the Aubin–Talenti solution of the energy-critical wave
equation in near field and to the confluent hypergeometric function in far
field. The shooting method gives the precise dependence of the eigenvalue
parameter versus the supremum norm.

###### Key words and phrases:

Gross–Pitaevskii equation, ground state, energy-critical case, shooting method

##  1\. Introduction

We consider the stationary Gross–Pitaevskii equation with a harmonic
potential,

$(-\Delta+|x|^{2})u-|u|^{2p}u=\lambda u,$ (1.1)

where $x\in\mathbb{R}^{d}$, $\lambda\in\mathbb{R}$, and $u\in\mathbb{R}$.
Existence of its ground state (a positive and radially decreasing solution)
has been addressed by using variational methods in the energy subcritical [11,
19] and critical [25, 26] cases, where the critical exponent is
$p=\frac{2}{d-2}$ if $d\geq 3$. Uniqueness of the ground state was proven in
the energy subcritical [16, 17] and critical [28] cases.

Variational methods are not applicable in the energy supercritical case
$p(d-2)>2$ if $d\geq 3$ for which a more efficient shooting method was
developed in our previous work [3, 22] (see also [8] for study of the
Schrödinger–Newton–Hooke model). The shooting method is based on the
reformulation of the existence problem after the Emden–Fowler transformation
[9, 18] and construction of two analytic families of solutions, one family
gives a bounded solution near $r=0$ with parameter
$b:=u(0)\equiv\|u\|_{L^{\infty}}$ and the other family gives a decaying
solution as $r\to\infty$ with parameter

$c:=\lim_{r\to\infty}u(r)e^{\frac{1}{2}r^{2}}r^{\frac{d-\lambda}{2}}.$ (1.2)

The shooting method gives robust results in the large-norm limit as
$b\to\infty$. In an asymptotic region, where both solution families coexist,
the matching condition gives a condition on $(c,\lambda)$ as a function of $b$
which determines the solution curve $\lambda=\lambda(b)$. The $c$-family of
solutions is defined in a local neighborhood of the limiting singular solution
constructed in [27] for $\lambda=\lambda_{\infty}\in(0,d)$ such that
$\lambda(b)\to\lambda_{\infty}$ as $b\to\infty$ along the solution curve. As
follows from the shooting method [3] under some non-degeneracy assumptions,
the convergence is oscillatory for $2+\frac{2}{p}<d<d_{*}(p)$ and monotone for
$d>d_{*}(p)$, where

$d_{*}(p):=6+\frac{2}{p}+2\sqrt{4+\frac{2}{p}}.$

The same shooting method was extended in [22] to compute the Morse index of
the ground state in the monotone case from the Morse index of the limiting
singular solution. It was recently proven in [23] by using comparison with the
stationary Schrödinger equation solvable in terms of the confluent
hypergeometric functions (see [1] for review) that the Morse index of the
limiting singular solution is infinite in the oscillatory case and is equal to
one in the monotone case for large values of $d>d_{*}(p)$.

Properties of the energy-supercritical Gross–Pitaevskii equation with a
harmonic potential are very similar to those for the energy-supercritical
nonlinear Schrödinger equation in a ball. See [5, 6, 7] for the developments
in the shooting method, [21] for convergence to the limiting singular
solution, and [12, 20] for computation of the Morse index.

The purpose of this work is to extend the shooting method to the energy-
critical case and to obtain the asymptotic representation of $\lambda(b)$ as
$b\to\infty$. As far as we are aware, the shooting method has not been
previously developed in the context of the energy-critical case, for which the
variational approximations are more common.

For the shooting method in the energy-critical case with $d=2+\frac{2}{p}$, we
introduce the same two analytic families of solutions, the $b$-family is
defined by parameter $b:=u(0)$ and the $c$-family is defined by parameter $c$
in the asymptotic behavior (1.2). Contrary to the energy-supercritical case,
the $c$-family exists in a local neighborhood of a spatially decaying solution
to the stationary Schrödinger equation

$V^{\prime\prime}(r)+\frac{d-1}{r}V^{\prime}(r)-r^{2}V(r)+\lambda V(r)=0,$

which is satisfied by
$V(r)=ce^{-\frac{1}{2}r^{2}}\mathfrak{U}(r^{2};\alpha,\beta)$, where
$c\in\mathbb{R}$ is arbitrary and $\mathfrak{U}(z;\alpha,\beta)$ is the
Tricomi function (see [1]) with

$\alpha:=\frac{p+1}{2p}-\frac{\lambda}{4},\qquad\beta:=1+\frac{1}{p}.$ (1.3)

Furthermore, contrary to the energy-supercritical case, the $b$-family exists
in a local neighborhood of the algebraic soliton

$\displaystyle
U_{b}(r)=\frac{b}{(1+\alpha_{p}b^{2p}r^{2})^{\frac{1}{p}}},\qquad\alpha_{p}:=\frac{p^{2}}{4(1+p)},$
(1.4)

where the parameter $b$ has been introduced from the condition $b=U_{b}(0)$.
The algebraic soliton (also called the Aubin-Talenti solution [2, 29])
satisfies the nonlinear wave equation $-\Delta U_{b}=U_{b}^{2p+1}$ for every
$b>0$. It has been used in many studies of the energy-critical wave equations
in bounded domains as in the pioneering work [4] and in follow-up works [10,
13, 14, 15, 24]. In the context of the stationary Gross–Pitaevskii equation
(1.1), it was used in [25] in order to obtain the lower bound on the
dependence $\lambda(b)$ from a variational method. Recently in [23], the
variational methods and the elliptic estimates were extended in order to get
the upper bound on the dependence $\lambda(b)$. We will use the shooting
method to justify the relevance of the algebraic soliton (1.4) for the
asymptotic behavior of $\lambda(b)$ as $b\to\infty$.

Figure 1.1. Ground states of the stationary equation (1.1) with $p=1$ and
$d=4$ compared with the algebraic soliton (1.4) for $b=2$ (left) and $b=10$
(right). Figure 1.2. The dependence $\lambda=\lambda(b)$ for $p=1$ and $d=4$.

Figure 1.1 shows the numerically obtained profile $u_{b}$ versus $r$ in
comparison with the profile $U_{b}$ for $b=2$ (left) and $b=10$ (right).
Visualization is given for $p=1$ (that corresponds to $d=4$). Results for
other values of $p\in(0,1)$ are similar. The two profiles are different for
$b=2$ but the discrepancy gets smaller for $b=10$ and becomes invisible for
larger values of $b$. The values of $\lambda$ are uniquely defined in terms of
$b$ along the curve $\lambda=\lambda(b)$ which is shown in Figure 1.2 for
$p=1$.

The following theorem presents outcomes of the shooting method, which is the
main result of this work. We use the following notations:

  * •

$\lambda(b)\sim\lambda_{0}(b)$ denotes the asymptotic equivalence in the sense
$\lim\limits_{b\to\infty}\lambda_{0}(b)^{-1}\lambda(b)=1$,

  * •

$\lambda(b)=\mathcal{O}(b^{q})$ denotes the order of magnitude in the sense
that $|\lambda(b)|\leq Cb^{q}$ for some $C>0$ and all sufficiently large $b$.

######  Theorem 1.1.

Fix $p=\frac{2}{d-2}\in(0,1)$ for $d>4$ and let $\lambda=\lambda(b)$ be the
solution curve for the ground state $u=u_{b}$ of the stationary
Gross–Pitaevskii equation (1.1) satisfying $u_{b}(0)=b$, $u_{b}^{\prime}(r)<0$
for $r>0$, and $u_{b}(r)\to 0$ as $r\to\infty$. Then,

$\lambda(b)\sim C_{p}\left\\{\begin{array}[]{ll}\displaystyle
b^{-2(1-p)},&\quad\frac{1}{2}<p<1,\\\ \displaystyle b^{-1}\log b,&\quad
p=\frac{1}{2},\\\ \displaystyle b^{-2p},&\quad 0<p<\frac{1}{2},\\\
\end{array}\right.\quad\mbox{\rm as}\quad b\to\infty,$ (1.5)

with

$C_{p}=\left\\{\begin{array}[]{ll}\displaystyle-\frac{\Gamma\left(\frac{p+1}{2p}\right)\Gamma\left(-\frac{1}{p}\right)\Gamma\left(\frac{2}{p}\right)}{(1+p)\Gamma\left(\frac{p-1}{2p}\right)\Gamma\left(\frac{1}{p}\right)\Gamma\left(\frac{1}{p}-1\right)\Gamma\left(\frac{1}{p}+1\right)}\left[\frac{4(1+p)}{p^{2}}\right]^{\frac{1}{p}},&\quad\frac{1}{2}<p<1,\\\
\displaystyle 144,&\quad p=\frac{1}{2},\\\
\displaystyle\frac{8(1+p)^{2}}{p^{2}(1-2p)},&\quad 0<p<\frac{1}{2}.\\\
\end{array}\right.$

Moreover, for every $a\in(0,\frac{p}{1+p})$, there exist $B_{a},C_{a}>0$ such
that for every $b\geq B_{a}$, we have

$\sup_{r\in[0,b^{-p(1-a)}]}b^{-1}\left|u_{b}(r)-U_{b}(r)\right|\leq
C_{a}b^{-2p(1-a)},$ (1.6)
$\sup_{r\in[b^{-p(1-a)},1]}r^{\frac{2}{p}}\left|u_{b}(r)-c(b)e^{-\frac{1}{2}r^{2}}\mathfrak{U}(r^{2};\alpha,\beta)\right|\leq
C_{a}|c(b)|^{2p+1}b^{2p(1-a)}$ (1.7)

and

$\sup_{r\in[1,\infty)}\left|e^{\frac{1}{2}r^{2}}r^{\frac{d-\lambda(b)}{2}}u_{b}(r)-c(b)r^{\frac{d-\lambda(b)}{2}}\mathfrak{U}(r^{2};\alpha,\beta)\right|\leq
C_{a}|c(b)|^{2p+1},$ (1.8)

where $c=c(b)\sim A_{p}b^{-1}$ as $b\to\infty$ for some $A_{p}>0$.

######  Remark 1.1.

The asymptotic result (1.5) coincides with Theorem 1.1 in [23] obtained by the
variational theory and elliptic estimates. It follows from Remark 1.2 in [23]
that there exists $C_{d}$ such that

$\lambda(\varepsilon)\sim C_{d}\left\\{\begin{array}[]{ll}\varepsilon&\quad
d=5,\\\ \varepsilon^{2}|\log\varepsilon|&\quad d=6,\\\ \varepsilon^{2}&\quad
d\geq 7,\\\ \end{array}\right.$ (1.9)

where $\varepsilon=b^{-p}$ is defined from the algebraic soliton (1.4). The
case $d\geq 7$ corresponds to $0<p<\frac{1}{2}$ as in (1.5). For $d=6$, we
have $p=\frac{1}{2}$ so that $\varepsilon^{2}|\log\varepsilon|\sim b^{-1}\log
b$ as in (1.5). For $d=5$, we have $p=\frac{2}{3}$ so that $\varepsilon\sim
b^{-2/3}=b^{-2(1-p)}$ as in (1.5).

######  Remark 1.2.

It also follows from Remark 1.2 in [23] that $\lambda(\varepsilon)\sim
C|\log\varepsilon|^{-1}$ for $d=4$ and $\lambda(\varepsilon)-1\sim
C\varepsilon$ for $d=3$. In our notations with $\varepsilon=b^{-p}$, this
would correspond to $\lambda(b)\sim C(\log b)^{-1}$ for $p=1$ and
$\lambda(b)-1\sim Cb^{-2}$ for $p=2$. However, we have found that the shooting
method based on the $b$-family and the $c$-family can be applied for
$p\in(0,1)$ but needs some further modifications for $p\geq 1$.

######  Remark 1.3.

Since $\mathfrak{U}(r^{2};\alpha,\beta)=\mathcal{O}(r^{-\frac{2}{p}})$ as
$r\to b^{-p(1-a)}$, bound (1.7) shows that $u_{b}(r)=\mathcal{O}(b^{1-2a})$ as
$r\to b^{-p(1-a)}$. This is smaller than $u_{b}(r)=\mathcal{O}(b)$ as $r\to 0$
in the bound (1.6). Since
$\mathfrak{U}(r^{2};\alpha,\beta)=\mathcal{O}(r^{-\frac{d-\lambda(b)}{p}})$ as
$r\to\infty$, bound (1.8) shows that $u_{b}(r)$ satisfies the asymptotic
behavior (1.2) with $c=c(b)=\mathcal{O}(b^{-1})$ as $b\to\infty$.

Figure 1.3 illustrates relevance of the asymptotic result (1.5) for the
solution curve $\lambda=\lambda(b)$. For a given dimension $d$ and the
critical exponent $p=\frac{2}{d-2}$, we numerically find $\lambda(b)$ and plot
it versus $b$ in comparison with the asymptotic dependence (1.5). The left and
right panels show the plots for $d=7$ when $p=\frac{2}{5}$ and $\lambda(b)\sim
C_{p}b^{-4/5}$ and for $d=5$ when $p=\frac{2}{3}$ and $\lambda(b)\sim
C_{p}b^{-2/3}$, where $C_{p}$ is obtained from the best least square fit. The
proximity between the numerical and analytical curves becomes obvious in the
log-log plot for larger values of $b$.

Figure 1.3. Log-log graphs of $\lambda(b)$ versus $b$ for the ground state of
the stationary equation (1.1) for $d=7$ (left) and $d=5$ (right) compared with
the analytical dependence given in (1.5).

Our strategy to prove Theorem 1.1 is as follows. Section 2 contains
preliminary results where the existence problem is reformulated after the
Emden–Fowler transformation and the two solution families and their truncated
limits are clearly identified. Section 3 gives analysis of the $b$-family in a
local neighborhood of the algebraic soliton (1.4) which becomes the
exponentially decaying soliton after the Emden–Fowler transformation. Section
4 describes analysis of the $c$-family in a local neighborhood of the
confluent hypergeometric functions. Theorem 1.1 is proven in Section 5 where
the two families are considered in the common asymptotic region with
parameters $c$ and $\lambda$ obtained uniquely in the asymptotic limit
$b\to\infty$. Besides the asymptotic dependence (1.5) which recovers
independently the result (1.9) obtained in [23] with different methods, the
main outcome of this work is the precise asymptotic construction of the ground
state with pointwise estimates (1.6), (1.7), and (1.8) near the Aubin-Talenti
solution and the confluent hypergeometric function.

##  2\. Preliminary results

As in our previous works [3, 22], we reformulate the existence problem for the
ground state of the stationary Gross–Pitaevskii equation (1.1) as the
following initial-value problem:

$\left\\{\begin{array}[]{ll}f^{\prime\prime}(r)+\frac{d-1}{r}f^{\prime}(r)-r^{2}f(r)+\lambda
f(r)+|f(r)|^{2p}f(r)=0,&\quad r>0,\\\ f(0)=b,\quad
f^{\prime}(0)=0,&\end{array}\right.$ (2.1)

where $b>0$ is the free parameter and $d=2+\frac{2}{p}$ is defined in terms of
$p>0$ in the energy-critical case. We say that the solution of the initial-
value problem (2.1) is a ground state if $f^{\prime}(r)<0$ for $r>0$ and
$f(r)\to 0$ as $r\to\infty$. Similarly to Lemmas 3.2 and 3.4 in [3] obtained
in the particular case $p=1$, the existence of a unique classical solution to
the initial-value problem (2.1) can be concluded by using the integral
equation formulation and the Lyapunov function method. We skip the proof since
it is standard and state that for every $p>0$, $\lambda\in\mathbb{R}$, and
$b>0$, there exists a unique classical solution $f\in C^{2}(0,\infty)$ to the
initial-value problem (2.1) satisfying the asymptotic behavior:

$f(r)=b-\frac{p(\lambda+b^{2p})}{4(p+1)}br^{2}+\mathcal{O}(r^{4}),\quad\text{as
}\;\;r\to 0.$ (2.2)

The singularity of the stationary equation (2.1) at $r=0$ is unfolded by
introducing the Emden-Fowler transformation:

$r=e^{t},\quad\Psi(t)=e^{\frac{t}{p}}f(e^{t}).$ (2.3)

After the transformation of variables, $\Psi$ satisfies the second-order
nonautonomous equation

$\Psi^{\prime\prime}(t)-\frac{1}{p^{2}}\Psi(t)+|\Psi(t)|^{2p}\Psi(t)=-\lambda
e^{2t}\Psi(t)+e^{4t}\Psi(t).$ (2.4)

We say that the $b$-family of solutions to Eq. (2.4) is defined by applying
the transformation (2.3) to the unique solution of the initial-value problem
(2.1). The corresponding $b$-solution, denoted as $\Psi_{b}(t)$, satisfies the
asymptotic behaviour

$\Psi_{b}(t)=be^{\frac{t}{p}}\left[1-\frac{p(\lambda+b^{2p})}{4(p+1)}e^{2t}+\mathcal{O}(e^{4t})\right],\quad\text{
as }\;\;t\to-\infty,$ (2.5)

which follows from (2.2). Thus, the $b$-family of solutions decays to zero as
$t\to-\infty$. We will show in Section 3 that the $b$-family stays close to
the positive homoclinic orbit of the truncated version of Eq. (2.4) given by
the second-order autonomous equation

$\Theta^{\prime\prime}(t)-\frac{1}{p^{2}}\Theta(t)+|\Theta(t)|^{2p}\Theta(t)=0.$
(2.6)

The second-order equation (2.6) is integrable with the first-order invariant

$\frac{1}{2}(\Theta^{\prime})^{2}-\frac{1}{2p^{2}}\Theta^{2}+\frac{1}{2(p+1)}\Theta^{2(p+1)}=E,$
(2.7)

where $E$ is constant along the classical solutions of Eq. (2.6). The origin
in the $(\Theta,\Theta^{\prime})$-plane is a saddle point. The unique (up to
translation) positive homoclinic orbit exists at the energy level $E=0$ for
every $p>0$. The homoclinic orbit can be found explicitly in the form

$\Theta_{h}(t+t_{0})=\frac{e^{\frac{t+t_{0}}{p}}}{(1+\alpha_{p}e^{2(t+t_{0})})^{\frac{1}{p}}},\qquad\alpha_{p}:=\frac{p^{2}}{4\left(1+p\right)},$
(2.8)

where $t_{0}\in\mathbb{R}$ is an arbitrary parameter of translation. Since

$\Theta_{h}(t)=\left\\{\begin{array}[]{ll}e^{\frac{t}{p}}[1+\mathcal{O}(e^{2t})]&\text{
as }\;\;t\to-\infty,\\\
\alpha_{p}^{-\frac{1}{p}}e^{-\frac{t}{p}}[1+\mathcal{O}(e^{-2t})]&\text{ as
}\;\;t\to+\infty,\end{array}\right.$ (2.9)

it follows by comparison with (2.5) that $\Psi_{b}(t)\sim b\Theta_{h}(t)$ as
$t\to-\infty$ for which the translation parameter $t_{0}$ in (2.8) is uniquely
selected as $t_{0}=p\log b$. With this choice for $t_{0}$, we observe that
$\Theta_{h}(t+p\log b)$ after transformation (2.3) coincides with the
algebraic soliton $U_{b}(r)$ given by (1.4).

Next, we introduce another analytical family of solutions to Eq. (2.4) that
decay to zero as $t\to+\infty$, which we call the $c$-family and denote as
$\Psi_{c}(t)$. We show in Section 4 that the $c$-family stays close to the
decaying solutions of the linearized version of Eq. (2.4) given by the linear
second-order nonautonomous equation

$\Upsilon^{\prime\prime}(t)-\frac{1}{p^{2}}\Upsilon(t)+\lambda
e^{2t}\Upsilon(t)-e^{4t}\Upsilon(t)=0.$ (2.10)

By using the change of variables

$z=e^{2t},\quad\Upsilon(t)=z^{\frac{1}{2p}}e^{-\frac{1}{2}z}u(z),$ (2.11)

the second-order equation (2.10) becomes the confluent hypergeometric equation
(also known as the Kummer equation):

$zu^{\prime\prime}(z)+\left(\beta-z\right)u^{\prime}(z)-\alpha u(z)=0,$ (2.12)

with parameters $\alpha$ and $\beta$ given by (1.3). Two special solutions of
the Kummer equation (2.12) are given by the Kummer function
$\mathfrak{M}(z;\alpha,\beta)$ and the Tricomi function
$\mathfrak{U}(z;\alpha,\beta)$, which are defined as follows [1]. The Kummer
function is defined by the power series

$\displaystyle\mathfrak{M}(z;\alpha,\beta)$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(\alpha)_{k}}{(\beta)_{k}}\;\frac{z^{k}}{k!}$
(2.13)
$\displaystyle=1+\frac{\alpha}{\beta}\frac{z}{1!}+\frac{\alpha(\alpha+1)}{\beta(\beta+1)}\frac{z^{2}}{2!}+\frac{\alpha(\alpha+1)(\alpha+2)}{\beta(\beta+1)(\beta+2)}\frac{z^{3}}{3!}+\ldots,$

hence it is bounded as $z\to 0$. The Tricomi function satisfies the asymptotic
behavior

$\mathfrak{U}(z;\alpha,\beta)\sim
z^{-\alpha}\left[1+\mathcal{O}(z^{-1})\right]\quad\mbox{\rm
as}\;\;z\to+\infty,$ (2.14)

hence it is decaying as $z\to+\infty$ if $\alpha>0$. In fact, for every
$\lambda\in(-\infty,d)$, we have

$\alpha=\frac{p+1}{2p}-\frac{\lambda}{4}>\frac{p+1}{2p}-\frac{d}{4}=\frac{p+1}{2p}-\frac{1}{2}-\frac{1}{2p}=0,$
(2.15)

so that $\alpha>0$ is satisfied in the energy-critical case. By [1, 13.1.3],
the Tricomi function can be represented in the superposition form

$\mathfrak{U}(z;\alpha,\beta)=\frac{\pi}{\sin\pi\beta}\left[\frac{\mathfrak{M}(z;\alpha,\beta)}{\Gamma(1+\alpha-\beta)\Gamma(\beta)}-z^{1-\beta}\frac{\mathfrak{M}(z;1+\alpha-\beta,2-\beta)}{\Gamma(\alpha)\Gamma(2-\beta)}\right],$
(2.16)

which is true for $\beta\notin\mathbb{Z}$ but can also be used in the limit
$\beta\to\mathbb{Z}$. By using the identity

$\frac{\pi}{\sin\pi z}=\Gamma(1-z)\Gamma(z),\quad z\notin\mathbb{Z}.$ (2.17)

we can rewrite (2.16) for $\beta\notin\mathbb{N}$ as

$\mathfrak{U}(z;\alpha,\beta)=\frac{\Gamma(1-\beta)}{\Gamma(1+\alpha-\beta)}\mathfrak{M}(z;\alpha,\beta)+z^{1-\beta}\frac{\Gamma(\beta-1)}{\Gamma(\alpha)}\mathfrak{M}(z;1+\alpha-\beta,2-\beta).$
(2.18)

By [1, 13.1.6], if $\beta=n+1\in\mathbb{N}$, then

$\displaystyle\mathfrak{U}(z;\alpha,n+1)$
$\displaystyle=\frac{(-1)^{n+1}}{n!\Gamma(\alpha-n)}(\mathfrak{M}(z;\alpha,n+1)\log
z$
$\displaystyle\qquad+\sum_{k=0}^{\infty}\frac{(\alpha)_{k}}{(n+1)_{k}}\;\frac{z^{k}}{k!}[\psi(\alpha+k)-\psi(1+k)-\psi(1+n+k)])$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{n}\frac{(k-1)!(1-\alpha+k)_{n-k}}{(n-k)!}z^{-k},$
(2.19)

where $\psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$.

By means of the transformation (2.11), Tricomi function determines a suitable
solution of the linear equation (2.10):

$\Upsilon_{h}(t)=e^{\frac{t}{p}}e^{-\frac{1}{2}e^{2t}}\mathfrak{U}(e^{2t};\alpha,\beta).$
(2.20)

This solution is considered to be the leading-order approximation of the
$c$-family such that $\Psi_{c}(t)\sim c\Upsilon_{h}(t)$ as $t\to+\infty$
satisfies the asymptotic behavior

$\Psi_{c}(t)\sim
ce^{-\frac{(2-\lambda)t}{2}}e^{-\frac{1}{2}e^{2t}}\quad\mbox{\rm
as}\;\;t\to+\infty.$ (2.21)

The ground state of Theorem 1.1 is the connection of the unique solution of
the initial-value problem (2.1) satisfying (2.2) with the unique solution
satisfying the decay behavior

$f(r)\sim
cr^{-(1+\frac{1}{p}-\frac{\lambda}{2})}e^{-\frac{1}{2}r^{2}}\quad\mbox{\rm
as}\;\;r\to\infty,$ (2.22)

which follows from (2.3) and (2.21). The connection between (2.2) and (2.22)
only exists for some specific values of $c=c(b)$ and $\lambda=\lambda(b)$.
Thus, the main question is to find and to justify the analytical expressions
for $c(b)$ and $\lambda(b)$ in the asymptotic limit $b\to\infty$.

##  3\. Persistence of the $b$-family of solutions

The $b$-family of solutions $\Psi_{b}$ of the differential equation (2.4)
satisfying (2.5) is considered in a neighborhood of the homoclinic orbit
$\Theta_{h}$ of the differential equation (2.6) satisfying (2.9). Since the
comparison gives $\Psi_{b}(t)\sim b\Theta_{h}(t)\sim\Theta_{h}(t+p\log b)$ as
$t\to-\infty$, we translate $\Psi_{b}(t)$ by $-p\log b$ and introduce the
perturbation term

$\gamma(t):=\Psi_{b}(t-p\log b)-\Theta_{h}(t),$

which satisfies

$\mathcal{L}\gamma=f_{b}(\Theta_{h}+\gamma)-N(\Theta_{h},\gamma),$ (3.1)

where $f_{b}(t):=-\lambda b^{-2p}e^{2t}+b^{-4p}e^{4t}$,

$\displaystyle(\mathcal{L}\gamma)(t):=\gamma^{\prime\prime}(t)-\frac{1}{p^{2}}\gamma(t)+(2p+1)|\Theta_{h}(t)|^{2p}\gamma(t),$

and

$\displaystyle
N(\Theta_{h},\gamma):=|\Theta_{h}+\gamma|^{2p}(\Theta_{h}+\gamma)-|\Theta_{h}|^{2p}\Theta_{h}-(2p+1)|\Theta_{h}|^{2p}\gamma.$

######  Remark 3.1.

Since $\Theta_{h}(t)$ is positive for all $t\in\mathbb{R}$ and
$\Theta_{h}(t)+\gamma(t)$ is shown to be positive in the region of $t$ where
we analyze persistence of the $b$-family of solutions, we can neglect writing
modulus signs in $\mathcal{L}\gamma$ and $N(\Theta_{h},\gamma)$.

The nonlinear term $N(\Theta_{h},\gamma)$ is superlinear in $\gamma$ if
$p\in(0,\frac{1}{2})$ and quadratic if $p\geq\frac{1}{2}$, according to the
following proposition.

######  Proposition 3.1.

Fix $p>0$ and $a>0$. If $F:[-a,a]\rightarrow\mathbb{R}$ is defined as

$F(x):=(a+x)^{2p+1}-a^{2p+1}-(2p+1)a^{2p}x,$

then there exists a positive constant $C>0$, such that for all $x\in[-a,a]$

$|F(x)|\leq\bigg{\\{}\begin{array}[]{ll}C|x|^{2p+1},&\text{if
}p\in(0,\frac{1}{2}),\\\ C|x|^{2},&\text{if
}p\in[\frac{1}{2},\infty).\end{array}$ (3.2)

###### Proof.

Without the loss of generality, we assume that $a=1$ due to the scaling
transformation:

$F(x)=a^{2p+1}\left[\left(1+\frac{x}{a}\right)^{2p+1}-1-\left(2p+1\right)\frac{x}{a}\right].$

Note that $F^{\prime\prime}(x)=2p(2p+1)(1+x)^{2p-1}$, so that if $2p-1\geq 0$,
then $F^{\prime\prime}$ is bounded on $[-1,1]$, and the second line of (3.2)
follows from Taylor’s theorem. If $2p-1<0$, then $F^{\prime\prime}$ is bounded
on $[-\frac{1}{2},\frac{1}{2}]$ so that

$|F(x)|\leq C|x|^{2}\leq C|x|^{2p+1},\quad
x\in\left[-\frac{1}{2},\frac{1}{2}\right],$

for some positive constant $C$. For $|x|\in[\frac{1}{2},1]$, we have
$\left(\frac{1}{2}\right)^{2p+1}\leq|x|^{2p+1}$ so that

$|F(x)|=\left(\frac{1}{2}\right)^{2p+1}2^{2p+1}|F(x)|\leq
C|x|^{2p+1},\quad|x|\in\left[\frac{1}{2},1\right],$

for another positive constant $C$. The two estimates above give the second
line of (3.2). ∎

The homogeneous equation $\mathcal{L}\gamma=0$ admits two linearly independent
solutions. The first one is given by $\Theta_{h}^{\prime}(t)$ due to the
translation symmetry of the autonomous equation (2.6). The other solution
denoted by $\Sigma(t)$ can be obtained from the Wronskian relation

$\Theta_{h}^{\prime}(t)\Sigma^{\prime}(t)-\Theta_{h}^{\prime\prime}(t)\Sigma(t)=\Sigma_{0},\qquad
t\in\mathbb{R},$ (3.3)

where $\Sigma_{0}\neq 0$ is constant. We take $\Sigma_{0}=1$ for normalizing
$\Sigma(t)$. Using (2.9) in (3.3), we obtain that

$\Sigma(t)=-\frac{p^{2}}{2}\left\\{\begin{array}[]{ll}e^{-\frac{t}{p}}[1+\mathcal{O}(e^{2t})]&\text{
as }\;\;t\to-\infty,\\\
\alpha_{p}^{\frac{1}{p}}e^{\frac{t}{p}}[1+\mathcal{O}(e^{-2t})]&\text{ as
}\;\;t\to+\infty.\end{array}\right.$ (3.4)

Using the two linearly independent solutions $\Theta_{h}^{\prime}$ and
$\Sigma$ of the homogeneous equation $\mathcal{L}\gamma=0$, we rewrite (3.1)
as an integral equation for $\gamma$:

$\gamma(t)=\int_{-\infty}^{t}\left(\Theta_{h}^{\prime}(t^{\prime})\Sigma(t)-\Theta_{h}^{\prime}(t)\Sigma(t^{\prime})\right)\left[f_{b}(t^{\prime})\left(\Theta_{h}(t^{\prime})+\gamma(t^{\prime})\right)-N\left(\Theta_{h}(t^{\prime}),\gamma(t^{\prime})\right)\right]dt^{\prime},$
(3.5)

where the free solution $c_{1}\Theta_{h}^{\prime}(t)+c_{2}\Sigma(t)$ has been
set to zero from the requirement that $\gamma(t)$ decays to zero as
$t\to-\infty$ faster than $\Theta_{h}^{\prime}(t)$.

The perturbation term $\gamma$ can be estimated to be small in the
$L^{\infty}$ norm on the semi-infinite interval $(-\infty,T+ap\log b]$ with
fixed $T>0$ and $a>0$, where the right end point diverges asymptotically to
$+\infty$ as $b\to\infty$. The following lemma gives the persistence result
for the solution $\Psi_{b}(t-p\log b)$ to stay close to the leading-order term
$\Theta_{h}(t)$ for $t\in(-\infty,T+ap\log b]$.

######  Lemma 3.1.

Fix $p\in(0,1]$ and $\lambda\in\mathbb{R}$. For any fixed $T>0$ and
$a\in(0,\frac{p}{1+p})$ there exist $b_{T,a}>0$ and $C_{T,a}>0$ such that the
unique solution $\Psi_{b}(t)$ to the second-order equation (2.4) with
asymptotic behaviour (2.5) satisfies for $t\in(-\infty,T+ap\log b]$ and all
$b\geq b_{T,a}$:

$|\Psi_{b}(t-p\log b)-\Theta_{h}(t)|\leq C_{T,a}b^{-2p(1-a)}e^{\frac{t}{p}},$
(3.6)

where the bound can be differentiated in $t$.

######  Remark 3.2.

The remainder term in the bound (3.6) is small on $[ap\log b,T+ap\log b]$ for
every $a\in(0,\frac{2p}{1+2p})$ for which $b^{-2p+a(2p+1)}\to 0$ as
$b\to\infty$. However, $\Theta_{h}(t)=\mathcal{O}(b^{-a})$ on the same
interval so that the remainder term $\gamma(t)$ is smaller than the leading-
order term $\Theta_{h}(t)$ for $t\in[ap\log b,T+ap\log b]$ if
$a\in(0,\frac{p}{1+p})$ for which $b^{-2p+a(2p+1)}\ll b^{-a}$ for sufficiently
large $b$.

###### Proof.

In order to eliminate the divergence of the integral kernel in the integral
equation (3.5) as $t\to-\infty$, we introduce a change of variables:
$\tilde{\Theta}_{h}(t):=e^{-\frac{t}{p}}\Theta_{h}(t)$ and
$\tilde{\gamma}(t):=e^{-\frac{t}{p}}\gamma(t)$. The new integral equation for
$\tilde{\gamma}$ can be considered as the fixed-point equation
$\tilde{\gamma}=A\tilde{\gamma}$, where

$(A\tilde{\gamma})(t):=\int_{-\infty}^{t}K(t,t^{\prime})\left[f_{b}(t^{\prime})(\tilde{\Theta}_{h}(t^{\prime})+\tilde{\gamma}(t^{\prime}))-e^{2t^{\prime}}N(\tilde{\Theta}_{h}(t^{\prime}),\tilde{\gamma}(t^{\prime}))\right]dt^{\prime},$
(3.7)

where the integral kernel is defined as

$K(t,t^{\prime}):=e^{\frac{t^{\prime}}{p}}\Theta_{h}^{\prime}(t^{\prime})e^{-\frac{t}{p}}\Sigma(t)-e^{-\frac{t}{p}}\Theta_{h}^{\prime}(t)e^{\frac{t^{\prime}}{p}}\Sigma(t^{\prime}),\quad
t^{\prime}\leq t.$

It follows from the asymptotic behaviours (2.9) and (3.4) for $\Theta_{h}$ and
$\Sigma$ that the integral kernel is bounded for every $t\in\mathbb{R}$ by

$|K(t,t^{\prime})|\leq C\left(1+e^{-\frac{2}{p}(t-t^{\prime})}\right),\quad
t^{\prime}\leq t,$

for some positive constant $C$. Thus, as the integration in (3.7) is done in
$t^{\prime}$ from $-\infty$ to $t$, the kernel $K(t,t^{\prime})$ is bounded.
In addition, the lower bound on $\tilde{\Theta}_{h}$ follows from (2.9):

$\tilde{\Theta}_{h}(t)\geq C_{T,a}b^{-2a},\quad t\in(-\infty,T+ap\log b),$

for some positive constant $C_{T,a}$ that depends on $T$ and $a$ for all large
$b$. We shall prove that the integral operator $A$ is a contraction in a small
closed ball in the Banach space $L^{\infty}(-\infty,T+ap\log b)$ equipped with
the norm $\|\cdot\|_{\infty}$.

Case $p\in(0,\frac{1}{2})$. Since $\tilde{\Theta}_{h}(t)$ is bounded from
below for $t\in(-\infty,T+ap\log b)$, it follows by Proposition 3.1 if
$\|\tilde{\gamma}\|_{\infty}\ll b^{-2a}$ for all large $b$, then

$\|N(\tilde{\Theta}_{h},\tilde{\gamma})\|_{\infty}\leq
C\|\tilde{\gamma}\|_{\infty}^{2p+1},$ (3.8)

for some positive constant $C$. We use
$f_{b}(t):=b^{-2p}e^{2t}[-\lambda+b^{-2p}e^{2t}]$ and estimate

$\displaystyle\|A\tilde{\gamma}\|_{\infty}$ $\displaystyle\leq
C\left[\left(1+\|\tilde{\gamma}\|_{\infty}\right)\int_{-\infty}^{T+ap\log
b}|f_{b}(t^{\prime})|dt^{\prime}+\|\tilde{\gamma}\|_{\infty}^{2p+1}\int_{-\infty}^{T+ap\log
b}e^{2t^{\prime}}dt^{\prime}\right]$ $\displaystyle\leq
C\left[\left(1+\|\tilde{\gamma}\|_{\infty}\right)b^{-2p(1-a)}+b^{2ap}\|\tilde{\gamma}\|_{\infty}^{2p+1}\right],$

where the positive constant $C$ can change from one line to the other line. If
$\|\tilde{\gamma}\|_{\infty}\leq 2Cb^{-2p(1-a)}$, then

$\displaystyle\|A\tilde{\gamma}\|_{\infty}$ $\displaystyle\leq
C\left[b^{-2p(1-a)}+2Cb^{-4p(1-a)}+(2C)^{2p+1}b^{-2p(1-2a(p+1)+2p)}\right]$
$\displaystyle\leq 2Cb^{-2p(1-a)},$

where we have used $2p(1-a)<2p(1-2a(p+1)+2p)$ if $a\in(0,\frac{2p}{1+2p})$.
Since $2a<2p(1-a)$ if $a\in(0,\frac{p}{1+p})$ with
$\frac{p}{1+p}<\frac{2p}{1+2p}$, the bound $\|\tilde{\gamma}\|_{\infty}\leq
2Cb^{-2p(1-a)}$ ensures validity of the bound $\|\tilde{\gamma}\|_{\infty}\ll
b^{-2a}$ for which the bound (3.8) can be used. Similar calculations show that
for two functions $\tilde{\gamma}$ and $\tilde{\gamma}^{\prime}$ in the same
small closed ball in $L^{\infty}(-\infty,T+ap\log b)$, we have

$\|A\tilde{\gamma}-A\tilde{\gamma}^{\prime}\|_{\infty}\leq
Cb^{-2p(1-a)}\|\tilde{\gamma}-\tilde{\gamma}^{\prime}\|_{\infty},$

so that $A$ is a contraction for sufficiently large $b$. By the Banach fixed-
point theorem, there exists a unique fixed point $\tilde{\gamma}$ of $A$ such
that

$\sup_{t\in(-\infty,T+ap\log b)}|\tilde{\gamma}(t)|\leq 2Cb^{-2p(1-a)}.$

Since $\gamma(t)=e^{\frac{t}{p}}\tilde{\gamma}(t)$, we obtain the bound (3.6)
for the unique solution $\gamma(t)$ to the integral equation (3.5).

Case $p\in[\frac{1}{2},1]$. The only difference in the proof is that, by
Proposition 3.1, the bound (3.8) is replaced by the bound

$\|N(\tilde{\Theta}_{h},\tilde{\gamma})\|_{\infty}\leq
C\|\tilde{\gamma}\|_{\infty}^{2},$ (3.9)

if $\|\tilde{\gamma}\|_{\infty}\ll b^{-2a}$ for all large $b$. In this case,
we get the estimate

$\displaystyle\|A\tilde{\gamma}\|_{\infty}$ $\displaystyle\leq
C\left[\left(1+\|\tilde{\gamma}\|_{\infty}\right)b^{-2p(1-a)}+b^{2ap}\|\tilde{\gamma}\|_{\infty}^{2}\right],$
$\displaystyle\leq
C\left[b^{-2p(1-a)}+2Cb^{-4p(1-a)}+(2C)^{2}b^{-2p(2-3a)}\right]$
$\displaystyle\leq 2Cb^{-2p(1-a)},$

where we have used $2p(1-a)<2p(2-3a)$ if $a\in(0,\frac{1}{2})$. Since
$2a<2p(1-a)$ if $a\in(0,\frac{p}{1+p})$ with $\frac{p}{1+p}\leq\frac{1}{2}$,
the bound $\|\tilde{\gamma}\|_{\infty}\leq 2Cb^{-2p(1-a)}$ ensures validity of
the bound $\|\tilde{\gamma}\|_{\infty}\ll b^{-2a}$ for which the bound (3.9)
can be used. The rest of the proof is verbatim to the case of
$p\in(0,\frac{1}{2})$. ∎

######  Remark 3.3.

The bound (3.6) can be extended for every $p\geq 1$ if the values of $a$ are
restricted to $a\in(0,\frac{1}{2})$ as follows from the proof of Lemma 3.1 in
the case of $p\in[\frac{1}{2},1]$.

The result of Lemma 3.1 allows us to justify the validity of

$\Psi_{b}(t-p\log
b)\sim\Theta_{h}(t)\sim\alpha_{p}^{-\frac{1}{p}}e^{-\frac{t}{p}},\quad
t\in[ap\log b,T+ap\log b]$

due to (2.9) and (3.6). In order to obtain the correction term which behaves
like $e^{\frac{t}{p}}$ in the same asymptotic region, we need to analyze
$\gamma$ in more details and obtain the leading-order part of $\gamma$. To do
so, we write $\gamma=\gamma_{h}+\delta$, where the leading-order term
$\gamma_{h}$ satisfies $\mathcal{L}\gamma_{h}=f_{b}\Theta_{h}$ and is given
explicitly by

$\gamma_{h}(t)=\Sigma(t)\int_{-\infty}^{t}f_{b}(t^{\prime})\Theta_{h}^{\prime}(t^{\prime})\Theta_{h}(t^{\prime})dt^{\prime}-\Theta_{h}^{\prime}(t)\int_{-\infty}^{t}f_{b}(t^{\prime})\Sigma(t^{\prime})\Theta_{h}(t^{\prime})dt^{\prime},$
(3.10)

whereas the correction term $\delta$ satisfies

$\mathcal{L}\delta=f_{b}(\gamma_{h}+\delta)-N(\Theta_{h},\gamma_{h}+\delta).$
(3.11)

The following lemma gives a sharper bound on $\gamma_{h}$ compared to the
bound (3.6). The sharper bound holds on $[ap\log b,T+ap\log b]$, where the
asymptotic behavior of $\Theta_{h}(t)$ and $\Sigma(t)$ as $t\to+\infty$ is
relevant.

######  Lemma 3.2.

Fix $p\in(0,1)$ and $\lambda\in\mathbb{R}$. For any fixed $T>0$ and
$a\in(0,\frac{p}{1+p})$ there exist $b_{T,a}>0$ and $C_{T,a}>0$ such that
$\gamma_{h}$ in (3.10) satisfies for $t\in[ap\log b,T+ap\log b]$ and all
$b\geq b_{T,a}$:

  * •

if $p\in(0,\frac{1}{2})$, then

$|\gamma_{h}(t)|\leq
C_{T,a}\left[(|\lambda|b^{-2p(1-a)}+b^{-4p(1-a)})e^{-\frac{t}{p}}+(|\lambda|b^{-2p}+b^{-4p})e^{\frac{t}{p}}\right],$
(3.12)

  * •

if $p\in[\frac{1}{2},1)$, then

$|\gamma_{h}(t)|\leq
C_{T,a}\left[(|\lambda|b^{-2p(1-a)}+b^{-4p(1-a)})e^{-\frac{t}{p}}+(|\lambda|b^{-2p}+b^{-4p(1-a)})e^{\frac{t}{p}}\right],$
(3.13)

where the bounds can be differentiated in $t$.

###### Proof.

Since $\Sigma(t)\Theta_{h}(t)$ is bounded for every $t\in\mathbb{R}$
independently of $b$, the second integral term in (3.10) is controlled by

$\displaystyle\left|\int_{-\infty}^{T+ap\log
b}f_{b}(t)\Sigma(t)\Theta_{h}(t)dt\right|$ $\displaystyle\leq
C\int_{-\infty}^{T+ap\log
b}\left(|\lambda|b^{-2p}e^{2t}+b^{-4p}e^{4t}\right)dt$ $\displaystyle\leq
C_{T,a}(|\lambda|b^{-2p(1-a)}+b^{-4p(1-a)}).$

This estimate yields the first term in the bounds (3.12) and (3.13) due to the
asymptotic behavior (2.9). On the other hand, since
$\Theta_{h}(t)^{2}=\mathcal{O}(e^{-\frac{2t}{p}})$ as $t\to+\infty$, the first
integral term in (3.10) is controlled by

$\displaystyle\left|\int_{-\infty}^{T+ap\log
b}f_{b}(t)\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt\right|$ $\displaystyle\leq
C\int_{-\infty}^{T+ap\log
b}\left(|\lambda|b^{-2p}e^{-\frac{2(1-p)}{p}t}+b^{-4p}e^{-\frac{2(1-2p)}{p}t}\right)dt$
$\displaystyle\leq C_{T,a}(|\lambda|b^{-2p}+b^{-4p+2a\nu_{p}}),$

where $\nu_{p}=0$ for $p\in(0,\frac{1}{2})$ and $\nu_{p}=2p-1$ for
$p\in[\frac{1}{2},1)$. This yields the second term in the bounds (3.12) and
(3.13) due to the asymptotic behavior (3.4), where we have also used that
$4p(1-a)<4p-2a(2p-1)$. ∎

The sharper bounds (3.12) and (3.13) are compatible with the bound (3.6) on
the semi-infinite interval $(-\infty,T+ap\log b]$, which can be rewritten in
the form:

$\displaystyle|\gamma_{h}(t)|\leq C_{T,a}b^{-2p(1-a)}e^{\frac{t}{p}},\qquad
t\in(-\infty,T+ap\log b].$ (3.14)

The correction term $\delta$ is estimated to be smaller than $\gamma_{h}$
according to the following lemma.

######  Lemma 3.3.

Fix $p\in(0,1]$ and $\lambda\in\mathbb{R}$. For any fixed $T>0$,
$a\in(0,\frac{p}{1+p})$, there exist $b_{T,a}>0$ and $C_{T,a}>0$ such that for
$t\in(-\infty,T+ap\log b]$ and all $b\geq b_{T,a}$:

  * •

if $p\in(0,\frac{1}{2})$, then

$|\Psi_{b}(t-p\log b)-\Theta_{h}(t)-\gamma_{h}(t)|\leq
C_{T,a}b^{-2p\left[(2p+1)(1-a)-a\right]}e^{\frac{t}{p}},$ (3.15)

  * •

if $p\in[\frac{1}{2},1]$, then

$|\Psi_{b}(t-p\log b)-\Theta_{h}(t)-\gamma_{h}(t)|\leq
C_{T,a}b^{-2p(2-3a)}e^{\frac{t}{p}},$ (3.16)

where the bounds can be differentiated in $t$.

######  Remark 3.4.

If $a\in(0,\frac{2p}{1+2p})$ for $p\in(0,\frac{1}{2})$, we have

$2p(1-a)<2p\left[(2p+1)(1-a)-a\right],$

so that comparison (3.14) and (3.15) shows that $\delta$ is smaller than
$\gamma_{h}$ for sufficiently large $b$. Similarly, if $a\in(0,\frac{1}{2})$
for $p\geq\frac{1}{2}$, we have

$2p(1-a)<2p(2-3a),$

so that the comparison of (3.14) and (3.16) shows that $\delta$ is smaller
than $\gamma_{h}$ for sufficiently large $b$. In both cases, by Lemma 3.1, we
also have $\gamma_{h}$ being smaller than $\Theta_{h}$ if
$a\in(0,\frac{p}{1+p})$, where
$\frac{p}{1+p}\leq\min\\{\frac{2p}{1+2p},\frac{1}{2}\\}$ if $p\in(0,1]$.

###### Proof.

Equation (3.11) for $\delta$ can be written similarly to (3.5) as the integral
equation

$\delta(t)=\int_{-\infty}^{t}\left(\Theta_{h}^{\prime}(t^{\prime})\Sigma(t)-\Theta_{h}^{\prime}(t)\Sigma(t^{\prime})\right)\left[f_{b}(t^{\prime})\left(\gamma_{h}(t^{\prime})+\delta(t^{\prime})\right)-N\left(\Theta_{h}(t^{\prime}),\gamma_{h}(t^{\prime})+\delta(t^{\prime})\right)\right]dt^{\prime}.$
(3.17)

We proceed in a similar way to the proof of Lemma 3.1. Using the change of
variables

$\tilde{\Theta}_{h}(t):=e^{-\frac{t}{p}}\Theta_{h}(t),\quad\tilde{\gamma}_{h}(t):=e^{-\frac{t}{p}}\gamma_{h}(t),\quad\tilde{\delta}(t):=e^{-\frac{t}{p}}\delta(t),$

we rewrite the integral equation for $\delta$ as the fixed-point equation
$\tilde{\delta}=B\tilde{\delta}$, where

$(B\tilde{\delta})(t):=\int_{-\infty}^{t}K(t,t^{\prime})\left[f_{b}(t^{\prime})(\tilde{\gamma}_{h}(t^{\prime})+\tilde{\delta}(t^{\prime}))-e^{2t^{\prime}}N(\tilde{\Theta}_{h}(t^{\prime}),\tilde{\gamma}_{h}(t^{\prime})+\tilde{\delta}(t^{\prime}))\right]dt^{\prime}.$
(3.18)

The only essential difference between $A$ in (3.7) and $B$ in (3.18) is the
source term which dictates the size of the closed ball in
$L^{\infty}(-\infty,T+ap\log b]$, where the fixed-point iterations are closed.
In (3.18), it consists of the linear term $f_{b}\tilde{\gamma}_{h}$ and the
contribution from nonlinearity term
$N(\tilde{\Theta}_{h},\tilde{\gamma}_{h})$. The linear term is estimated from
(3.14) as

$\left|\int_{-\infty}^{T+ap\log b}f_{b}(t)\tilde{\gamma}_{h}(t)dt\right|\leq
C\|\tilde{\gamma}_{h}\|_{\infty}\int_{-\infty}^{T+ap\log b}|f_{b}(t)|dt\leq
Cb^{-4p(1-a)}.$ (3.19)

Estimates for the nonlinear term depend on the value of $p$. To proceed with
the estimates, we decompose

$N(\tilde{\Theta}_{h},\tilde{\gamma}_{h}+\tilde{\delta})=N(\tilde{\Theta}_{h},\tilde{\gamma}_{h})+\tilde{N}(\tilde{\Theta}_{h},\tilde{\gamma}_{h},\tilde{\delta}),$

where

$\displaystyle\tilde{N}(\tilde{\Theta}_{h},\tilde{\gamma}_{h},\tilde{\delta})$
$\displaystyle:=N(\tilde{\Theta}_{h},\tilde{\gamma}_{h}+\tilde{\delta})-N(\tilde{\Theta}_{h},\tilde{\gamma}_{h})$
$\displaystyle=(\tilde{\Theta}_{h}+\tilde{\gamma}_{h}+\tilde{\delta})^{2p+1}-(\tilde{\Theta}_{h}+\tilde{\gamma}_{h})^{2p+1}-(2p+1)(\tilde{\Theta}_{h})^{2p}\tilde{\delta}.$

Case $p\in(0,\frac{1}{2})$. By Proposition 3.1, we have

$\|N(\tilde{\Theta}_{h},\tilde{\gamma}_{h})\|_{\infty}\leq
C^{\prime}\|\tilde{\gamma}_{h}\|_{\infty}^{2p+1}.$ (3.20)

Since $\tilde{N}|_{\delta=0}=0$ and

$\frac{\partial\tilde{N}}{\partial\tilde{\delta}}\bigg{|}_{\tilde{\delta}=0}=(2p+1)(\tilde{\Theta}_{h}+\tilde{\gamma}_{h})^{2p}-(2p+1)(\tilde{\Theta}_{h})^{2p},$

we obtain by a minor modification of the proof of Proposition 3.1 that

$\|\tilde{N}(\tilde{\Theta}_{h},\tilde{\gamma}_{h},\tilde{\delta})\|_{\infty}\leq
C\|\tilde{\gamma}_{h}\|_{\infty}^{2p}\|\tilde{\delta}\|_{\infty}.$ (3.21)

Putting together estimates (3.19), (3.20), and (3.21), we obtain that

$\displaystyle\|B\tilde{\delta}\|_{\infty}\leq
C\left(b^{-4p(1-a)}+b^{-2p(1-a)}\|\tilde{\delta}\|_{\infty}+b^{-2p[(2p+1)(1-a)-a]}+b^{-2p\left[2p(1-a)-a\right]}\|\tilde{\delta}\|_{\infty}\right).$

Since $4p(1-a)>2p[(2p+1)(1-a)-a]$ for every $p\in(0,\frac{1}{2})$, we have
$b^{-4p(1-a)}\ll b^{-2p[(2p+1)(1-a)-a]}$ for sufficiently large $b$, hence the
source term coming from the nonlinearity
$N(\tilde{\Theta}_{h},\tilde{\gamma}_{h})$ is much larger than the source term
coming from $f_{b}\tilde{\gamma}_{h}$ as $b\to\infty$. As a result, if
$\|\tilde{\delta}\|_{\infty}\leq 2Cb^{-2p[2p(1-a)-a]}$, then
$\|B\tilde{\delta}\|_{\infty}\leq 2Cb^{-2p[2p(1-a)-a]}$. Moreover, $B$ is a
contraction in the same small closed ball in $L^{\infty}(-\infty,T+ap\log b)$
for sufficiently large $b$. Hence, there exists a unique fixed point
$\tilde{\delta}$ of $B$ satisfying $\|\tilde{\delta}\|_{\infty}\leq
2Cb^{-2p[(2p+1)(1-a)-a]}$, which yields (3.15) for
$\delta(t)=e^{\frac{t}{p}}\tilde{\delta}(t)$.

Case $p\in[\frac{1}{2},1]$. By Proposition 3.1, we have

$\|N(\tilde{\Theta}_{h},\tilde{\gamma}_{h})\|_{\infty}\leq
C\|\tilde{\gamma}_{h}\|_{\infty}^{2},$

and similarly,

$\|\tilde{N}(\tilde{\Theta}_{h},\tilde{\gamma}_{h},\tilde{\delta})\|_{\infty}\leq
C\|\tilde{\gamma}_{h}\|_{\infty}\|\tilde{\delta}\|_{\infty}.$

Proceeding similarly to the previous computations, we obtain

$\displaystyle\|B\tilde{\delta}\|_{\infty}\leq
C^{\prime}\left(b^{-4p(1-a)}+b^{-2p(1-a)}\|\tilde{\delta}\|_{\infty}+b^{-2p(2-3a)}+b^{-2p(1-2a)}\|\tilde{\delta}\|_{\infty}\right).$

Since $4p(1-a)>2p(2-3a)$, we have $b^{-4p(1-a)}\ll b^{-2p(2-3a)}$ for
sufficiently large $b$, hence again the source term coming from the
nonlinearity $N(\tilde{\Theta}_{h},\tilde{\gamma}_{h})$ is much larger than
the source term coming from $f_{b}\tilde{\gamma}_{h}$ as $b\to\infty$.
Proceeding similarly, for sufficiently large $b$, there exists a unique fixed
point $\tilde{\delta}$ of $B$ satifying $\|\tilde{\delta}\|_{\infty}\leq
2Cb^{-2p(2-3a)}$, which yields (3.16) for
$\delta(t)=e^{\frac{t}{p}}\tilde{\delta}(t)$. ∎

Similarly to Lemma 3.2, we can find a sharper bound on $\delta$ compared to
the bounds (3.15) and (3.16). This is given by the following lemma, the proof
of which follows from the estimates obtained in Lemma 3.3.

######  Lemma 3.4.

Fix $p\in(0,1)$ and $\lambda\in\mathbb{R}$. For any fixed $T>0$ and
$a\in(0,\frac{p}{1+p})$ there exist $b_{T,a}>0$ and $C_{T,a}>0$ such that
$\delta$ in (3.17) satisfies for $t\in[ap\log b,T+ap\log b]$ and all $b\geq
b_{T,a}$:

  * •

if $p\in(0,\frac{1}{2})$, then

$|\delta(t)|\leq
C_{T,a}(|\lambda|b^{-2p}+b^{-4p})^{2p+1}b^{2ap}e^{\frac{t}{p}},$ (3.22)

  * •

if $p\in[\frac{1}{2},1)$, then

$|\gamma_{h}(t)|\leq
C_{T,a}(|\lambda|b^{-2p}+b^{-4p(1-a)})^{2}b^{2ap}e^{\frac{t}{p}},$ (3.23)

where the bounds can be differentiated in $t$.

###### Proof.

For $p\in(0,\frac{1}{2})$, the first term in the bound (3.12) with
$e^{-\frac{t}{p}}$ is much smaller than the second term with $e^{\frac{t}{p}}$
on $[ap\log b,T+ap\log b]$. As a result, it can be neglected. On the other
hand, the second term in the bound (3.12) can be extended for the semi-
infinite interval $(-\infty,T+ap\log b]$ such that the sharper bound compared
to (3.6) can be written in the form

$|\gamma_{h}(t)|\leq C_{T,a}(|\lambda|b^{-2p}+b^{-4p})e^{\frac{t}{p}},\quad
t\in(-\infty,T+ap\log b].$

The bound (3.22) follows from analysis of the integral equation (3.17) by
using the transformation to the tilde variables in the proof of Lemma 3.3 and
the estimate (3.20) on the nonlinear term $N$ which is much larger than the
source term from $f_{b}$.

For $p\in[\frac{1}{2},1)$, the proof is analogous but we use

$|\gamma_{h}(t)|\leq
C_{T,a}(|\lambda|b^{-2p}+b^{-4p(1-a)})e^{\frac{t}{p}},\quad
t\in(-\infty,T+ap\log b].$

and the estimate (3.21) on the nonlinear term $N$ which is still much larger
than the source term from $f_{b}$. ∎

##  4\. Persistence of the $c$-family of solutions

The $c$-family of solutions $\Psi_{c}$ of the differential equation (2.4)
satisfying (2.21) is considered near the solution $\Upsilon_{h}$ of the linear
equation (2.10) given by (2.20). The comparison gives $\Psi_{c}(t)\sim
c\Upsilon_{h}(t)$ as $t\to+\infty$. The correction term
$\eta(t):=\Psi_{c}(t)-c\Upsilon_{h}(t)$ satisfies

$\mathcal{M}\eta=-\left|c\Upsilon_{h}+\eta\right|^{2p}(c\Upsilon_{h}+\eta),$
(4.1)

where

$\left(\mathcal{M}\eta\right)(t):=\eta^{\prime\prime}(t)-\frac{1}{p^{2}}\eta(t)+\lambda
e^{2t}\eta(t)-e^{4t}\eta(t).$

The homogeneous equation $\mathcal{M}\eta=0$ has two linearly independent
solutions. One solution is $\Upsilon_{h}$ given by (2.20). The other solution,
denoted as $\Upsilon_{g}$, can be obtained from the normalized Wronskian
relation

$\Upsilon_{h}(t)\Upsilon_{g}^{\prime}(t)-\Upsilon_{h}^{\prime}(t)\Upsilon_{g}(t)=1.$
(4.2)

Since it follows from (2.21) that

$\Upsilon_{h}(t)\sim
e^{-\left(1-\frac{\lambda}{2}\right)t}e^{-\frac{1}{2}e^{2t}},\quad\text{ as
}t\to+\infty,$ (4.3)

integrating the Wronskian relation (4.2) yields

$\Upsilon_{g}(t)\sim\frac{1}{2}e^{-\left(1+\frac{\lambda}{2}\right)t}e^{\frac{1}{2}e^{2t}},\quad\text{
as }t\to+\infty.$ (4.4)

With two linearly independent solutions $\Upsilon_{h}$ and $\Upsilon_{g}$, we
rewrite (4.1) as an integral equation for $\eta$:

$\eta(t)=\int_{t}^{\infty}\left(\Upsilon_{h}(t^{\prime})\Upsilon_{g}(t)-\Upsilon_{h}(t)\Upsilon_{g}(t^{\prime})\right)|c\Upsilon_{h}(t^{\prime})+\eta(t^{\prime})|^{2p}(c\Upsilon_{h}(t^{\prime})+\eta(t^{\prime}))dt^{\prime},$
(4.5)

where the free solution $c_{1}\Upsilon_{h}+c_{2}\Upsilon_{g}$ has been set to
zero in order to guarantee that $\eta(t)$ decays to zero as $t\to+\infty$
faster than $\Upsilon_{h}(t)$.

The following lemma describes the size of $\eta(t)$ for $t\in[0,\infty)$.

######  Lemma 4.1.

Fix $\lambda\in(-\infty,2]$ and $p>0$. Then, there exist some constants $C>0$
and $c_{0}>0$, such that for $t\in[0,\infty)$ and $c\in(-c_{0},c_{0})$, we
have

$|\Psi_{c}(t)-c\Upsilon_{h}(t)|\leq
C|c|^{2p+1}e^{-(1-\frac{\lambda}{2})t}e^{-\frac{1}{2}e^{2t}},$ (4.6)

where the bound can be differentiated term by term.

###### Proof.

In order to obtain a bounded kernel in the integral equation (4.5), we first
introduce the change of variables

$\tilde{\Upsilon}_{h}(t):=e^{(1-\frac{\lambda}{2})t}e^{\frac{1}{2}e^{2t}}\Upsilon_{h}(t),\qquad\tilde{\eta}(t):=e^{(1-\frac{\lambda}{2})t}e^{\frac{1}{2}e^{2t}}\eta(t),$

which applied to (4.5) results in the integral equation
$\tilde{\eta}=E\tilde{\eta}$, where

$(E\tilde{\eta})(t):=\int_{t}^{\infty}\hat{K}(t,t^{\prime})e^{-p(2-\lambda)t^{\prime}-pe^{2t^{\prime}}-2t^{\prime}}\hat{N}(c\tilde{\Upsilon}_{h}(t^{\prime}),\tilde{\eta}(t^{\prime}))dt^{\prime},$
(4.7)

and where the kernel $\hat{K}$ and the nonlinearity $\hat{N}$ are given by

$\hat{K}(t,t^{\prime}):=e^{-(1-\frac{\lambda}{2})(t^{\prime}-t)+2t^{\prime}+\frac{1}{2}(e^{2t}-e^{2t^{\prime}})}(\Upsilon_{h}(t^{\prime})\Upsilon_{g}(t)-\Upsilon_{h}(t)\Upsilon_{g}(t^{\prime})),$

and

$\hat{N}(c\tilde{\Upsilon}_{h},\tilde{\eta}):=|c\tilde{\Upsilon}_{h}+\tilde{\eta}|^{2p}(c\tilde{\Upsilon}_{h}+\tilde{\eta}).$

Using asymptotic behaviours (4.3) and (4.4), we get that there exists $C>0$
such that

$|\hat{K}(t,t^{\prime})|\leq C(1+e^{\phi(t,t^{\prime})}),\qquad 0\leq t\leq
t^{\prime}.$ (4.8)

where
$\phi(t,t^{\prime}):=\lambda(t^{\prime}-t)-e^{2t^{\prime}}(1-e^{-2(t^{\prime}-t)})$.
Since $\phi(t,t^{\prime})\to-\infty$ as $t^{\prime}\to+\infty$ and
$\phi(t,t^{\prime})$ has an extremum in $t^{\prime}$ at
$\lambda-2e^{2t^{\prime}}=0$, which does not belong to $\mathbb{R}$ if
$\lambda\in(-\infty,0]$ and is located on $\mathbb{R}_{-}$ if
$\lambda\in(0,2)$, we conclude that
$\max\limits_{t^{\prime}\in[t,\infty)}e^{\phi(t,t^{\prime})}=e^{\phi(t,t)}=1$
for $t\geq 0$. Hence, the kernel $\hat{K}(t,t^{\prime})$ is bounded for every
$0\leq t\leq t^{\prime}<\infty$. On the other hand, since
$\hat{N}(c\Upsilon_{h},\tilde{\eta})$ is a $C^{1}$ function for every $p>0$,
the nonlinear term satisfies the following bound:

$|\hat{N}(c\tilde{\Upsilon}_{h},\tilde{\eta})|\leq
C|c|^{2p}(|c|+|\tilde{\eta}|),\quad\mbox{\rm as long as }\;|\tilde{\eta}|\leq
C,$ (4.9)

where the constant $C>0$ is independent of $c$.

In order to use the Banach fixed-point theorem, we first estimate the size of
$E\tilde{\eta}$ for $\tilde{\eta}$ in a small closed ball in
$L^{\infty}(0,\infty)$. Since $e^{-p(2-\lambda)t-pe^{2t}-2t}$ in (4.7) is
absolutely integrable on $[0,\infty)$, we obtain by using (4.8) and (4.9) that

$\|E\tilde{\eta}\|_{\infty}\leq C|c|^{2p}(|c|+\|\tilde{\eta}\|_{\infty}),$

where the constant $C>0$ is independent of $c$. Thus, the operator $E$ maps a
closed ball of radius $2C|c|^{2p+1}$ into itself as long as $|c|$ is chosen
sufficiently small.

Similarly, by Proposition 3.1, we get for every $\tilde{\eta}_{1}$ and
$\tilde{\eta}_{2}$ in the same small closed ball in $L^{\infty}(0,\infty)$
that $\hat{N}$ is a Lipschitz function satisfying

$\|\hat{N}(c\tilde{\Upsilon}_{h},\tilde{\eta}_{1})-\hat{N}(c\tilde{\Upsilon}_{h},\tilde{\eta}_{2})\|_{\infty}\leq
C|c|^{2p}\|\tilde{\eta}_{1}-\tilde{\eta}_{2}\|_{\infty},$ (4.10)

which yields

$\|E\tilde{\eta}_{1}-E\tilde{\eta}_{2}\|_{\infty}\leq
C|c|^{2p}\|\tilde{\eta}_{1}-\tilde{\eta}_{2}\|_{\infty},$

so that the operator $E$ is a contraction for sufficiently small values of
$|c|$. By the Banach fixed-point theorem, there exists a unique solution
$\tilde{\eta}(t)\in L^{\infty}(0,\infty)$ of the integral equation
$\tilde{\eta}=E\tilde{\eta}$ satisfying $\|\tilde{\eta}\|_{\infty}\leq
2C|c|^{2p+1}$. This estimate yields (4.6) after unfolding the transformation
for $\eta(t)$. ∎

Using the result of Lemma 4.1, we can now extend the estimates for $\eta(t)$
for large negative values of $t$.

######  Lemma 4.2.

Fix $\lambda\in(-\infty,2]$, $p>0$, and $a\in(0,1)$. Then, there exist
$b_{0}>0$ and $C>0$, such that for every $b\geq b_{0}$ there exists $c_{0}>0$,
such that for $t\in[-(1-a)p\log b,0]$ and
$c\in(-c_{0}b^{-(1-a)},c_{0}b^{-(1-a)})$, we have

$|\Psi_{c}(t)-c\Upsilon_{h}(t)|\leq C|c|^{2p+1}b^{2p(1-a)}e^{-\frac{t}{p}},$
(4.11)

where the bound can be differentiated term by term.

###### Proof.

We rewrite equation (4.1) as an integral equation for $t\in[-(1-a)p\log b,0]$:

$\eta(t)=\left(\Upsilon_{g}^{\prime}(0)\Upsilon_{h}(t)-\Upsilon_{h}^{\prime}(0)\Upsilon_{g}(t)\right)\eta(0)+\left(\Upsilon_{h}(0)\Upsilon_{g}(t)-\Upsilon_{g}(0)\Upsilon_{h}(t)\right)\eta^{\prime}(0)\\\
+\int_{t}^{0}\left(\Upsilon_{h}(t^{\prime})\Upsilon_{g}(t)-\Upsilon_{h}(t)\Upsilon_{g}(t^{\prime})\right)\left(c\Upsilon_{h}(t^{\prime})+\eta(t^{\prime})\right)\left|c\Upsilon_{h}(t^{\prime})+\eta(t^{\prime})\right|^{2p}dt^{\prime},$
(4.12)

where $|\eta(0)|+|\eta^{\prime}(0)|\leq C|c|^{2p+1}$ by Lemma 4.1. By using
the scattering relation (2.18) and the transformation (2.20), we obtain the
following asymptotic behavior for $\Upsilon_{h}(t)$:

$\Upsilon_{h}(t)\sim\frac{\Gamma\left(\frac{1}{p}\right)}{\Gamma(\alpha)}e^{-\frac{t}{p}},\quad\text{
as }t\to-\infty,$ (4.13)

where $\alpha>0$ by (2.15). Wronskian relation (4.2) implies the following
asymptotic behavior for $\Upsilon_{g}(t)$:

$\Upsilon_{g}(t)\sim\frac{p\Gamma(\alpha)}{2\Gamma\left(\frac{1}{p}\right)}e^{\frac{t}{p}},\quad\text{
as }t\to-\infty.$ (4.14)

The divergent behaviour of $\Upsilon_{h}(t)$ as $t\to-\infty$ dictates the
correct form of the transformation to use, which in this case is given by:

$\tilde{\Upsilon}_{h}(t):=e^{\frac{t}{p}}\Upsilon_{h}(t),\quad\tilde{\Upsilon}_{g}(t):=e^{\frac{t}{p}}\Upsilon_{g}(t),\quad\tilde{\eta}(t):=e^{\frac{t}{p}}\eta(t).$

Applying it to the integral equation (4.12) results in the fixed point
equation $\tilde{\eta}=\mathcal{F}\tilde{\eta}$, where

$(\mathcal{F}\tilde{\eta})(t):=\left(\Upsilon_{g}^{\prime}(0)\tilde{\Upsilon}_{h}(t)-\Upsilon_{h}^{\prime}(0)\tilde{\Upsilon}_{g}(t)\right)\eta(0)+\left(\tilde{\Upsilon}_{h}(0)\Upsilon_{g}(t)-\Upsilon_{g}(0)\tilde{\Upsilon}_{h}(t)\right)\eta^{\prime}(0)\\\
+\int_{t}^{0}e^{-2t^{\prime}}\tilde{K}(t,t^{\prime})\hat{N}(c\tilde{\Upsilon}_{h}(t^{\prime}),\tilde{\eta}(t^{\prime}))dt^{\prime},$
(4.15)

where

$\tilde{K}(t,t^{\prime}):=e^{\frac{t-t^{\prime}}{p}}\left(\Upsilon_{h}(t^{\prime})\Upsilon_{g}(t)-\Upsilon_{h}(t)\Upsilon_{g}(t^{\prime})\right)$

and $\hat{N}$ is the same as in the proof of Lemma 4.1.

We proceed by estimating each term of $\mathcal{F}\tilde{\eta}$ in the space
$L^{\infty}(-(1-a)p\log b,0)$. Since $\tilde{\Upsilon}_{h}$ and
$\tilde{\Upsilon}_{g}$ are bounded for $t\in(-\infty,0]$ and
$|\eta(0)|+|\eta^{\prime}(0)|\leq C|c|^{2p+1}$, we obtain

$\left|\left(\Upsilon_{g}^{\prime}(0)\tilde{\Upsilon}_{h}(t)-\Upsilon_{h}^{\prime}(0)\tilde{\Upsilon}_{g}(t)\right)\eta(0)+\left(\tilde{\Upsilon}_{h}(0)\Upsilon_{g}(t)-\Upsilon_{g}(0)\tilde{\Upsilon}_{h}(t)\right)\eta^{\prime}(0)\right|\leq
C|c|^{2p+1}.$

Furthermore, if $t\ll-1$, asymptotics (4.13) and (4.14) allow us to estimate
size of the last term in (4.15) as

$\displaystyle\left|\int_{t}^{0}e^{-2t^{\prime}}\tilde{K}(t,t^{\prime})\hat{N}(c\tilde{\Upsilon}_{h}(t^{\prime}),\tilde{\eta}(t^{\prime}))dt^{\prime}\right|$
$\displaystyle\leq
C|c|^{2p}\int_{t}^{0}e^{-2t^{\prime}}(1+e^{-\frac{2}{p}(t^{\prime}-t)})(|c|+|\tilde{\eta}(t^{\prime})|)dt^{\prime}$
$\displaystyle\leq C|c|^{2p}b^{2p(1-a)}(|c|+\|\tilde{\eta}\|_{\infty}),$

where we have used the $C^{1}$ property of
$\hat{N}(c\tilde{\Upsilon}_{h},\tilde{\eta})$ satisfying (4.9). These two
estimates yield

$\|\mathcal{F}\tilde{\eta}\|_{\infty}\leq
C|c|^{2p}b^{2p(1-a)}\left(|c|+\|\tilde{\eta}\|_{\infty}\right),$

for sufficiently large values of $b$. The divergent behaviour of $b^{2p(1-a)}$
for large $b$ is controlled by appropriately reducing the value of $|c|$
satisfying $|c|<c_{0}b^{-(1-a)}$ for sufficiently small $c_{0}>0$. Thus, we
see that the operator $\mathcal{F}$ maps the closed ball of the radius
$2C|c|^{2p+1}b^{2p(1-a)}$ in $L^{\infty}(-(1-a)p\log b,0)$ into itself.
Moreover, since $\hat{N}$ is a Lipschitz function satisfying (4.10), we get
that if $\tilde{\eta}_{1},\tilde{\eta}_{2}$ belong to the same ball, then

$\|\mathcal{F}\eta_{1}-\mathcal{F}\eta_{2}\|_{\infty}\leq
C|c|^{2p}b^{2p(1-a)}\|\tilde{\eta}_{1}-\tilde{\eta}_{2}\|_{\infty},$

so that the operator $\mathcal{F}$ is a contraction as long as
$|c|<c_{0}b^{-(1-a)}$ for sufficiently small $c_{0}>0$. By the Banach fixed-
point theorem, there exists a unique $\tilde{\eta}\in L^{\infty}(-(1-a)p\log
b,0)$ satisfying

$\sup_{t\in[-(1-a)p\log b,0]}|\tilde{\eta}(t)|\leq C|c|^{2p+1}b^{2p(1-a)},$

which yields the bound (4.11) due to the transformation
$\eta(t)=e^{-\frac{t}{p}}\tilde{\eta}(t)$. ∎

##  5\. Matching solutions: the proof of Theorem 1.1

We are now equipped with all the necessary estimates to prove Theorem 1.1. The
ground state $u=u_{b}$ of the stationary Gross–Pitaevskii equation (1.1) in
the Emden-Fowler variables (2.3) exhibits decaying behaviour both as
$t\to-\infty$ and as $t\to+\infty$ for every $b>0$ if $\lambda=\lambda(b)$. In
other words, it appears at the intersection of the two solution families with

$\Psi_{b}(t)=\Psi_{c(b)}(t),\qquad t\in\mathbb{R}$ (5.1)

for some $\lambda=\lambda(b)$ and $c=c(b)$. This allows us to use the
asymptotic behaviours (3.15) and (3.16) for $\Psi_{b}(t)$, and the asymptotic
behavior (4.11) for $\Psi_{c}(t)$ at the times $t=T-(1-a)p\log b$ with varying
$T>0$ and sufficiently large values of $b$. Equaling the asymptotic behaviors
due to (5.1) yields two implicit equations for parameters $\lambda$ and $c$.

Bound (1.6) follows from the bound (3.6) with $T=0$ after the transformation
(2.3). Bounds (1.7) and (1.8) follow from the bounds (4.6) and (4.11) in the
reversed order after the transformation (2.3). The proof of Theorem 1.1 is
completed after obtaining the asymptotic representation for $\lambda(b)$ and
$c(b)$ for large $b$.

We fix $p\in(0,1)$, $T>0$, and $a\in(0,\frac{p}{1+p})$. For sufficiently large
$b\geq b_{T,a}$, we consider $(\lambda,c)$ in the rectangle
$[0,2]\times[0,c_{0}b^{-(1-a)}]$ for which both Lemmas 3.4 and 4.2 can be
applied.

By Lemma 3.4, we have for $t\in[ap\log b,T+ap\log b]$,

$\Psi_{b}(t-p\log
b)=\Theta_{h}(t)+\gamma_{h}(t)+\mathcal{O}(F(\lambda,b)b^{2ap}e^{\frac{t}{p}}),$
(5.2)

where

$F(\lambda,b):=\left\\{\begin{array}[]{ll}(|\lambda|b^{-2p}+b^{-4p})^{2p+1},&\quad
p\in(0,\frac{1}{2}),\\\ (|\lambda|b^{-2p}+b^{-4p(1-a)})^{2},&\quad
p\in[\frac{1}{2},1),\end{array}\right.$

and the asymptotic expansion can be differentiated in $t$. Using (3.10), we
can write (5.2) as

$\displaystyle\Psi_{b}(t-p\log b)=$
$\displaystyle\Theta_{h}(t)+\Sigma(t)\int_{-\infty}^{t}f_{b}(t^{\prime})\Theta_{h}^{\prime}(t^{\prime})\Theta(t^{\prime})dt^{\prime}$
$\displaystyle-\Theta_{h}^{\prime}(t)\int_{-\infty}^{t}f_{b}(t^{\prime})\Sigma(t^{\prime})\Theta_{h}(t^{\prime})dt^{\prime}+\mathcal{O}(F(\lambda,b)b^{2ap}e^{\frac{t}{p}}).$

Evaluating these expressions at $t=T+ap\log b$ and using the asymptotic
relations (2.9) and (3.4) for $\Theta_{h}(t)$ and $\Sigma(t)$ as
$t\to+\infty$, we obtain

$\displaystyle\Psi_{b}(T-(1-a)p\log b)=$
$\displaystyle\alpha_{p}^{-1/p}e^{-\frac{T}{p}}b^{-a}[1+\mathcal{O}(b^{-2ap})]$
$\displaystyle-\frac{1}{2}p^{2}\alpha_{p}^{1/p}e^{\frac{T}{p}}b^{a}[1+\mathcal{O}(b^{-2ap})]\int_{-\infty}^{T+ap\log
b}f_{b}(t)\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt$
$\displaystyle+\frac{1}{p}\alpha_{p}^{-1/p}e^{-\frac{T}{p}}b^{-a}[1+\mathcal{O}(b^{-2ap})]\int_{-\infty}^{T+ap\log
b}f_{b}(t)\Sigma(t)\Theta_{h}(t)dt$
$\displaystyle+\mathcal{O}(F(\lambda,b)b^{2ap+a}e^{\frac{T}{p}}),$

where $f_{b}(t)=-\lambda b^{-2p}e^{2t}+b^{-4p}e^{4t}$. Since

$\displaystyle\left|\int_{-\infty}^{T+ap\log
b}f_{b}(t)\Sigma(t)\Theta_{h}(t)dt\right|\leq C_{T,a}b^{-2p(1-a)}$

is obtained in the proof of Lemma 3.2, we finally obtain the asymptotic
formula:

$\displaystyle\Psi_{b}(T-(1-a)p\log
b)=\alpha_{p}^{-1/p}e^{-\frac{T}{p}}b^{-a}[1+\mathcal{O}(b^{-2ap},b^{-2p(1-a)})]$
$\displaystyle\qquad-\frac{1}{2}p^{2}\alpha_{p}^{1/p}e^{\frac{T}{p}}b^{a}\left[\int_{-\infty}^{T+ap\log
b}f_{b}(t)\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-2ap})]+\mathcal{O}(F(\lambda,b)b^{2ap})\right].$
(5.3)

By Lemma 4.2, we have for $t\in(-(1-a)p\log b,0]$,

$\Psi_{c}(t)=c\Upsilon_{h}(t)+\mathcal{O}(|c|^{2p+1}b^{2p(1-a)}e^{-\frac{t}{p}}),$
(5.4)

where $\Upsilon_{h}$ is given by (2.20) and the asymptotic expansion can be
differentiated in $t$. Since the expansion (5.4) is used for $t\to-\infty$, we
can use either (2.18) or (2.19) for asymptotic expansions of the Tricomi
function $\mathfrak{U}(e^{2t};\alpha,\beta)$ in (2.20), where
$\alpha=\frac{p+1}{2p}-\frac{\lambda}{4}>0$ due to (2.15) and
$\beta=1+\frac{1}{p}$. If $p\neq\frac{1}{n}$ with $n\in\mathbb{N}$, then the
asymptotic formula for the solution $\Psi_{c}$ evaluated at $t=T-(1-a)p\log b$
is obtained with the help of (2.18), (2.20), and (5.4) in the form:

$\displaystyle\Psi_{c}(T-(1-a)p\log b)=$ $\displaystyle
ce^{\frac{T}{p}}\frac{\Gamma\left(-\frac{1}{p}\right)}{\Gamma\left(\frac{p-1}{2p}-\frac{\lambda}{4}\right)}b^{-(1-a)}[1+\mathcal{O}(b^{-2p(1-a)})]$
$\displaystyle+ce^{-\frac{T}{p}}\frac{\Gamma\left(\frac{1}{p}\right)}{\Gamma\left(\frac{p+1}{2p}-\frac{\lambda}{4}\right)}b^{1-a}[1+\mathcal{O}(b^{-2p(1-a)},|c|^{2p}b^{2p(1-a)})].$
(5.5)

If $p=\frac{1}{n}$ with $n\in\mathbb{N}$, then the asymptotic formula for the
solution $\Psi_{c}$ evaluated at $t=T-(1-a)p\log b$ is obtained with the help
of (2.19), (2.20), and (5.4) in the form:

$\displaystyle\Psi_{c}(T-(1-a)p\log b)=$ $\displaystyle
ce^{nT}\frac{2(-1)^{n+1}}{n!\Gamma\left(\frac{1-n}{2}-\frac{\lambda}{4}\right)}b^{-(1-a)}[(T-(1-a)p\log
b)[1+\mathcal{O}(b^{-2p(1-a)})]+\mathcal{O}(1)]$
$\displaystyle+ce^{-nT}\frac{(n-1)!}{\Gamma(\alpha)}b^{1-a}[1+\mathcal{O}(b^{-2p(1-a)},|c|^{2p}b^{2p(1-a)})],$
(5.6)

where $\alpha=\frac{1+n}{2}-\frac{\lambda}{4}>0$.

When we use the connection equation (5.1), it sets up the system of two
equations for two unknowns $\lambda$ and $c$. These two equations can be
obtained by equaling $\Psi_{b}(t)$ and $\Psi_{c}(t)$ as well as their first
derivatives at the time $t=T-(1-a)p\log b$. Alternatively, since the
asymptotic expansions are differentiable in $t$ term by term, we can set up
the system by equaling coefficients in front of the exponential functions
$e^{\frac{T}{p}}$ and $e^{-\frac{T}{p}}$. Equaling the coefficients for the
$e^{-\frac{T}{p}}$ terms in (5.3) with either (5.5) or (5.6) yields the
following equation:

$\alpha_{p}^{-1/p}b^{-a}[1+\mathcal{O}(b^{-2ap},b^{-2p(1-a)})]=c\frac{\Gamma\left(\frac{1}{p}\right)}{\Gamma(\alpha)}b^{1-a}[1+\mathcal{O}(b^{-2p(1-a)},|c|^{2p}b^{2p(1-a)})].$
(5.7)

The nonlinear equation (5.7) is defined for
$(\lambda,c)\in[0,2]\times[0,c_{0}b^{-(1-a)}]$ and the remainder terms are
$C^{1}$ functions with respect to $(\lambda,c)$. Since the leading-order part
of the nonlinear equation (5.7) is linear in $c$ and suggests the solution
$c=\mathcal{O}(b^{-1})$, which clearly exists inside $|c|\leq
c_{0}b^{-(1-a)}$, we have by an application of the implicit function theorem
the existence of a $C^{1}$ function $c=c(\lambda,b)$ for $\lambda\in[0,2]$ and
sufficiently large $b\geq b_{T,a}$, which is given asymptotically as

$c(\lambda,b)=\alpha_{p}^{-1/p}\frac{\Gamma(\alpha)}{\Gamma\left(\frac{1}{p}\right)}b^{-1}[1+\mathcal{O}(b^{-2ap},b^{-2p(1-a)})].$
(5.8)

Equaling the coefficients for the $e^{\frac{T}{p}}$ terms in (5.3) with either
(5.5) or (5.6) and substituting the expression (5.8) for $c$ yields a
nonlinear equation for $\lambda$, which we can also solve with an application
of the implicit function theorem. However, details of computations depend on
the value of $p\in(0,1)$ and hence are reported separately for different
values of $p$.

Case $p\in(0,\frac{1}{2})$. If $p\neq\frac{1}{n}$ for every $n\in\mathbb{N}$,
we use (5.3) and (5.5) in (5.1), equal the coefficients for the
$e^{\frac{T}{p}}$ terms, and substitute the expression (5.8) for
$c=c(\lambda,b)$. This yields the nonlinear equation for $\lambda$:

$\displaystyle\frac{1}{2}p^{2}\alpha_{p}^{1/p}b^{a}\big{[}\lambda
b^{-2p}\int_{-\infty}^{T+ap\log
b}e^{2t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-2ap})]$
$\displaystyle-b^{-4p}\int_{-\infty}^{T+ap\log
b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-2ap})]+\mathcal{O}((|\lambda|b^{-2p}+b^{-4p})^{2p+1}b^{2ap})\big{]}$
$\displaystyle=\alpha_{p}^{-1/p}\frac{\Gamma\left(\frac{p+1}{2p}-\frac{\lambda}{4}\right)\Gamma\left(-\frac{1}{p}\right)}{\Gamma\left(\frac{p-1}{2p}-\frac{\lambda}{4}\right)\Gamma\left(\frac{1}{p}\right)}b^{-2+a}[1+\mathcal{O}(b^{-2p(1-a)})].$
(5.9)

If $p\in(0,\frac{1}{2})$, both integrals in the left-hand-side of (5.9)
converge due to the asymptotic expansion (2.9) so that they can be expanded as

$\displaystyle\int_{-\infty}^{T+ap\log
b}e^{2t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt$
$\displaystyle=-\int_{-\infty}^{+\infty}e^{2t}\Theta_{h}(t)^{2}dt+\mathcal{O}(b^{-2a(1-p)}),$
$\displaystyle\int_{-\infty}^{T+ap\log
b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt$
$\displaystyle=-2\int_{-\infty}^{+\infty}e^{4t}\Theta_{h}(t)^{2}dt+\mathcal{O}(b^{-2a(1-2p)}),$

which implies that the nonlinear equation (5.9) for $\lambda$ can be rewritten
in the equivalent form:

$\displaystyle\lambda\int_{-\infty}^{+\infty}e^{2t}\Theta_{h}(t)^{2}dt[1+\mathcal{O}(b^{-2ap},b^{-2a(1-p)})]$
$\displaystyle-2b^{-2p}\int_{-\infty}^{+\infty}e^{4t}\Theta_{h}(t)^{2}dt[1+\mathcal{O}(b^{-2ap},b^{-2a(1-2p)})]+\mathcal{O}((|\lambda|b^{-2p}+b^{-4p})^{2p+1}b^{2p(1+a)})$
$\displaystyle=-2p^{-2}\alpha_{p}^{-2/p}\frac{\Gamma\left(\frac{p+1}{2p}-\frac{\lambda}{4}\right)\Gamma\left(-\frac{1}{p}\right)}{\Gamma\left(\frac{p-1}{2p}-\frac{\lambda}{4}\right)\Gamma\left(\frac{1}{p}\right)}b^{-2(1-p)}[1+\mathcal{O}(b^{-2p(1-a)})].$
(5.10)

If $p\neq\frac{1}{n}$ for every $n\in\mathbb{N}$, then
$\frac{p-1}{2p},-\frac{1}{p}\neq-m$ for every $m\in\mathbb{N}$ so that the
arguments of the Gamma functions at $\lambda=0$ are away from their pole
singularities. Hence, all terms of the nonlinear equation (5.10) are $C^{1}$
functions of $\lambda$ at $\lambda=0$. For $p\in(0,\frac{1}{2})$,
$b^{-2(1-p)}\ll b^{-2p}$ for sufficiently large $b$. Since the leading-order
part of the nonlinear equation (5.10) is linear in $\lambda$ and suggests the
solution $\lambda=\mathcal{O}(b^{-2p})$, we have by an application of the
implicit function theorem the existence of a $C^{1}$ function
$\lambda=\lambda(b)$ for sufficiently large $b$ which is given asymptotically
by

$\lambda(b)=2\frac{\int_{-\infty}^{+\infty}e^{4t}\Theta_{h}(t)^{2}dt}{\int_{-\infty}^{+\infty}e^{2t}\Theta_{h}(t)^{2}dt}b^{-2p}+\mathcal{O}(b^{-2(1-p)},b^{-2p(1+a)},b^{-2(p+a(1-2p))},b^{-2p(4p+1-a)}).$
(5.11)

The integrals in (5.11) can be computed by using the explicit expression for
$\Theta_{h}$ given by (2.8) with $t_{0}=0$. Using the substitution
$s=(1+\alpha_{p}e^{2t})^{-1}$ we express the integrals in terms of the Beta
function

$B(z_{1},z_{2}):=\int_{0}^{1}s^{z_{1}-1}(1-s)^{z_{2}-1}ds=\frac{\Gamma(z_{1})\Gamma(z_{2})}{\Gamma(z_{1}+z_{2})},\quad
z_{1},z_{2}>0$

and obtain

$\displaystyle\int_{-\infty}^{+\infty}e^{2t}\Theta_{h}(t)^{2}dt$
$\displaystyle=\frac{1}{2\alpha_{p}^{1+1/p}}\int_{0}^{1}s^{\frac{1}{p}-2}(1-s)^{\frac{1}{p}}ds=\frac{\Gamma\left(\frac{1}{p}-1\right)\Gamma\left(\frac{1}{p}+1\right)}{2\alpha_{p}^{1+1/p}\Gamma\left(\frac{2}{p}\right)},$
$\displaystyle\int_{-\infty}^{+\infty}e^{4t}\Theta_{h}(t)^{2}dt$
$\displaystyle=\frac{1}{2\alpha_{p}^{2+1/p}}\int_{0}^{1}s^{\frac{1}{p}-3}(1-s)^{\frac{1}{p}+1}ds=\frac{\Gamma\left(\frac{1}{p}-2\right)\Gamma\left(\frac{1}{p}+2\right)}{2\alpha_{p}^{2+1/p}\Gamma\left(\frac{2}{p}\right)}.$

Substituting these expressions into (5.11) yields the final formula for
$p\in(0,\frac{1}{2})$ and $p\neq\frac{1}{n}$ for every $n\in\mathbb{N}$:

$\lambda(b)=\frac{2(1+p)}{\alpha_{p}(1-2p)}b^{-2p}+\mathcal{O}(b^{-2(1-p)},b^{-2p(1+a)},b^{-2(p+a(1-2p))},b^{-2p(4p+1-a)}),$
(5.12)

where we have used the property $\Gamma(z+1)=z\Gamma(z)$.

If $p=\frac{1}{n}$ for some $n\in\mathbb{N}\backslash\\{1,2\\}$, we use (5.3)
and (5.6) in (5.1), equal the coefficients for the $e^{\frac{T}{p}}$ terms,
and substitute the expression (5.8) for $c=c(\lambda,b)$. This yields the
nonlinear equation for $\lambda$:

$\displaystyle\frac{1}{2}p^{2}\alpha_{p}^{1/p}b^{a}\big{[}\lambda
b^{-2p}\int_{-\infty}^{T+ap\log
b}e^{2t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-2ap})]$
$\displaystyle-b^{-4p}\int_{-\infty}^{T+ap\log
b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-2ap})]+\mathcal{O}((|\lambda|b^{-2p}+b^{-4p})^{2p+1}b^{2ap})\big{]}$
$\displaystyle=\frac{2(-1)^{n+1}\Gamma\left(\frac{1+n}{2}-\frac{\lambda}{4}\right)}{\alpha_{p}^{n}n!(n-1)!\Gamma\left(\frac{1-n}{2}-\frac{\lambda}{4}\right)}b^{-2+a}[(T-(1-a)p\log
b)[1+\mathcal{O}(b^{-2p(1-a)})]+\mathcal{O}(1)].$ (5.13)

After dividing it by $b^{a-2p}$, this equation can be rewritten in the form
(5.10), where the right-hand side has the order of

$\frac{\log b\;b^{-2(1-p)}}{\Gamma(\frac{1-n}{2}-\frac{\lambda}{4})},$

which is much smaller than the leading-order term of the order of
$\mathcal{O}(b^{-2p})$ for $p\in(0,\frac{1}{2})$. For even $n$, the final
formula (5.11) for $\lambda(b)$ is modified as follows:

$\lambda(b)=\frac{2(1+p)}{\alpha_{p}(1-2p)}b^{-2p}+\mathcal{O}(\log
bb^{-2(1-p)},b^{-2p(1+a)},b^{-2(p+a(1-2p))},b^{-2p(4p+1-a)}).$

For odd $n$, we also have $\Gamma(\frac{1-n}{2})=\infty$. Since

$\Gamma(z)=\frac{(-1)^{n}}{n!(z+n)}+\mathcal{O}(1)\quad\mbox{\rm
as}\;\;z\to-n$

and $\lambda(b)=\mathcal{O}(b^{-2p})$, we have

$\frac{1}{\Gamma(\frac{1-n}{2}-\frac{\lambda}{4})}=\mathcal{O}(\lambda)=\mathcal{O}(b^{-2p}),$

which modifies the final formula (5.11) for $\lambda(b)$ according to

$\lambda(b)=\frac{2(1+p)}{\alpha_{p}(1-2p)}b^{-2p}+\mathcal{O}(\log
bb^{-2},b^{-2p(1+a)},b^{-2(p+a(1-2p))},b^{-2p(4p+1-a)}).$

In both formulas for $\lambda=\lambda(b)$, we have $p=\frac{1}{n}$ with either
even or odd $n\in\mathbb{N}\backslash\\{1,2\\}$. In all cases, $\lambda(b)>0$
for suffiently large values of $b$.

Case $p\in(\frac{1}{2},1)$. Since $p\neq\frac{1}{n}$ for every
$n\in\mathbb{N}$ if $p\in(\frac{1}{2},1)$, the nonlinear equation (5.9) can be
used. However, the integral $\int_{-\infty}^{T+ap\log
b}e^{4t}\Theta_{h}(t)^{2}dt$ diverges exponentially in the upper limit since
$\Theta_{h}(t)^{2}=\mathcal{O}(e^{-\frac{2t}{p}})$ as $t\to+\infty$.
Consequently, there is a positive constant $C_{T,a}$ such that for all $b\geq
b_{T,a}$, we have

$\left|\int_{-\infty}^{T+ap\log
b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt\right|\leq C_{T,a}b^{2a(2p-1)}.$

The nonlinear equation (5.9) can be rewritten in the equivalent form:

$\displaystyle\lambda\int_{-\infty}^{+\infty}e^{2t}\Theta_{h}(t)^{2}dt[1+\mathcal{O}(b^{-2ap},b^{-2a(1-p)})]$
$\displaystyle+b^{-2p}\int_{-\infty}^{T+ap\log
b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-2ap})]+\mathcal{O}((|\lambda|b^{-2p}+b^{-4p(1-a)})^{2}b^{2p(1+a)})$
$\displaystyle=-2p^{-2}\alpha_{p}^{-2/p}\frac{\Gamma\left(\frac{p+1}{2p}-\frac{\lambda}{4}\right)\Gamma\left(-\frac{1}{p}\right)}{\Gamma\left(\frac{p-1}{2p}-\frac{\lambda}{4}\right)\Gamma\left(\frac{1}{p}\right)}b^{-2(1-p)}[1+\mathcal{O}(b^{-2p(1-a)})],$
(5.14)

where the second term on the left-hand side is of the order of
$b^{-2p+2a(2p-1)}\to 0$ as $b\to\infty$ since $-2p+2a(2p-1)<0$ for
$a<\frac{p}{2p-1}$, which is satisfied automatically, since $\frac{p}{2p-1}>1$
for $p\in(\frac{1}{2},1)$. Moreover, since $b^{-2p+2a(2p-1)}\ll b^{-2(1-p)}$
for $p\in\left(\frac{1}{2},1\right)$ and $a\in(0,1)$, the right-hand side
dominates in the nonlinear equation (5.14). Solving the nonlinear equation
(5.14) by an application of the implicit function theorem, we have the
existence of a $C^{1}$ function $\lambda=\lambda(b)$ for sufficiently large
$b$ which is given asymptotically by

$\displaystyle\lambda(b)=$
$\displaystyle-\frac{4\alpha_{p}^{1-1/p}\Gamma\left(\frac{p+1}{2p}\right)\Gamma\left(-\frac{1}{p}\right)\Gamma\left(\frac{2}{p}\right)}{p^{2}\Gamma\left(\frac{p-1}{2p}\right)\Gamma\left(\frac{1}{p}\right)\Gamma\left(\frac{1}{p}-1\right)\Gamma\left(\frac{1}{p}+1\right)}b^{-2(1-p)}$
$\displaystyle\qquad+\mathcal{O}(b^{-2p+2a(2p-1)},b^{-2(1-p)-2ap},b^{-2(1+a)(1-p)},b^{-2(1-pa)},b^{-2p(3-5a)}).$
(5.15)

Since $-\frac{1}{p}\in(-2,-1)$ and $\frac{p-1}{2p}\in(-\frac{1}{2},0)$ if
$p\in(\frac{1}{2},1)$, we have $\Gamma\left(-\frac{1}{p}\right)>0$ and
$\Gamma\left(\frac{p-1}{2p}\right)<0$. Hence, $\lambda(b)>0$ for sufficiently
large values of $b$.

Case $p=\frac{1}{2}$. This case corresponds to $n=2$ in the nonlinear equation
(5.13), which we can rewrite in the equivalent form:

$\displaystyle\lambda\int_{-\infty}^{+\infty}e^{2t}\Theta_{h}(t)^{2}dt[1+\mathcal{O}(b^{-a})]$
$\displaystyle+b^{-1}\int_{-\infty}^{T+ap\log
b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-a})]+\mathcal{O}((|\lambda|b^{-1}+b^{-2}\log
b)^{2}b^{a})$
$\displaystyle=\frac{8\Gamma\left(\frac{3}{2}-\frac{\lambda}{4}\right)}{\alpha_{p}^{4}\Gamma\left(-\frac{1}{2}-\frac{\lambda}{4}\right)}b^{-1}[(T-(1-a)p\log
b)[1+\mathcal{O}(b^{-(1-a)})]+\mathcal{O}(1)],$ (5.16)

where $\alpha_{p=\frac{1}{2}}=\frac{1}{4!}$. The second integral
$\int_{-\infty}^{T+ap\log b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt$
diverges linearly in the upper limit since
$\Theta_{h}(t)^{2}=\mathcal{O}(e^{-4t})$ as $t\to+\infty$. The exact
computations with the help of the explicit formula (2.8) yield the following
asymptotic expression:

$\displaystyle\int_{-\infty}^{T+ap\log
b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt$
$\displaystyle=\frac{1}{2}e^{4t}\Theta_{h}(t)^{2}\biggr{|}^{t=T+ap\log
b}_{t\to-\infty}-2\int_{-\infty}^{T+ap\log b}e^{4t}\Theta_{h}(t)^{2}dt$
$\displaystyle=\frac{1}{2\alpha_{p}^{4}}-\frac{1}{\alpha_{p}^{4}}\left[2(T+ap\log
b)+\log(\alpha_{p})-\frac{11}{6}+\mathcal{O}(b^{-2ap})\right]$
$\displaystyle=-\frac{2}{\alpha_{p}^{4}}(T+ap\log
b)+\frac{7}{3\alpha_{p}^{4}}-\frac{\log(\alpha_{p})}{\alpha_{p}^{4}}+\mathcal{O}(b^{-2ap}).$

On the other hand, we use (2.17) and obtain

$\frac{8\Gamma\left(\frac{3}{2}\right)}{\Gamma\left(-\frac{1}{2}\right)}=-\frac{8}{\pi}\Gamma^{2}\left(\frac{3}{2}\right)=-2,$

so that the leading-order terms of the nonlinear equation (5.16) can be
collected together as

$\displaystyle\lambda\int_{-\infty}^{+\infty}e^{2t}\Theta_{h}(t)^{2}dt[1+\mathcal{O}(b^{-a})]+\mathcal{O}(b^{-1},\log
b\;b^{-1-a})+\mathcal{O}((|\lambda|b^{-1}+b^{-2}\log b)^{2}b^{a})$
$\displaystyle=\frac{1}{\alpha_{p}^{4}}\log
b\;b^{-1}[1+\mathcal{O}(\lambda)][1+\mathcal{O}(b^{-(1-a)})]+\mathcal{O}(b^{-1}).$

By using the implicit function theorem, we have the existence of a $C^{1}$
function $\lambda=\lambda(b)$ for sufficiently large $b$ which is given
asymptotically by

$\lambda(b)=144\log b\;b^{-1}+\mathcal{O}(b^{-1},\log b\;b^{-1-a},(\log
b)^{2}\;b^{-2}),$ (5.17)

where we have used $p=\frac{1}{2}$, $\alpha_{p=\frac{1}{2}}=\frac{1}{4!}$, and

$\int_{-\infty}^{+\infty}e^{2t}\Theta_{h}(t)^{2}dt=\frac{1}{6\alpha_{p}^{3}}.$

Hence, $\lambda(b)>0$ for sufficiently large values of $b$.

Theorem 1.1 is proven. For details in Remark 1.2, we give the following
computations.

Case $p=1$. This case corresponds to $n=1$ in the nonlinear equation (5.13),
which we can rewrite in the equivalent form:

$\displaystyle\lambda\int_{-\infty}^{T+a\log
b}e^{2t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-2a})]$
$\displaystyle-b^{-2}\int_{-\infty}^{T+a\log
b}e^{4t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt[1+\mathcal{O}(b^{-2a})]+\mathcal{O}((|\lambda|b^{-2}\log
b+b^{-4(1-a)})^{2}b^{2a})$
$\displaystyle=\frac{4\Gamma\left(1-\frac{\lambda}{4}\right)}{\alpha_{p}^{2}\Gamma\left(-\frac{\lambda}{4}\right)}[(T-(1-a)\log
b)[1+\mathcal{O}(b^{-2(1-a)})]+\mathcal{O}(1)],$

where $\alpha_{p=1}=\frac{1}{8}$. Since

$\Gamma(z)=\frac{1}{z}+\mathcal{O}(1)\quad\mbox{\rm as}\;\;z\to 0$

and

$\displaystyle\int_{-\infty}^{T+a\log
b}e^{2t}\Theta_{h}^{\prime}(t)\Theta_{h}(t)dt=-\frac{1}{\alpha_{p}^{2}}(T+a\log
b-1)-\frac{1}{2\alpha_{p}^{2}}\log(\alpha_{p})+\mathcal{O}(b^{-2a}),$

the leading-order terms contain only $\lambda\log b$, which are not balanced
by the terms of the order of $\mathcal{O}(1)$ to get the asymptotic balance
$\lambda=\mathcal{O}((\log b)^{-1})$ according to Remark 1.2. This failure of
the shooting method is due to only one exponential term that appears in (5.6)
for $n=1$ and $\lambda=0$. The way to handle the asymptotic balance is to
obtain the second exponential terms from the higher-order (nonlinear) terms of
the expansion for $\Psi_{c}(t)$ beyond the leading order. However, this adds
complexity to the shooting method beyond the scopes of this work.

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