Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case

1.   Introduction and main results

This presentation will investigate the influence of the lower term of nonlinearity $f$ on the asymptotic behavior and uniqueness of large solutions $u\in W^{1, p}_{\rm loc}(\Omega)\cap C(\Omega)$ to the following quasilinear elliptic equation

where $\Delta_{p}u: = \mbox{div}(|\nabla u|^{p-2}\nabla u)$ stands for $p$\, -Laplacian operator with $1<p<N$ and $\Omega\subseteq\mathbb{R}^{N}$ $(N\geq3)$ is an exterior domain with compact smooth boundary. We say that $u\in W_{\rm loc}^{1, p}(\Omega)$ is a local weak solution to Eq. (1) in $\Omega$ means that for every sub-domain $D\Subset\Omega$, it holds

A local weak solution $u$ is said to be a large solution if $u\in C(\Omega)$ and satisfies

In this paper we assume that the nonlinearity $f$ satisfies the following hypotheses:

$\mathbf{(f_{1}):}$ $f\in C[0, \infty)$, $f(0) = 0,$ $f(t)>0$, $t>0$, and $f$ is increasing on $[0, \infty)$;

$\mathbf{(f_{2}):}$ the following generalized Keller-Osserman condition holds,

The weight $b$ satisfies the following hypotheses, not necessarily simultaneously:

$\mathbf{(b_{1}):}$ $b\in C(\Omega)$ is non-negative in $\Omega,$ where $\Omega$ is an exterior domain with compact smooth boundary;

$\mathbf{(b_{2}):}$ there exist $\theta\in \Lambda_{1}$ and positive constant $b_{0}$ such that

where $\Lambda_{1}$ denotes the set of all positive non-increasing functions $\theta\in C^{1}[R_{0}, \infty) \cap L^{1}[R_{0}, \infty)$ $(R_{0}>0)$ which satisfy

$\mathbf{(b_{3}):}$ there exist $k\in\Lambda_{2}$ and positive constant $b_{1}$ such that

where $\Lambda_{2}$ denotes the set of all positive monotonic functions $k\in C^{1}(0, \delta_{0})\cap L^{1}(0, \delta_{0})$ $(\delta_{0}>0)$ which satisfy

$\Lambda_{2}$ was first introduced by Cîrstea and Rǎdulescu [1] for non-decreasing functions and by Mohammed [11] for non-increasing functions.

When $\Omega$ is a bounded domain, $b\equiv1$ in $\Omega$ and $f(u) = u^{\gamma}$ with $\gamma>p-1$, Diaz and Letelier [3] first studied the existence of large solutions to Eq. (1). Then, when $f$ satisfies $\mathbf{(f_{1})}$ (or $f\in C(\mathbb{R})$ is positive and increasing on $\mathbb{R}$) and $\mathbf{(f_{2})}$, the existence and boundary behavior of large solutions to Eq. (1) are further investigated by Matero [9] in a bounded domain $\Omega\subseteq \mathbb{R}^{N}$ $(N\geq2)$ with a $C^{2}$-boundary. In particular, the author obtained the following boundary behavior:

where $\Phi$ is given by

If $f$ further satisfies

then $u$ satisfies

where $\Psi$ is the inverse of $\Phi$, i.e., $\Psi$ is given by

Gladiali and Porru [6] showed that if $b\equiv1$ in bounded domain $\Omega\subseteq\mathbb{R}^{N}$, $f$ satisfies $\mathbf{(f_{1})}$ and $t\mapsto F(t)t^{-p}$ is increasing for large $t$, then any large solution $u$ to Eq. (1) satisfies

Furthermore, under the additional assumption $F(t)t^{-2p}\rightarrow\infty$ as $t\rightarrow\infty$, they obtained

Based on a comparison principle, Du and Guo [4] discussed the existence, uniqueness and asymptotic behavior of various boundary blow-up solutions for a class of quasilinear elliptic equations. Let $\Omega$ be a bounded domain, $b$ satisfy $\mathbf{(b_{1})}$ and

$\mathbf{(C):}$ If there exists $x_{0}\in \Omega$ such that $b(x_{0}) = 0$, then there exists a bounded domain $\Omega_{0}$ ($\Omega_{0}\Subset\Omega$) containing $x_{0}$ such that $b(x)>0$ for all $x\in\partial\Omega_{0}$,

$f$ satisfy $\mathbf{(f_{1})}$ and $\mathbf{(f_{2})}$. Mohammed [10] showed that if the Poisson problem

has a weak solution, then Eq. (1) admits a non-negative boundary blow-up solution. Moreover, the author further established the asymptotic boundary estimates of such blow-up solutions. Later, Mohammed [11] showed that if $\Omega\subseteq\mathbb{R}^{N}$ is a bounded domain with $C^{2}$-boundary, $b$ satisfies $\mathbf{(b_{1})}$ and $\mathbf{(b_{3})}$ ($k$ is non-increasing on $(0, \delta_{0})$), $f$ satisfies $\mathbf{(f_{1})}$ and $f\in RV_{\rho+1}$ (please refer to Definition 2.1) with $\rho>p-2$, then any large solution $u\in W^{1,\,p}_{\rm loc}(\Omega)\cap C(\Omega)$ to Eq. (1) satisfies

where $\Psi$ is given by (3). By introducing some structure condition, the result of Mohammed in [11] was extended by Zhang [18] from $f\in RV_{\rho+1}$ with $\rho>p-2$ to the case that $f\in RV_{\rho+1}$ with $\rho\geq p-2$ or $f$ is rapidly varying at infinity. Moreover, the author also studied the boundary behavior of large solutions to Eq. (1) when $b$ is critical singular on the boundary. Inspired by the above results, by using Karamata regular varying theory, Wan [14] investigated the asymptotic behavior and uniqueness of entire large solutions to Eq. (1) in $\mathbb{R}^{N}$. For other related insight on Eq. (1.1), please refer to [2], [8]-[7], [15]-[16].

In this paper, by structuring a comparison function, we establish the new asymptotic behavior of large solutions to problem (1)-(2) (including the case of $p = 2$) when $f\in NRV_{p-1}$. Our results imply that the lower term of $f$ has an important influence on the asymptotic behavior of large solutions to the above exterior domain problem. Then, we further establish the uniqueness of the solutions to problem (1)-(2).

To obtain our results, we further assume that $f$ satisfies

$\mathbf{(f_{3}):}$ there exist some constant $t_{0}>0$ and two functions $f_{1}$ and $f_{2}$ such that

where $f_{1}\in C^{2}[t_{0}, \infty)$. If we denote $g(t): = \frac{tf'_{1}(t)}{f_{1}(t)}-(p-1),\,t\geq t_{0}$ and $g$ and $f_{2}$ satisfy the following conditions:

$\mathbf{(f_{4}):}$

$\mathbf{(f_{5}):}$ for any $\xi>0$

and there exists $E_{1}\neq0$ such that

or

$\mathbf{(f_{6}):}$

and there exists $\mu\leq p-1$ such that for any $\xi>0$

Our results are summarized as follows.

Theorem 1.1. Let $f$ satisfy $\mathbf{(f_{1})}$-$\mathbf{(f_{6})}$, $b$ satisfy $\mathbf{(b_{1})}$-$\mathbf{(b_{2})}$. If

then any solution $u$ to problem (1)-(2) satisfies

where $\psi$ is uniquely determined by

and

with

Theorem 1.2. Let $f$ satisfy $\mathbf{(f_{1})}$-$\mathbf{(f_{6})}$, $b$ satisfy $\mathbf{(b_{1})}$ and $\mathbf{(b_{3})}$ with one of the following conditions:

$\mathbf{(I):}$ $k$ is non-decreasing on $(0, \delta_{0});$

$\mathbf{(II):}$ $k$ is non-increasing on $(0, \delta_{0})$ with $D_{k}>1$.

When $\mathbf{(II)}$ holds, we further assume $\frac{1}{p}+\kappa_{g}>0$. Then any solution $u$ to problem (1)-(2) satisfies

where $\psi$ is uniquely determined by (5) and

where $E_{2}$ is defined by (6).

Remark 1.3. If we replace $D_{k}>1$ by $D_{k} = 1$ in $\mathbf{(II)}$ of Theorem 1.2 and further assume that

where $\tilde{C}$ is a positive constant and $H$ is the mean curvature of $\partial\Omega$, then Theorem 1.2 still holds.

Remark 1.4. In Theorems 1.1 and 1.2, the comparison function $\psi$ given by (5) can not be replaced by $\Psi$ given by the following integral equation

Remark 1.5. Some basic examples which satisfy all of our requirements in Theorems 1.1-1.2 are the following:

$\mathbf{(i):}$ $f(t) = c_{1}t^{p-1}(\ln t)^{p\alpha}+c_{2}t^{\alpha_{1}}(\ln t)^{\alpha_{2}},\,t\geq t_{0},$ where $\alpha>1,\,c_{1}>0,\,\alpha_{1}\leq p-1$ and $c_{2},\,\alpha_{2}\in\mathbb{R}$. By a straightforward calculation, we obtain that

and

$\mathbf{(ii):}$ $f(t) = c_{1}t^{p-1}\exp((\ln t)^{q})+c_{2}t^{\alpha_{1}}(\ln t)^{\alpha_{2}}\exp((\ln t)^{\alpha_{3}}),\,t\geq t_{0}$, where $c_{1}>0$, $\alpha_{1}\leq p-1$, $q,\,\alpha_{3}\in(0, 1)$, $c_{2},\,\alpha_{2}\in\mathbb{R}$. By a straightforward calculation, we obtain that

and

$\mathbf{(iii):}$ $f(t) = c_{1}t^{p-1}(\ln t)^{p}(\ln(\ln t))^{p\alpha}+c_{2}t^{\alpha_{1}}(\ln t)^{\alpha_{2}}(\ln(\ln t))^{\alpha_{3}},\,t\geq t_{0},$ where $c_{1}>0,\,\alpha>1$ and $\alpha_{1}\leq p-1,$ $c_{2},\,\alpha_{2},\,\alpha_{3}\in\mathbb{R}$. By a straightforward calculation, we obtain that

and

Theorem 1.6. Let $f$ satisfy $\mathbf{(f_{1})}$ and

$\mathbf{(f_{8}):}$ $t\mapsto f(t)t^{1-p}$ is non-decreasing on $(0, \infty)$,

$b$ satisfy $\mathbf{(b_{1})}$, and $u_{1}, u_{2}$ be arbitrary positive solutions to problem (1)-(2) and satisfy

then $u_{1} = u_{2}$ in $\Omega$.

Corollary 1.7. If $b, f$ satisfy the hypotheses in Theorems 1.1-1.2 and $\mathbf{(f_{8})}$ holds, then the solution to problem (1)-(2) is unique.

The paper is organized as follows. In Section 2, we give some bases of Karamata regular variation theory. In Section 3, we collect some preliminary considerations. The proofs of our Theorems are given in Sections 4-6, respectively.

2.   Some basic facts from Karamata regular variation theory

In this section, we introduce some preliminaries of Karamata regular variation theory which come from [12]-[13].

Definition 2.1. A positive continuous function $f$ defined on $[a,\infty)$, for some $a>0$, is called regularly varying at infinity with index $\mu$, denoted by $f \in RV_{\mu}$, if for each $\xi>0$ and some $\mu \in \mathbb R$,

In particular, when $\mu = 0$, $f$ is called slowly varying at infinity.

Clearly, if $f\in RV_\mu$, then $L(t): = f(t)/{t^\mu}$ is slowly varying at infinity.

We also see that a positive continuous function $h$ defined on $(0,a)$ for some $a>0$, is regularly varying at zero with index $\mu$ (written as $h \in RVZ_\mu$) if $t\rightarrow h(1/t)\in RV_{-\mu}$.

Proposition 2.1. $(\mathit{\mbox{Uniform Convergence Theorem}})$ If $f\in RV_\mu$, then (8) holds uniformly for $\xi \in [c_1, c_2]$ with $0<c_1<c_2$.

Proposition 2.2. $(\mathit{\mbox{Representation Theorem}})$ A function $L$ is slowly varying at infinity if and only if it may be written in the form

for some $a_1\geq a$, where the functions $\varphi$ and $y$ are continuous and for $t \rightarrow \infty$, $y(t)\rightarrow 0$ and $\varphi(t)\rightarrow c_0$, with $c_0>0$. If $\varphi\equiv c_{0},$ then $L$ is called normalized slowly varying at infinity and

is called normalized regularly varying at infinity with index $\mu$ $(\mathit{\mbox{written as}}\;f\in NRV_\mu).$

A function $f\in C^1[a_1, \infty)\,\mbox{ for some } a_1>0$ belongs to $NRV_\mu$ if and only if

Similarly, $h\in C^{1}(0, a_{1}] \mbox{ for some } a_{1}>0$ belongs to $NRVZ_{\mu}$ if and only if

3.   Auxiliary results

In this section, we collect some useful results.

Lemma 3.1. Let $\theta\in\Lambda_{1}$, then

$\mathbf{(i):}$ $\Theta'(t) = -\theta(t),\, t\geq R_{0}$, $\lim_{t\rightarrow\infty}\frac{\Theta(t)}{t\theta(t)} = D_{\theta}$, i.e., $\Theta\in NRV_{-1/D_{\theta}}$ and $\lim_{t\rightarrow\infty}\frac{\Theta(t)}{\theta(t)} = \infty;$

$\mathbf{(ii):}$ $\lim_{t\rightarrow\infty}\frac{\Theta(t)\theta'(t)}{\theta^{2}(t)} = -1-D_{\theta}$ and $\theta\in NRV_{-(1+D_{\theta})/D_{\theta}}$.

Proof. $\mathbf{(i)}$ By the definition of $\Lambda_{1}$ and the l'Hospital's rule, we obtain $\mathbf{(i)}$ holds.

$\mathbf{(ii)}$ A straightforward calculation shows that $\lim_{t\rightarrow\infty}\frac{\Theta(t)\theta'(t)}{\theta^{2}(t)} = -1-D_{\theta}.$ This combined with $\mathbf{(i)}$ implies that $\theta\in NRV_{-(1+D_{\theta})/D_{\theta}}$.

Lemma 3.2. $($[17] , Lemma 2.1$)$ Let $k\in \Lambda_{2}$, then

$\mathbf{(i):}$ $\lim_{t\rightarrow0^{+}}\frac{K(t)}{k(t)} = 0$ and $\lim_{t\rightarrow0^{+}}\frac{K(t)k'(t)}{k^{2}(t)} = 1-D_{k};$

$\mathbf{(ii):}$ when $k$ is non-decreasing, $D_{k}\in[0, 1]$; when $k$ is non-increasing, $D_{k}\geq1$;

$\mathbf{(iii):}$ when $D_{k}>0$, $k\in NRVZ_{(1-D_{k})/D_{k}};$

$\mathbf{(iv):}$ when $D_{k} = 0$, then $\lim_{t\rightarrow0^{+}}t^{-m}K(t) = 0$ for any $m>0$.

Lemma 3.3. Let $f$ satisfy $\mathbf{(f_{1})}$-$\mathbf{(f_{6})}$, $\psi$ be the unique solution of (5), then

$\mathbf{(i):}$ $\psi'(t) = -\frac{1}{(p-1)^{1/p}}(\psi(t)f_{1}(\psi(t)))^{1/p},\,t>0$ and $\lim_{t\rightarrow0^{+}}\psi(t) = \infty;$

$\mathbf{(ii):}$ $(-\psi'(t))^{p-2}\psi''(t) = \frac{1}{p(p-1)}\big(f_{1}(\psi(t))+\psi(t)f_{1}'(\psi(t))\big),\,t>0;$

$\mathbf{(iii):}$ $\lim_{t\rightarrow\infty}(g(t))^{-1}\big(\frac{f_{1}(\xi t)}{\xi^{p-1}f_{1}(t)}-1\big) = \ln \xi,\,\xi>0;$

$\mathbf{(iv):}$ $\lim_{t\rightarrow\infty}\frac{f_{2}(\xi t)}{\xi^{p-1} g(t)f_{1}(t)} = E_{2},\,\xi>0;$

$\mathbf{(v):}$ $\lim_{t\rightarrow\infty}\frac{1}{p-1}\frac{(tf_{1}(t))^{(p-1)/p}}{g(t)f_{1}(t)\int_{t}^{\infty}(sf_{1}(s))^{-1/p}ds} = \frac{1}{p-1}\big(\frac{1}{p}+\kappa_{g}\big);$

$\mathbf{(vi):}$ $\lim_{t\rightarrow0^{+}}(g(\psi(t)))^{-1}\bigg[\frac{1}{p}\bigg(1+\frac{\psi(t)f'_{1}(\psi(t))}{f_{1}(\psi(t))}\bigg)-\frac{f_{1}(\xi\psi(t))}{\xi^{p-1}f_{1}(\psi(t))}\bigg] = \frac{1}{p}-\ln\xi,\,\xi>0;$

$\mathbf{(vii):}$ $\lim_{t\rightarrow0^{+}}\frac{f_{2}(\xi\psi(t))}{\xi^{p-1} g(\psi(t))f_{1}(\psi(t))} = E_{2},\,\xi>0;$

$\mathbf{(viii):}$ $\lim_{t\rightarrow0^{+}}\frac{1}{(p-1)^{(p-1)/p}}\frac{(\psi(t)f_{1}(\psi(t)))^{(p-1)/p}}{tg(\psi(t))f_{1}(\psi(t))} = \frac{1}{p-1}\big(\frac{1}{p}+\kappa_{g}\big).$

Proof. $\mathbf{(i)}$ By the definition of $\psi$ and a direct calculation, we see that $\mathbf{(i)}$-$\mathbf{(ii)}$ hold.

$\mathbf{(iii)}$ If $\xi = 1$, the result is obvious. Otherwise, by $f\in NRV_{p-1}$, we have

It follows by $\mathbf{(f_{4})}$ and Proposition 2.1 that $\lim_{t\rightarrow\infty}\frac{g(ts)}{s} = 0 \mbox{ and }\lim_{t\rightarrow\infty}\frac{g(ts)}{g(t)} = 1$ uniformly with respect to $s\in[c_{1}, c_{2}]$. Hence, we have

By the Lebesgue's dominated convergence theorem, we obtain

On the other hand, we see that

and

It follows by (9)-(12) that $\mathbf{(iii)}$ holds.

$\mathbf{(iv)}$ Since

we see that if $\mathbf{(f_{5})}$ holds, then

if $\mathbf{(f_{6})}$ holds, then

$\mathbf{(v)}$ By $\mathbf{(f_{3})}$-$\mathbf{(f_{4})}$ and the l'Hospital's rule, we obtain

$\mathbf{(vi)}$-$\mathbf{(viii)}$ We conclude by $\mathbf{(f_{3})}$ and $\mathbf{(iii)}$-$\mathbf{(v)}$ that $\mathbf{(vi)}$-$\mathbf{(viii)}$ hold.

Lemma 3.4. $($[11], Lemma 2.2$)$ Let $\Omega$ be a bounded domain and $G:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be non-increasing in the second variable and continuous. Let $u,\,w\in W^{1,\, p}(\Omega)$ satisfy the respective inequalities

for all non-negative $\varphi\in W_{0}^{1,\, p}(\Omega)$. Then the inequality $u\leq w$ on $\partial \Omega$ implies $u\leq w$ in $\Omega$.

4.   Proof of Theorem 1.1

Proof. Take $\varepsilon\in (0, \min\{\xi_{0}, b_{0}\}/2)$ and

A simple calculation shows that

For any constant $R>R_{0}$, we define

where $R_{0}$ is given by the definition $\Lambda_{1}$ in $\mathbf{(b_{2})}$.

By Lemma 3.1 and Lemma 3.3 $\mathbf{(vi)}$-$\mathbf{(viii)}$, we see that

where

This implies that there exist a large constant $R_{\varepsilon}>R_{0}$ and a small constant $\delta_{\varepsilon}>0$ corresponding to $\varepsilon$ such that

and for any $(t, x)\in(0, 2\delta_{\varepsilon})\times\Omega_{R_{\varepsilon}}$, the following hold

In fact, we can always adjust $R_{\varepsilon}$ such that for any $x\in\Omega_{R_{\varepsilon}}$, it holds

Let $u$ be the solution of problem (1)-(2) and take

Set

where

and

where $r_{0}$ is a large enough constant such that $D_{R_{\varepsilon}+}^{\sigma}$ is an annular domain. Moreover, by the definition of $\Omega_{R_{\varepsilon}-}^{\sigma}$, we see that $D_{R_{\varepsilon}-}^{\sigma}$ is also an annular domain.

Define

By a straightforward calculation, we have for any $x\in D_{R_{\varepsilon}-}^{\sigma}$,

i.e., $\overline{u}_{\varepsilon}$ is an upper solution to Eq. (1) in $D_{R_{\varepsilon}-}^{\sigma}$. In a similar way, we can show that $\underline{u}_{\varepsilon}$ is a lower solution to Eq. (1) in $D_{R_{\varepsilon}+}^{\sigma}$.

We can choose a positive constant $M$ independent of $\sigma$ such that

Next, we prove

and

Since

we take a small enough positive constant $\rho$ such that

where

and

By (15) and (18), we have

On the other hand, combining (14) and (15), we obtain

Since $f$ is increasing on $[0, \infty)$, we see that $\overline{u}_{\varepsilon}+M$ and $u+M$ are both upper solutions in $\tilde{D}^{\sigma}_{R_{\varepsilon}-}$ and $D_{R_{\varepsilon}+}^{\sigma}$, respectively. By Lemma 3.4, we have

and

By (18)-(19), we obtain (16) holds. By (14) and (20), we obtain (17) holds. So, passing to $\sigma\rightarrow0$, we have for $x\in\Omega_{R_{\varepsilon}}$,

We obtain by Lemma 3.3 $\mathbf{(i)}$ that

Passing to $\varepsilon\rightarrow0$, we obtain (4).

5.   Proof of Theorem 1.2

Proof. Take $\varepsilon\in (0, \min\{\xi_{1}, b_{1}\}/2)$ and

A simple calculation shows that

For any $\delta>0$, we define

Since $\Omega$ is a smooth exterior domain in $\mathbb{R}^{N}$, there exists $\delta_{1}>0$ such that (please refer to Lemmas 14.16 and 14.17 in [5])

where for all $x\in\mathfrak{D}_{\delta_{1}}$ near the boundary of $\Omega$, $\bar{x}\in\partial\Omega$ is the nearest point to $x$, and $H(\bar{x})$ denotes the mean curvature of $\partial\Omega$ at $\bar{x}$.

Case 1. $k$ is non-decreasing on $(0, \delta_{0})$. By Lemma 3.2 $\mathbf{(i)}$ and Lemma 3.3 $\mathbf{(vi)}$-$\mathbf{(viii)}$, we see that

where

This implies that there exists a sufficiently small constant $0<\delta_{\varepsilon}<\frac{\delta_{1}}{2}$ corresponding to $\varepsilon$ such that for any $x\in\mathfrak{D}_{2\delta_{\varepsilon}}$, the following hold

As before, we can always adjust $\delta_{\varepsilon}$ such that for any $x\in \mathfrak{D}_{2\delta_{\varepsilon}}$, it holds

Set $\sigma\in (0, \delta_{\varepsilon})$ and define

and

Let

By a straightforward calculation, we obtain that for any $x\in D_{-}^{\sigma}$,

i.e., $\overline{u}_{\varepsilon}$ is an upper solution to Eq. (1) in $D_{-}^{\sigma}$. In a similar way, we can show that $\underline{u}_{\varepsilon}$ is a lower solution to Eq. (1) in $D_{+}^{\sigma}$.

Case 2. $k$ is non-increasing on $(0, \delta_{0})$. As before, by Lemma 3.2 $\mathbf{(i)}$ and Lemma 3.3 $\mathbf{(vi)}$-$\mathbf{(viii)}$, we obtain

where

with

By (24), we see that there exists a small enough constant $\delta_{\varepsilon}\in (0, \delta_{1}/2)$ corresponding to $\varepsilon$ such that for any $(r, x)\in (0, \delta_{\varepsilon})\times \mathfrak{D}_{2\delta_{\varepsilon}}$, the following hold

and (22) holds here for any $x\in \mathfrak{D}_{2\delta_{\varepsilon}}$.

Take $\sigma\in (0, \delta_{\varepsilon})$ and let

where $D_{\mp}^{\sigma}$ are defined as (23). A straightforward calculation shows that for any $x\in D_{-}^{\sigma}$,

i.e., $\overline{u}_{\varepsilon}$ is an upper solution to Eq. (1) in $D_{-}^{\sigma}$. In a similar way, we can show that $\underline{u}_{\varepsilon}$ is a lower solution to Eq. (1) in $D_{+}^{\sigma}$.

For case 1 and case 2, let $u$ be an arbitrary solution of problem (1)-(2). Next, we prove that there exists a large constant $M>0$ such that

Obviously, we can always take a constant $M>0$ independent of $\sigma$ such that

On the other hand, we have

This implies that we can take a small enough positive constant $\rho$ with $0<\rho<\delta_{\varepsilon}$ such that

where

Since $f$ is increasing on $[0, \infty)$, we see that $\overline{u}_{\varepsilon}+M$ and $u+M$ are both upper solutions in $\tilde{D}_{-}^{\sigma}$ and $\tilde{D}_{+}^{\sigma}$, respectively. We conclude by (26)-(27) and Lemma 3.4 that

This fact, combined with (27), shows that (25) holds. So, passing to $\sigma\rightarrow0$, we have for $x\in \mathfrak{D}_{2\delta_{\varepsilon}}$,

We obtain by Lemma 3.3 $\mathbf{(i)}$ that

Passing to $\varepsilon\rightarrow0$, we obtain (7).

6.   Proof of Theorem 1.6

Proof. Let $u_{1}$ and $u_{2}$ be two positive solutions of problem (1)-(2). By

we see that for fixed $\varepsilon>0$, there exist a sufficiently large constant $R_{\varepsilon}$ and a sufficiently small constant $\delta_{\varepsilon}$ such that

and

where $\Omega_{R_{\varepsilon}}$ and $\mathfrak{D}_{\delta_{\varepsilon}}$ are defined as (13) and (21), respectively.

Let

The condition $\mathbf{(f_{8})}$ implies that

Assume that $u_{0}$ is the unique solution for

where

It follows by Lemma 3.4 that

Since $u_{0}\equiv u_{1}$ in $\Omega_{0}$, by (28)-(29) we have

It follows by passing to $\varepsilon\rightarrow0$ that $u_{1} = u_{2}$ in $\Omega$.

Acknowledgments

The authors are greatly indebted to the editor and the anonymous referees for the very valuable suggestions and comments which improved the quality of the presentation.