Nontrivial boundary structure in a Neumann problem on balls with radii tending to infinity

Abstract

This note introduces a class of nonlinear Neumann problems on balls expanding with the radii tending toward infinity. Performing singular perturbation arguments, we establish the corresponding concentration phenomenon and refined asymptotic expansions with the precise first two-order terms. In doing so, we obtain the nontrivial boundary structure of solutions with effects coming from the nonlinear Neumann boundary condition and the boundary mean curvature varied with expanding domains.

1 Introduction

This work is motivated by some stationary reaction–diffusion models and electrochemistry models in a reactor of macroscopic length scale involving nonlinear adsorption process on the surface [2, 6, 11, 15, 18]. In such a situation, the region for a chemical substance to diffuse across is much larger compared with a reaction process [3, 5, 19].

Mathematically, one considers the related differential equations with nonlinear Neumann boundary conditions in expanding domains, where the nonlinear source describes the absorption process, and the boundary effect is associated with the adsorption process; see, e.g., [16]. Here, the expanding domain means that the diameter of a large domain keeps increasing toward infinity. Such expanding domains may formally approach the entire space, the half space or an unbounded exterior domain. However, due to the nonlinear boundary effect, the asymptotic behavior of solutions varied with the expanding domain is totally different from the entire solutions. Since the domain keeps getting large, let us imagine in mind firstly that as the domain boundary expands out with the same distance along the outward normal direction, the corresponding solutions asymptotically vary with the expanding domain, and its asymptotics remains to be strongly affected by nonlinear boundary conditions [1, 4]. Essentially, such a phenomenon can be investigated under appropriate scales related to the diameter of the domain. Accordingly, the problem is equivalently transformed into singularly perturbed equations in finite domains. For the large domain with diameter tending to infinity, an important issue arises about the optimal upper bounds and the asymptotic behavior of solutions with respect to the domain geometry.

To basically understand the influence of expanding domains on solutions, we focus on the domain $B_R$ a ball of large radius $R\gg 1$ centered at the origin in $\mathbb {R}^N$, $N\ge 2$. We shall investigate a class of semilinear elliptic equations which are more general than models in [16]. The model reads

$$\begin{aligned}&\nabla \cdot (\pmb {\alpha }(|x|)\nabla \mathtt {u}(x))=\pmb {\beta }(|x|)f(\mathtt {u}(x))\,\,\,\,\text{ in }\,\quad \,B_R, \end{aligned}$$

(1.1)

$$\begin{aligned}&\frac{\partial \mathtt {u}}{\partial {\vec {\nu }}}(x)=\pmb {\eta }(\mathtt {u}(x))\,\,\,\,\,\,\,\,\,\qquad \qquad \qquad \qquad \text{ on }\,\,\,\,\partial {B}_R, \end{aligned}$$

(1.2)

where $\nabla$ and $\nabla \cdot$ are the gradient and the divergence operators, respectively. |x| denotes the standard N-dimensional Euclidean norm, $\vec {\nu }=\vec {\nu }(x)$ is the unit outward normal vector to $\partial {B_R}$ at x, $\frac{\partial }{\partial {\vec {\nu }}}$ is the unit outward normal derivative, and functions f and $\pmb {\eta }$ admit the following assumptions:

(A1):

$f\in \text{ C }_{\mathrm {loc}}^{1,\tau }(\mathbb {R})$ with $\tau \in (0,1)$, $\displaystyle \inf _{\mathbb {R}}f'>0$ and $f(\theta _0)=0$ for some $\theta _0\in \mathbb {R}$.

(A2):

$\pmb {\eta }\in \text{ C }_{\mathrm {loc}}^{1,\tau }(\mathbb {R})$ is monotonically decreasing and strictly positive in $\mathbb {R}$.

Equation (1.1) has many practical applications in the fields of physics, chemistry and biology, where $\pmb {\alpha }$ characterizes the diffusion, $\pmb {\beta }$ is regarded as a spatially inhomogeneous reaction term for the absorption f, and $\pmb {\eta }$ admitting (A2) models a degradation process in $B_R$ which is compensated by adsorption through $\partial {B_R}$. For a simplified case $\pmb {\alpha }\equiv 1$ and $\pmb {\beta }\equiv 1$, we refer the reader to [16, (2a) and (2b)] for a typical model obeying assumptions (A1) and (A2). In this work, $\pmb {\alpha }$ and $\pmb {\beta }$ are treated in more general settings as follows:

(A3):

$\pmb {\alpha }\in \text{ C }_{\mathrm {loc}}^{2,\tau }([0,\infty ))$ and $\pmb {\beta }\in \text{ C }_{\mathrm {loc}}^{1,\tau }([0,\infty ))$ are bounded above and have positive infima, and

$$\begin{aligned} \pmb {\beta }(r)r^{N-1}~ \hbox {is}~ \hbox {increasing}~ \hbox {to}~ r>0. \end{aligned}$$

Moreover, for $\pmb {\alpha }_R(r):=\pmb {\alpha }(r)\chi _{[0,R]}(r)$ and $\pmb {\beta }_R(r):=\pmb {\beta }(r)\chi _{[0,R]}(r)$ restricted in the domain [0, R] with sufficiently large R, there exists $k^*\in (0,1)$ independent of R such that

$$\begin{aligned} \lim _{R\rightarrow \infty }\sup _{r\in [k^*R,R)}\Big ({R}\left( {\left| \pmb {\alpha }_R'(r)\right| }+{\left| \pmb {\beta }_R'(r)\right| }\right) +R^{2}{|\pmb {\alpha }_R''(r)|}\Big )\in (0,\infty ). \end{aligned}$$

(1.3)

As an example in (A3), we introduce a smooth function $\pmb {\alpha }_R=\pmb {\alpha }\chi _{[0,R]}$ satisfying property (1.3) with $\pmb {\alpha }(r)=k^*$ for $r\in [0,k^*R]$, $\pmb {\alpha }(r)\in [k^*,1]$ for $r\in [k^*R,kR]$, and $\pmb {\alpha }(r)=1$ for $r\in [kR,\infty )$, where $k^*\in (0,1)$ and $k>1$ are constants independent of R.

For (1.1), one naturally considers the boundary condition $\pmb {\alpha }(|x|)\frac{\partial \mathtt {u}}{\partial {\vec {\nu }}}(x)=\pmb {\eta }(\mathtt {u}(x))$. Here, we use (1.2) since $\pmb {\alpha }$ is a positive constant on $\partial {B}_R$. In the related issues, some previous works have been traced back to [4, 16]. Let us mention [4, 16], where the optimal bounds for solutions of (1.1)–(1.2) with $\pmb {\alpha }\equiv 1$ and $\pmb {\beta }\equiv 1$ have been investigated. However, at the best of our knowledge, only partial results for the structure of solutions have been obtained. One of main difficulties lies on unknown boundary behavior of $\mathtt {u}$ and $\frac{\partial \mathtt {u}}{\partial \vec {\nu }}$ which interact with each other in the nonlinear boundary condition (1.2).

Starting with an interior estimate, we prove that for any $R_0\in (0,R)$,

$$\begin{aligned} \max _{B_{R_0}}\left( \left| \mathtt {u}(x)-\theta _0\right| +\left( \frac{|x|}{R}\right) ^{N-1}\left| \nabla \mathtt {u}(x)\right| \right) \le \mathtt {L}_0e^{-{{\mathtt {M}}_0}(R-R_0)}, \end{aligned}$$

(1.4)

where $\mathtt {L}_0$ and $\mathtt {M}_0$ are positive constants independent of R and $R_0$ [cf. (2.6)]. As a consequence, $\mathtt {u}$ behaves as a flat core (converges to $\theta _0$ exponentially) in any compact subset K of $B_R$ as ${\mathrm {dist}}(\partial {K},\partial {B_R}) {\mathop {\longrightarrow }\limits ^{R\rightarrow \infty }}\infty$. Since $\theta _0$ does not satisfy the boundary condition (1.2), $\mathtt {u}$ is nontrivial near the boundary. To deal with the boundary asymptotics, one can observe that under the scale $x=R\widetilde{x}$, (1.1) becomes a singularly perturbed model in the domain $B_1:=\{\widetilde{x}\in \mathbb {R}^N:|\widetilde{x}|<1\}$ with a parameter $\frac{1}{R^2}\rightarrow 0$, and on the boundary $\partial {B_1}$, the outward normal derivative in (1.2) has a parameter $\frac{1}{R}\rightarrow 0$ [see, e.g., (2.17) and Equation (2.19)–(2.20)]. Hence, the singularity of $|\nabla \mathtt {u}|$ near $\partial {B_R}$ introduces additional difficulties when trying to implement the standard technique of matching asymptotic expansions that do work for singularly perturbed semilinear elliptic problems. In this work, we are devoted to refined boundary asymptotics of $\mathtt {u}$ as $R\gg 1$. We propose a new analysis technique based on arguments in [7, 10, 12,13,14, 17] and [9, Proposition 2]. For the fist situation, we assume that the perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$ with respect to $R\gg 1$ is sufficiently small in the sense

$$\begin{aligned} \lim _{R\rightarrow \infty }R\left( \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right) =0, \end{aligned}$$

(1.5)

where $\mu _0$ is a positive constant independent of R. Then, the boundary asymptotic expansions at each boundary point $x_{\mathrm {bd}}\in \partial {B}_R$ can be formally depicted as follows [see (2.7)–(2.9] for the rigorous versions):

$$\begin{aligned}&\mathtt {u}(x_{\mathrm {bd}})\,\varvec{{\mathop {\approx }\limits ^{R\gg 1}}}\quad \,\,\,{p_0}+\frac{\displaystyle \int _{\theta _0}^{p_0}\sqrt{\frac{F(t)-F(\theta _0)}{F(p_0)-F(\theta _0)}}\,\mathrm {d}t}{\displaystyle \mu _0\frac{f(p_0)}{\pmb {\eta }(p_0)}-{\pmb {\eta }'(p_0)}}\left( \frac{N-1}{R}+\frac{\pmb {\alpha }'(R)}{2\pmb {\alpha }(R)}+\frac{\pmb {\beta }'(R)}{2\pmb {\beta }(R)}\right) , \end{aligned}$$

(1.6)

$$\begin{aligned}&\frac{\partial \mathtt {u}}{\partial \vec {\nu }}(x_{\mathrm {bd}}) \,\varvec{{\mathop {\approx }\limits ^{R\gg 1}}}\,\pmb {\eta }(p_0)+\frac{\displaystyle \pmb {\eta }'(p_0)\int _{\theta _0}^{p_0}\sqrt{\frac{F(t)-F(\theta _0)}{F(p_0)-F(\theta _0)}}\,\mathrm {d}t}{\displaystyle \mu _0\frac{f(p_0)}{\pmb {\eta }(p_0)}-{\pmb {\eta }'(p_0)}}\left( \frac{N-1}{R}+\frac{\pmb {\alpha }'(R)}{2\pmb {\alpha }(R)}+\frac{\pmb {\beta }'(R)}{2\pmb {\beta }(R)}\right) , \end{aligned}$$

(1.7)

where $\mathtt {a}\varvec{{\mathop {\approx }\limits ^{R\gg 1}}}\mathtt {b}$ means $R(\mathtt {a}-\mathtt {b})\rightarrow 0$ as $R\rightarrow \infty$, and

$$\begin{aligned} F(t)=\int _0^tf(s)\,\mathrm {d}s \end{aligned}$$

(1.8)

is the primitive of f, and $p_0>\theta _0$ is uniquely determined by $\pmb {\eta }(p_0)=\sqrt{{2}{\mu _0}(F(p_0)-F(\theta _0))}$ [cf. (2.10)]. It is clear that even if R is large, $\mathtt {u}$ is strongly influenced by the nonlinear effect of (1.2) on the boundary. We stress that the asymptotics (1.6) and (1.7) are obtained under assumption (1.5), i.e., $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}\varvec{{\mathop {\approx }\limits ^{R\gg 1}}}\mu _0$. In light of (1.6) and (1.7), solutions asymptotically expand as the radius of the domain $B_R$ tends to infinity, and $\pmb {\alpha }$, $\pmb {\alpha }'$, $\pmb {\beta }$, $\pmb {\beta }'$, $\pmb {\eta }$, $\pmb {\eta }'$ and the curvature $\frac{1}{R}$ have significant influence on the structure of solutions. Note also that even if $|x_{\mathrm {bd}}|=R\rightarrow \infty$, both $\mathtt {u}(x_{\mathrm {bd}})$ and $\frac{\partial \mathtt {u}}{\partial \vec {\nu }}(x_{\mathrm {bd}})$ remain finite and positive. Hence, $\mathtt {u}$ forms a boundary layer with the concentration phenomenon near the boundary $\partial {B}_R$. The rigorous boundary asymptotic expansions of $\mathtt {u}$ and $\frac{\partial \mathtt {u}}{\partial \vec {\nu }}$ will be presented in Theorem 2.1. For an application of such asymptotics, we refer the reader to Corollary 2.2. To describe the related boundary concentration phenomena of the solution $\mathtt {u}$ via a theoretical perspective, we show that $R(\mathtt {u}(x)-\theta _0)$ and $R|\nabla \mathtt {u}(x)|^2$ weakly converge to Dirac measures concentrating at infinity as R tends toward infinity. Such phenomena will be described in Theorem 2.3.

Despite the crucial roles of $\mu _0$ and $p_0$ in asymptotics (1.6) and (1.7), assumption (1.5) implies that the perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}$ with respect to $\mu _0$ is actually rather small than the curvature of $\partial {B}_R$ as R is sufficiently large. To study further the influence of small perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$ on asymptotic expansions of $\mathtt {u}(x_{\mathrm {bd}})$ and $\frac{\partial \mathtt {u}}{\partial \vec {\nu }}(x_{\mathrm {bd}})$, we shall consider the situation $\displaystyle \liminf \nolimits _{R\rightarrow \infty }$ $R\Big |\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\Big |>0$ instead of (1.5). In the final Sect. 4, we will establish the corresponding boundary asymptotic expansions in Corollary 4.1 which are more complicated than (1.6) and (1.7). As an application of Corollary 4.1, we focus particularly on the case

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}=\mu _0\,\,\,{\mathrm {and}}\,\,\, \lim _{R\rightarrow \infty }R^{\tau _*}\left| \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right| \in (0,\infty )\,\,{\mathrm {for}}\,\,{\mathrm {some}}\,\,\tau _*>0. \end{aligned}$$

(1.9)

For doing so, the effects of boundary curvature $\frac{1}{R}$ and the perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$ on boundary asymptotics of $\mathtt {u}$ and $\frac{\partial \mathtt {u}}{\partial \vec {\nu }}$ will be classified via three situations $\tau _*\in (0,1)$, $\tau _*=1$ and $\tau _*\ge 1$. Such a result can be found in Remark 3.

2 Statement of the main results

The associated energy functional of (1.1)–(1.2) is defined by

$$\begin{aligned} \mathcal {E}[\mathtt {v}]=\int _{B_R}\frac{\pmb {\alpha }(|x|)}{2}|\nabla \mathtt {v}|^2+\pmb {\beta }(|x|)F(\mathtt {v})\,{\mathrm {d}}{x}\,-\pmb {\alpha }(R)\int _{\partial {B_R}}\int _{\theta _0}^{\mathtt {v}}\pmb {\eta }(t)\,{\mathrm {d}}{t}\,{\mathrm {d}}\sigma _{x},\,\,\mathtt {v}\in \mathrm {H}^1(B_R). \end{aligned}$$

Let us fix $R>0$. Since $\displaystyle \min _{\mathbb {R}}{F}={F}(\theta _0)$ [by (A1)], together with (A2)–(A3) we verify that $\mathcal {E}$ is bounded below over $\mathrm {H}^1(B_R)$. Thus, applying the standard direct method to $\mathcal {E}$, one immediately obtains the existence of weak solutions to (1.1)–(1.2). Thanks again to (A1)–(A3), for each fixed $R>0$ we can further follow the standard argument consisting of the maximum principle and the elliptic regularity theorem [cf. [8]] to show that (1.1)–(1.2) has a unique solution $\mathtt {u}\in \text{ C }^1(\overline{B_R})\cap \text{ C }^{\infty }(B_R)$ satisfying $\mathtt {u}(x)\ge \theta _0$, $\forall {x}\in \overline{B_R}$. In particular, the uniqueness implies that $\mathtt {u}(x)=\pmb {\mathrm {U}}(|x|)$ is radially symmetric in $B_R$, where $\pmb {\mathrm {U}}$ is the unique solution of

$$\begin{aligned} \left( r^{N-1}\pmb {\alpha }(r)\pmb {\mathrm {U}}'(r)\right) '&=r^{N-1}\pmb {\beta }(r)f\left( \pmb {\mathrm {U}}(r)\right) ,\,\,r\in (0,R), \end{aligned}$$

(2.1)

$$\begin{aligned} \pmb {\mathrm {U}}'(0)&=0,\,\,\pmb {\mathrm {U}}'(R)=\pmb {\eta }(\pmb {\mathrm {U}}(R)), \end{aligned}$$

(2.2)

and satisfies

$$\begin{aligned} \pmb {\mathrm {U}}(r)\ge \theta _0\,\,\text{ in }\,\,[0,R]. \end{aligned}$$

(2.3)

This along with (A1) yields $f(\pmb {\mathrm {U}}(r))\ge 0$ in [0, R]. Notice also that $\pmb {\alpha }(r)$ and $\pmb {\beta }(r)$ are positive in (0, R). Since $\pmb {\mathrm {U}}$ solves (2.1) and satisfies $\pmb {\mathrm {U}}'(0)=0$, we know that $r^{N-1}\pmb {\alpha }(r)\pmb {\mathrm {U}}'(r)$ is increasing to r and, consequently,

$$\begin{aligned} \pmb {\mathrm {U}}'(r)\ge 0\,\,\text{ in }\,\,[0,R]. \end{aligned}$$

(2.4)

Accordingly, $\mathtt {u}$ is monotonically increasing in the sense that $\mathtt {u}(x)\ge \mathtt {u}(y)$ if $|x|\ge |y|$. It should also be mentioned that $\mathtt {u}$ is stable since the second variation of $\mathcal {E}[\mathtt {u}]$ with respect to compactly supported smooth perturbations $\xi$ is nonnegative, i.e.,

$$\begin{aligned}&Q_{\mathtt {u}}[\xi ]:=\int _{B_R}{\pmb {\alpha }(|x|)}|\nabla \xi |^2+\pmb {\beta }(|x|)f'(\mathtt {u})\xi ^2\,{\mathrm {d}}{x}\\&\,\,\qquad \qquad -\pmb {\alpha }(R)\int _{\partial {B_R}}\pmb {\eta }'(\mathtt {u})\xi ^2\,{\mathrm {d}}\sigma _{x}\ge 0,\,\,\forall \,\xi \in \mathrm {C}_c^1({B_R}) \end{aligned}$$

[trivially due to (A1)–(A3)].

2.1 Boundary structure and concentration phenomena

The main goal of this work is to establish asymptotic behavior of solution $\pmb {\mathrm {U}}$ as R goes to infinity. Later on we will prove that both $\pmb {\mathrm {U}}$ and $\pmb {\mathrm {U}}'$ are uniformly bounded in [0, R] for all $R>0$. To establish the refined asymptotics, asymptotic expansions of $\pmb {\alpha }(R)$ and $\pmb {\beta }(R)$ with respect to $R\gg 1$ are required. In what follows, we continue along the relation (1.3) to further assume that as $R\rightarrow \infty$, $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}$ approaches a positive constant $\mu _0$ in the sense described in (1.5), i.e.,

$$\begin{aligned} \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}=\mu _0+\frac{o(1)}{R},\,\,\text{ as }\,\,R\gg 1, \end{aligned}$$

(2.5)

where o(1) denotes the quantity approaching zero as R goes to infinity. The first result is about an interior estimate of $\pmb {\mathrm {U}}$ and $\pmb {\mathrm {U}}'$ and refined, precise asymptotics for $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$. Particularly, the boundary asymptotic expansions involve the domain geometry and the behavior of $\pmb {\alpha }'(R)$ and $\pmb {\beta }'(R)$.

Theorem 2.1

(Interior and boundary asymptotics). Assume (A1)–(A3). For $N\ge 2$ and $R>0$, let $\pmb {\mathrm {U}}\in \mathrm {C}^1((0,R])\cap \mathrm {C}^{\infty }((0,R))$ be the unique solution of (2.1)–(2.2). Then, $\pmb {\mathrm {U}}$ is monotonically increasing in [0, R]. As $R\gg 1$, $\pmb {\mathrm {U}}$ is strictly convex near the boundary, and there exist positive constants ${\mathtt {L}_0}$ and ${\mathtt {M}_0}$ independent of R such that for $r\in [0,R]$,

$$\begin{aligned} |\pmb {\mathrm {U}}(r)-\theta _0|+\left( \frac{r}{R}\right) ^{N-1}|\pmb {\mathrm {U}}'(r)|\le {{\mathtt {L}_0}}e^{-{\mathtt {M}_0}(R-r)}. \end{aligned}$$

(2.6)

Moreover, if (2.5) is satisfied, then the boundary asymptotics of $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ involving the effects of $\pmb {\alpha }'(R)$, $\pmb {\beta }'(R)$ and the curvature $\frac{1}{R}$ are depicted as

$$\begin{aligned} \pmb {\mathrm {U}}(R)&=\,{p_0}+\varvec{\mathtt {C}_{0}}\pmb {\mathcal {H}}(R)+\frac{o(1)}{R}, \end{aligned}$$

(2.7)

$$\begin{aligned} \pmb {\mathrm {U}}'(R)&=\,\pmb {\eta }(p_0)+\pmb {\eta }'(p_0)\varvec{\mathtt {C}_{0}}\pmb {\mathcal {H}}(R)+\frac{o(1)}{R}, \end{aligned}$$

(2.8)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}{\varvec{\mathtt {C}}}_{\varvec{0}}\,=\,\displaystyle \left( \mu _0\frac{f(p_0)}{\pmb {\eta }(p_0)}-{\pmb {\eta }'(p_0)}\right) ^{-1}{\int _{\theta _0}^{p_0}\sqrt{\frac{F(t)-F(\theta _0)}{F(p_0)-F(\theta _0)}}\,\mathrm {d}t},\\ \\ &{}\pmb {\mathcal {H}}(R)\,=\,\displaystyle \frac{N-1}{R}+\frac{1}{2}\left( \frac{\pmb {\alpha }'(R)}{\pmb {\alpha }(R)}+\frac{\pmb {\beta }'(R)}{\pmb {\beta }(R)}\right) . \end{array}\right. } \end{aligned}$$

(2.9)

Here, $p_0>\theta _0$ is uniquely determined by the nonlinear algebraic equation

$$\begin{aligned} \pmb {\eta }(p_0)=\sqrt{{2}{\mu _0}(F(p_0)-F(\theta _0))}, \end{aligned}$$

(2.10)

and F is defined in (1.8).

Note that $\mathtt {C}_{0}$ is a positive coefficient independent of R [cf. (A1) and (A2)]. The uniqueness of equation (2.10) is trivially due to the fact that $\pmb {\eta }$ is a decreasing function and F is strictly increasing in $(\theta _0,\infty )$ [by (A1) and (A2)].

Equations (2.7) and (2.8) provide fruitful information for the effects of $\pmb {\alpha }$ and $\pmb {\beta }$ on boundary asymptotics of $\pmb {\mathrm {U}}$. It should be mentioned a case

$$\begin{aligned} \frac{N-1}{R}+\frac{1}{2}\left( \frac{\pmb {\alpha }'(R)}{\pmb {\alpha }(R)}+\frac{\pmb {\beta }'(R)}{\pmb {\beta }(R)}\right) =\frac{o(1)}{R}\,\,{\mathrm {as}}\,\,R\gg 1; \end{aligned}$$

for example, $\pmb {\alpha }(r)=\frac{N-1}{R}(R-r)+1$ and $\pmb {\beta }(r)=\mu _0\pmb {\alpha }(r)$ for $r\in [0,R]$. Then, we have

$$\begin{aligned} \pmb {\mathrm {U}}(R)=p_0+\frac{o(1)}{R}\,\,\text {and}\,\, \pmb {\mathrm {U}}'(R)=\pmb {\eta }(p_0)+\frac{o(1)}{R}, \end{aligned}$$

and conclude that the effect of the domain size on solution $\pmb {\mathrm {U}}$ is inconspicuous. Let us consider another special case where $\pmb {\alpha }(r)\pmb {\beta }(r)$ is a constant value as $r\ge {r}_0$ for some $r_0>0$. Then, as $R\gg 1$, (2.9) implies $\pmb {\mathcal {H}}(R)=\frac{N-1}{R}$. In this case, $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ are indeed varied with the boundary curvature, but the effect of $\pmb {\alpha }$ and $\pmb {\beta }$ on $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ is quite slight.

We shall also stress the importance of second-order terms of (2.7) and (2.8). Note that $\displaystyle \max _{[0,R]}\pmb {\mathrm {U}}=\pmb {\mathrm {U}}(R)\sim {p}$ and $\pmb {\mathrm {U}}'(R)\sim \pmb {\eta }(p_0)$ as $R\gg 1$. When $\pmb {\eta }'(p_0)<0$ [cf. (A2)], by the second-order terms of (2.7) and (2.8) one further gets

$$\begin{aligned}&\displaystyle \frac{N-1}{R}+\frac{1}{2}\left( \frac{\pmb {\alpha }'(R)}{\pmb {\alpha }(R)}+\frac{\pmb {\beta }'(R)}{\pmb {\beta }(R)}\right)>0~ \hbox {as}~ R\gg 1 \pmb {\pmb {\Longleftrightarrow }} \pmb {\mathrm {U}}(R)>p_0~ \hbox {and}\\&\pmb {\mathrm {U}}'(R)<\pmb {\eta }(p_0)~ \hbox {as}~ R\gg 1. \end{aligned}$$

In particular, if $\pmb {\alpha }(r)=\alpha _1$ and $\pmb {\beta }(r)=\beta _1$ are constants as r is close to R, then for sufficiently large R, $\pmb {\mathcal {H}}(R)=\frac{N-1}{R}$, and $\pmb {\mathrm {U}}(R)>p_0$ and $0<\pmb {\mathrm {U}}'(R)<\pmb {\eta }(p_0)$. Moreover, some monotone properties for boundary asymptotics of $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ with respect to $\pmb {\alpha }'(R)$, $\pmb {\beta }'(R)$ and the sufficiently large radius R of the domain $B_R$ are stated as follows:

Corollary 2.2

Under the same hypotheses as in Theorem 2.1, let $\pmb {\alpha }_i\in \text{ C }_{\mathrm {loc}}^{2,\tau }([0,\infty ))$ and $\pmb {\beta }_i\in \text{ C }_{\mathrm {loc}}^{1,\tau }([0,\infty ))$ satisfy (A3). Then, we have

• (I) Let $\pmb {\mathrm {U}}_{\pmb {\alpha }_i,\pmb {\beta }_i}$ be the unique solution of (2.1)–(2.2) with $(R,\pmb {\alpha },\pmb {\beta })=(R_i,\pmb {\alpha }_i,\pmb {\beta }_i)$, $i=1,2$, where $1<{R}_1<R_2$ and $\displaystyle \sup _{R_1\gg 1} \frac{R_2}{R_1}<\infty$. If $\frac{\pmb {\beta }_i(R_i)}{\pmb {\alpha }_i(R_i)}$ satisfies (2.5) and

  $$\begin{aligned} \left( \frac{\pmb {\alpha }_1'(R_1)}{\pmb {\alpha }_1(R_1)}+\frac{\pmb {\beta }_1'(R_1)}{\pmb {\beta }_1(R_1)}\right) -\left( \frac{\pmb {\alpha }_2'(R_2)}{\pmb {\alpha }_2(R_2)}+\frac{\pmb {\beta }_2'(R_2)}{\pmb {\beta }_2(R_2)}\right) =\frac{o(1)}{R_1}. \end{aligned}$$

  Then, as $R_1$ is sufficiently large, there hold

  $$\begin{aligned} \pmb {\mathrm {U}}_{\pmb {\alpha }_1,\pmb {\beta }_1}(R_1)>\pmb {\mathrm {U}}_{\pmb {\alpha }_2,\pmb {\beta }_2}(R_2)>\theta _0 \quad {\mathrm {and}}\quad 0<\pmb {\mathrm {U}}_{\pmb {\alpha }_1,\pmb {\beta }_1}'(R_1)\le \pmb {\mathrm {U}}_{\pmb {\alpha }_2,\pmb {\beta }_2}'(R_2). \end{aligned}$$

  Moreover, when $\pmb {\eta }'(p_0)<0$, we have $0<\pmb {\mathrm {U}}_{\pmb {\alpha }_1,\pmb {\beta }_1}'(R_1)<\pmb {\mathrm {U}}_{\pmb {\alpha }_2,\pmb {\beta }_2}'(R_2)$ as $1\ll {R}_1<R_2$.

• (II) Let $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_i,\pmb {\beta }_i}$ be the unique solution of (2.1)–(2.2) in (0, R) with $(\pmb {\alpha },\pmb {\beta })=(\pmb {\alpha }_i,\pmb {\beta }_i)$, $i=1,2$. Assume further that

  $$\begin{aligned} \frac{\pmb {\beta }_1(R)}{\pmb {\alpha }_1(R)}\,\,and\,\,\frac{\pmb {\beta }_2(R)}{\pmb {\alpha }_2(R)}\,\,are\,\,positive\,\,constants\,\,independent\,\,of\,\,R, \end{aligned}$$

  and one of the following assumptions holds:

  • (i) $\displaystyle \frac{\pmb {\beta }_1(R)}{\pmb {\alpha }_1(R)}<\frac{\pmb {\beta }_2(R)}{\pmb {\alpha }_2(R)}$;

  • (ii) $\displaystyle \frac{\pmb {\beta }_1(R)}{\pmb {\alpha }_1(R)}=\frac{\pmb {\beta }_2(R)}{\pmb {\alpha }_2(R)}$, $\displaystyle \frac{\pmb {\alpha }_1'(R)}{\pmb {\alpha }_1(R)}+\frac{\pmb {\beta }_1'(R)}{\pmb {\beta }_1(R)}>\frac{\pmb {\alpha }_2'(R)}{\pmb {\alpha }_2(R)}+\frac{\pmb {\beta }_2'(R)}{\pmb {\beta }_2(R)}$ and $\pmb {\eta }'(p_0)<0$,

  then $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}(R)>\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}(R)>\theta _0$ and $0<\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}'(R)<\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}'(R)$ as $R\gg 1$.

A discussion on Corollary 2.2(II) is stated as follows:

Remark 1

It seems that the standard comparison is difficult to imply Corollary 2.2(II). Let us consider another situation that $\pmb {\alpha }_i$ and $\pmb {\beta }_i$ satisfy

$$\begin{aligned} \frac{\pmb {\beta }_1(r)}{\pmb {\alpha }_1(r)}\le \frac{\pmb {\beta }_2(r)}{\pmb {\alpha }_2(r)}\,\,\,and\,\,\,\frac{\pmb {\alpha }_1'(r)}{\pmb {\alpha }_1(r)}\ge \frac{\pmb {\alpha }_2'(r)}{\pmb {\alpha }_2(r)},\,\,\forall \,r\in [0,R]. \end{aligned}$$

(2.11)

Then, applying the standard PDE comparison to (2.1)–(2.2) and using (2.3)–(2.4), one obtains $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}\ge \widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}\ge \theta _0$ in [0, R]. In particular, if $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}\not =\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}$ at an interior point, then $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}(R)>\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}(R)>\theta _0$. This is the same as the corresponding result in Corollary 2.2(II), but the conditions (i) and (ii) are far weaker than condition (2.11).

Let us return to Theorem 2.1 which establishes refined asymptotics of $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ under a strong assumption (2.5). It should be stressed that if $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}\rightarrow \mu _0$ but it does not satisfy (2.5), then the effect of the perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$ cannot be ignored. We will establish asymptotics of $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ involving the effect of the perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$ in Sect. 4; see (4.3)–(4.4).

To see the concentration phenomenon of $\pmb {\mathrm {U}}$ near the boundary $r=R$ as $R\rightarrow \infty$, let us introduce a Dirac measure $\delta ^{\infty }$ defined in the interval of nonnegative extended real numbers, which satisfies $\delta ^{\infty }(r)=0$ for $r\in (0,\infty )$ and $\int _0^{\infty }\delta ^{\infty }(r)\,\mathrm {d}r=1$. We focus on the behavior of $\pmb {\mathrm {U}}$ in the region $(k^*R,R)$ and define

$$\begin{aligned} \delta _{R(\pmb {\mathrm {U}}-\theta _0)}(r)= {\left\{ \begin{array}{ll} R(\pmb {\mathrm {U}}(r)-\theta _0),\,\,&{}\text{ for }\,\,r\in (k^*R,R),\\ 0,\,\,&{}\text{ for }\,\,r\in [0,k^*R]\cup [R,\infty ), \end{array}\right. } \end{aligned}$$

(2.12)

and

$$\begin{aligned} \delta _{R\pmb {\mathrm {U}}'^2}(r)= {\left\{ \begin{array}{ll} R\pmb {\mathrm {U}}'^2(r),\,\,&{}\text{ for }\,\,r\in (k^*R,R),\\ 0,\,\,&{}\text{ for }\,\,r\in [0,k^*R]\cup [R,\infty ), \end{array}\right. } \end{aligned}$$

(2.13)

where $k^*$ is defined in (A3). The following theorem confirms that $\delta _{R(\pmb {\mathrm {U}}-\theta _0)}$ and $\delta _{R\pmb {\mathrm {U}}'^2}$ behave as Dirac measures at infinity in the following weak sense:

$$\begin{aligned} \delta _{R(\pmb {\mathrm {U}}-\theta _0)}{\mathop {\rightharpoonup }\limits ^{R\rightarrow \infty }}&\left( \frac{1}{\sqrt{\mu _0}}\int _{\theta _0}^{p_0}\frac{t-\theta _0}{\sqrt{2(F(t)-F(\theta _0))}}\,\mathrm {d}t\right) \delta ^{\infty },\\ \delta _{R\pmb {\mathrm {U}}'^2}{\mathop {\rightharpoonup }\limits ^{R\rightarrow \infty }}&\left( \sqrt{\mu _0}\int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t\right) \delta ^{\infty }. \end{aligned}$$

Theorem 2.3

(Boundary concentrations). Under the same hypotheses as in Theorem 2.1, as $R\rightarrow \infty$, for any $r\in [0,\infty )$, there hold

$$\begin{aligned} \delta _{R(\pmb {\mathrm {U}}-\theta _0)}(r)\rightarrow 0\quad {and}\quad \delta _{R\pmb {\mathrm {U}}'^2}(r)\rightarrow 0\quad {as}\,\,{R\rightarrow \infty }, \end{aligned}$$

(2.14)

and

$$\begin{aligned} \lim _{R\rightarrow \infty }\int _0^{\infty }\delta _{R(\pmb {\mathrm {U}}-\theta _0)}(r)\,\mathrm {d}r&=\,\frac{1}{\sqrt{\mu _0}}\int _{\theta _0}^{p_0}\frac{t-\theta _0}{\sqrt{2(F(t)-F(\theta _0))}}\,\mathrm {d}t, \end{aligned}$$

(2.15)

$$\begin{aligned} \lim _{R\rightarrow \infty }\int _0^{\infty }\delta _{R\pmb {\mathrm {U}}'^2}(r)\,\mathrm {d}r&=\,\sqrt{\mu _0}\int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t. \end{aligned}$$

(2.16)

Remark 2

We shall stress that (2.15) is well defined. Indeed, by (A1) it is easy to obtain

$$\begin{aligned} \frac{1}{p_0-\theta _0}\int _{\theta _0}^{p_0}\frac{t-\theta _0}{\sqrt{2(F(t)-F(\theta _0))}}\,\mathrm {d}t \in \left[ \left( \displaystyle \max _{[\theta _0,p_0]}f'\right) ^{-1/2},\left( {\displaystyle \min _{[\theta _0,p_0]}f'}\right) ^{-1/2}\right] . \end{aligned}$$

2.2 A significant idea

To study the asymptotic behavior of $\pmb {\mathrm {U}}$ as $R\rightarrow \infty$, we consider a change of variables

$$\begin{aligned} \epsilon =\frac{1}{R}\rightarrow 0+,\,\,s=\epsilon {r}\in (0,1],\,\,{u_{\epsilon }}(s)=\pmb {\mathrm {U}}(r),\,\,{\alpha _{\epsilon }}(s)=\pmb {\alpha }(r),\,\,{\beta _{\epsilon }}(s)=\pmb {\beta }(r). \end{aligned}$$

(2.17)

In what follows, we use the symbol

$$\begin{aligned} \varvec{\mathrm {D}}:=\frac{\text{ d }}{\text{ d }s} \end{aligned}$$

for the derivative with respect to the variable s rather than $'$ to avoid the notation confusion with the prime notation $'$ for the derivative with respect to the variable r. Then, we have

$$\begin{aligned} (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)=\epsilon ^{-1}{\pmb {\mathrm {U}}'(r)}=R\pmb {\mathrm {U}}'(r),\,\,(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)=R\pmb {\alpha }'(r),\,\,(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(s)=R\pmb {\beta }'(r), \end{aligned}$$

(2.18)

and (2.1)–(2.2) is equivalent to the following singularly perturbed equation with small parameter $\epsilon$:

$$\begin{aligned}&\epsilon ^2\left( (\varvec{\mathrm {D}}^2{{u_{\epsilon }}})(s)+\left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right) \nonumber \\&\qquad =\,\frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}f({u_{\epsilon }}(s)),\,\,s\in (0,1), \end{aligned}$$

(2.19)

$$\begin{aligned}&(\varvec{\mathrm {D}}{{u_{\epsilon }}})(0)=0,\,\,\epsilon (\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)=\,\pmb {\eta }({u_{\epsilon }}(1)). \end{aligned}$$

(2.20)

Hence, Equation (2.1) in the domain (0, R) with $R\rightarrow \infty$ becomes a singularly perturbed equation (2.19) with $\epsilon \downarrow 0$ in a finite domain (0, 1). To deal with asymptotics of ${u_{\epsilon }}$, one can multiply (2.19) by $\varvec{\mathrm {D}}{{u_{\epsilon }}}$ and make simple calculations to obtain a first-order ODE

$$\begin{aligned}&\frac{\epsilon ^2}{2}\left( (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2-\frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}F({u_{\epsilon }}(s))\right) \nonumber \\&\quad =-\int _{k^*}^s\left[ \epsilon ^2\left( \frac{N-1}{t}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(t)}{{\alpha _{\epsilon }}(t)}\right) ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(t))^2+F({u_{\epsilon }}(t))\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(t)}{{\alpha _{\epsilon }}(t)}\right) \right] \text{ d }t \nonumber \\&\qquad +C_{k^*,\epsilon },\,\,s\in [k^*,1], \end{aligned}$$

(2.21)

with

$$\begin{aligned} C_{k^*,\epsilon }=\frac{\epsilon ^2}{2}((\varvec{\mathrm {D}}{{u_{\epsilon }}})(k^*))^2-\frac{{\beta _{\epsilon }}(k^*)}{{\alpha _{\epsilon }}(k^*)}F({u_{\epsilon }}(k^*)), \end{aligned}$$

(2.22)

where $\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}}{{\alpha _{\epsilon }}}\right) :=\frac{\text{ d }}{\text{ d }t}\left( \frac{{\beta _{\epsilon }}}{{\alpha _{\epsilon }}}\right)$ and F is defined in (1.8). In particular, (2.21) together with the boundary condition (2.20) implies

$$\begin{aligned}&-\frac{1}{2}\big (\pmb {\eta }({u_{\epsilon }}(1))\big )^2+\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}F({u_{\epsilon }}(1))\nonumber \\&\quad =\int _{k^*}^1\left[ \epsilon ^2\left( \frac{N-1}{t}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(t)}{{\alpha _{\epsilon }}(t)}\right) ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(t))^2+F({u_{\epsilon }}(t))\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(t)}{{\alpha _{\epsilon }}(t)}\right) \right] \text{ d }t-C_{k^*,\epsilon }. \end{aligned}$$

(2.23)

We will show that the right-hand side of (2.23) tends to zero as $\epsilon \downarrow 0$. Its precise leading term plays a key role in the asymptotics of ${u_{\epsilon }}(1)$.

The remainder of the paper proceeds as follows: In the next section, we will establish the interior and gradient estimate of ${u_{\epsilon }}$ in Lemmas 3.1 and 3.2, which give the precise leading order term of the expression in the right-hand side of (2.23). In particular, by (2.5), (2.17) and (2.23), we obtain

$$\begin{aligned} (\pmb {\eta }({u_{\epsilon }}(1)))^2=2\mu _0\left( F({u_{\epsilon }}(1))-F(\theta _0)\right) +o_{\epsilon }(1)\,\,{\mathrm {as}}\,\,\epsilon \downarrow 0. \end{aligned}$$

(2.24)

As will be mentioned later on, the interior estimate (3.1) and the gradient estimate (3.2) show that if $\displaystyle \lim _{\epsilon \downarrow 0}\frac{1-s_{\epsilon }}{\epsilon }=\infty$, there still hold ${u_{\epsilon }}(s_{\epsilon })\rightarrow \theta _0$ and $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s_{\epsilon })\rightarrow 0$ exponentially as $\epsilon$ goes to zero. Furthermore, in Theorem 3.3, we combine (2.23) with (3.26)–(3.27) to establish the precise leading order terms of (2.24) as follows [see (3.37) also]:

$$\begin{aligned} \frac{1}{\epsilon }&\left( -\frac{1}{2}(\pmb {\eta }({u_{\epsilon }}(1)))^2+\mu _0\left( F({u_{\epsilon }}(1))-F(\theta _0)\right) \right) \\&\quad =\sqrt{\mu _0}\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{2{\alpha _{\epsilon }}(1)}+\frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{2{\beta _{\epsilon }}(1)}\right) \int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t+o_{\epsilon }(1), \end{aligned}$$

which will determine the precise first two-order terms of ${u_{\epsilon }}(1)$ and $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)$ with respect to small $\epsilon >0$. We shall highlight here that Theorem 3.3 plays a key role in the proof of the main theorems. The proof of Theorems 2.1 and 2.3 and Corollary 2.2 will be stated in Sect. 3.3. To see the effect of the perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}$ around $\mu _0$ on solution asymptotics, in the final Sect. 4 we replace the strong assumption (2.5) with $\displaystyle \liminf \nolimits _{R\rightarrow \infty }$ $R(\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0)>0$ which includes the situation (1.9). Then, we establish in Corollary 4.1 the precise effect of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$ on asymptotics of $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$.

3 Proof of the main results

In this section, we first investigate asymptotics for solutions ${u_{\epsilon }}$ of Eq. (2.19)–(2.20) and establish the corresponding boundary gradient asymptotic expansions as $\epsilon$ tends to zero. Such asymptotics play a crucial role in the asymptotic expansions of $\pmb {\mathrm {U}}$ and $\pmb {\mathrm {U}}'$ as R approaches infinity. In Sect. 3.3, we shall complete the proof of Theorems 2.1 and 2.3 and Corollary 2.2.

3.1 Interior estimates

To go further, let us state some properties which can be obtained directly from (A1)–(A3), (2.3), (2.4) and (2.17)–(2.20).

(P1):

As $\epsilon >0$ is sufficiently small, we have

$$\begin{aligned} \frac{{\alpha _{\epsilon }}(s)}{{\beta _{\epsilon }}(s)}\ge \frac{1}{2}\lim _{R\rightarrow \infty }\inf _{[0,R]}\frac{\pmb {\alpha }(r)}{\pmb {\beta }(r)}\quad {\mathrm {and}}\quad \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\ge \frac{1}{2}\lim _{R\rightarrow \infty }\inf _{[0,R]}\frac{\pmb {\beta }(r)}{\pmb {\alpha }(r)},\,\,\forall \,s\in [0,1]. \end{aligned}$$

Henceforth, we set $\displaystyle {C}_1:=\frac{1}{2}\min \left\{ \lim _{R\rightarrow \infty }\inf _{[0,R]}\frac{\pmb {\alpha }(r)}{\pmb {\beta }(r)},\lim _{R\rightarrow \infty }\inf _{[0,R]}\frac{\pmb {\beta }(r)}{\pmb {\alpha }(r)}\right\} >0$. This along with (A3) gives

$$\begin{aligned} \min _{s\in [0,1]}\frac{{\alpha _{\epsilon }}(s)}{{\beta _{\epsilon }}(s)}\ge {C}_1,\,\,\min _{s\in [0,1]}\frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\ge {C}_1\,\,\text{ as }\,\,0<\epsilon \ll 1. \end{aligned}$$

(P2):

As $\epsilon >0$ is sufficiently small,

$$\begin{aligned} \sup _{s\in [k^*,1]}\left( \frac{\left| (\varvec{\mathrm {D}}{\alpha _{\epsilon }})(s)\right| }{{\alpha _{\epsilon }}(s)}+\frac{\left| (\varvec{\mathrm {D}}{\beta _{\epsilon }})(s)\right| }{{\beta _{\epsilon }}(s)}+\frac{\left| (\varvec{\mathrm {D}}^2{\alpha _{\epsilon }})(s)\right| }{\alpha _{\epsilon }^2(s)}\right) \le {C}_2, \end{aligned}$$

where $k^*\in (0,1)$ is defined in (A3) and $C_2$ is a positive constant independent of $\epsilon$.

(P3):

${u_{\epsilon }}-\theta _0$ and $\varvec{\mathrm {D}}{u_{\epsilon }}$ are nonnegative in [0, 1]. Moreover, by (A1) we have

$$\begin{aligned} f'({u_{\epsilon }}(s))\ge {C}_3\,\,{\mathrm {and}}\,\,f({u_{\epsilon }}(s))({u_{\epsilon }}(s)-\theta _0)\ge {C}_3({u_{\epsilon }}(s)-\theta _0)^2,\quad \forall \,s\in [0,1], \end{aligned}$$

where $C_3$ is a positive constant independent of $\epsilon$.

(P4):

By (1.2) and (A2), we have

$$\begin{aligned} {\epsilon }^{-1}{\pmb {\eta }\left( \max _{[0,1]}{u_{\epsilon }}\right) }\le (\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)\le {\epsilon }^{-1}{\pmb {\eta }(\theta _0)}. \end{aligned}$$

(P5):

By (2.19) and ${u_{\epsilon }}\ge \theta _0$, we have

$$\begin{aligned} \varvec{\mathrm {D}}\left( s^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right) =s^{N-1}{\beta _{\epsilon }}(s)f({u_{\epsilon }}(s))\ge 0,\,\,\forall \,s\in (0,1). \end{aligned}$$

Hence, $s^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)$ is increasing to $s\in [0,1]$.

Moreover, we have the following estimates of ${u_{\epsilon }}$ and $\varvec{\mathrm {D}}{{u_{\epsilon }}}$ with respect to sufficiently small $\epsilon >0$.

Lemma 3.1

Assume that (A1)–(A3) hold. For $\epsilon >0$ and ${\alpha _{\epsilon }}$ and ${\beta _{\epsilon }}$ satisfying (2.17), let ${u_{\epsilon }}\in \mathrm {C}^1((0,1])\cap \mathrm {C}^{\infty }((0,1))$ be the unique solution of (2.19)–(2.20). Then, there exist positive constants $\epsilon ^*$ and $M^*$ independent of $\epsilon$ such that as $0<\epsilon <\epsilon ^*$,

$$\begin{aligned} 0\le {u_{\epsilon }}(s)-\theta _0\le 2({u_{\epsilon }}(1)-\theta _0)e^{-\frac{M^*}{\epsilon }(1-s)}, \end{aligned}$$

(3.1)

and

$$\begin{aligned} 0\le {s}^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\le \frac{2}{\epsilon }{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)e^{-\frac{M^*}{\epsilon }(1-s)}, \end{aligned}$$

(3.2)

for $s\in [0,1]$.

Proof

We first deal with the estimate of ${u_{\epsilon }}(s)-\theta _0$. Multiplying (2.19) by ${u_{\epsilon }}(s)-\theta _0$ and using (P1) and (P3), we obtain

$$\begin{aligned} \epsilon ^2\left( (\varvec{\mathrm {D}}^2{{u_{\epsilon }}})(s)+\left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right) ({u_{\epsilon }}(s)-\theta _0) \ge \,C_1C_3({u_{\epsilon }}(s)-\theta _0)^2. \end{aligned}$$

(3.3)

One can further check that, for $s\in [k^*,1]$,

$$\begin{aligned}&\left( (\varvec{\mathrm {D}}^2{{u_{\epsilon }}})(s)+\left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right) ({u_{\epsilon }}(s)-\theta _0)\nonumber \\&\quad =\frac{1}{2}\varvec{\mathrm {D}}^2(({u_{\epsilon }}(s)-\theta _0)^2)-\left( \varvec{\mathrm {D}}({u_{\epsilon }}(s)-\theta _0)\right) ^2\nonumber \\&\qquad +\left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) \left( \varvec{\mathrm {D}}({u_{\epsilon }}(s)-\theta _0)\right) ({u_{\epsilon }}(s)-\theta _0)\nonumber \\&\quad \le \frac{1}{2}\varvec{\mathrm {D}}^2(({u_{\epsilon }}(s)-\theta _0)^2)+\frac{1}{4}\left( \frac{N-1}{k^*}+C_2\right) ^2({u_{\epsilon }}(s)-\theta _0)^2. \end{aligned}$$

(3.4)

Here, we have used (P2), (P3) and ${u_{\epsilon }}(s)\ge \theta _0$ to deal with the last inequality of (3.4). Combining (3.3) with (3.4), one finds

$$\begin{aligned} \epsilon ^2\varvec{\mathrm {D}}^2(({u_{\epsilon }}(s)-\theta _0)^2)&\ge \left[ 2C_1C_3-\frac{\epsilon ^2}{2}\left( \frac{N-1}{k^*}+C_2\right) ^2\right] ({u_{\epsilon }}(s)-\theta _0)^2\nonumber \\&\ge \,C_1C_3({u_{\epsilon }}(s)-\theta _0)^2,\,\,s\in [k^*,1], \end{aligned}$$

(3.5)

as

$$\begin{aligned} 0<\epsilon \le {\sqrt{2C_1C_3}}\left( \frac{N-1}{k^*}+C_2\right) ^{-1}. \end{aligned}$$

Consequently, applying the standard PDE comparison theorem to (3.5), we may arrive at the estimate

$$\begin{aligned} 0\le {u_{\epsilon }}(s)-\theta _0\le ({u_{\epsilon }}(1)-\theta _0)\left( e^{-\frac{\sqrt{C_1C_3}}{2\epsilon }(s-k^*)}+e^{-\frac{\sqrt{C_1C_3}}{2\epsilon }(1-s)}\right) ,\,\,\forall \,s\in [k^*,1]. \end{aligned}$$

(3.6)

Now we shall refine the estimate (3.6). Firstly, we assume $s\in [\frac{k^*+1}{2},1]$, i.e., $s-k^*\ge 1-s$. Then, (3.6) implies

$$\begin{aligned} 0\le {u_{\epsilon }}(s)-\theta _0\le 2({u_{\epsilon }}(1)-\theta _0)e^{-\frac{\sqrt{C_1C_3}}{2\epsilon }(1-s)}. \end{aligned}$$

(3.7)

On the other hand, for $s\in [0,\frac{k^*+1}{2}]$, by the property $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\ge 0$ and (3.6) we have

$$\begin{aligned} 0\le {u_{\epsilon }}(s)-\theta _0\le&{u_{\epsilon }}\left( \frac{k^*+1}{2}\right) -\theta _0\le \,2({u_{\epsilon }}(1)-\theta _0)e^{-\frac{\sqrt{C_1C_3}}{4\epsilon }(1-k^*)}\nonumber \\ \le&\,2({u_{\epsilon }}(1)-\theta _0)e^{-\frac{(1-k^*)\sqrt{C_1C_3}}{4\epsilon }(1-s)}. \end{aligned}$$

(3.8)

It therefore follows from (3.7) and (3.8) that

$$\begin{aligned} 0\le {u_{\epsilon }}(s)-\theta _0\le 2({u_{\epsilon }}(1)-\theta _0)e^{-\frac{(1-k^*)\sqrt{C_1C_3}}{4\epsilon }(1-s)},\,\,\forall \,s\in [0,1]. \end{aligned}$$

(3.9)

Now we shall deal with the estimate of $\varvec{\mathrm {D}}{u_{\epsilon }}$. Multiplying (2.19) by $s^{N-1}{\alpha _{\epsilon }}(s)$ and taking the derivative of the expression with respect to the variable s, one arrives at

$$\begin{aligned} \epsilon ^2\varvec{\mathrm {D}}^2\left( s^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right) =\varvec{\mathrm {D}}\left( s^{N-1}{\beta _{\epsilon }}(s)\right) f({u_{\epsilon }}(s))+s^{N-1}{\beta _{\epsilon }}(s)f'({u_{\epsilon }}(s))(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s). \end{aligned}$$

(3.10)

To deal with the left-hand side of (3.10), we first notice $\varvec{\mathrm {D}}\left( s^{N-1}{\beta _{\epsilon }}(s)\right) \ge 0$ [by (A3)]. Thanks to (P1) and (P3), we arrive at a differential inequality

$$\begin{aligned} \epsilon ^2\varvec{\mathrm {D}}^2\left( s^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right)&\ge \,{C_3}\left( \inf _{s\in [0,1]}\frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \left( s^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right) \nonumber \\&\ge \,{C_1C_3}s^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s),\,\,\text{ in }\,\,(0,1). \end{aligned}$$

(3.11)

Applying the standard PDE comparison theorem to (3.11) and using (P4) immediately give

$$\begin{aligned} 0\le {s}^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\le \frac{{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)}{\epsilon }\left( e^{-\frac{\sqrt{C_1C_3}}{\epsilon }s}+e^{-\frac{\sqrt{C_1C_3}}{\epsilon }(1-s)}\right) . \end{aligned}$$

(3.12)

Along with the fact that ${s}^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)$ is increasing to s [see (P5)], we may follow the similar argument as in (3.6)–(3.9) to obtain

$$\begin{aligned} 0\le {s}^{N-1}{\alpha _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\le \frac{2}{\epsilon }{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)e^{-\frac{(1-k^*)\sqrt{C_1C_3}}{2\epsilon }(1-s)}. \end{aligned}$$

(3.13)

Let us set $M^*=\frac{(1-k^*)\sqrt{C_1C_3}}{4}$. Then, (3.1) and (3.2) follow from (3.9) and (3.13), respectively. This completes the proof of Lemma 3.1. $\square$

The following result states the uniform boundedness of ${u_{\epsilon }}$ and the leading order terms of ${u_{\epsilon }}(1)$ and $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)$ with respect to $0<\epsilon \ll 1$.

Lemma 3.2

Under the same hypotheses as in Lemma 3.1, $\displaystyle \max \nolimits _{[0,1]}{u_{\epsilon }}={u_{\epsilon }}(1)$ is uniformly bounded as $\epsilon >0$ is sufficiently small. In particular, as $\epsilon \downarrow 0$, for each $s\in [0,1)$ independent of $\epsilon$, $|{u_{\epsilon }}(s)-\theta _0|+\epsilon |(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)|\rightarrow 0$ exponentially, and

$$\begin{aligned} {u_{\epsilon }}(1)\rightarrow {p}\,\,and\,\,\epsilon (\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)\rightarrow \pmb {\eta }(p_0), \end{aligned}$$

(3.14)

where p is the unique root of (2.10). Moreover,

$$\begin{aligned} \left| \epsilon (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)-\sqrt{\frac{2{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\left( F({u_{\epsilon }}(s))-F(\theta _0)\right) }\right| \le \widetilde{C}\epsilon ^{1/2},\,\,for\,\,s\in [k^*,1], \end{aligned}$$

(3.15)

where $\widetilde{C}$ is a positive constant independent of $\epsilon$.

Proof

We first claim $\displaystyle \limsup _{\epsilon \downarrow 0}{u_{\epsilon }}(1)<\infty$. Integrating (3.2) over the interval $(k^*,1)$, one obtains

$$\begin{aligned} \displaystyle (k^*)^{N-1}\left( \min _{[k^*,1]}{\alpha _{\epsilon }}\right) ({u_{\epsilon }}(1)-{u_{\epsilon }}(k^*))\le \frac{{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)}{M^*}. \end{aligned}$$

Along with (3.1), one arrives at

$$\begin{aligned} {u_{\epsilon }}(1)-\frac{{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)}{\displaystyle {M}^*(k^*)^{N-1}\min \nolimits _{[k^*,1]}{\alpha _{\epsilon }}}\le {u_{\epsilon }}(k^*)\le \theta _0+2({u_{\epsilon }}(1)-\theta _0)e^{-\frac{M^*}{\epsilon }(1-k^*)}. \end{aligned}$$

Because $M^*>0$, $k^*<1$ and $\frac{{\alpha _{\epsilon }}(1)}{\displaystyle \min _{[k^*,1]}{\alpha _{\epsilon }}}$ is uniformly bounded to $0<\epsilon \ll 1$ [by (A3) and (P1)], the above inequality implies

$$\begin{aligned} \limsup _{\epsilon \downarrow 0}{u_{\epsilon }}(1)\le \theta _0+\frac{{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)}{\displaystyle {M}^*(k^*)^{N-1}\min _{[k^*,1]}{\alpha _{\epsilon }}}<\infty . \end{aligned}$$

(3.16)

Since ${u_{\epsilon }}(1)$ is uniformly bounded as $0<\epsilon \ll 1$, and $\theta _0\le {u_{\epsilon }}(s)\le {u_{\epsilon }}(1)$, we immediately obtain the uniform boundedness of ${u_{\epsilon }}$ as $0<\epsilon \ll 1$. Moreover, (3.1) can be improved by

$$\begin{aligned} 0\le {u_{\epsilon }}(s)-\theta _0\le {L}_{\epsilon }e^{-\frac{M^*}{\epsilon }(1-s)}, \end{aligned}$$

(3.17)

as $0<\epsilon \ll 1$, where

$$\begin{aligned} {L}_{\epsilon }:=1+\theta _0+\frac{{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)}{\displaystyle {M}^*(k^*)^{N-1}\min _{[k^*,1]}{\alpha _{\epsilon }}}. \end{aligned}$$

(3.18)

Note that $L_{\epsilon }$ is uniformly bounded to $\epsilon >0$. Consequently, by (3.1) and (3.17), we show that for each $s\in [0,1)$ independent of $\epsilon$, both $|{u_{\epsilon }}(s)-\theta _0|$ and $\epsilon |(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)|$ decay to zero exponentially as $\epsilon$ approaches zero.

To prove (3.14), we shall obtain the precise leading order terms of ${u_{\epsilon }}(1)$ and $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)$ with respect to small $\epsilon$. Let us first deal with the terms in the right-hand side of (2.23). Firstly, by (P2) and (3.2) one may check that, as $0<\epsilon \ll 1$,

$$\begin{aligned}&\int _{k^*}^1\epsilon ^2\left( \frac{N-1}{t}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(t)}{{\alpha _{\epsilon }}(t)}\right) ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(t))^2\text{ d }t\nonumber \\&\quad \le \int _{k^*}^1\left( \frac{N-1}{t}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(t)}{{\alpha _{\epsilon }}(t)}\right) \left( \frac{{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)}{{t}^{N-1}{\alpha _{\epsilon }}(t)}\right) ^2\left( e^{-\frac{M^*}{\epsilon }t}+e^{-\frac{M^*}{\epsilon }(1-t)}\right) ^2\text{ d }t \nonumber \\&\quad \le \,\frac{2}{M^*}\left( \frac{{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)}{\displaystyle {(k^*)}^{N-1}\min _{[k^*,1]}{\alpha _{\epsilon }}}\right) ^2\left( \frac{N-1}{k^*}+C_2\right) \epsilon :=C_4\epsilon . \end{aligned}$$

(3.19)

Note that $C_4$ is a positive constant independent of $\epsilon$ due to (A3) and (2.18). Next, we shall claim

$$\begin{aligned} \int _{k^*}^1F({u_{\epsilon }}(t))\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(t)}{{\alpha _{\epsilon }}(t)}\right) \text{ d }t-C_{k^*,\epsilon }\sim \,F(\theta _0)\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)},\,\,{\mathrm {as}}\,\,0<\epsilon \ll 1. \end{aligned}$$

By using (1.8), (2.22) (P1)–(P3), (3.2) and (3.17), we have

$$\begin{aligned}&\left| \int _{k^*}^1F({u_{\epsilon }}(t))\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(t)}{{\alpha _{\epsilon }}(t)}\right) \text{ d }t-C_{k^*,\epsilon }-F(\theta _0)\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right| \nonumber \\&\quad \le \left| C_{k^*,\epsilon }+\frac{{\beta _{\epsilon }}(k^*)}{{\alpha _{\epsilon }}(k^*)}F(\theta _0)\right| +\left| \int _{k^*}^1F({u_{\epsilon }}(t))\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(t)}{{\alpha _{\epsilon }}(t)}\right) \text{ d }t-F(\theta _0)\left( \frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}-\frac{{\beta _{\epsilon }}(k^*)}{{\alpha _{\epsilon }}(k^*)}\right) \right| \nonumber \\&\quad \le \left| C_{k^*,\epsilon }+\frac{{\beta _{\epsilon }}(k^*)}{{\alpha _{\epsilon }}(k^*)}F(\theta _0)\right| +\int _{k^*}^1\left| F({u_{\epsilon }}(t))-F(\theta _0)\right| \left| \varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(t)}{{\alpha _{\epsilon }}(t)}\right) \right| \text{ d }t \nonumber \\&\quad \le \,{C}_5\left( e^{-\frac{M^*}{\epsilon }k^*}+e^{-\frac{M^*}{\epsilon }(1-k^*)}\right) +{C}_2\left( 1+\max _{t\in [0,1]}\frac{{\alpha _{\epsilon }}(t)}{{\beta _{\epsilon }}(t)}\right) f({u_{\epsilon }}(1))\int _{k^*}^1({u_{\epsilon }}(t)-\theta _0)\text{ d }t\nonumber \\&\quad \le \,{C}_5\left( e^{-\frac{M^*}{\epsilon }k^*}+e^{-\frac{M^*}{\epsilon }(1-k^*)}\right) +C_6\epsilon , \end{aligned}$$

(3.20)

as $0<\epsilon \ll 1$, where $C_5$ is a positive constant independent of $\epsilon$, and $C_6$ can be any large positive constant satisfying

$$\begin{aligned} C_6>\frac{2C_2}{M^*}\limsup _{\epsilon \downarrow 0}\left\{ L_{\epsilon }\left( 1+\max _{t\in [0,1]}\frac{{\alpha _{\epsilon }}(t)}{{\beta _{\epsilon }}(t)}\right) f(L_{\epsilon }+\theta _0)\right\} . \end{aligned}$$

Here, we have used (2.22), (3.2) and (3.17) to get

$$\begin{aligned} \left| C_{k^*,\epsilon }+\frac{{\beta _{\epsilon }}(k^*)}{{\alpha _{\epsilon }}(k^*)}F(\theta _0)\right| \le {C}_5\left( e^{-\frac{M^*}{\epsilon }k^*}+e^{-\frac{M^*}{\epsilon }(1-k^*)}\right) \end{aligned}$$

(3.21)

which verifies the last second line of (3.20). Combining (2.23) with (3.20) yields

$$\begin{aligned} {(\pmb {\eta }({u_{\epsilon }}(1)))^2}-\frac{{2}{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}(F({u_{\epsilon }}(1))-F(\theta _0)){\mathop {\longrightarrow }\limits ^{\epsilon \downarrow 0}}0. \end{aligned}$$

(3.22)

On the other hand, by (2.5) and (2.17), we have $\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\rightarrow \mu _0$ as $\epsilon \downarrow 0$. Note that F is strictly increasing in $(\theta _0,\infty )$. Since ${u_{\epsilon }}(1)\ge \theta _0$ and $\pmb {\eta }>0$ is a decreasing function [cf. (A2)], we obtain $\lim _{\epsilon \downarrow 0}{u_{\epsilon }}(1)=p$ which uniquely solves (2.10). Moreover, by this with the boundary condition (2.20), we have $\lim _{\epsilon \downarrow 0}\epsilon (\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)=\pmb {\eta }(p_0)$. Therefore, we obtain (3.14).

It remains to prove (3.15). Let $s\in [k^*,1]$. Following the similar arguments as in (3.19) and (3.20), we can get estimates

$$\begin{aligned} \int _{k^*}^s\epsilon ^2&\left( \frac{N-1}{t}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(t)}{{\alpha _{\epsilon }}(t)}\right) ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(t))^2\text{ d }t \le \,C_7\left( \frac{N-1}{k^*}+C_2\right) \epsilon \end{aligned}$$

(3.23)

and

$$\begin{aligned} \left| \int _{k^*}^sF({u_{\epsilon }}(t))\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(t)}{{\alpha _{\epsilon }}(t)}\right) \text{ d }t-C_{k^*,\epsilon }-F(\theta _0)\frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right| \le \,C_8\epsilon , \end{aligned}$$

(3.24)

as $0<\epsilon \ll 1$, where $C_7,\,C_8>0$ independent of s and $\epsilon$. Then, by (2.21) and (3.23)–(3.24), we arrive at

$$\begin{aligned} \left| \epsilon ^2\left( (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right) ^2-\frac{2{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\left( F({u_{\epsilon }}(s))-F(\theta _0)\right) \right| \le {\widetilde{C}}^2\epsilon \end{aligned}$$

(3.25)

with a positive constant $\widetilde{C}$ independent of s and $\epsilon$. Since $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\ge 0$ and $F({u_{\epsilon }}(s))\ge {F}(\theta _0)$, $\forall {s}\in [0,1]$ [see (P3)], together with (3.25) we immediately get (3.15) and complete the proof Lemma 3.2. $\square$

3.2 Boundary asymptotics with precise first two-order terms

Recall that (3.19) and (3.20) imply

$$\begin{aligned}&\sup _{0<\epsilon \ll 1}\epsilon \int _{k^*}^1\left| \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right| \left( (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\right) ^2\mathrm {d}s<\infty ,\\&\sup _{0<\epsilon \ll 1}\frac{1}{\epsilon }\left| \int _{k^*}^1F({u_{\epsilon }}(t)) \left( \varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}}{{\alpha _{\epsilon }}}\right) \right) (t)\,\text{ d }t-C_{k^*,\epsilon }-F(\theta _0)\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right| <\infty . \end{aligned}$$

To obtain the structure of the solution ${u_{\epsilon }}$, we further establish their precise leading order terms which play a crucial role in the refined asymptotics of ${u_{\epsilon }}(1)$ and $\left( \varvec{\mathrm {D}}{u_{\epsilon }}\right) (1)$. The asymptotics are stated as follows:

Theorem 3.3

Under the same hypotheses as in Lemma 3.1, for $\epsilon {>}0$ sufficiently small, we have

$$\begin{aligned}&\epsilon \int _{k^*}^1\left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\mathrm {d}s\nonumber \\&\quad =\sqrt{\mu _0}\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{{\alpha _{\epsilon }}(1)}\right) \int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t+o_{\epsilon }(1), \end{aligned}$$

(3.26)

and

$$\begin{aligned}&\frac{1}{\epsilon }\left( \int _{k^*}^1\right. \left. F({u_{\epsilon }}(s)) \varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \mathrm {d}s-C_{k^*,\epsilon }-F(\theta _0)\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right) \nonumber \\&\quad =\frac{1}{2{\alpha _{\epsilon }}(1)}\left( \frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{\sqrt{\mu _0}}-\sqrt{\mu _0}(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)\right) \int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t+o_{\epsilon }(1), \end{aligned}$$

(3.27)

where $o_{\epsilon }(1)$ denotes the quantity approaching zero as $\epsilon \downarrow 0$.

Proof

Let us fix a number $\tau _a\in (0,1)$ independent of $\epsilon$. By (P1) and (P2), we obtain

$$\begin{aligned}&\sup _{s\in [1-\epsilon ^{\tau _a},1]}\left| \frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}-\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{{\alpha _{\epsilon }}(1)}\right| \le \epsilon ^{\tau _a}\sup _{[1-\epsilon ^{\tau _a},1]}\left| \varvec{\mathrm {D}}\left( \frac{\varvec{\mathrm {D}}{\alpha _{\epsilon }}}{{\alpha _{\epsilon }}}\right) \right| \nonumber \\&\quad \le \left( C_2^2+C_2\sup _{[1-\epsilon ^{\tau _a},1]}{\alpha _{\epsilon }}\right) \epsilon ^{\tau _a}{\mathop {\longrightarrow }\limits ^{\epsilon \downarrow 0}}0, \end{aligned}$$

(3.28)

$$\begin{aligned}&\sup _{s\in [1-\epsilon ^{\tau _a},1]}\left| \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}-\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right| \le \epsilon ^{\tau _a}\sup _{[1-\epsilon ^{\tau _a},1]}\left| \varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}}{{\alpha _{\epsilon }}}\right) \right| \nonumber \\&\quad \le \left( 1+\frac{1}{C_1}\right) C_2\epsilon ^{\tau _a}{\mathop {\longrightarrow }\limits ^{\epsilon \downarrow 0}}0. \end{aligned}$$

(3.29)

Hence, for $0<\epsilon \ll 1$, we consider a decomposition

$$\begin{aligned}&\int _{k^*}^1\epsilon ^2\left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\text{ d }s\nonumber \\&\quad =\int _{k^*}^{1-{\epsilon }^{\tau _a}}\epsilon ^2\left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\text{ d }t\nonumber \\&\qquad +\int _{1-{\epsilon }^{\tau _a}}^1\epsilon ^2\left[ \left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) -\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{{\alpha _{\epsilon }}(1)}\right) \right] ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\text{ d }s\nonumber \\&\qquad +\epsilon ^2\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{{\alpha _{\epsilon }}(1)}\right) \int _{1-{\epsilon }^{\tau _a}}^1((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\text{ d }s. \end{aligned}$$

(3.30)

Using the gradient estimate (3.2) and (3.28), we may follow the similar argument as in (3.19) to

$$\begin{aligned}&\left| \int _{k^*}^{1-{\epsilon }^{\tau _a}}\epsilon ^2\left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\,\text{ d }t\right| \\&\quad \le 2\left( \frac{N-1}{k^*}+C_2\right) \left( \frac{{\alpha _{\epsilon }}(1)\pmb {\eta }(\theta _0)}{\displaystyle {(k^*)}^{N-1}\min _{[k^*,1]}{\alpha _{\epsilon }}}\right) ^2 \int _{k^*}^{1-{\epsilon }^{\tau _a}}\left( e^{-\frac{2M^*}{\epsilon }s}+e^{-\frac{2M^*}{\epsilon }(1-s)}\right) \text{ d }s \\&\qquad (\text{ due }\,\,\text{ to }\,\,\tau _a\in (0,1))\,\le {C}_9\left( \frac{N-1}{k^*}+C_2\right) {\epsilon }e^{-{2M^*}{\epsilon ^{\tau _a-1}}}\ll \epsilon \,\,\text{ as }\,\,\,\,0<\epsilon \ll 1, \end{aligned}$$

and

$$\begin{aligned}&\left| \int _{1-{\epsilon }^{\tau _a}}^1\epsilon ^2\left[ \left( \frac{N-1}{s}+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(s)}{{\alpha _{\epsilon }}(s)}\right) -\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{{\alpha _{\epsilon }}(1)}\right) \right] ((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\,\text{ d }s\right| \\&\quad \le \,C_{10}\epsilon ^{\tau _a}\int _{1-{\epsilon }^{\tau _a}}^1\epsilon ^2((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\,\text{ d }s\ll \epsilon \,\,\text{ as }\,\,0<\epsilon \ll 1, \end{aligned}$$

where $C_9$ and $C_{10}$ are positive constants independent of $\epsilon$.

To deal with the last term of (3.30), let us rewrite (3.15) as

$$\begin{aligned} \epsilon (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)=\sqrt{\frac{2{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\left( F({u_{\epsilon }}(s))-F(\theta _0)\right) }+{\gamma _{\epsilon }}(s)\,\,\,{\mathrm {and}}\,\,\,|{\gamma _{\epsilon }}(s)|\le \widetilde{C}\epsilon ^{1/2},\,\,\,\forall \,s\in [k^*,1]. \end{aligned}$$

(3.31)

Then, by (3.29) and (3.31) one may check that

$$\begin{aligned}&\epsilon \int _{1-{\epsilon }^{\tau _a}}^1((\varvec{\mathrm {D}}{{u_{\epsilon }}})(s))^2\,\text{ d }s\nonumber \\&\quad =\int _{1-{\epsilon }^{\tau _a}}^1\left( \sqrt{\frac{2{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\left( F({u_{\epsilon }}(s))-F(\theta _0)\right) }+{\gamma _{\epsilon }}(s)\right) (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\text{ d }s\nonumber \\&\quad =\sqrt{\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}}\int _{{u_{\epsilon }}(1-{\epsilon }^{\tau _a})}^{{u_{\epsilon }}(1)}\sqrt{2(F(t)-F(\theta _0))}\,\text{ d }t+o_{\epsilon }(1)\nonumber \\&\quad =\sqrt{\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}}\left\{ \int _{{u_{\epsilon }}(1-{\epsilon }^{\tau _a})}^{\theta _0}+\int _{\theta _0}^{p_0}+\int _{p}^{{u_{\epsilon }}(1)}\right\} \sqrt{2(F(t)-F(\theta _0))}\,\text{ d }t+o_{\epsilon }(1)\nonumber \\&\quad =\sqrt{\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}}\int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\text{ d }t+o_{\epsilon }(1). \end{aligned}$$

(3.32)

Here, we have used the following three estimates to deal with (3.32):

$$\begin{aligned}&\left| \int _{1-{\epsilon }^{\tau _a}}^1{\gamma _{\epsilon }}(s)(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\text{ d }s\right| \le \widetilde{C}\epsilon ^{1/2}\int _{1-{\epsilon }^{\tau _a}}^1(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\text{ d }s\le \widetilde{C}\epsilon ^{1/2}({u_{\epsilon }}(1)-\theta _0)\lesssim \epsilon ^{1/2}, \\&\left| \int _{1-{\epsilon }^{\tau _a}}^1\sqrt{2\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}-\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right) \left( F({u_{\epsilon }}(s))-F(\theta _0)\right) }(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\text{ d }s\right| \\&\quad (\text{ by }\,\,(3.29))\,\,\le \sqrt{2\left( 1+\frac{1}{C_1}\right) C_2}\epsilon ^{{\tau _a}/{2}}\left( F({u_{\epsilon }}(1))-F(\theta _0)\right) ({u_{\epsilon }}(1)\,-\theta _0)\lesssim \epsilon ^{{\tau _a}/{2}}, \end{aligned}$$

and

$$\begin{aligned}&\left| \left\{ \int _{{u_{\epsilon }}(1-{\epsilon }^{\tau _a})}^{\theta _0}+\int _{p}^{{u_{\epsilon }}(1)}\right\} \sqrt{F(t)-F(\theta _0)}\,\text{ d }t\right| \\&\quad (\text{ by }\,\,(3.14)\,\,\text{ and }\,\,(3.17))\,\,\le \sqrt{F({u_{\epsilon }}(1))-F(\theta _0)}\big (|{u_{\epsilon }}(1- \,{\epsilon }^{\tau _a})-\theta _0|+|{u_{\epsilon }}(1)-p|\big ){\mathop {\longrightarrow }\limits ^{\epsilon \downarrow 0}}0. \end{aligned}$$

Since $\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\rightarrow \mu _0$ as $\epsilon \downarrow 0$, (3.32) immediately implies (3.26).

Now we shall prove (3.27). From the first three lines of (3.20), we obtain

$$\begin{aligned}&\frac{1}{\epsilon }\left| \left( \int _{k^*}^1F({u_{\epsilon }}(s))\varvec{\mathrm {D}}\right. \right. \left. \left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \mathrm {d}s-C_{k^*,\epsilon }-F(\theta _0)\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right) \nonumber \\&\quad \left. -\int _{k^*}^1\left( F({u_{\epsilon }}(s))-F(\theta _0)\right) \varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \text{ d }s\right| \lesssim \frac{1}{\epsilon }\left( e^{-\frac{M^*}{\epsilon }k^*}+e^{-\frac{M^*}{\epsilon }(1-k^*)}\right) {\mathop {\longrightarrow }\limits ^{\epsilon \downarrow 0}}0, \end{aligned}$$

(3.33)

Hence, by (3.31) and (3.33), one finds

$$\begin{aligned}&\frac{1}{\epsilon }\left( \int _{k^*}^1F({u_{\epsilon }}(s)) \varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \,\mathrm {d}s-C_{k^*,\epsilon }-F(\theta _0)\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right) \nonumber \\&\quad =\frac{1}{\epsilon }\int _{k^*}^1\sqrt{F({u_{\epsilon }}(s))-F(\theta _0)}\left( \epsilon (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)-\gamma _{\epsilon }(s)\right) \sqrt{\frac{{\alpha _{\epsilon }}(s)}{2{\beta _{\epsilon }}(s)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \,\text{ d }s+o_{\epsilon }(1)\nonumber \\&\quad =\int _{k^*}^1\sqrt{F({u_{\epsilon }}(s))-F(\theta _0)}\,(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\sqrt{\frac{{\alpha _{\epsilon }}(s)}{2{\beta _{\epsilon }}(s)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \,\text{ d }s+o_{\epsilon }(1). \end{aligned}$$

(3.34)

Here, we have used (P1)–(P2), $|{\gamma _{\epsilon }}(s)|\le \widetilde{C}\epsilon ^{1/2}$ and the interior estimate (3.17) to verify

$$\begin{aligned} \frac{1}{\epsilon }\left| \int _{k^*}^1\sqrt{F({u_{\epsilon }}(s))-F(\theta _0)}\,\gamma _{\epsilon }(s)\sqrt{\frac{{\alpha _{\epsilon }}(s)}{2{\beta _{\epsilon }}(s)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \text{ d }s\right| \ll 1. \end{aligned}$$

On the other hand, notice that $\sqrt{\frac{{\alpha _{\epsilon }}(s)}{2{\beta _{\epsilon }}(s)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \in \text{ C }_{\mathrm {loc}}^{0,\tau }([0,\infty ))$. Thus, by (P1) and (P2), we have

$$\begin{aligned} \left| \sqrt{\frac{{\alpha _{\epsilon }}(s)}{2{\beta _{\epsilon }}(s)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) -\sqrt{\frac{{\alpha _{\epsilon }}(1)}{2{\beta _{\epsilon }}(1)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right) \right| \le \,C_{11}|s-1|^{\tau }, \end{aligned}$$

where $C_{11}$ is a positive constant independent of $\epsilon$. Let us also recall $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\ge 0$ and $\tau \in (0,1)$. Hence, following the similar argument as in (3.32) arrives at the precise leading order term of the expansion in the last line of (3.34):

$$\begin{aligned}&\int _{k^*}^1\sqrt{F({u_{\epsilon }}(s))-F(\theta _0)}\,(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\sqrt{\frac{{\alpha _{\epsilon }}(s)}{2{\beta _{\epsilon }}(s)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \,\text{ d }s\nonumber \\&\quad =\int _{1-\epsilon ^{1/2}}^1\sqrt{F({u_{\epsilon }}(s))-F(\theta _0)}\,(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\sqrt{\frac{{\alpha _{\epsilon }}(s)}{2{\beta _{\epsilon }}(s)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) \,\text{ d }s+o_{\epsilon }(1)\nonumber \\&\quad =\sqrt{\frac{{\alpha _{\epsilon }}(1)}{2{\beta _{\epsilon }}(1)}}\varvec{\mathrm {D}}\left( \frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\right) \int _{1-\epsilon ^{1/2}}^1\sqrt{F({u_{\epsilon }}(s))-F(\theta _0)}\,(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\,\text{ d }s+o_{\epsilon }(1) \nonumber \\&\quad =\frac{1}{{\alpha _{\epsilon }}(1)}\left( \sqrt{\frac{{\alpha _{\epsilon }}(1)}{2{\beta _{\epsilon }}(1)}}(\varvec{\mathrm {D}}{\beta _{\epsilon }})(1)-\sqrt{\frac{{\beta _{\epsilon }}(1)}{2{\alpha _{\epsilon }}(1)}}(\varvec{\mathrm {D}}{\alpha _{\epsilon }})(1)\right) \nonumber \\&\qquad \times \int _{{u_{\epsilon }}(1-\epsilon ^{1/2})}^{{u_{\epsilon }}(1)}\sqrt{F(t)-F(\theta _0)}\,\text{ d }t+o_{\epsilon }(1)\nonumber \\&\quad =\frac{1}{{\alpha _{\epsilon }}(1)}\left( \sqrt{\frac{{\alpha _{\epsilon }}(1)}{2{\beta _{\epsilon }}(1)}}(\varvec{\mathrm {D}}{\beta _{\epsilon }})(1)-\sqrt{\frac{{\beta _{\epsilon }}(1)}{2{\alpha _{\epsilon }}(1)}}(\varvec{\mathrm {D}}{\alpha _{\epsilon }})(1)\right) \nonumber \\&\qquad \times \int _{\theta _0}^{p_0}\sqrt{F(t)-F(\theta _0)}\,\text{ d }t+o_{\epsilon }(1). \end{aligned}$$

(3.35)

Since $\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}=\mu _0+o_{\epsilon }(1)$, by (3.34) and (3.35), we obtain (3.27) and complete the proof of Theorem 3.3. $\square$

Thanks to Theorem 3.3, now we shall establish the precise first two-order terms of ${u_{\epsilon }}(1)$ and $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)$ with respect to sufficiently small $\epsilon$. Note that by (2.5) and (2.17), we have

$$\begin{aligned} \frac{1}{\epsilon }\left( \frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}-\mu _0\right) \ll 1,\,\,\text{ as }\,\,0<\epsilon \ll 1. \end{aligned}$$

(3.36)

Combining (2.23) with (3.26)–(3.27), one may obtain

$$\begin{aligned}&-\frac{(\pmb {\eta }({u_{\epsilon }}(1)))^2}{2}+\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\left( F({u_{\epsilon }}(1))-F(\theta _0)\right) \nonumber \\&\quad =\,\epsilon \sqrt{\mu _0}\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{{\alpha _{\epsilon }}(1)}\right) \left( \int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t+o_{\epsilon }(1)\right) \nonumber \\&\qquad +\frac{\epsilon }{2{\alpha _{\epsilon }}(1)}\left( \frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{\sqrt{\mu _0}}-\sqrt{\mu _0}(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)\right) \nonumber \\&\qquad \times \left( \int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t+o_{\epsilon }(1)\right) \nonumber \\&\quad =\,\epsilon \sqrt{\mu _0}\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{2{\alpha _{\epsilon }}(1)}+\frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{2{\beta _{\epsilon }}(1)}\right) \nonumber \\&\qquad \times \left( \int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t+o_{\epsilon }(1)\right) . \end{aligned}$$

(3.37)

The next task at hand is to deal with the first two terms of ${u_{\epsilon }}(1)$ and $(\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)$. By (3.14), we obtain

$$\begin{aligned} {u_{\epsilon }}(1)=p+q_{\epsilon }\,\,\text{ with }\,\,\lim _{\epsilon \downarrow 0}q_{\epsilon }=0. \end{aligned}$$

(3.38)

Combining the boundary condition (2.20) with (3.38) gives the asymptotics

$$\begin{aligned} \epsilon (\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)=\pmb {\eta }(p_0)+q_{\epsilon }\pmb {\eta }'(p_0)(1+o_{\epsilon }(1)). \end{aligned}$$

(3.39)

On the other hand, by (2.10) and (3.38), we have, for $0<\epsilon \ll 1$, that

$$\begin{aligned}&-\frac{1}{2}(\pmb {\eta }({u_{\epsilon }}(1)))^2+\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\left( F({u_{\epsilon }}(1))-F(\theta _0)\right) \nonumber \\&\quad =\,-\frac{1}{2}\left[ \pmb {\eta }(p_0)+q_{\epsilon }\pmb {\eta }'(p_0)(1+o_{\epsilon }(1))\right] ^2\nonumber \\&\qquad +\frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}\left[ F(p_0)-F(\theta _0)+q_{\epsilon }f(p_0)(1+o_{\epsilon }(1))\right] \nonumber \\&\quad =\,q_{\epsilon }\left[ -\pmb {\eta }(p_0)\pmb {\eta }'(p_0)+\mu _0f(p_0)+o_{\epsilon }(1)\right] \nonumber \\&\qquad +\left( \frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}-\mu _0\right) (F(p_0)-F(\theta _0)). \end{aligned}$$

(3.40)

As a consequence, by (2.10), (3.36), (3.37) and (3.40) one may check that

$$\begin{aligned} \frac{q_{\epsilon }}{\epsilon }=&\,({\displaystyle -\pmb {\eta }(p_0)\pmb {\eta }'(p_0)+\mu _0f(p_0)})^{-1}\left[ \displaystyle \sqrt{\mu _0}\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{2{\alpha _{\epsilon }}(1)}+\frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{2{\beta _{\epsilon }}(1)}\right) \right. \nonumber \\&\left. \times \int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t+\frac{1}{\epsilon }\left( \frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}-\mu _0\right) (F(p_0)-F(\theta _0))\right] +o_{\epsilon }(1)\nonumber \\ =&\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{2{\alpha _{\epsilon }}(1)}+\frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{2{\beta _{\epsilon }}(1)}\right) \frac{\displaystyle \int _{\theta _0}^{p_0}\sqrt{\frac{F(t)-F(\theta _0)}{F(p_0)-F(\theta _0)}}\,\mathrm {d}t}{\displaystyle -\pmb {\eta }'(p_0)+\mu _0\frac{f(p_0)}{\pmb {\eta }(p_0)}}+o_{\epsilon }(1). \end{aligned}$$

(3.41)

Here, we have used (3.36) to verify the second equality. By (3.38) and (3.41), it yields the precise first two-order terms of ${u_{\epsilon }}(1)$ with respect to small $\epsilon$:

$$\begin{aligned} {u_{\epsilon }}(1)=p_0+\epsilon \mathtt {C}_{0}\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{2{\alpha _{\epsilon }}(1)}+\frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{2{\beta _{\epsilon }}(1)}+o_{\epsilon }(1)\right) , \end{aligned}$$

(3.42)

where $\mathtt {C}_{0}=\left( -{\pmb {\eta }'(p_0)}+\mu _0\frac{f(p_0)}{\pmb {\eta }(p_0)}\right) ^{-1}{\int _{\theta _0}^{p_0}\sqrt{\frac{F(t)-F(\theta _0)}{F(p_0)-F(\theta _0)}}\,\mathrm {d}t}$ is defined in Theorem 2.1. Finally, (3.39) and (3.41) imply

$$\begin{aligned} (\varvec{\mathrm {D}}{{u_{\epsilon }}})(1)=\frac{\pmb {\eta }(p_0)}{\epsilon }+\pmb {\eta }'(p_0)\mathtt {C}_{0}\left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{2{\alpha _{\epsilon }}(1)}+\frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{2{\beta _{\epsilon }}(1)}\right) +o_{\epsilon }(1). \end{aligned}$$

(3.43)

3.3 Completion of the proofs

Proof of Theorem 2.1

The monotonic increase of $\pmb {\mathrm {U}}$ follows immediately from (2.4). To deal with the convexness of $\pmb {\mathrm {U}}$ as $R\gg 1$, let us recall (2.19), (P2), (P3) and Lemma 3.1. Firstly, we choose $k_{\epsilon }\in [k^*,1)$ such that ${u_{\epsilon }}(k_{\epsilon })=\frac{\theta _0+p_0}{2}\in (\theta _0,p_0)$. Then, by (3.17) and (3.18) we have $0<\frac{p_0-\theta _0}{2}\le {L}_{\epsilon }e^{-\frac{M^*}{\epsilon }(1-k_{\epsilon })}$ with $0{\mathop {\longleftarrow }\limits ^{\epsilon \downarrow 0}}\epsilon \log \frac{p-\theta _0}{2L_{\epsilon }}\le {-{M^*}(1-k_{\epsilon })}<0$, implying

$$\begin{aligned} k^*<1+\frac{\epsilon }{M^*}\log \frac{p_0-\theta _0}{2L_{\epsilon }}\le {k}_{\epsilon }<1\,\,\text{ as }\,\,0<\epsilon \ll 1. \end{aligned}$$

(3.44)

Moreover, we have $\displaystyle \min \nolimits _{[k_{\epsilon },1]}u_{\epsilon }\ge (\theta _0+p_0)/2$ for any $\epsilon >0$. Hence, by (2.19), (P2), (P3) and (3.44) we obtain, for sufficiently small $\epsilon >0$, that

$$\begin{aligned} \epsilon ^2(\varvec{\mathrm {D}}^2{{u_{\epsilon }}})(s)&\ge \,-\epsilon ^2\left( \frac{N-1}{k^*}+\sup _{[k^*,1]}\left| \frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})}{{\alpha _{\epsilon }}}\right| \right) (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)+\left( \inf _{[k^*,1]}\frac{{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\right) f({u_{\epsilon }}(k^*))\nonumber \\&\ge \,-\epsilon ^2\left( \frac{N-1}{k^*}+C_2\right) (\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)+C_1f(\frac{\theta _0+p}{2})\ge \frac{C_1}{2}f\left( \frac{\theta _0+p}{2}\right) >0 \end{aligned}$$

since $\displaystyle \lim \nolimits _{\epsilon \downarrow 0}\sup _{[k^*,1]}\epsilon (\varvec{\mathrm {D}}{{u_{\epsilon }}})<\infty$. As a consequence,

$$\begin{aligned} (\varvec{\mathrm {D}}^2{{u_{\epsilon }}})(s)>0\,\,{\mathrm {in}}\,\,[k_{\epsilon },1]\,\,{\mathrm {as}}\,\,0<\epsilon \ll 1. \end{aligned}$$

This along with (2.17) gives $\pmb {\mathrm {U}}''>0$ in $[\widetilde{k}_{R},R]$ as $R\gg 1$, where $\widetilde{k}_{R}:=k_{1/R}R\,(=k_{\epsilon }R)$ admits

$$\begin{aligned} R+\frac{1}{M^*}\log \frac{p-\theta _0}{2L_{\epsilon }}\le \widetilde{k}_{R}<{R}. \end{aligned}$$

Hence, we obtain the convexness of $\pmb {\mathrm {U}}$ near the boundary $r=R$ as $R\gg 1$.

It remains to deal with (2.6). By (2.17)–(2.18), (3.2) and (3.17), one arrives at

$$\begin{aligned}&0\le \left( \frac{r}{R}\right) ^{N-1}\pmb {\mathrm {U}}'(r)\le \frac{2\pmb {\eta }(\theta _0)\pmb {\alpha }(R)}{\displaystyle \min _{[0,R]}\pmb {\alpha }}e^{-{M^*}(R-r)},\\&0\le \pmb {\mathrm {U}}(r)-\theta _0\le \,{L}_{\epsilon }e^{-{M^*}(R-r)}, \end{aligned}$$

for $r\in [0,R]$. Consequently, we prove (2.6) with $\mathtt {M}_0=M^*$ and $\displaystyle \mathtt {L}_0={\displaystyle 2\pmb {\eta }(\theta _0)\max \nolimits _{[0,R]}\pmb {\alpha }}\left( \displaystyle \min \nolimits _{[0,R]}\pmb {\alpha }\right) ^{-1}+\sup \nolimits _{0<\epsilon \ll 1}{L}_{\epsilon }$ which are positive constants independent of R. Finally, by (2.17), (2.18), (3.42) and (3.43), we immediately obtain (2.7) and (2.8). Therefore, we complete the proof of Theorem 2.1. $\square$

Proof of Corollary 2.2

Corollary 2.2(I) follows directly from (2.7)–(2.9) so we omit the proof. We are now in a position to prove Corollary 2.2(II). Assume firstly that (i) is satisfied. Setting $\mu _i=\frac{\pmb {\beta }_i(R)}{\pmb {\alpha }_i(R)}$, $i=1,2$, which are independent of R, we denote $p_i=p(\mu _{i})$ the unique root of (2.10) with $\mu _0=\mu _i$, $i=1,2$. Notice that $F(p_0)$ is strictly increasing to $p_0\in (\theta _0,\infty )$ and $\pmb {\eta }(p_0)$ is decreasing to $p_0\in (\theta _0,\infty )$ [see (A1) and (A2)]. Hence, from (2.10) it is easy to check that $p_0=p_0(\mu _0)$ is strictly decreasing to $\mu _0>0$. As a consequence, the assumption $\mu _1<\mu _2$ implies

$$\begin{aligned} p_1>p_2>\theta _0~ \hbox {and}~ 0<\pmb {\eta }(p_1)<\pmb {\eta }(p_2). \end{aligned}$$

Accordingly, the leading order terms in (2.7) and (2.8) immediately imply

$$\begin{aligned} \widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}(R)>\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}(R)>\theta _0 ~ \hbox {and}~ 0<\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}'(R)<\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}'(R)~ \hbox {as}~ R\gg 1. \end{aligned}$$

Now we assume that (ii) is satisfied. Then, as $R\rightarrow \infty$, by (2.7) we know that $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}(R)$ and $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}(R)$ have the same leading order term, and by (2.8), $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}'(R)$ and $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}'(R)$ have the same leading order term. Due to the fact that the second and third conditions in (ii) exactly appear in the second-order terms of (2.7) and (2.8), a simple comparison immediately shows $\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}(R)>\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}(R)>\theta _0$ and $0<\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_1,\pmb {\beta }_1}'(R)<\widetilde{\pmb {\mathrm {U}}}_{\pmb {\alpha }_2,\pmb {\beta }_2}'(R)$ as $R\gg 1$. Therefore, we complete the proof of Corollary 2.2(II). $\square$

Proof of Theorem 2.3

It suffices to prove

$$\begin{aligned} \lim _{R\rightarrow \infty }R\int _{k^*R}^{R}(\pmb {\mathrm {U}}(r)-\theta _0)\,\mathrm {d}r=&\frac{1}{\sqrt{\mu _0}}\int _{\theta _0}^{p_0}\frac{t-\theta _0}{\sqrt{2(F(t)-F(\theta _0))}}\,\mathrm {d}t,\\ \lim _{R\rightarrow \infty }R\int _{k^*R}^{R}\pmb {\mathrm {U}}'^2(r)\,\mathrm {d}r=&\sqrt{\mu _0}\int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t, \end{aligned}$$

which are equivalent to claiming

$$\begin{aligned} \lim _{\epsilon \downarrow 0}\int _{k^*}^{1}\frac{{u_{\epsilon }}(s)-\theta _0}{\epsilon }\,\mathrm {d}s=&\frac{1}{\sqrt{\mu _0}}\int _{\theta _0}^{p_0}\frac{t-\theta _0}{\sqrt{2(F(t)-F(\theta _0))}}\,\mathrm {d}t, \end{aligned}$$

(3.45)

$$\begin{aligned} \lim _{\epsilon \downarrow 0}\epsilon \int _{k^*}^{1}(\varvec{\mathrm {D}}{{u_{\epsilon }}})^2(s)\,\mathrm {d}s=&\sqrt{\mu _0}\int _{\theta _0}^{p_0}\sqrt{2(F(t)-F(\theta _0))}\,\mathrm {d}t, \end{aligned}$$

(3.46)

respectively. Firstly, by following the similar argument as the proof of (3.26), (3.46) can be obtained straightforwardly so we omit the detailed proof. It remains to prove (3.45).

To deal with (3.45), we first consider the decomposition

$$\begin{aligned} \int _{k^*}^{1}\frac{{u_{\epsilon }}(s)-\theta _0}{\epsilon }\,\mathrm {d}s=\left\{ \int _{k^*}^{1-{\epsilon }^{\tau _a}}+\int _{1-{\epsilon }^{\tau _a}}^{1}\right\} \frac{{u_{\epsilon }}(s)-\theta _0}{\epsilon }\,\mathrm {d}s, \end{aligned}$$

where $\tau _a\in (0,1)$ has already been used in the proof of Theorem 3.3. Due to the interior estimate (3.17), we have

$$\begin{aligned} \left| \int _{k^*}^{1-{\epsilon }^{\tau _a}}\frac{{u_{\epsilon }}(s)-\theta _0}{\epsilon }\,\text{ d }s\right| \ll 1,\,\,\text{ as }\,\,0<\epsilon \ll 1. \end{aligned}$$

(3.47)

Utilizing (3.15) and following the similar argument as the proof of (3.27), we can deal with the second integral as follows:

$$\begin{aligned} \int _{1-{\epsilon }^{\tau _a}}^1\frac{{u_{\epsilon }}(s)-\theta _0}{\epsilon }\,\text{ d }s =&\int _{1-{\epsilon }^{\tau _a}}^1\frac{{u_{\epsilon }}(s)-\theta _0}{\sqrt{\frac{2{\beta _{\epsilon }}(s)}{{\alpha _{\epsilon }}(s)}\left( F({u_{\epsilon }}(s))-F(\theta _0)\right) }+o_{\epsilon }(1)}(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\,\text{ d }s\nonumber \\ =&\frac{1}{\sqrt{\mu _{0}}}\int _{1-{\epsilon }^{\tau _a}}^1\frac{{u_{\epsilon }}(s)-\theta _0}{\sqrt{2\left( F({u_{\epsilon }}(s))-F(\theta _0)\right) }+o_{\epsilon }(1)}(\varvec{\mathrm {D}}{{u_{\epsilon }}})(s)\,\text{ d }s+o_{\epsilon }(1) \nonumber \\ =&\frac{1}{\sqrt{\mu _{0}}}\int _{{u_{\epsilon }}(1-{\epsilon }^{\tau _a})}^{{u_{\epsilon }}(1)}\frac{t-\theta _0}{\sqrt{2\left( F(t)-F(\theta _0)\right) }+o_{\epsilon }(1)}\text{ d }t+o_{\epsilon }(1)\nonumber \\ =&\frac{1}{\sqrt{\mu _{0}}}\int _{\theta _0}^{p_0}\frac{t-\theta _0}{\sqrt{2\left( F(t)-F(\theta _0)\right) }}\text{ d }t+o_{\epsilon }(1). \end{aligned}$$

(3.48)

Here, we have used (3.29), ${u_{\epsilon }}(1)\rightarrow {p}$, ${u_{\epsilon }}(1-{\epsilon }^{\tau _a})\rightarrow \theta _0$ and the fact that $\int _{\theta _0}^{p_0}\frac{t-\theta _0}{\sqrt{2\left( F(t)-F(\theta _0)\right) }}\text{ d }t$ is finite [cf. Remark 2] to verify (3.48). Therefore, (3.45) follows from (3.47)–(3.48). The proof of Theorem 2.3 is done. $\square$

4 Final remark: how strongly does the small perturbation of $\varvec{\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}}$ affect the boundary structure of $\varvec{\pmb {\mathrm {U}}}$?

In Theorem 2.1, we have established refined asymptotics of $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ under a strong assumption (2.5). The situation shows that, on the boundary asymptotics of $\pmb {\mathrm {U}}$, the effect of the perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$ with respect to $R\gg 1$ is far smaller than the effect of boundary curvature $\frac{1}{R}$ since $\Big |\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\Big | \ll |\pmb {\mathcal {H}}(R)| \sim \frac{1}{R}$ as $R\gg 1$.

With regard to the small perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$, particularly for including its significant effect on boundary structure of $\pmb {\mathrm {U}}$, we shall pay attention to the situation

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}=\mu _0\,\,{\mathrm {and}}\,\,\liminf _{R\rightarrow \infty }R\left| \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right| >0. \end{aligned}$$

(4.1)

The main difference between (2.5) and (4.1) comes from the fact that (4.1) implies

$$\begin{aligned} \left| \pmb {\mathcal {H}}(R)\right| \lesssim \left| \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right| \,\,{\mathrm {as}}\,\,R\gg 1. \end{aligned}$$

(4.2)

Accordingly, the perturbation of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}$ around $\mu _0$ plays a crucial role in asymptotic behaviors of $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ and is undoubtedly not to be ignored. Note also that (4.1) includes (1.9). Hence, (3.36) is no longer satisfied, and the asymptotic expansions of $\pmb {\mathrm {U}}(R)$ and $\pmb {\mathrm {U}}'(R)$ are more complicated than the corresponding results in Theorem 2.1. Such a result is stated as follows:

Corollary 4.1

Under the hypotheses as in Theorem 2.1, we replace (2.5) with (4.1). Then, as $R\gg 1$, we have

$$\begin{aligned} \pmb {\mathrm {U}}(R)=&\,{p_0}+\frac{\sqrt{F(p_0)-F(\theta _0)}}{\mu _0f(p_0)-\pmb {\eta }(p_0)\pmb {\eta }'(p_0)}\left( \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right) +\varvec{{\mathtt {C}}_{0}}\pmb {\mathcal {H}}(R)+\frac{o(1)}{R}, \end{aligned}$$

(4.3)

$$\begin{aligned} \pmb {\mathrm {U}}'(R)=&\,\pmb {\eta }(p_0)+\pmb {\eta }'(p_0)\left( \frac{\sqrt{F(p_0)-F(\theta _0)}}{\mu _0f(p_0)-\pmb {\eta }(p_0)\pmb {\eta }'(p_0)}\left( \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right) +\varvec{{\mathtt {C}}_{0}}\pmb {\mathcal {H}}(R)\right) \nonumber \\&+\,\frac{o(1)}{R}. \end{aligned}$$

(4.4)

Proof

The argument is similar to (3.41)–(3.43), where we should note that the second equality of (3.41) is obtained from (3.36) [which is equivalent to (2.5)]. Note also that (2.10) and the first equality of (3.41) still hold under assumption (4.1). Since (4.1) cannot imply (3.36), we shall use the first equality of (3.41) and (2.10) to obtain that, as $\epsilon =\frac{1}{R}\rightarrow \infty$,

$$\begin{aligned} q_{\epsilon } =&\frac{1}{R}\left( \displaystyle -\pmb {\eta }'(p_0)+\mu _0\frac{f(p_0)}{\pmb {\eta }(p_0)}\right) ^{-1}\left[ \left( (N-1)+\frac{(\varvec{\mathrm {D}}{{\alpha _{\epsilon }}})(1)}{2{\alpha _{\epsilon }}(1)}+\frac{(\varvec{\mathrm {D}}{{\beta _{\epsilon }}})(1)}{2{\beta _{\epsilon }}(1)}\right) \right. \nonumber \\&\left. \times \int _{\theta _0}^{p_0}\sqrt{\frac{F(t)-F(\theta _0)}{F(p_0)-F(\theta _0)}}\,\mathrm {d}t+\frac{1}{\sqrt{2\mu _0}\epsilon }\left( \frac{{\beta _{\epsilon }}(1)}{{\alpha _{\epsilon }}(1)}-\mu _0\right) +o_{\epsilon }(1)\right] \nonumber \\ =&\frac{\sqrt{F(p_0)-F(\theta _0)}}{\mu _0f(p_0)-\pmb {\eta }(p_0)\pmb {\eta }'(p_0)}\left( \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right) +\varvec{{\mathtt {C}}_{0}}\pmb {\mathcal {H}}(R)+\frac{o(1)}{R}. \end{aligned}$$

(4.5)

As a consequence, by (2.17)–(2.18), (3.42)–(3.43) and (4.5), we get (4.3) and (4.4) and end the proof of Corollary 4.1. $\square$

At the end of this note, we take a holistic viewpoint to answer the question on the title of this section.

Remark 3

To see the effect of $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0$ on asymptotics of $\pmb {\mathrm {U}}$, we may assume $\frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0=\mu _*R^{-\tau _*}$ with $\mu _*\ne 0$ and $\tau _*>0$. We stress that different $\tau _*$ results in the various asymptotics of $\pmb {\mathrm {U}}$. More precisely, by (4.3)–(4.4) we have

$$\begin{aligned} \displaystyle |\pmb {\mathrm {U}}(R)-{p_0}|+|\pmb {\mathrm {U}}'(R)-\pmb {\eta }(p_0)|\lesssim {R^{-\min \{1,\tau _*\}}}~ \hbox {as}~ R\gg 1. \end{aligned}$$

Moreover,

• If $0<\tau _*<1$, then $\left| \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right| \gg |\pmb {\mathcal {H}}(R)|$ and

  $$\begin{aligned} \pmb {\mathrm {U}}(R)=&\,{p_0}+\frac{\mu _*\sqrt{F(p_0)-F(\theta _0)}}{\mu _0f(p_0)-\pmb {\eta }(p_0)\pmb {\eta }'(p_0)}\frac{1}{R^{\tau _*}}+\frac{o(1)}{R^{\tau _*}},\\ \pmb {\mathrm {U}}'(R)=&\,\pmb {\eta }(p_0)+\frac{\mu _*\pmb {\eta }'(p_0)\sqrt{F(p_0)-F(\theta _0)}}{\mu _0f(p_0)-\pmb {\eta }(p_0)\pmb {\eta }'(p_0)}\frac{1}{R^{\tau _*}}+\frac{o(1)}{R^{\tau _*}}. \end{aligned}$$

  Note also that if $\mu _*<0$ (resp., $>0$), there holds $\pmb {\mathrm {U}}(R)<p_0$ (resp., $>p_0$) and $\pmb {\mathrm {U}}'(R)>\pmb {\eta }(p_0)$ (resp., $<\pmb {\eta }(p_0)$) as $R\gg 1$.

• If $\tau _*=1$, then $\left| \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right| \sim |\pmb {\mathcal {H}}(R)|$ and

  $$\begin{aligned} \pmb {\mathrm {U}}(R)=&\,{p_0}+\left( \frac{\mu _*\sqrt{F(p_0)-F(\theta _0)}}{\mu _0f(p_0)-\pmb {\eta }(p_0)\pmb {\eta }'(p_0)}\frac{1}{R}+\varvec{\mathtt {C}_{0}}\pmb {\mathcal {H}}(R)\right) +\frac{o(1)}{R},\\ \pmb {\mathrm {U}}'(R)=&\,\pmb {\eta }(p_0)+\pmb {\eta }'(p_0)\left( \frac{\mu _*\sqrt{F(p_0)-F(\theta _0)}}{\mu _0f(p_0)-\pmb {\eta }(p_0)\pmb {\eta }'(p_0)}\frac{1}{R}+\varvec{\mathtt {C}_{0}}\pmb {\mathcal {H}}(R)\right) +\frac{o(1)}{R}. \end{aligned}$$

• If $\tau _*>1$, then $\left| \frac{\pmb {\beta }(R)}{\pmb {\alpha }(R)}-\mu _0\right| \ll |\pmb {\mathcal {H}}(R)|$ and

  $$\begin{aligned} \pmb {\mathrm {U}}(R)=&\,{p_0}+\varvec{\mathtt {C}_{0}}\pmb {\mathcal {H}}(R)+\frac{o(1)}{R},\\ \pmb {\mathrm {U}}'(R)=&\,\pmb {\eta }(p_0)+\pmb {\eta }'(p_0)\varvec{\mathtt {C}_{0}}\pmb {\mathcal {H}}(R)+\frac{o(1)}{R}. \end{aligned}$$