Boundary behavior of large solutions to a class of Hessian equations

Abstract

In this paper, we consider the m-Hessian equation $S_{m} [D^{2} u] = b (x) f (u) > 0$ in Ω, subject to the singular boundary condition $u = \infty$ on $\partial Ω$ . We give estimates of the asymptotic behavior of such solutions near $\partial Ω$ when the nonlinear term f satisfies a new structure condition.

Keywords

Hessian equations singular boundary condition asymptotic behavior

1. Introduction and the main results

Let Ω be a bounded strictly convex smooth domain in $R^{N}$ ( $N ⩾ 2$ ). In this paper, we consider the asymptotic behavior near the boundary to the following boundary blow-up problem for the m-Hessian equation of the form $\begin{matrix} (1.1) & S_{m} [D^{2} u] = σ_{m} (λ_{1}, \dots, λ_{N}) = b (x) f (u), u > 0, x \in Ω, u |_{\partial Ω} = \infty, \end{matrix}$ where the last condition means that $u (x) \to + \infty$ as $d (x) : = dist (x, \partial Ω) \to 0$ , the operator $D^{2} u (x) = (\frac{\partial^{2} u (x)}{\partial x_{i} \partial x_{j}})$ denotes the Hessian of $u (x)$ , $λ_{1}, \dots, λ_{N}$ are eigenvalues of $D^{2} u$ and $σ_{m}$ is defined as $\begin{matrix} σ_{m} (λ_{1}, \dots, λ_{N}) = \sum_{1 ⩽ i_{1} < \dots < i_{m} ⩽ N} λ_{i_{1}} \dots λ_{i_{m}} . \end{matrix}$

It is obvious that the Laplace operator and the Monge–Ampère operator are well-known examples of Hessian operators corresponding to $m = 1$ and $m = N$ , respectively.

Suppose b satisfies

$b \in C^{\infty} (\bar{Ω})$ is positive in $\bar{Ω}$

and f satisfies

$f \in C [0, \infty) \cap C^{\infty} (0, \infty)$ is positive increasing such that $f (0) = 0$ .

A natural class of functions for the solutions to (1.1) is m-convex functions. For $m \in {1, \dots, N}$ , recall that a function $u \in C^{2} (Ω)$ is called m-convex (or strictly m-convex) if $(λ_{1}, \dots, λ_{N}) \in {\bar{Γ}}_{m}$ (or $(λ_{1}, \dots, λ_{N}) \in Γ_{m}$ ) for every $x \in Ω$ , where $λ_{1}, \dots, λ_{N}$ are the eigenvalue of $D^{2} u (x)$ and $Γ_{m}$ is the connected component ${λ \in R^{N} : σ_{m} (λ) > 0}$ containing the positive cone $\begin{matrix} Γ^{+} = {λ = (λ_{1}, \dots, λ_{N}) \in R^{N} : λ_{i} > 0, i = 1, 2, \dots, N} . \end{matrix}$ It follows from [1] that $\begin{matrix} Γ^{+} = Γ_{N} \subset \dots \subset Γ_{m + 1} \subset Γ_{m} \subset \dots \subset Γ_{1} . \end{matrix}$

For an open bounded subset Ω of $R^{N}$ with boundary of class $C^{2}$ and every $\bar{x} \in \partial Ω$ , we denote by $κ_{1} (\bar{x}), \dots, κ_{N - 1} (\bar{x})$ the principal curvatures of $\partial Ω$ at $\bar{x}$ .

For $m \in {1, 2, \dots, N - 1}$ , we define the mth curvature $ω_{m} (\bar{x})$ at $\bar{x}$ by $\begin{matrix} (1.2) & ω_{m} (\bar{x}) = σ_{m} (κ_{1} (\bar{x}), \dots, κ_{N - 1} (\bar{x})) = \sum_{1 ⩽ i_{1} < \dots < i_{m} ⩽ N - 1} κ_{i_{1}} (\bar{x}) \dots κ_{i_{m}} (\bar{x}) . \end{matrix}$

Recall that Ω is said to be m-convex if $ω_{j} (\bar{x}) ⩾ 0$ for every $\bar{x} \in \partial Ω$ and every $j \in {1, \dots, m}$ , and it is called strictly m-convex if it is m-convex and $ω_{m} (\bar{x}) > 0$ for every $\bar{x} \in \partial Ω$ . In particular, the (strict) $(N - 1)$ -convexity for domains is equivalent to the usual (strict) convexity.

When Ω is a bounded strictly convex smooth domain, one can see that there exists $R_{1} > 0$ such that for any $\bar{x} \in \partial Ω$ there is a ball $B_{1} \subset Ω$ of radius $R_{1}$ with $\bar{x} \in \partial B_{1}$ . Furthermore, there exists a constant $R_{2} > 0$ such that for any $\bar{x} \in \partial Ω$ it is possible to find a ball $B_{2}$ of radius $R_{2}$ which is tangent to Ω at $\bar{x}$ such that $Ω \subset B_{2}$ .

In the past years, increasing attention has been paid to the study of boundary blow-up problems by various approaches.

First, let us review the following blow-up problem involving the classical Laplace operator Δ, i.e. $\begin{matrix} (1.3) & Δ u = b (x) f (u), u > 0, x \in Ω, u |_{\partial Ω} = \infty, \end{matrix}$

Problem (1.3) arises in Riemannian geometry, mathematical physics or population dynamics, and was considered for the first time by Bieberbach [3]. With $N = 2$ , $b (x) = 1$ and $f (u) = e^{u}$ , the author showed that problem (1.3) has a unique solution $u (x) \in C^{2} (Ω)$ such that $u (x) + 2 ln (d (x))$ is bounded as $d (x) \to 0^{+}$ . Later, for $b (x) \equiv 1$ , Keller–Osserman [12,21] gave a necessary and sufficient condition $\begin{matrix} (1.4) & \int_{1}^{\infty} \frac{d ν}{\sqrt{2 F (ν)}} < \infty, F (ν) = \int_{0}^{ν} f (s) d s, \end{matrix}$ for the existence of solutions of problem (1.3).

It is worthwhile to point out that Cîrstea and Rǎdulescu [5–7], Rǎdulescu [23], Rǎdulescu and Ghergu [9] introduced the Karamata regular variation theory to study the boundary behavior and uniqueness of solutions for problem (1.3) and obtained a series of rich and significant information about the boundary behavior of the blow-up solutions.

Next, let us review the following blow-up problem for Monge–Ampère equation: $\begin{matrix} (1.5) & det D^{2} u = b (x) f (u), u > 0, x \in Ω, u |_{\partial Ω} = \infty . \end{matrix}$

Monge–Ampère problems which can describe Weingarten curvature or reflector shape design have been widely considered by various approaches in the past few years.

For $b \in C^{\infty} (\bar{Ω})$ with $b > 0$ on $\bar{Ω}$ and $f (u) = exp (u)$ or $f (u) = u^{p}$ with $p > N$ , Lazer and McKenna [13] showed that problem (1.5) has a unique solution $u \in C^{\infty} (\bar{Ω})$ .

When $b \in C^{\infty} (\bar{Ω})$ is positive on $\bar{Ω}$ , $f \in C^{\infty}$ meets (f₁),

the Keller–Osserman type condition $\begin{matrix} Ψ (t) = \int_{t}^{\infty} {[(N + 1) F (s)]}^{- \frac{1}{N + 1}} d s < \infty, \forall t > 0, F (s) = \int_{0}^{s} f (ν) d ν, \end{matrix}$ and $\begin{matrix} \int_{1}^{\infty} \frac{d t}{\sqrt{N \int_{0}^{t} {(f (s))}^{1 / N} d s}} < \infty . \end{matrix}$

Matero [16] proved that there exists a strictly convex solution

u \in C^{\infty} (Ω)

to problem (1.5). Moreover, for any solution u to problem (1.5), it holds

\begin{matrix} c_{1}^{1 / (N + 1)} R_{1}^{(N - 1) / (N + 1)} ⩽ lim_{d (x) \to 0} \frac{Ψ (u (x))}{d (x)} ⩽ c_{2}^{1 / (N + 1)} R_{2}^{(N - 1) / (N + 1)}, \end{matrix}

where Ψ is given as in (

f_{2}^{'}

c_{1}

and

c_{2}

are positive constants.

Later, using regular variation theory, Cîstea and Trombetti [8] established the existence of positive strictly convex solutions to problem (1.5) and gave asymptotic estimates of the behavior of such solutions near $\partial Ω$ . For more results about problem (1.5), please see [17,29,32,34] and the reference therein.

Now let us return to problem (1.1).

The problem (1.1) is an important class of fully nonlinear elliptic equations which is closely related to a geometric problem (see [27,28]). Hence, the Hessian operator is not introduced as a straightforward generalization of the Laplace or Monge–Ampère operator.

When $b \in C^{\infty} (\bar{Ω})$ is positive on $\bar{Ω}$ , $f \in C^{\infty}$ meets (f₁) and

the Keller–Osserman type condition $\begin{matrix} ψ_{m} (t) = \int_{t}^{\infty} {[(m + 1) F (s)]}^{- \frac{1}{m + 1}} d s < \infty, \forall t > 0, F (s) = \int_{0}^{s} f (ν) d ν . \end{matrix}$

Salani [25] proved that there exists a k-convex viscosity solution of

u \in C^{\infty} (Ω)

to problem (1.1). Moreover, for every solution u to problem (1.1), it holds

\begin{matrix} c_{1}^{1 / (m + 1)} p (R_{1}) ⩽ \underset{d (x) \to 0}{lim inf} \frac{ψ_{m} (u (x))}{d (x)} ⩽ \underset{d (x) \to 0}{lim sup} \frac{ψ_{m} (u (x))}{d (x)} ⩽ c_{2}^{1 / (m + 1)} p (R_{2}), \end{matrix}

where

ψ_{m}

is given as in (F₂) and

p (R) = {(\begin{matrix} N - 1 \\ m - 1 \end{matrix})}^{- 1 / (m + 1)} R^{(m - 1) / (m + 1)}

Later, under suitable condition on $b (x)$ and $f (u)$ , Ma and Li [14], Zhang and Peng [30,31], Huang [10] further derived accurately the blow-up rate of solutions to problem (1.1). For more results about blow-up solutions for fully nonlinear equations, please see [2,18–20,22] and the reference therein.

Inspired by the above works, in this paper, by Karamata regular variation theory and the method of lower and upper solutions, we investigate the new boundary asymptotic behavior of solutions to problem (1.1) when the nonlinear term f satisfies the following new structure condition

there exists $C_{f} \in (0, \infty)$ such that $\begin{matrix} lim_{s \to \infty} H^{'} (s) \int_{s}^{\infty} \frac{1}{H (ν)} d ν = C_{f}, where H (s) = f^{\frac{1}{m}} (s) . \end{matrix}$

A complete characterization of f in (f₂) is provided in the following Lemma 3.2.

The key to our estimates in this paper is the solution to the problem $\begin{matrix} (1.6) & \int_{ϕ (t)}^{\infty} \frac{d s}{{(f (s))}^{\frac{1}{m}}} = t, t > 0 . \end{matrix}$

Our main results are summarized as follows.

Theorem 1.1.
Let Ω be a bounded strictly convex smooth domain in $R^{N}$ ( $N ⩾ 2$ ), f satisfy (f₁)–(f₂), b satisfy (b₁) and
there exist $k \in Λ$ and positive constants $b_{i}$ ( $i = 1, 2$ ) such that $\begin{matrix} b_{1} : = \underset{d (x) \to 0}{lim inf} \frac{b (x)}{k^{m + 1} (d (x))} ⩽ b_{2} : = \underset{d (x) \to 0}{lim sup} \frac{b (x)}{k^{m + 1} (d (x))}, \end{matrix}$ where Λ denote the set of all positive non-decreasing functions $k \in C^{1} (0, δ_{0})$ which satisfy $\begin{matrix} lim_{t \to 0^{+}} \frac{d}{d t} (\frac{K (t)}{k (t)}) = C_{k}, where K (t) = \int_{0}^{t} k (s) d s . \end{matrix}$
If either $\begin{matrix} (1.7) & C_{f} > 1, \end{matrix}$ or $\begin{matrix} (1.8) & C_{f} = 1 and C_{k} > 0, \end{matrix}$ then for every strictly m-convex solution u of problem ( 1.1 ), it holds that $\begin{matrix} (1.9) & 1 ⩽ \underset{d (x) \to 0}{lim inf} \frac{u (x)}{ϕ (ξ_{2} K^{\frac{m + 1}{m}} (d (x)))} and \underset{d (x) \to 0}{lim sup} \frac{u (x)}{ϕ (ξ_{1} K^{\frac{m + 1}{m}} (d (x)))} ⩽ 1, \end{matrix}$ where ϕ is uniquely determined by ( 1.6 ), $\begin{matrix} (1.10) & ξ_{1} = \frac{m}{m + 1} {(\frac{m b_{1}}{M_{0} (m (C_{f} - 1 + C_{k}) + (C_{f} - 1))})}^{\frac{1}{m}} \end{matrix}$ and $\begin{matrix} (1.11) & ξ_{2} = \frac{m}{m + 1} {(\frac{m b_{2}}{m_{0} (m (C_{f} - 1 + C_{k}) + (C_{f} - 1))})}^{\frac{1}{m}}, \end{matrix}$ where $\begin{matrix} (1.12) & M_{0} : = max_{\bar{x} \in \partial Ω} ω_{m - 1} (\bar{x}) and m_{0} : = min_{\bar{x} \in \partial Ω} ω_{m - 1} (\bar{x}) . \end{matrix}$
Remark 1.1.
We note that for each $k \in Λ$ , $\begin{matrix} lim_{t \to 0^{+}} \frac{K (t)}{k (t)} = 0 and C_{k} \in [0, 1] . \end{matrix}$
Remark 1.2.
Some basic examples of the functions which satisfy (f₂) are
When $f (s) = s^{γ}$ , $s ⩾ 0$ with $γ > m$ , $C_{f} = \frac{γ}{γ - m}$ , $\begin{matrix} ϕ (t) = {(((m - γ) t) / m)}^{\frac{m}{m - γ}}, \forall t > 0 . \end{matrix}$

When $f (s) = e^{s}$ , $s ⩾ S_{0}$ for some large $S_{0} > 0$ , $C_{f} = 1$ , $\begin{matrix} ϕ (t) = m ln m - m ln t, for sufficiently small t > 0 . \end{matrix}$

When $f (s) = c_{0}^{m + 1} s^{1 + γ} \hat{L} (s)$ , $c_{0} > 0$ , $γ + 1 > m s ⩾ S_{0}$ for some large $S_{0} > 0$ and $\hat{L} (s)$ is normalized slowly varying at infinity (see Section 2), $C_{f} = \frac{γ + 1}{γ + 1 - m}$ .

The outline of this paper is as follows. In Sections 2–3, we give some preparation that will be used in the next section. The proof of Theorem 1.1 will be given in Section 4.
2. Preparation

In this section, we first give a brief account of the definition and properties of regularly varying functions involved in our papers (see [4,15,24,26]).

Definition 2.1.
A positive measurable function f defined on $[a, \infty)$ , for some $a > 0$ , is called regularly varying at infinity with index ρ, written as $f \in {RV}_{ρ}$ , if for each $ξ > 0$ and some $ρ \in R$ , $\begin{matrix} (2.1) & lim_{s \to \infty} \frac{f (ξ s)}{f (s)} = ξ^{ρ} . \end{matrix}$ In particular, when $ρ = 0$ , f is called slowly varying at infinity.

Clearly, if $f \in {RV}_{ρ}$ , then $L (s) : = f (s) / s^{ρ}$ is slowly varying at infinity.
Definition 2.2.
A positive measurable function f defined on $[a, \infty)$ , for some $a > 0$ , is called rapidly varying at infinity if for each $ρ > 1$ $\begin{matrix} (2.2) & lim_{s \to \infty} \frac{f (s)}{s^{ρ}} = \infty . \end{matrix}$

We also see that a positive measurable function g defined on $(0, a)$ for some $a > 0$ , is regularly varying at zero with index σ (written as $g \in {RVZ}_{σ}$ ) if $t \to g (1 / t)$ belongs to ${RV}_{- σ}$ . Similarly, g is called rapidly varying at zero if $t \to g (1 / t)$ is rapidly varying at infinity.
Proposition 2.1 (Uniform convergence theorem).

If $f \in {RV}_{ρ}$ , then (2.1) holds uniformly for $ξ \in [c_{1}, c_{2}]$ with $0 < c_{1} < c_{2}$ . Moreover, if $ρ < 0$ , then uniform convergence holds on intervals of the form $(a_{1}, \infty)$ with $a_{1} > 0$ ; if $ρ > 0$ , then uniform convergence holds on intervals $(0, a_{1}]$ provided f is bounded on $(0, a_{1}]$ for all $a_{1} > 0$ .

Proposition 2.2 (Representation theorem).

A function L is slowly varying at infinity if and only if it may be written in the form $\begin{matrix} (2.3) & L (s) = φ (s) exp (\int_{a_{1}}^{s} \frac{y (τ)}{τ} d τ), s ⩾ a_{1}, \end{matrix}$ for some $a_{1} ⩾ a$ , where the functions φ and y are measurable and for $s \to \infty$ , $y (s) \to 0$ and $φ (s) \to c_{0}$ , with $c_{0} > 0$ .

We call that $\begin{matrix} (2.4) & \hat{L} (s) = c_{0} exp (\int_{a_{1}}^{s} \frac{y (τ)}{τ} d τ), s ⩾ a_{1}, \end{matrix}$ is normalized slowly varying at infinity and $\begin{matrix} (2.5) & f (s) = c_{0} s^{ρ} \hat{L} (s), s ⩾ a_{1}, \end{matrix}$ is normalized regularly varying at infinity with index ρ (and written as $f \in {NRV}_{ρ}$ ).

Similarly, g is called normalized regularly varying at zero with index ρ, written as $g \in {NRVZ}_{ρ}$ if $t \to g (1 / t)$ belongs to ${NRV}_{- ρ}$ .

A function $f \in {RV}_{ρ}$ belongs to ${NRV}_{ρ}$ if and only if $\begin{matrix} (2.6) & f \in C^{1} [a_{1}, \infty) for some a_{1} > 0 and lim_{s \to \infty} \frac{s f^{'} (s)}{f (s)} = ρ . \end{matrix}$

Proposition 2.3.
If functions L, $L_{1}$ are slowly varying at infinity, then
$L^{σ}$ for every $σ \in R$ , $c_{1} L + c_{2} L_{1}$ ( $c_{1} ⩾ 0$ , $c_{2} ⩾ 0$ with $c_{1} + c_{2} > 0$ ), $L \circ L_{1}$ (if $L_{1} (t) \to + \infty$ as $t \to + \infty$ ), are also slowly varying at infinity.

For every $θ > 0$ and $t \to + \infty$ , $t^{θ} L (t) \to + \infty$ , $t^{- θ} L (t) \to 0$ .

For $ρ \in R$ and $t \to + \infty$ , $\frac{ln (L (t))}{ln t} \to 0$ and $\frac{ln (t^{ρ} L (t))}{ln t} \to ρ$ .

Proposition 2.4.

If $f_{1} \in R V_{ρ_{1}}$ , $f_{2} \in R V_{ρ_{2}}$ with ${lim}_{t \to \infty} f_{2} (t) = \infty$ , then $f_{1} \circ f_{2} \in R V_{ρ_{1} ρ_{2}}$ .

If $f \in {RV}_{ρ}$ , then $f^{α} \in {RV}_{ρ α}$ for every $α \in R$ .

Proposition 2.5 (Asymptotic behavior).

If a function L is slowly varying at infinity, then for $a ⩾ 0$ and $t \to \infty$ ,

$\int_{a}^{t} s^{ρ} L (s) d s ≅ {(ρ + 1)}^{- 1} t^{1 + ρ} L (t)$ , for $ρ > - 1$ ;

$\int_{t}^{\infty} s^{ρ} L (s) d s ≅ {(- ρ - 1)}^{- 1} t^{1 + ρ} L (t)$ , for $ρ < - 1$ .

3. Some auxiliary results

In this section, we collect some useful results that will be used in the proof of the theorem.

Lemma 3.1 (See [33]).

Let $k \in Λ$ . Then

${lim}_{t \to 0^{+}} \frac{K (t)}{k (t)} = 0$ , ${lim}_{t \to 0^{+}} \frac{K (t)}{t k (t)} = C_{k}$ ;

${lim}_{t \to 0^{+}} \frac{K (t) k^{'} (t)}{k^{2} (t)} = 1 - C_{k}$ ;

when $C_{k} \in (0, 1)$ , ${lim}_{t \to 0^{+}} \frac{K (t)}{k (t)} ln K (t) = 0$ ;

when $C_{k} \in (0, 1)$ , $k \in {NRVZ}_{(1 - C_{k}) / C_{k}}$ .

Lemma 3.2.
Let f satisfy (f₁).
If f satisfies (f₂), then $C_{f} ⩾ 1$ ;

if f satisfies (f₂), then for some $a > 0$ $\begin{matrix} \int_{a}^{\infty} \frac{1}{H (ν)} d ν < \infty; \end{matrix}$

if f satisfies (f₂), then f satisfies the Keller–Osserman type condition (F₂);

(f₂) holds for $C_{f} \in (1, \infty)$ if and only if $f \in {NRV}_{γ}$ with $γ > 0$ . In the case $γ = m C_{f} / (1 - C_{f})$ ;

if (f₂) holds with $C_{f} = 1$ , then f is rapidly varying at infinity;

if $\begin{matrix} (3.1) & lim_{s \to 0^{+}} \frac{f^{″} (s) f (s)}{{(f^{'} (s))}^{2}} = 1, \end{matrix}$ then f satisfies (f₂) with $C_{f} = 1$ .

Proof.
(i) Let $\begin{matrix} I (s) = \frac{1}{m} f^{\frac{1}{m} - 1} (s) f^{'} (s) \int_{s}^{\infty} f^{- 1 / m} (ν) d ν, \forall s \in (0, \infty) . \end{matrix}$

Integrate $I (t)$ from a ( $a > 0$ ) to s and integrate by parts, we obtain that $\begin{matrix} \int_{a}^{s} I (t) d t = f^{\frac{1}{m}} (s) \int_{s}^{\infty} \frac{1}{f^{\frac{1}{m}} (ν)} d ν - f^{\frac{1}{m}} (a) \int_{a}^{\infty} \frac{1}{f^{\frac{1}{m}} (ν)} d ν + s - a, \forall s \in (a, \infty) . \end{matrix}$ It follows from the l’Hospital’s rule that $\begin{matrix} (3.2) & 0 ⩽ lim_{s \to \infty} \frac{f^{1 / m} (s)}{s} \int_{s}^{\infty} \frac{1}{f^{\frac{1}{m}} (ν)} d ν = lim_{s \to \infty} \frac{1}{s} \int_{a}^{s} I (t) d t - 1 = C_{f} - 1, \end{matrix}$ i.e., $C_{f} ⩾ 1$ . So (i) holds.

(ii) Let $\begin{matrix} I (s) = H^{'} (s) \int_{s}^{\infty} H (ν) d ν = \frac{1}{m} f^{\frac{1}{m} - 1} (s) f^{'} (s) \int_{s}^{\infty} f^{- 1 / m} (ν) d ν, \forall s \in (0, \infty) . \end{matrix}$ Integrate $I (t)$ from a ( $a > 0$ ) to s and integrate by parts, we obtain that $\begin{matrix} \int_{a}^{s} I (t) d t = f^{\frac{1}{m}} (s) \int_{s}^{\infty} \frac{1}{f^{\frac{1}{m}} (ν)} d ν - f^{\frac{1}{m}} (a) \int_{a}^{\infty} \frac{1}{f^{\frac{1}{m}} (ν)} d ν + s - a, \forall s \in (a, \infty), \end{matrix}$ It follows from l’Hospital’s rule that $\begin{matrix} (3.3) & 0 ⩽ lim_{s \to \infty} \frac{f^{1 / m} (s)}{s} \int_{s}^{\infty} \frac{1}{f^{\frac{1}{m}} (ν)} d ν = lim_{s \to \infty} \frac{1}{s} \int_{a}^{s} I (t) d t - 1 = C_{f} - 1 . \end{matrix}$ For any $q \in (1, \frac{C_{f}}{C_{f} - 1})$ , by (3.2), one can see that $\begin{matrix} (3.4) & lim_{s \to \infty} (H^{'} (s) - q \frac{H (s)}{s}) \int_{s}^{\infty} \frac{1}{H (ν)} d ν = C_{f} - q (C_{f} - 1) . \end{matrix}$ Then there exists $S_{0} > 0$ such that $\begin{matrix} {(\frac{f^{\frac{1}{m}} (s)}{s^{q}})}^{'} = \frac{1}{s^{q}} (H^{'} (s) - q \frac{H (s)}{s}) > 0, \forall s > S_{0}, \end{matrix}$ i.e., $\frac{f^{\frac{1}{m}} (s)}{s^{q}}$ is increasing on $[S_{0}, \infty)$ . So does $\frac{f (s)}{s^{m q}}$ . Hence, there exists $C (q) \in (0, \infty]$ such that $\begin{matrix} (3.5) & lim_{s \to \infty} \frac{f (s)}{s^{m q}} = C (q) . \end{matrix}$ Consequently, there exist $S_{1} > 0$ and $c (q) \in (0, C (q))$ such that $\begin{matrix} \frac{f (s)}{s^{m q}} > c (q), \forall s > S_{1} . \end{matrix}$ Since $q > 1$ , we can get for some $a > 0$ $\begin{matrix} \int_{a}^{\infty} \frac{1}{H (ν)} d ν < \infty . \end{matrix}$

(iii) It follows by (3.5) that there exist $S_{2} > 0$ such that $\begin{matrix} F (s) ⩾ \frac{c (q)}{2 m q + 1} s^{m q + 1}, \forall s > S_{2}, \end{matrix}$ i.e. the Keller–Osserman type condition (F₂) holds.

(iv) When (f₂) holds with $C_{f} \in (1, \infty)$ , it follows by (3.2) that $\begin{matrix} (3.6) & lim_{s \to \infty} \frac{f (s)}{s f^{'} (s)} = lim_{s \to \infty} \frac{f^{1 / m} (s) \int_{s}^{\infty} \frac{1}{f^{1 / m} (ν)} d ν}{s f^{'} (s) \int_{s}^{\infty} \frac{1}{f^{1 / m} (ν)} d ν f^{\frac{1}{m} - 1} (s)} = \frac{C_{f} - 1}{m C_{f}}, \end{matrix}$ i.e., $f \in {NRV}_{m C_{f} / (1 - C_{f})}$ .

Conversely, when $f \in {NRV}_{γ}$ with $γ > 0$ , i.e., ${lim}_{s \to \infty} \frac{s f^{'} (s)}{f (s)} = γ$ and there exist sufficiently large $S_{0}$ such that $\begin{matrix} f (s) = c_{0} s^{γ} \hat{L} (s), s ⩾ S_{0}, \end{matrix}$ where $\hat{L}$ is normalized slowly varying at infinity.

It follows by (2.6) and Proposition 2.5(i) that $\begin{array}{rcl} lim_{s \to \infty} \frac{1}{m f^{1 - \frac{1}{m}} (s)} f^{'} (s) \int_{s}^{\infty} f^{- 1 / m} (ν) d ν & = & \frac{1}{m} lim_{s \to \infty} \frac{s f^{'} (s)}{f (s)} lim_{s \to \infty} \frac{f^{1 / m} (s)}{s} \int_{s}^{\infty} f^{- 1 / m} (ν) d ν \\ = & \frac{γ}{m} lim_{s \to \infty} s^{\frac{γ}{m} - 1} {(\hat{L} (s))}^{\frac{1}{m}} \int_{s}^{\infty} ν^{- \frac{γ}{m}} {(\hat{L} (ν))}^{- \frac{1}{m}} d ν \\ = & \frac{γ}{γ - m} = C_{f} . \end{array}$

(v) By $C_{f} = 1$ and the proof of (ii), we see that ${lim}_{s \to \infty} \frac{f (s)}{s f^{'} (s)} = 0$ , i.e., ${lim}_{s \to \infty} \frac{s f^{'} (s)}{f (s)} = \infty$ . Consequently, for an arbitrary $ρ > 1$ , there exists $S_{0} > 0$ such that $\begin{matrix} \frac{f^{'} (s)}{f (s)} > (ρ + 1) s^{- 1}, \forall s > S_{0} . \end{matrix}$ Integrating the above inequality from $S_{0}$ to s, we obtain $\begin{matrix} ln (f (s)) - ln (f (S_{0})) > (ρ + 1) (ln s - ln S_{0}), \forall s > S_{0}, \end{matrix}$ i.e., $\begin{matrix} \frac{f (s)}{s^{ρ}} > \frac{f (S_{0}) s}{S_{0}^{ρ}}, \forall s > S_{0} . \end{matrix}$ Letting $s \to \infty$ , we see by Definition 2.2 that f is rapidly varying at infinity.

(vi) By (3.1) and the l’Hospital’s rule, we obtain that $\begin{matrix} (3.7) & lim_{s \to \infty} \frac{f (s)}{s f^{'} (s)} = lim_{s \to \infty} \frac{\frac{f (s)}{f^{'} (s)}}{s} = lim_{s \to \infty} \frac{d}{d s} (\frac{f (s)}{f^{'} (s)}) = 1 - lim_{s \to \infty} \frac{f (s) f^{″} (s)}{{(f^{'} (s))}^{2}} = 0 . \end{matrix}$ Consequently, for an arbitrary $p > m$ , there exists $S_{1} > 0$ such that $\begin{matrix} \frac{f^{'} (s)}{f (s)} > \frac{p + m}{s}, s ⩾ S_{1} . \end{matrix}$ Integrating from $S_{1}$ to s, we obtain $\begin{matrix} f (s) > \frac{f (S_{1})}{S_{1}^{p}} s^{p + m}, s ⩾ S_{1} . \end{matrix}$ Hence, $\begin{matrix} {(f (s))}^{\frac{1}{m}} > {(\frac{f (S_{1})}{S_{1}^{p}})}^{\frac{1}{m}} s^{\frac{p}{m} + 1}, s ⩾ S_{1} \end{matrix}$ and $\begin{matrix} lim_{s \to \infty} \frac{f^{\frac{1}{m}} (s)}{s} = \infty . \end{matrix}$ It follows by (3.7) that $\begin{matrix} (3.8) & lim_{s \to \infty} \frac{f^{1 - \frac{1}{m}} (s)}{f^{'} (s)} = lim_{s \to \infty} \frac{f (s)}{s f^{'} (s)} \frac{s}{f^{\frac{1}{m}} (s)} = lim_{s \to \infty} \frac{f (s)}{s f^{'} (s)} lim_{s \to \infty} \frac{s}{f^{\frac{1}{m}} (s)} = 0 . \end{matrix}$ By the l’Hospital’s rule and (3.8), we get that $\begin{array}{l} lim_{s \to \infty} \frac{1}{m} f^{\frac{1}{m} - 1} (s) f^{'} (s) \int_{s}^{\infty} f^{- 1 / m} (ν) d ν \\ = lim_{s \to \infty} \frac{1}{m} \frac{\int_{s}^{\infty} f^{- 1 / m} (ν) d ν}{\frac{f^{1 - \frac{1}{m}} (s)}{f^{'} (s)}} = lim_{s \to \infty} \frac{1}{m} \frac{- 1}{1 - \frac{1}{m} - \frac{f^{″} (s) f (s)}{{(f^{'} (s))}^{2}}} \\ = 1, \end{array}$ i.e. $C_{f} = 1$ . □
Lemma 3.3.
Let f satisfy (f₁)–(f₂) and ϕ be the solution to the problem $\begin{matrix} \int_{ϕ (t)}^{\infty} \frac{d s}{{(f (s))}^{\frac{1}{m}}} = t, \forall t > 0 . \end{matrix}$ Then
$ϕ^{'} (t) = - {(f (ϕ (t)))}^{\frac{1}{m}}$ , $ϕ (t) > 0$ , $t > 0$ , $ϕ (0) : = {lim}_{t \to 0^{+}} ϕ (t) = + \infty$ and $ϕ^{″} (t) = \frac{1}{m} {(f (ϕ (t)))}^{\frac{2}{m} - 1} f^{'} (ϕ (t))$ , $t > 0$ ;

$ϕ \in {NRVZ}_{1 - C_{f}}$ and $ϕ^{'} \in {NRVZ}_{- C_{f}}$ ;

${lim}_{t \to 0^{+}} \frac{1}{ϕ (ξ K (t))} = 0$ uniformly for $ξ \in [c_{1}, c_{2}]$ with $0 < c_{1} < c_{2}$ .

Proof.
By the definition of ϕ and a direct calculation, we show that (i) holds.

(ii) It follows from (i), (3.2) and (f₂) that $\begin{array}{rcl} lim_{t \to 0^{+}} \frac{t ϕ^{'} (t)}{ϕ (t)} & = & lim_{t \to 0^{+}} \frac{- t {(f (ϕ (t)))}^{\frac{1}{m}}}{ϕ (t)} \\ = & - lim_{s \to \infty} \frac{{(f (s))}^{\frac{1}{m}} \int_{s}^{\infty} \frac{d ν}{{(f (ν))}^{\frac{1}{m}}}}{s} = 1 - C_{f}, \end{array}$ i.e., $ϕ \in {NRVZ}_{1 - C_{f}}$ , and $\begin{array}{rcl} lim_{t \to 0^{+}} \frac{t ϕ^{″} (t)}{ϕ^{'} (t)} & = & - \frac{1}{m} lim_{t \to 0^{+}} \frac{f^{'} (ϕ (t)) {(f (ϕ (t)))}^{\frac{1}{m}} \int_{ϕ (t)}^{\infty} {(f (ν))}^{- \frac{1}{m}} d ν}{f (ϕ (t))} \\ = & - \frac{1}{m} lim_{s \to \infty} \frac{f^{'} (s) {(f (s))}^{\frac{1}{m}} \int_{s}^{\infty} {(f (ν))}^{- \frac{1}{m}} d ν}{f (s)} \\ = & - C_{f} . \end{array}$

(iii) Since $K (0) = 0$ , the result follows by (i). □

4. Proof of the theorem

In this section, we prove Theorem 1.1.

First, we need the following result.

Lemma 4.1 (The comparison principle, [11], Lemma 2.1).

Suppose that $u, v \in C^{2} (Ω)$ are m-convex satisfying $\begin{matrix} S_{m} (D^{2} u) ⩾ ψ (x, u, D u) and S_{m} (D^{2} v) ⩽ φ (x, v, D v) in Ω, \end{matrix}$ where $ψ, φ \in C^{1} (Ω \times R \times R^{N})$ , such that $\begin{matrix} ψ (x, z, p) ⩾ φ (x, z, p), \forall (x, z, p) \in (Ω \times R \times R^{N}) . \end{matrix}$ Suppose that $Ω \subset R^{N}$ is a bounded domain, and $u, v \in C (\bar{Ω})$ satisfy $u ⩽ v$ on $\partial Ω$ . If either $ψ_{z} (x, z, p) > 0$ or $φ_{z} (x, z, p) > 0$ for any $(x, z, p) \in (Ω \times R \times R^{N})$ , then $u ⩽ v$ in Ω.

Lemma 4.2 ([10], Proposition 2.1).

Let Ω be an open subset of $R^{N}$ with $N ⩾ 2$ . If $h \in C^{2} (R)$ and $g \in C^{2} (R)$ then $\begin{array}{rcl} σ_{m} (D^{2} h (g (x))) & = & {(h^{'} (g (x)))}^{m - 1} h^{″} (g (x)) σ_{m - 1} (D^{2} g |_{i, j}) g_{i} g_{j} \\ + {(h^{'} (g (x)))}^{m} σ_{m} (D^{2} g), for all x \in Ω, \end{array}$ where $D^{2} g |_{i, j}$ is the cofactor of the $(i, j)$ th entry of the symmetric matrix $D^{2} g (x)$ .

First fix $ε > 0$ . For any $δ > 0$ , we define $Ω_{δ} = {x \in Ω : 0 < d (x) < δ}$ . Since Ω is $C^{l}$ -smooth for $l ⩾ 2$ , choose $δ_{1} \in (0, δ_{0})$ ( $δ_{0}$ is given as in Λ) such that $d \in C^{l} (Ω_{δ_{1}})$ and $| \nabla d (x) | = 1$ , $\forall x \in Ω_{δ_{1}}$ .

Let $\bar{x}$ be the projection of the point $x \in Ω_{δ_{1}}$ to $\partial Ω$ , and $κ_{i} (\bar{x})$ , $i = 1, \dots, N - 1$ are the principal curvatures of $\partial Ω$ at $\bar{x}$ , then $\begin{matrix} D^{2} (d (x)) = diag [\frac{- κ_{1} (\bar{x})}{1 - d (x) κ_{1} (\bar{x})}, \dots, \frac{- κ_{N - 1} (\bar{x})}{1 - d (x) κ_{N - 1} (\bar{x})}, 0] . \end{matrix}$

Lemma 4.3 ([10], Corollary 2.3).

Let Ω be bounded with $\partial Ω \in C^{l}$ for $l ⩾ 2$ . Assume that $δ_{1} > 0$ is small such that $d \in C^{2} (Ω_{δ_{1}})$ and h is a $C^{2}$ -function on $(0, δ_{1})$ . Let $x \in Ω_{δ_{1}} ∖ \partial Ω$ and $\bar{x} \in \partial Ω$ be such that $| x - \bar{x} | = d (x)$ . Then, we have $\begin{array}{rcl} σ_{m} (D^{2} h (d (x))) & = & {(- h^{'} (d (x)))}^{m - 1} h^{″} (d (x)) σ_{m - 1} (ζ_{1}, \dots, ζ_{N - 1}) \\ + {(- h^{'} (d (x)))}^{m} σ_{m} (ζ_{1}, \dots, ζ_{N - 1}), for all x \in Ω, \end{array}$ where $\begin{matrix} ζ_{i} = \frac{κ_{i} (\bar{x})}{1 - d (x) κ_{i} (\bar{x})} . \end{matrix}$

Proof of Theorem 1.1.

For an arbitrary $ε \in (0, min {\frac{1}{2}, \frac{b_{1}}{2}})$ , let $\begin{array}{l} τ_{1} = \frac{m}{m + 1} {(\frac{m (b_{1} - ε) (1 - ε) - ε}{M_{0} (m (C_{f} - 1 + C_{k}) + (C_{f} - 1))})}^{\frac{1}{m}}; \\ τ_{2} = \frac{m}{m + 1} {(\frac{m (b_{2} + ε) (1 + ε) + ε}{m_{0} (m (C_{f} - 1 + C_{k}) + (C_{f} - 1))})}^{\frac{1}{m}}, \end{array}$ where $m_{0}$ , $M_{0}$ are given as in (1.12), $b_{1}$ and $b_{2}$ are given as in (b₂).

By using Lemma 3.1(ii) and Lemma 3.3(ii), we get that $\begin{array}{l} lim_{d (x) \to 0^{+}} \frac{K (d (x))}{k (d (x))} = 0; \\ lim_{d (x) \to 0^{+}} \frac{K (d (x)) k^{'} (d (x))}{k^{2} (d (x))} = 1 - C_{k}; \\ lim_{d (x) \to 0^{+}} \frac{(τ_{i} K^{\frac{m + 1}{m}} (d (x))) ϕ^{″} (τ_{i} K^{\frac{m + 1}{m}} (d (x)))}{- ϕ^{'} (τ_{i} K^{\frac{m + 1}{m}} (d (x)))} = C_{f}, i = 1, 2 . \end{array}$ It follows by (b₁)–(b₂) and (1.7)–(1.8) that there is $δ_{ε} \in (0, δ_{1} / 2)$ (which is corresponding to ε) sufficiently small such that

$(b_{1} - ε) k^{m + 1} (d (x) - ρ) ⩽ (b_{1} - ε) k^{m + 1} (d (x)) < b (x)$ , $x \in D_{ρ}^{-} = Ω_{2 δ_{ε}} / {\bar{Ω}}_{ρ}$ ; $b (x) < (b_{2} + ε) k^{m + 1} (d (x)) ⩽ (b_{2} + ε) k^{m + 1} (d (x) + ρ)$ , $x \in D_{ρ}^{+} = Ω_{2 δ_{ε} - ρ}$ , where $ρ \in (0, δ_{ε})$ .

$\begin{matrix} 1 - ε < \prod_{i = 1}^{i = m - 1} (1 - d (x) κ_{i} (\bar{x})) < 1 + ε, \forall x \in Ω_{2 δ_{ε}}, \end{matrix}$ and for $x \in Ω_{2 δ_{ε}}$ $\begin{array}{l} \frac{M_{0}}{1 - ε} [τ_{1}^{m} {(\frac{m + 1}{m})}^{m + 1} \frac{ϕ^{″} (τ_{1} K^{\frac{m + 1}{m}} (d (x))) τ_{1} K^{\frac{m + 1}{m}} (d (x))}{- ϕ^{'} (τ_{1} K^{\frac{m + 1}{m}} (d (x)))} - τ_{1}^{m} \frac{1}{m} {(\frac{m + 1}{m})}^{m} \\ - τ_{1}^{m} {(\frac{m + 1}{m})}^{m} \frac{K (d (x)) k^{'} (d (x))}{k^{2} (d (x))}] + τ_{1}^{m} {(\frac{m + 1}{m})}^{m} \frac{K (d (x))}{k (d (x))} σ_{m} (ζ_{1} \dots ζ_{N - 1}) \\ (4.1) & - (b_{1} - ε) < 0; \\ \frac{m_{0}}{1 + ε} [τ_{2}^{m} {(\frac{m + 1}{m})}^{m + 1} \frac{ϕ^{″} (τ_{2} K^{\frac{m + 1}{m}} (d (x))) τ_{2} K^{\frac{m + 1}{m}} (d (x))}{- ϕ^{'} (τ_{2} K^{\frac{m + 1}{m}} (d (x)))} - τ_{2}^{m} \frac{1}{m} {(\frac{m + 1}{m})}^{m} \\ - τ_{2}^{m} {(\frac{m + 1}{m})}^{m} \frac{K (d (x)) k^{'} (d (x))}{k^{2} (d (x))}] + τ_{2}^{m} {(\frac{m + 1}{m})}^{m} \frac{K (d (x))}{k (d (x))} σ_{m} (ζ_{1} \dots ζ_{N - 1}) \\ (4.2) & - (b_{2} + ε) > 0; \end{array}$

for $x \in Ω_{2 δ_{ε}}$ $\begin{matrix} \frac{(τ_{i} K^{\frac{m + 1}{m}} (d (x))) ϕ^{″} (τ_{i} K^{\frac{m + 1}{m}} (d (x)))}{- ϕ^{'} (τ_{i} K^{\frac{m + 1}{m}} (d (x)))} \frac{m + 1}{m} - \frac{1}{m} - \frac{k^{'} (d (x)) K (d (x))}{k^{2} (d (x))} > 0 . \end{matrix}$

Let

\begin{array}{l} (4.3) & d_{1} (x) = d (x) - ρ, d_{2} (x) = d (x) + ρ, \\ (4.4) & {\bar{u}}_{ε} = ϕ (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x))), x \in D_{ρ}^{-} and {\underline{u}}_{ε} = ϕ (τ_{2} K^{\frac{m + 1}{m}} (d_{2} (x))), x \in D_{ρ}^{+} . \end{array}

It follows by Lemma 3.3 (

h = ϕ (τ_{1} K^{\frac{m + 1}{m}} (t))

) and a direct computation that for

x \in D_{ρ}^{-}

\begin{array}{l} S_{m} [D^{2} {\bar{u}}_{ε}] - (b_{1} - ε) k^{m + 1} (d_{1} (x)) f (ϕ (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x)))) \\ = k^{m + 1} (d_{1} (x)) {(- ϕ^{'} (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x))))}^{m} \\ \times {[{(\frac{m + 1}{m} τ_{1})}^{m + 1} {(- ϕ^{'} (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x))))}^{- 1} (ϕ^{″} (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x)))) K^{\frac{m + 1}{m}} (d_{1} (x)) \\ - τ_{1}^{m} \frac{1}{m} {(\frac{m + 1}{m})}^{m} - τ_{1}^{m} {(\frac{m + 1}{m})}^{m} \frac{K (d_{1} (x)) k^{'} (d_{1} (x))}{k^{2} (d_{1} (x))}] σ_{m - 1} (ζ_{1} \dots ζ_{N - 1}) \\ + τ_{1}^{m} {(\frac{m + 1}{m})}^{m} \frac{K (d_{1} (x))}{k (d_{1} (x))} σ_{m} (ζ_{1} \dots ζ_{N - 1}) - (b_{1} - ε)} \\ ⩽ 0, \end{array}

i.e.,

{\bar{u}}_{ε}

is a supersolution of equation (1.1) in

D_{ρ}^{-}

Moreover, by (r₃), we can get that for $x \in D_{ρ}^{-}$ $\begin{array}{l} S_{i} [D^{2} {\bar{u}}_{ε}] \\ = k^{i + 1} (d_{1} (x)) {(- ϕ^{'} (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x))))}^{i} K^{\frac{i}{m} - 1} (d_{1} (x)) \\ \times {[{(\frac{m + 1}{m} τ_{1})}^{i + 1} {(- ϕ^{'} (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x))))}^{- 1} (ϕ^{″} (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x)))) K^{\frac{m + 1}{m}} (d_{1} (x)) \\ - \frac{1}{m} {(\frac{m + 1}{m} τ_{1})}^{i} - {(\frac{m + 1}{m} τ_{1})}^{i} \frac{K (d_{1} (x)) k^{'} (d_{1} (x))}{k^{2} (d_{1} (x))}] σ_{i - 1} (ζ_{1} \dots ζ_{N - 1}) \\ + {(τ_{1} \frac{m + 1}{m})}^{i} \frac{K (d_{1} (x))}{k (d_{1} (x))} σ_{i} (ζ_{1} \dots ζ_{N - 1})} \\ > 0, i = 1, 2, \dots, N . \end{array}$ This implies that ${\bar{u}}_{ε}$ is strictly convex in $D_{ρ}^{-}$ .

Hence, ${\bar{u}}_{ε}$ is a strictly convex supersolution of equation (1.1) in $D_{ρ}^{-}$ .

In a similar way, we can show that $\begin{matrix} {\underline{u}}_{ε} = ϕ (τ_{2} K^{\frac{m + 1}{m}} (d_{2} (x))) \end{matrix}$ is a strictly convex subsolution of equation (1.1) in $D_{ρ}^{+}$ .

Let $u \in C^{\infty} (Ω)$ be an arbitrary strictly m-convex solution to problem (1.1). Choose M large enough such that $\begin{matrix} (4.5) & u (x) ⩽ M + {\bar{u}}_{ε} (x) on d (x) = 2 δ_{ε} and {\underline{u}}_{ε} (x) ⩽ u (x) + M on d (x) = 2 δ_{ε} - ρ . \end{matrix}$ We observe that ${\bar{u}}_{ε} (x) \to \infty$ as $d (x) \to ρ$ and $u |_{\partial Ω} = \infty > {\underline{u}}_{ε} |_{\partial Ω}$ . It follows from Lemma 4.1 that $\begin{matrix} u (x) ⩽ M + {\bar{u}}_{ε} in D_{ρ}^{-} and {\underline{u}}_{ε} ⩽ u + M in D_{ρ}^{+} . \end{matrix}$ i.e. $\begin{matrix} 1 - \frac{M}{ϕ (τ_{2} K^{\frac{m + 1}{m}} (d_{2} (x)))} ⩽ \frac{u (x)}{ϕ (τ_{2} K^{\frac{m + 1}{m}} (d_{2} (x)))}, x \in D_{ρ}^{+} \end{matrix}$ and $\begin{matrix} \frac{u (x)}{ϕ (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x)))} ⩽ 1 + \frac{M}{ϕ (τ_{1} K^{\frac{m + 1}{m}} (d_{1} (x)))}, x \in D_{ρ}^{-} . \end{matrix}$ Letting $ρ \to 0$ for $x \in D_{ρ}^{-} \cap D_{ρ}^{+}$ , we obtain $\begin{matrix} 1 - \frac{M}{ϕ (τ_{2} K^{\frac{m + 1}{m}} (d (x)))} ⩽ \frac{u (x)}{ϕ (τ_{2} K^{\frac{m + 1}{m}} (d (x)))}, \end{matrix}$ and $\begin{matrix} \frac{u (x)}{ϕ (τ_{1} K^{\frac{m + 1}{m}} (d (x)))} ⩽ 1 + \frac{M}{ϕ (τ_{1} K^{\frac{m + 1}{m}} (d (x)))} . \end{matrix}$ Then, combining with Lemma 3.3(iii), we obtain $\begin{matrix} 1 ⩽ \underset{d (x) \to 0}{lim inf} \frac{u (x)}{ϕ (τ_{2} K^{\frac{m + 1}{m}} (d (x)))} and \underset{d (x) \to 0}{lim sup} \frac{u (x)}{ϕ (τ_{1} K^{\frac{m + 1}{m}} (d (x)))} ⩽ 1 . \end{matrix}$

Thus, letting $ε \to 0$ , we obtain (1.9). □

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Funding

This work was partially supported by NSF of China (Grant no. 11771196).

Footnotes

Acknowledgements

The authors express the sincere gratitude to the editors and referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.

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