Boundary behavior of large solutions for semilinear elliptic equations with weights

Abstract

This paper is concerned with boundary blow-up elliptic problems $△ u = b (x) f (u)$ , $x \in Ω$ , $u |_{\partial Ω} = + \infty$ , where Ω is a bounded domain with smooth boundary in $R^{N}$ , and $b \in C_{loc}^{α} (Ω)$ for some $α \in (0, 1)$ which is nonnegative nontrivial in Ω, but may be vanishing or appropriate singular (including critical singular) on $\partial Ω$ . Under a new structure condition on f near infinity, we study the exact boundary behavior of such solutions.

Keywords

semilinear elliptic equations boundary blow-up the first expansions of solutions near the boundary

1. Introduction and the main results

In this paper we analyze the boundary behavior of solutions to the following boundary blow-up elliptic problem $\begin{matrix} (1.1) & △ u = b (x) f (u), x \in Ω, u |_{\partial Ω} = + \infty, \end{matrix}$ where the last condition means that $u (x) \to + \infty$ as $d (x) = dist (x, \partial Ω) \to 0$ , Ω is a bounded domain with smooth boundary in $R^{N}$ , f satisfies

$f \in C [0, \infty)$ , $f (0) = 0$ , $f (s) > 0$ , $s > 0$ , and $f (s)$ is nondecreasing on $[0, \infty)$ (or ( $f_{01}$ ) $f \in C (R)$ , $f (s) > 0$ , $\forall s \in R$ , f is nondecreasing on $R$ );

the Keller–Osserman condition [25,38] $\begin{matrix} Ψ (r) : = \int_{r}^{\infty} \frac{d τ}{\sqrt{2 F (τ)}} < \infty \forall r > 0, F (τ) = \int_{0}^{τ} f (υ) d υ; \end{matrix}$

and b satisfies

$b \in C_{loc}^{α} (Ω)$ for some $α \in (0, 1)$ , is nonnegative in Ω, and satisfies the condition that if there exists $x_{0} \in Ω$ such that $b (x_{0}) = 0$ , then there exists a sub-domain $Ω_{0} \subset \subset Ω$ containing $x_{0}$ such that $b (x) > 0$ for all $x \in \partial Ω_{0}$ .

Problem (1.1) arises from many branches of mathematics and applied mathematics and has been discussed by many authors in many contexts, see, for instance, [1–6,8–21,23–35,37–45,47–52] and the references therein.

For $b \equiv 1$ in Ω:

When $f (u) = exp (u)$ and $N = 2$ , problem (1.1) was first considered by Bieberbach [6] in 1916. Rademacher [41], using the ideas of Bieberbach, showed that if $N = 3$ , then problem (1.1) has a unique solution $u \in C^{2} (Ω)$ such that $\begin{matrix} | u (x) + 2 ln (d (x)) | is bounded on Ω . \end{matrix}$

When f satisfies ( $f_{1}$ ) (or ( $f_{01}$ )), Keller–Osserman [25,38] first supplied a necessary and sufficient condition ( $f_{2}$ ) for the existence of solutions to problem (1.1).

Loewner and Nirenberg [31] showed that if

f (u) = u^{p_{0}}

with

p_{0} = (N + 2) / (N - 2)

N > 2

, then problem (1.1) has a unique positive solution u which satisfies

\begin{matrix} lim_{d (x) \to 0} u (x) {(d (x))}^{(N - 2) / 2} = {(N (N - 2) / 4)}^{(N - 2) / 4} . \end{matrix}

Their results were extended to

f (u) = u^{p}

with

p > 1

by Kondratév and Nikishkin [26].

Bandle and Marcus [5] established the following results: if f satisfies ( $f_{1}$ ), ( $f_{2}$ ) and the condition that

$\begin{matrix} lim_{r \to \infty} inf \frac{Ψ (λ r)}{Ψ (r)} > 1 \forall λ \in (0, 1), \end{matrix}$ then for any solution u to problem (1.1), it holds $\begin{matrix} (1.2) & \frac{u (x)}{ψ (d (x))} \to 1 as d (x) \to 0, \end{matrix}$ where ψ is the inverse of Ψ, i.e., ψ satisfies $\begin{matrix} (1.3) & \int_{ψ (t)}^{\infty} \frac{d τ}{\sqrt{2 F (τ)}} = t \forall t > 0 . \end{matrix}$

If f further satisfies

$f (s) / s$ is increasing on $(0, \infty)$ ,

then problem (1.1) has a unique solution.

Next Lazer–McKenna [30] showed that if f satisfies ( $f_{1}$ ) (or ( $f_{01}$ )) and the condition that

there exists $S_{0} > 0$ such that $f^{'}$ is nondecreasing on $[S_{0}, \infty)$ , and $\begin{matrix} lim_{s \to \infty} \frac{f^{'} (s)}{\sqrt{F (s)}} = \infty, \end{matrix}$ then for any solution u to problem (1.1), it holds $\begin{matrix} (1.4) & u (x) - ψ (d (x)) \to 0 as d (x) \to 0 . \end{matrix}$

For convenience, we introduce three kinds of functions.

First, we denote by $K_{\infty}$ the set of all functions $\hat{L}$ which are normalized slowly varying at infinity defined on $(0, \infty)$ by $\begin{matrix} (1.5) & \hat{L} (s) = c_{0} exp (\int_{S_{0}}^{s} \frac{y (τ)}{τ} d τ), s > S_{0} for some S_{0} > 0, \end{matrix}$ where $c_{0} > 0$ and the function $y \in C [S_{0}, \infty)$ with ${lim}_{s \to \infty} y (s) = 0$ .

Analogously, we denote by $K_{0}$ the set of all functions ${\hat{L}}_{0}$ which are normalized slowly varying at zero defined on $(0, s_{0}]$ for some $s_{0} > 0$ by $\begin{matrix} (1.6) & {\hat{L}}_{0} (s) = c_{0} exp (\int_{s}^{s_{0}} \frac{y (τ)}{τ} d τ), s \in (0, s_{0}], \end{matrix}$ where $c_{0} > 0$ and the function $y \in C [0, s_{0}]$ with $y (0) = 0$ .

Thirdly let Λ denote the set of all positive monotonic functions k in $C^{1} (0, δ_{0}) \cap L^{1} (0, δ_{0})$ ( $δ_{0} > 0$ ) which satisfy $\begin{matrix} (1.7) & lim_{t \to 0^{+}} \frac{d}{d t} (\frac{K (t)}{k (t)}) : = D_{k} \in [0, \infty), K (t) : = \int_{0}^{t} k (υ) d υ . \end{matrix}$ The set Λ was first introduced by Cîrstea and Rǎdulescu [11] for nondecreasing functions and by Mohammed [37] for nonincreasing functions.

By using a perturbation method, Karamata regularly varying theory and constructing comparison functions, Cîrstea and Rǎdulescu [11–13], Cîrstea and Du [9], Cîrstea [8] and Mohammed [37] showed the uniqueness and exact asymptotical behavior of solutions near the boundary to problem (1.1). A basic result is that

If f satisfies ( $f_{1}$ ) and

there exist $p > 1$ and $\hat{L} \in K_{\infty}$ such that $\begin{matrix} f (s) = s^{p} \hat{L} (s) \forall s > S_{0}, \end{matrix}$ b satisfies ( $b_{1}$ ) and there exist $b_{0} > 0$ and $k \in Λ$ such that

$\begin{matrix} lim_{d (x) \to 0} \frac{b (x)}{k^{2} (d (x))} = b_{0}, \end{matrix}$ then any solution u to problem (1.1) satisfies $\begin{matrix} (1.8) & lim_{d (x) \to 0} \frac{u (x)}{ψ (K (d (x)))} = ξ_{0}, \end{matrix}$ where $ξ_{0} = {(\frac{2 + D_{k} (p - 1)}{b_{0} (p + 1)})}^{1 / (p - 1)}$ .

Subsequently, for the critical case $\begin{matrix} (1.9) & lim_{r \to \infty} \frac{Ψ (λ r)}{Ψ (r)} = 1 \forall λ \in (0, 1), \end{matrix}$ when $b \equiv 1$ in Ω and f satisfies

f is locally Lipschitz continuous and nonnegative on $[0, \infty)$ , and $f (s) / s$ is increasing on $(0, \infty)$ ;

$\begin{matrix} f (s) = c_{1}^{2} s {(ln s)}^{2 α} + c_{2} s {(ln s)}^{2 α - 1} (1 + o (1)) as s \to \infty \end{matrix}$ with $c_{1} > 0$ , $α > 1$ and $c_{2} \in R$ .

Cîrstea and Du [10] first showed that problem (1.1) has a unique solution

u \in C^{2} (Ω)

satisfying

\begin{matrix} (1.10) & lim_{d (x) \to \infty} \frac{u (x)}{exp ({(c_{1} (α - 1) d (x))}^{- 1 / (α - 1)})} = exp (ξ_{0}), \end{matrix}

where

\begin{matrix} (1.11) & ξ_{0} = \frac{1}{2} - \frac{c_{2}}{2 α c_{1}^{2}} . \end{matrix}

Moreover, they extended the above result to weight b which satisfies (

b_{2}

It is very worthwhile to point out that (1.10) depends not only on $c_{1}^{2} s {(ln s)}^{2 α}$ but also on the lower term $c_{2} s {(ln s)}^{2 α - 1}$ in ( $f_{07}$ ). This is completely different from the case $f (s) = s^{p} (c_{1} + o (1))$ as $s \to \infty$ for some $p > 1$ , since problem (1.1) has a unique positive solution u which satisfies $\begin{matrix} lim_{d (x) \to 0} u (x) {(d (x))}^{2 / (p - 1)} = {(\frac{2 (p + 1)}{c_{1} {(p - 1)}^{2}})}^{1 / (p - 1)} \end{matrix}$ in such a situation and $b \equiv 1$ in Ω.

Inspired by the above works, in this paper we investigate the new boundary behavior of solutions to problem (1.1) under the following structure condition on f:

there exists $C_{f} \in (0, \infty]$ such that $\begin{matrix} lim_{s \to + \infty} g^{'} (s) \int_{s}^{\infty} \frac{d τ}{g (τ)} = C_{f}, g (s) : = \sqrt{2 F (s)} \forall s > 0 . \end{matrix}$

A complete characterization of f in (

f_{3}

) is provided in Lemma 2.2.

Our main results are summarized as follows.

Theorem 1.1.
Let f satisfy ( $f_{1}$ ) ( or ( $f_{01}$ ) ) , ( $f_{2}$ ) , ( $f_{3}$ ), b satisfy ( $b_{1}$ ) and the condition that
there exist $k \in Λ$ and positive constant $b_{i}$ ( $i = 1, 2$ ) such that $\begin{matrix} b_{1} : = lim_{d (x) \to 0} inf \frac{b (x)}{k^{2} (d (x))} ⩽ b_{2} : = lim_{d (x) \to 0} sup \frac{b (x)}{k^{2} (d (x))} . \end{matrix}$
If $\begin{matrix} (1.12) & C_{f} \in (1, \infty], \end{matrix}$ or $\begin{matrix} (1.13) & C_{f} = 1 and D_{k} > 0, \end{matrix}$ then for any solution u to problem (1.1), it holds $\begin{matrix} (1.14) & 1 ⩽ lim_{d (x) \to 0} inf \frac{u (x)}{ψ (ξ_{2} K (d (x)))} and lim_{d (x) \to 0} sup \frac{u (x)}{ψ (ξ_{1} K (d (x)))} ⩽ 1, \end{matrix}$ where ψ is given as (1.3) and $\begin{matrix} (1.15) & ξ_{1} = {(\frac{b_{1}}{1 - C_{f}^{- 1} (1 - D_{k})})}^{1 / 2}, ξ_{2} = {(\frac{b_{2}}{1 - C_{f}^{- 1} (1 - D_{k})})}^{1 / 2} . \end{matrix}$ In particular,
when $C_{f} = 1$ , u verifies $\begin{matrix} (1.16) & lim_{d (x) \to 0} \frac{u (x)}{ψ (K (d (x)))} = 1; \end{matrix}$

when $C_{f} \in (1, \infty)$ and $b_{1} = b_{2} = b_{0}$ in ( $b_{3}$ ), u verifies $\begin{matrix} (1.17) & lim_{d (x) \to 0} \frac{u (x)}{ψ (K (d (x)))} = {(\frac{b_{0}}{1 - C_{f}^{- 1} (1 - D_{k})})}^{(1 - C_{f}) / 2}; \end{matrix}$

when $C_{f} = \infty$ and $b_{1} = b_{2} = b_{0}$ in ( $b_{3}$ ), u verifies $\begin{matrix} (1.18) & lim_{d (x) \to 0} \frac{u (x)}{ψ (\sqrt{b_{0}} K (d (x)))} = 1 . \end{matrix}$

When b is a borderline case, we have Theorem 1.2.
Let f satisfy ( $f_{1}$ ) ( or ( $f_{01}$ ) ) , ( $f_{2}$ ) , ( $f_{3}$ ), b satisfy ( $b_{1}$ ) and the condition that
there exist positive constant $b_{i}$ ( $i = 1, 2$ ) and ${\hat{L}}_{0} \in K_{0}$ with $\begin{matrix} (1.19) & h (t) : = \int_{0}^{t} \frac{{\hat{L}}_{0} (υ)}{υ} d υ < \infty \end{matrix}$ such that $\begin{matrix} b_{1} : = lim_{d (x) \to 0} inf \frac{b (x)}{{(d (x))}^{- 2} {\hat{L}}_{0} (d (x))} ⩽ b_{2} : = lim_{d (x) \to 0} sup \frac{b (x)}{{(d (x))}^{- 2} {\hat{L}}_{0} (d (x))} . \end{matrix}$
If $\begin{matrix} (1.20) & C_{f} \in [1, \infty), \end{matrix}$ then for any solution u to problem (1.1), it holds $\begin{matrix} (1.21) & 1 ⩽ lim_{d (x) \to 0} inf \frac{u (x)}{ψ (ξ_{4} \sqrt{h (d (x))})} and lim_{d (x) \to 0} sup \frac{u (x)}{ψ (ξ_{3} \sqrt{h (d (x))})} ⩽ 1, \end{matrix}$ where $\begin{matrix} (1.22) & ξ_{3} = \sqrt{2 b_{1} C_{f}}, ξ_{4} = \sqrt{2 b_{2} C_{f}} . \end{matrix}$ In particular,
when $C_{f} = 1$ , u verifies $\begin{matrix} (1.23) & lim_{d (x) \to 0} \frac{u (x)}{ψ (\sqrt{h (d (x))})} = 1; \end{matrix}$

when $C_{f} \in (1, \infty)$ and $b_{1} = b_{2} = b_{0}$ in ( $b_{4}$ ), u verifies $\begin{matrix} (1.24) & lim_{d (x) \to 0} \frac{u (x)}{ψ (\sqrt{h (d (x))})} = {(2 b_{0} C_{f})}^{(1 - C_{f}) / 2} . \end{matrix}$

Remark 1.1.
One can see in Lemma 2.1 that $\begin{matrix} k (t) = t^{- 1} {({\hat{L}}_{0} (t))}^{1 / 2} does not belong to Λ . \end{matrix}$
Remark 1.2.
For the existence of the minimal solution to problem (1.1), see [27] and [43].
Remark 1.3.
Lemma 2.2 ensures that $C_{f} \in [1, \infty]$ . Hence (1.12) or (1.13) implies $C_{f} > 1 - D_{k}$ .
Remark 1.4.
Some basic examples of functions which satisfy ( $f_{3}$ ) are
$C_{f} = 1$ ,

$f (s) = exp ({(ln s)}^{q})$ , $q > 1$ , $s ⩾ S_{0}$ , where $S_{0} > 0$ is a large constant;

$f (s) = exp (s^{q})$ , $q > 0$ , $s ⩾ S_{0}$ ;

$f (s) = exp (s ln s)$ , $s ⩾ S_{0}$ ;

$f \in C^{1} (S_{0}, \infty)$ for some $S_{0} > 0$ and ${lim}_{s \to \infty} \frac{f^{'} (s) F (s)}{f^{2} (s)} = 1$ (Lemma 2.2);

$F (s) = c_{0} exp (\int_{S_{0}}^{s} \frac{d τ}{ϕ (τ)})$ , $s > S_{0}$ , ϕ is a positive $C^{1}$ -function on $[S_{0}, \infty)$ and ${lim}_{τ \to \infty} ϕ^{'} (τ) = 0$ (Lemma 2.2).

$C_{f} = \infty$ ,

$F (s) = 2 c_{1}^{2} s^{2} {(ln s)}^{2 q}$ , $q > 1$ , $c_{1} > 0$ , $s ⩾ S_{0}$ ;

$F (s) = 2 c_{1}^{2} s^{2} exp ({(ln s)}^{q})$ , $q \in (0, 1)$ , $s ⩾ S_{0}$ ;

$F (s) = 2 c_{1}^{2} s^{2} {(ln s)}^{2} {(ln (ln s))}^{2 q}$ , $q > 1$ , $s ⩾ S_{0}$ .

$C_{f} \in (1, \infty)$ ,

$F (s) = 2 s^{p + 1} \hat{L} (s)$ , $s ⩾ S_{0}$ , where $p > 1$ and $\hat{L} \in K_{\infty}$ . In this case, $C_{f} = \frac{p + 1}{p - 1}$ .

Remark 1.5.
When ${\hat{L}}_{0} (t) = {(- ln t)}^{- γ}$ with $γ > 1$ in ( $b_{4}$ ), $\begin{matrix} \sqrt{h (t)} = {(γ - 1)}^{- 1 / 2} {(- ln t)}^{- (γ - 1) / 2} . \end{matrix}$
The outline of this paper is as follows. In Section 2 we give some preliminary considerations. The proofs of Theorems 1.1 and 1.2 are provided in Section 3.
2. Preliminaries

Our approach relies on Karamata regular variation theory established by Karamata in 1930 and is a basic tool in stochastic process (see Bingham, Goldie and Teugels [7], Maric [36] and the references therein).

In this section, we present some basic facts from Karamata regular variation theory and some preliminaries.

Definition 2.1.
A positive continuous function f defined on $[a, \infty)$ , for some $a > 0$ , is called regularly varying at infinity with index ρ, denoted by $f \in {RV}_{ρ}$ , 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.

Some basic examples of slowly varying functions at infinity are
every continuous function on $[a, \infty)$ which has a positive limit at infinity;

${(ln s)}^{q}$ and ${(ln (ln s))}^{q}$ , $q \in R$ ;

$exp ({(ln s)}^{q})$ , $0 < q < 1$ .
We also say that a positive continuous function k defined on $(0, a)$ for some $a > 0$ , is regularly varying at zero with index ρ (and denoted by $k \in {RVZ}_{ρ}$ ) if $t \to k (1 / t)$ belongs to ${RV}_{- ρ}$ .
Definition 2.2.
A positive continuous function f defined on $[a, \infty)$ , for some $a > 0$ , is called rapidly varying at infinity if for each $ξ > 1$ $\begin{matrix} (2.2) & lim_{s \to \infty} \frac{f (ξ s)}{f (s)} = \infty . \end{matrix}$
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 $(c_{1}, \infty)$ with $c_{1} > 0$ ; if $ρ > 0$ , then uniform convergence holds on intervals $(0, c_{2}]$ provided f is bounded on $(0, c_{2}]$ for all $c_{2} > 0$ .

Proposition 2.2 (Representation theorem).

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

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

A function $f \in {NRV}_{ρ}$ if and only if $\begin{matrix} (2.6) & f \in C^{1} [S_{0}, \infty) for some S_{0} > 0 and lim_{s \to \infty} \frac{s f^{'} (s)}{f (s)} = ρ . \end{matrix}$ Similarly, k is called normalized regularly varying at zero with index ρ, denoted by $k \in {NRVZ}_{ρ}$ if $t \to k (1 / t)$ belongs to ${NRV}_{- ρ}$ .

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

For every $ρ > 0$ and $s \to \infty$ , $\begin{matrix} s^{ρ} L (s) \to \infty, s^{- ρ} L (s) \to 0 . \end{matrix}$

For $ρ \in R$ and $s \to \infty$ , $ln (L (s)) / ln s \to 0$ and $ln (s^{ρ} L (s)) / ln s \to ρ$ .

Proposition 2.4 (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 $ρ < - 1$ .

Proposition 2.5.
Let $ρ > 0$ . If $f \in {NRV}_{ρ}$ , then $F \in {NRV}_{ρ + 1}$ ; and if $F \in {NRV}_{ρ + 1}$ , then $f \in {RV}_{ρ}$ .
Proposition 2.6 ((Asymptotic behavior) (Karamata’s theorem in [7])).

If a function $h \in {RV}_{- 1}$ and $\begin{matrix} \int_{t}^{\infty} h (s) d s < \infty, t > 0, \end{matrix}$ then $\int_{t}^{\infty} h (s) d s$ is slowly varying at infinity and $\begin{matrix} lim_{t \to \infty} \frac{t h (t)}{\int_{t}^{\infty} h (s) d s} = 0 . \end{matrix}$

Proposition 2.7 ([46, Lemma 2.3]).

Let ${\hat{L}}_{0}$ be defined on $(0, η]$ and be normalized slowly varying at zero. Then we have $\begin{matrix} lim_{t \to 0^{+}} \frac{{\hat{L}}_{0} (t)}{\int_{t}^{η} \frac{{\hat{L}}_{0} (τ)}{τ} d τ} = 0 . \end{matrix}$ If further $\int_{0}^{η} \frac{{\hat{L}}_{0} (τ)}{τ} d τ$ converges, then we have $\begin{matrix} lim_{t \to 0^{+}} \frac{{\hat{L}}_{0} (t)}{\int_{0}^{t} \frac{{\hat{L}}_{0} (τ)}{τ} d τ} = 0 . \end{matrix}$

Lemma 2.1 ([52, Theorem 2.1]).

Let $k \in Λ$ . We have

When k is nondecreasing, $D_{k} \in [0, 1]$ ; and $D_{k} ⩾ 1$ provided k is nonincreasing;

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

when $D_{k} > 0$ , $k \in {NRVZ}_{(1 - D_{k}) / D_{k}}$ and $K \in {NRVZ}_{1 / D_{k}}$ . In particular, when $D_{k} = 1$ , k is normalized slowly varying at zero;

when $D_{k} = 0$ , k is rapidly varying to zero at zero.

Our results in the section are summarized as follows.

Let us recall that $\begin{matrix} (2.7) & Ψ (s) = \int_{s}^{\infty} \frac{d τ}{g (τ)}, g (s) = \sqrt{2 F (s)} \forall s > 0 . \end{matrix}$

Lemma 2.2.
Let f satisfy ( $f_{1}$ ) ( or ( $f_{01}$ ) ) and ( $f_{2}$ ). We have
if f satisfies ( $f_{3}$ ), then $C_{f} ⩾ 1$ ;

f satisfies ( $f_{3}$ ) with $C_{f} \in (1, \infty)$ if and only if $F \in {NRV}_{2 p}$ with $p > 1$ . In this case $p = C_{f} / (C_{f} - 1)$ ;

if f satisfies ( $f_{3}$ ) with $C_{f} = 1$ , then F is rapidly varying at infinity;

if $f \in C^{1} (S_{0}, \infty)$ for some $S_{0} > 0$ and $\begin{matrix} (2.8) & lim_{s \to \infty} \frac{f^{'} (s) F (s)}{f^{2} (s)} = 1, \end{matrix}$ then f satisfies ( $f_{3}$ ) with $C_{f} = 1$ ;

if $F (s) = c_{0} exp (\int_{S_{0}}^{s} \frac{d τ}{ϕ (τ)})$ , $s > S_{0}$ , where $S_{0} > 0$ is a large constant, ϕ is a positive $C^{1}$ -function on $[S_{0}, \infty)$ and $\begin{matrix} (2.9) & lim_{τ \to \infty} ϕ^{'} (τ) = 0, \end{matrix}$ then f satisfies ( $f_{3}$ ) with $C_{f} = 1$ ;

if $F \in {NRV}_{2}$ , then f satisfies ( $f_{3}$ ) with $C_{f} = \infty$ ;

if f satisfies ( $f_{3}$ ), then $\begin{matrix} lim_{s \to \infty} \frac{{(2 F (s))}^{1 / 2}}{f (s) Ψ (s)} = C_{f}^{- 1} . \end{matrix}$

Proof.
( $i_{1}$ ) Let $C_{f} \in (0, \infty]$ and $\begin{matrix} I (s) = g^{'} (s) \int_{s}^{\infty} \frac{d τ}{g (τ)} \forall s > 0 . \end{matrix}$ Integrating $I (υ)$ from a ( $a > 0$ ) to s and integrate by parts, we obtain $\begin{matrix} \int_{a}^{s} I (υ) d υ = g (s) \int_{s}^{\infty} \frac{d τ}{g (τ)} - g (a) \int_{a}^{\infty} \frac{d τ}{g (τ)} + s - a \forall s > a . \end{matrix}$ It follows from the l’Hospital’s rule that $\begin{matrix} (2.10) & 0 ⩽ lim_{s \to \infty} \frac{g (s) \int_{s}^{\infty} \frac{d τ}{g (τ)}}{s} = lim_{s \to \infty} \frac{\int_{a}^{s} I (υ) d υ}{s} - 1 = lim_{s \to \infty} I (s) - 1 = C_{f} - 1, \end{matrix}$ i.e., $C_{f} ⩾ 1$ .

( $i_{2}$ ) For $C_{f} \in (1, \infty)$ .

Necessity. By ( $f_{3}$ ) and (2.10), we see that $\begin{matrix} (2.11) & lim_{s \to \infty} \frac{g (s)}{s g^{'} (s)} = lim_{s \to \infty} \frac{g (s) \int_{s}^{\infty} \frac{d τ}{g (τ)}}{s g^{'} (s) \int_{s}^{\infty} \frac{d τ}{g (τ)}} = \frac{1}{C_{f}} lim_{s \to \infty} \frac{g (s) \int_{s}^{\infty} \frac{d τ}{g (τ)}}{s} = \frac{C_{f} - 1}{C_{f}}, \end{matrix}$ i.e., $g \in {NRV}_{C_{f} / (C_{f} - 1)}$ . Then $F \in {NRV}_{2 C_{f} / (C_{f} - 1)}$ .

Sufficiency. When $F \in {NRV}_{2 p}$ with $p > 1$ , i.e., $g \in {NRV}_{p}$ and $\begin{matrix} lim_{s \to \infty} \frac{s g^{'} (s)}{g (s)} = p, g (s) = s^{p} \hat{L} (s) \forall s ⩾ S_{0} \end{matrix}$ for sufficiently large $S_{0}$ , where $\hat{L}$ is normalized slowly varying at infinity. It follows from Proposition 2.4(ii) that $\begin{array}{rcl} lim_{s \to \infty} g^{'} (s) \int_{s}^{\infty} \frac{d τ}{g (τ)} & = & lim_{s \to \infty} \frac{s g^{'} (s)}{g (s)} lim_{s \to \infty} \frac{g (s)}{s} \int_{s}^{\infty} \frac{d τ}{g (τ)} \\ = & p lim_{s \to \infty} s^{p - 1} \hat{L} (s) \int_{s}^{\infty} τ^{- p} {(\hat{L} (τ))}^{- 1} d τ = \frac{p}{p - 1} = C_{f} . \end{array}$ ( $i_{3}$ ) When $C_{f} = 1$ , let $\begin{matrix} \frac{s g^{'} (s)}{g (s)} : = y (s) \forall s > 0, \end{matrix}$ i.e., $\begin{matrix} \frac{g^{'} (s)}{g (s)} = \frac{y (s)}{s} \forall s > 0 . \end{matrix}$ Integrating the above inequality from $S_{0}$ to s, we obtain $\begin{matrix} g (s) = c_{0} exp (\int_{s_{0}}^{s} \frac{y (τ)}{τ} d τ), s > S_{0}, \end{matrix}$ where $c_{0} = g (S_{0})$ .

By (2.11), we see that ${lim}_{s \to \infty} y (s) = + \infty$ and for each $ξ > 1$ , $\begin{matrix} \frac{g (ξ s)}{g (s)} = exp (\int_{s}^{ξ s} \frac{y (τ)}{τ} d τ) = exp (\int_{1}^{ξ} \frac{y (s υ)}{υ} d υ) \to + \infty as s \to \infty, \end{matrix}$ i.e., g is rapidly varying at infinity by Definition 2.2. So does F.

( $i_{4}$ ) By (2.8) and the l’Hospital’s rule, we see that $\begin{matrix} (2.12) & lim_{s \to \infty} \frac{F (s)}{s f (s)} = lim_{s \to \infty} \frac{\frac{F (s)}{f (s)}}{s} = lim_{s \to \infty} \frac{d}{d s} (\frac{F (s)}{f (s)}) = 1 - lim_{s \to \infty} \frac{F (s) f^{'} (s)}{f^{2} (s)} = 0 . \end{matrix}$ Consequently, for an arbitrary $p > 3$ , there exists $S_{p} > 0$ such that $\begin{matrix} \frac{f (s)}{F (s)} > \frac{p + 2}{s}, s ⩾ S_{p} . \end{matrix}$ Integrating from $S_{p}$ to s, we obtain $\begin{matrix} F (s) ⩾ \frac{F (S_{p})}{S_{p}^{p + 2}} s^{p + 2}, s > S_{p} . \end{matrix}$ So, $\begin{matrix} f (s) ⩾ \frac{(p + 2) F (S_{p})}{S_{p}^{p + 2}} s^{p + 1}, s > S_{p} \end{matrix}$ and $\begin{matrix} lim_{s \to \infty} \frac{f (s)}{s^{p}} = \infty . \end{matrix}$ Next, since $\begin{matrix} F (s) = \int_{0}^{s} f (τ) d τ ⩽ s f (s), s > 0, \end{matrix}$ we have that $\begin{matrix} 0 < \frac{\sqrt{2 F (s)}}{f (s)} ⩽ {(\frac{2 s}{f (s)})}^{1 / 2}, s > 0 . \end{matrix}$ So, $\begin{matrix} lim_{s \to \infty} \frac{\sqrt{2 F (s)}}{f (s)} = 0 . \end{matrix}$ Since $g (s) = \sqrt{2 F (s)}$ , we see that $\begin{matrix} g^{'} (s) = \frac{f (s)}{{(2 F (s))}^{1 / 2}}, \frac{f (s) Ψ (s)}{{(2 F (s))}^{1 / 2}} = g^{'} (s) \int_{s}^{\infty} \frac{d τ}{g (τ)} \forall s > 0 . \end{matrix}$ It follows by the l’Hospital’s rule and (2.8) that $\begin{matrix} lim_{s \to \infty} \frac{Ψ (s)}{\frac{\sqrt{2 F (s)}}{f (s)}} = lim_{s \to \infty} \frac{f^{2} (s)}{2 F (s) f^{'} (s) - f^{2} (s)} = lim_{s \to \infty} \frac{1}{2 \frac{F (s) f^{'} (s)}{f^{2} (s)} - 1} = 1 . \end{matrix}$ ( $i_{5}$ ) By a direct calculation, we see that (2.8) holds in this case.

( $i_{6}$ ) When $F \in {NRV}_{2}$ , we see that $g \in {NRV}_{1}$ and $g^{- 1} \in {NRV}_{- 1}$ . It follows by ( $f_{2}$ ) and Proposition 2.6 that $\begin{matrix} lim_{s \to \infty} \frac{s}{g (s) \int_{s}^{\infty} \frac{d τ}{g (τ)}} = 0 . \end{matrix}$ Thus $\begin{matrix} lim_{s \to \infty} g^{'} (s) \int_{s}^{\infty} \frac{d τ}{g (τ)} = lim_{s \to \infty} \frac{s g^{'} (s)}{g (s)} lim_{s \to \infty} \frac{g (s) \int_{s}^{\infty} \frac{d τ}{g (τ)}}{s} = \infty . \end{matrix}$ ( $i_{7}$ ) is obvious, since ( $f_{3}$ ) is precisely $\begin{matrix} lim_{s \to \infty} \frac{{(2 F (s))}^{1 / 2}}{f (s) Ψ (s)} = C_{f}^{- 1} . \end{matrix}$ □
Lemma 2.3.
If f satisfies ( $f_{1}$ ) ( or ( $f_{01}$ ) ) , ( $f_{2}$ ) and ( $f_{3}$ ), then
$- ψ^{'} (t) = {(2 F (ψ (t)))}^{1 / 2}$ , $t > 0$ , $ψ^{″} (t) = f (ψ (t))$ , $t > 0$ , and $ψ (t) > 0$ , $t > 0$ , $ψ (0) : = {lim}_{t \to 0^{+}} ψ (t) = + \infty$ ;

${lim}_{t \to 0} \frac{ψ^{'} (t)}{t ψ^{″} (t)} = - C_{f}^{- 1}$ ;

when $C_{f} \in (1, \infty)$ , $ψ \in {NRVZ}_{1 - C_{f}}$ and $- ψ^{'} \in {NRVZ}_{- C_{f}}$ ;

when $C_{f} = \infty$ , ψ is rapidly varying at zero.

Proof.
( $i_{1}$ ) follows by the definition of ψ and a direct calculation.

( $i_{2}$ ) Let $s = ψ (t)$ . It follows from Lemma 2.2( $i_{7}$ ) that $\begin{matrix} lim_{t \to 0} \frac{ψ^{'} (t)}{t ψ^{″} (t)} = - lim_{s \to \infty} \frac{{(2 F (s))}^{1 / 2}}{f (s) Ψ (s)} = - C_{f}^{- 1} . \end{matrix}$ ( $i_{3}$ ) For $C_{f} \in [1, \infty)$ , let $s = ψ (t)$ . It follows from (2.10) and ( $f_{3}$ ) that $\begin{matrix} (2.13) & lim_{t \to 0^{+}} \frac{t ψ^{'} (t)}{ψ (t)} = - lim_{t \to 0^{+}} \frac{t g (ψ (t))}{ψ (t)} = - lim_{s \to \infty} \frac{g (s)}{s} \int_{s}^{\infty} \frac{d τ}{g (τ)} = - (C_{f} - 1), \end{matrix}$ and $\begin{matrix} (2.14) & lim_{t \to 0^{+}} \frac{t ψ^{″} (t)}{ψ^{'} (t)} = - lim_{t \to 0^{+}} t g^{'} (ψ (t)) = - lim_{s \to \infty} g^{'} (s) \int_{s}^{\infty} \frac{d τ}{g (τ)} = - C_{f}, \end{matrix}$ i.e., $ψ \in {NRVZ}_{1 - C_{f}}$ and $- ψ^{'} \in {NRVZ}_{- C_{f}}$ .

( $i_{4}$ ) When $C_{f} = \infty$ , we see by using the l’Hospital’s rule that $\begin{matrix} lim_{s \to \infty} \frac{g (s) \int_{s}^{\infty} \frac{d τ}{g (τ)}}{s} = lim_{s \to \infty} g^{'} (s) \int_{s}^{\infty} \frac{d τ}{g (τ)} - 1 = + \infty, \end{matrix}$ i.e., $\begin{matrix} lim_{t \to 0^{+}} \frac{t ψ^{'} (t)}{ψ (t)} = - \infty . \end{matrix}$ In a similar proof to ( $i_{3}$ ) in Lemma 2.2, we can show that ψ is rapidly varying at zero. □

3. Proof of Theorems 1.1 and 1.2

In this section, we prove Theorems 1.1 and 1.2.

Lemma 3.1 ([16, Lemma 2.4] (The comparison principle)).

Let $Ω \subset R^{N}$ be a bounded domain, f be an nondecreasing function and let b satisfy ( $b_{1}$ ). Assume that $u_{1}, u_{2} \in C^{2} (Ω)$ satisfy $\begin{matrix} Δ u_{1} ⩾ b (x) f (u_{1}) and Δ u_{2} ⩽ b (x) f (u_{2}) in Ω . \end{matrix}$ If $\begin{matrix} \underset{x \to \partial Ω}{lim inf} (u_{2} - u_{1}) (x) ⩾ 0, \end{matrix}$ then $\begin{matrix} u_{2} ⩾ u_{1} in Ω . \end{matrix}$

Proof of Theorem 1.1.

Recall that for any $δ > 0$ , $\begin{matrix} Ω_{δ} = {x \in Ω : 0 < d (x) < δ} . \end{matrix}$ Since Ω is $C^{2}$ -smooth, choose $δ_{1} \in (0, δ_{0})$ ( $δ_{0}$ is given as in the definition of Λ) such that (see, 14.6. Appendix: Boundary Curvatures and the Distance Function in [22]) $d \in C^{2} (Ω_{δ_{1}})$ and $\begin{matrix} (3.1) & | \nabla d (x) | = 1 and Δ d (x) = - (N - 1) H (\bar{x}) + o (1) \forall x \in Ω_{δ_{1}}, \end{matrix}$ where $\bar{x}$ is the projection of the point x to $\partial Ω$ .

Now let $v_{0} \in C^{2 + α} (Ω) \cap C^{1} (\bar{Ω})$ be the unique solution to the problem $\begin{matrix} (3.2) & - Δ v_{0} = 1, v_{0} > 0, x \in Ω, v_{0} |_{\partial Ω} = 0 . \end{matrix}$ By the Höpf maximum principle in [22], we see that $\begin{matrix} (3.3) & \nabla v_{0} (x) \neq 0 \forall x \in \partial Ω and c_{1} d (x) ⩽ v_{0} (x) ⩽ c_{2} d (x) \forall x \in Ω, \end{matrix}$ where $c_{1}$ , $c_{2}$ are positive constants.

For an arbitrary $ε \in (0, b_{1} / 4)$ , let $\begin{matrix} (3.4) & ξ_{+ ε} = {(\frac{b_{2} + 2 ε}{1 - C_{f}^{- 1} (1 - D_{k})})}^{1 / 2}; ξ_{- ε} = {(\frac{b_{1} - 2 ε}{1 - C_{f}^{- 1} (1 - D_{k})})}^{1 / 2}, \end{matrix}$ where $b_{1}$ and $b_{2}$ are given as in ( $b_{3}$ ).

Then $\begin{matrix} \frac{1}{2} ξ_{1} < ξ_{- ε} < ξ_{+ ε} < 2 ξ_{2}, \end{matrix}$ where $ξ_{1}$ and $ξ_{2}$ are given as in (1.15).

By using (1.7), Lemma 2.1, Lemma 2.3 and $\begin{matrix} (3.5) & \begin{matrix} K \in C^{1} (0, δ_{0}) \cap C [0, δ_{0}), K (0) = 0, \\ K (d (x)) = \int_{ψ (K (d (x)))}^{\infty} \frac{d τ}{\sqrt{2 F (τ)}}, \end{matrix} \end{matrix}$ we see that $\begin{array}{l} lim_{d (x) \to 0} \frac{K (d (x)) k^{'} (d (x))}{k^{2} (d (x))} = 1 - D_{k}; \\ lim_{d (x) \to 0} \frac{{(2 F (ψ (K (d (x)))))}^{1 / 2}}{K (d (x)) f (ψ (K (d (x))))} = C_{f}^{- 1}; \\ (1 - C_{f}^{- 1} (1 - D_{k})) ξ_{+ ε}^{2} - (b_{2} + ε) = ε; \\ (1 - C_{f}^{- 1} (1 - D_{k})) ξ_{- ε}^{2} - (b_{1} - ε) = - ε . \end{array}$

(I) When k is nondecreasing, it follows by ( $b_{3}$ ) that there is $δ_{ε} \in (0, min {1, δ_{1} / 2})$ (which is corresponding to ε) sufficiently small such that for $σ \in (0, δ_{ε})$ $\begin{array}{l} (3.6) & (b_{1} - ε) k^{2} (d (x) - σ) ⩽ (b_{1} - ε) k^{2} (d (x)) < b (x), x \in D_{σ}^{-} = Ω_{2 δ_{ε}} / {\bar{Ω}}_{σ}; \\ (3.7) & b (x) < (b_{2} + ε) k^{2} (d (x)) ⩽ (b_{2} + ε) k^{2} (d (x) + σ), x \in D_{σ}^{+} = Ω_{2 δ_{ε} - σ}, \end{array}$ and for $x \in Ω_{2 δ_{ε}}$ $\begin{array}{l} ξ_{- ε}^{2} - ξ_{- ε}^{2} \frac{{(2 F (ψ (ξ_{- ε} K (d (x)))))}^{1 / 2}}{ξ_{- ε} K (d (x)) f (ψ (ξ_{- ε} K (d (x))))} (\frac{K (d (x)) k^{'} (d (x))}{k^{2} (d (x))} + \frac{K (d (x))}{k (d (x))} Δ d (x)) \\ - (b_{1} - ε) < 0; \\ ξ_{+ ε}^{2} - ξ_{+ ε}^{2} \frac{{(2 F (ψ (ξ_{+ ε} K (d (x)))))}^{1 / 2}}{ξ_{+ ε} K (d (x)) f (ψ (ξ_{+ ε} K (d (x))))} (\frac{K (d (x)) k^{'} (d (x))}{k^{2} (d (x))} + \frac{K (d (x))}{k (d (x))} Δ d (x)) \\ - (b_{2} + ε) > 0 . \end{array}$ Let $\begin{matrix} {\bar{u}}_{ε} (x) = ψ (ξ_{- ε} K (d_{1} (x))), x \in D_{σ}^{-} with d_{1} (x) : = d (x) - σ . \end{matrix}$ It follows that for $x \in D_{σ}^{-}$ $\begin{array}{l} △ {\bar{u}}_{ε} (x) - b (x) f ({\bar{u}}_{ε} (x)) \\ = ψ^{″} (ξ_{- ε} K (d_{1} (x))) ξ_{- ε}^{2} k^{2} (d_{1} (x)) + ξ_{- ε} ψ^{'} (ξ_{- ε} K (d_{1} (x))) \\ \times (k^{'} (d_{1} (x)) + k (d_{1} (x)) Δ d (x)) - b (x) f (ψ (ξ_{- ε} K (d_{1} (x)))) \\ ⩽ f (ψ (ξ_{- ε} K (d_{1} (x)))) k^{2} (d_{1} (x)) (ξ_{- ε}^{2} + ξ_{- ε}^{2} \frac{ψ^{'} (ξ_{- ε} K (d_{1} (x)))}{ξ_{- ε} K (d_{1} (x)) ψ^{″} (ξ_{- ε} K (d_{1} (x)))} \\ \times (\frac{k^{'} (d_{1} (x)) K (d_{1} (x))}{k^{2} (d_{1} (x))} + \frac{K (d_{1} (x))}{k (d_{1} (x))} Δ d (x)) - (b_{1} - ε)) \\ ⩽ 0, \end{array}$ i.e., ${\bar{u}}_{ε}$ is a supersolution to Eq. (1.1) in $D_{σ}^{-}$ .

In a similar way, we can show that $\begin{matrix} {\underline{u}}_{ε} = ψ (ξ_{+ ε} K (d_{2} (x))), x \in D_{σ}^{-} with d_{2} (x) : = d (x) + σ \end{matrix}$ is a subsolution of Eq. (1.1) in $D_{σ}^{+}$ .

Now let u be an arbitrary solution to problem (1.1). We assert that there exists a positive constant M such that $\begin{array}{l} (3.8) & u ⩽ M v_{0} (x) + {\bar{u}}_{ε}, x \in D_{σ}^{-}; \\ (3.9) & {\underline{u}}_{ε} ⩽ u + M v_{0} (x), x \in D_{σ}^{+}, \end{array}$ where $v_{0}$ is the solution to problem (3.2).

In fact, we can choose a large M such that $\begin{matrix} u ⩽ M v_{0} (x) + {\bar{u}}_{ε} on Γ_{2 δ_{ε}} : = {x \in Ω : d (x) = 2 δ_{ε}} . \end{matrix}$ By ( $f_{1}$ ) or ( $f_{01}$ ), we see that ${\bar{u}}_{ε} + M v_{0}$ is also a supersolution of Eq. (1.1) in $D_{σ}^{-}$ . Since $u < {\bar{u}}_{ε}$ on $Γ_{σ} : = {x \in Ω : d (x) = σ}$ , (3.8) follows by Lemma 3.1.

In a similar way, we can show (3.9).

Hence, $x \in D_{σ}^{-} \cap D_{σ}^{+}$ , by letting $σ \to 0$ , we have $\begin{matrix} 1 - \frac{M v_{0} (x)}{ψ (ξ_{+ ε} K (d (x)))} ⩽ \frac{u (x)}{ψ (ξ_{+ ε} K (d (x)))} and \frac{u (x)}{ψ (ξ_{- ε} K (d (x)))} ⩽ 1 + \frac{M v_{0} (x)}{ψ (ξ_{- ε} K (d (x)))} . \end{matrix}$ Consequently, $\begin{matrix} (3.10) & 1 ⩽ lim_{d (x) \to 0} inf \frac{u (x)}{ψ (ξ_{+ ε} K (d (x)))} and lim_{d (x) \to 0} sup \frac{u (x)}{ψ (ξ_{- ε} K (d (x)))} ⩽ 1 . \end{matrix}$ Thus letting $ε \to 0$ in (3.10), we obtain (1.14). Moreover, when $C_{f} \in [1, \infty)$ , one can see by Lemma 2.3 that $\begin{matrix} lim_{d (x) \to 0} \frac{ψ (ξ_{1} K (d (x)))}{ψ (K (d (x)))} = ξ_{1}^{1 - C_{f}}, lim_{d (x) \to 0} \frac{ψ (ξ_{2} K (d (x)))}{ψ (K (d (x)))} = ξ_{2}^{1 - C_{f}} . \end{matrix}$

(II) When k is nonincreasing, let $ξ_{- ε}$ and $ξ_{+ ε}$ be as in (3.4), $\begin{matrix} h_{1} (d (x)) : = K (d (x)) - K (σ), d (x) > σ > 0 . \end{matrix}$ By using ( $b_{3}$ ) and Lemma 2.3, we see that $\begin{matrix} (3.11) & lim_{(d (x), σ) \to (0, 0)} \frac{ψ^{'} (ξ_{- ε} h_{1} (d (x)))}{ξ_{- ε} h_{1} (d (x)) ψ^{″} (ξ_{- ε} h_{1} (d (x)))} = - C_{f}^{- 1}, \end{matrix}$ and there is $δ_{ε} \in (0, min {1, δ_{1} / 2})$ (which is corresponding to ε) sufficiently small such that $\begin{matrix} (3.12) & (b_{1} - ε) k^{2} (d (x)) ⩽ b (x) ⩽ (b_{2} + ε) k^{2} (d (x)), x \in Ω_{2 δ_{ε}} . \end{matrix}$ Let $σ \in (0, δ_{ε})$ and $\begin{matrix} {\bar{u}}_{ε} (x) = ψ (ξ_{- ε} h_{1} (d (x))), x \in D_{σ}^{-}, \end{matrix}$ by using Lemma 2.1, (3.5), (3.11) and $\begin{array}{l} 0 < \frac{K (d (x)) - K (σ)}{K (d (x))} < 1, \\ lim_{d (x) \to 0} (\frac{k^{'} (d (x)) K (d (x))}{k^{2} (d (x))} + \frac{K (d (x))}{k (d (x))} Δ d (x)) = 1 - D_{k} ⩽ 0, \end{array}$ we have for $x \in D_{σ}^{-}$ $\begin{array}{l} △ {\bar{u}}_{ε} (x) - b (x) f ({\bar{u}}_{ε} (x)) \\ = ψ^{″} (ξ_{- ε} h_{1} (d (x))) ξ_{- ε}^{2} k^{2} (d (x)) + ξ_{- ε} ψ^{'} (ξ_{- ε} h_{1} (d (x))) \\ \times (k^{'} (d (x)) + k (d (x)) Δ d (x)) - b (x) f (ψ (ξ_{- ε} h_{1} (d (x)))) \\ ⩽ f (ψ (ξ_{- ε} h_{1} (d (x)))) k^{2} (d (x)) (ξ_{- ε}^{2} + ξ_{- ε}^{2} \frac{ψ^{'} (ξ_{- ε} h_{1} (d (x)))}{ξ_{- ε} h_{1} (d (x)) ψ^{″} (ξ_{- ε} h_{1} (d (x)))} \\ \times \frac{K (d (x)) - K (σ)}{K (d (x))} (\frac{k^{'} (d (x)) K (d (x))}{k^{2} (d (x))} + \frac{K (d (x))}{k (d (x))} Δ d (x)) - (b_{1} - ε)) \\ ⩽ 0, \end{array}$ i.e., ${\bar{u}}_{ε}$ is a supersolution to Eq. (1.1) in $D_{σ}^{-}$ .

In a similar way, we can show that $\begin{matrix} {\underline{u}}_{ε} = ψ (ξ_{+ ε} h_{2} (d (x))), h_{2} (d (x)) : = K (d (x)) + K (σ), x \in D_{σ}^{-} \end{matrix}$ is a subsolution of Eq. (1.1) in $D_{σ}^{+}$ .

Using a similar way as in (I), we can show that the result of Theorem 1.1 holds. Here, we omit its proof.

This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2.

Let $ε \in (0, b_{1} / 4)$ , $\begin{matrix} ζ_{- ε} = \sqrt{2 C_{f} (b_{1} - 2 ε)}, ζ_{+ ε} = \sqrt{2 C_{f} (b_{2} + 2 ε)}, \end{matrix}$ where $b_{1}$ and $b_{2}$ are given as in ( $b_{4}$ ).

Then $\begin{matrix} \frac{ξ_{3}}{2} < ξ_{- ε} < ξ_{+ ε} < 2 ξ_{4}, \end{matrix}$ where $ξ_{3}$ and $ξ_{4}$ are given as in (1.22).

Let $\begin{matrix} {\bar{u}}_{ε} (x) = ψ (ζ_{- ε} h_{3} (d (x))), h_{3} (d (x)) : = \sqrt{h (d (x))} - \sqrt{h (σ)}, d (x) > σ > 0, \end{matrix}$ where h is given as in (1.19).

We see by ( $b_{4}$ ) that there is $δ_{ε} \in (0, min {1, δ_{1} / 2})$ (which is corresponding to ε) sufficiently small such that $\begin{matrix} (3.13) & (b_{1} - ε) {(d (x))}^{- 2} {\hat{L}}_{0} (d (x)) ⩽ b (x) ⩽ (b_{2} + ε) {(d (x))}^{- 2} {\hat{L}}_{0} (d (x)), x \in Ω_{2 δ_{ε}} . \end{matrix}$ By Proposition 2.7, (3.1) and ${\hat{L}}_{0}$ is normalized regularly varying at zero, we see that $\begin{array}{l} lim_{d (x) \to 0} \frac{{\hat{L}}_{0} (d (x))}{h (d (x))} = 0; \\ lim_{d (x) \to 0} \frac{1}{2} (\frac{d (x) {\hat{L}}_{0}^{'} (d (x))}{{\hat{L}}_{0} (d (x))} - 1 + d (x) Δ d (x)) = - \frac{1}{2}; \\ lim_{(d (x), σ) \to (0, 0)} \frac{ψ^{'} (ζ_{- ε} h_{3} (d (x)))}{ζ_{- ε} h_{3} (d (x)) ψ^{″} (ζ_{- ε} h_{3} (d (x)))} = - C_{f}^{- 1}, \end{array}$ we see that for $σ \in (0, δ_{ε})$ and $x \in D_{σ}^{-}$ $\begin{matrix} 0 < \frac{\sqrt{h (d (x))} - \sqrt{h (σ)}}{\sqrt{h (d (x))}} < 1, \end{matrix}$ and $\begin{array}{l} △ {\bar{u}}_{ε} (x) - b (x) f ({\bar{u}}_{ε} (x)) \\ ⩽ △ {\bar{u}}_{ε} (x) - (b_{1} - ε) {(d (x))}^{- 2} {\hat{L}}_{0} (d (x)) f ({\bar{u}}_{ε} (x)) \\ = ψ^{″} (ζ_{- ε} h_{3} (d (x))) ζ_{- ε}^{2} {(h_{3}^{'} (d (x)))}^{2} + ζ_{- ε} ψ^{'} (ζ_{- ε} h_{3} (d (x))) \\ \times (h_{3}^{″} (d (x)) + h_{3}^{'} (d (x)) Δ d (x)) \\ - (b_{1} - ε) {(d (x))}^{- 2} {\hat{L}}_{0} (d (x)) f (ψ (ζ_{- ε} h_{3} (d (x)))) \\ = f (ψ (ζ_{- ε} h_{3} (d (x)))) {(d (x))}^{- 2} {\hat{L}}_{0} (d (x)) (\frac{ζ_{- ε}^{2}}{4} \frac{{\hat{L}}_{0} (d (x))}{h (d (x))} \\ + ζ_{- ε}^{2} \frac{\sqrt{h (d (x))} - \sqrt{h (σ)}}{\sqrt{h (d (x))}} \frac{ψ^{'} (ζ_{- ε} h_{3} (d (x)))}{ζ_{- ε} h_{3} (d (x)) ψ^{″} (ζ_{- ε} h_{3} (d (x)))} \\ \times (- \frac{{\hat{L}}_{0} (d (x))}{4 h (d (x))} + \frac{1}{2} (\frac{d (x) {\hat{L}}_{0}^{'} (d (x))}{{\hat{L}}_{0} (d (x))} - 1 + d (x) Δ d (x))) \\ - (b_{1} - ε)) \\ ⩽ 0, \end{array}$ i.e., ${\bar{u}}_{ε}$ is a supersolution to Eq. (1.1) in $D_{σ}^{-}$ .

In a similar way, we can show that $\begin{matrix} {\underline{u}}_{ε} = ψ (ζ_{+ ε} h_{4} (d (x))), h_{4} (d (x)) : = \sqrt{h (d (x))} + \sqrt{h (σ)}, x \in D_{σ}^{+}, \end{matrix}$ is a subsolution of Eq. (1.1) in $D_{σ}^{+}$ .

The conclusion follows as in the proof of Theorem 1.1(I) and the proof is omitted.

This completes the proof of Theorem 1.2. □

Footnotes

Acknowledgement

This work is supported in part by NSF of P. R. China under grant 11571295.

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Boundary behavior of large solutions for semilinear elliptic equations with weights

Abstract

Keywords

1. Introduction and the main results

Proposition 2.2 (Representation theorem).

Proposition 2.5. Let ρ > 0 . If f ∈ NRV ρ , then F ∈ NRV ρ + 1 ; and if F ∈ NRV ρ + 1 , then f ∈ RV ρ . Proposition 2.6 ((Asymptotic behavior) (Karamata’s theorem in [7])).

Proposition 2.7 ([46, Lemma 2.3]).

Lemma 2.1 ([52, Theorem 2.1]).

Lemma 3.1 ([16, Lemma 2.4] (The comparison principle)).

Footnotes

Acknowledgement

References

Proposition 2.5.
Let $ρ > 0$ . If $f \in {NRV}_{ρ}$ , then $F \in {NRV}_{ρ + 1}$ ; and if $F \in {NRV}_{ρ + 1}$ , then $f \in {RV}_{ρ}$ .
Proposition 2.6 ((Asymptotic behavior) (Karamata’s theorem in [7])).