Exact boundary behavior for the solutions to a class of infinity Laplace equations

Exact boundary behavior for the solutions to a class of infinity Laplace equations

Ling Mi

School of Science, Linyi University, Linyi, Shandong, 276005, P.R. China Received 5 December 2015, appeared 23 May 2016

Communicated by Michal Feˇckan

Abstract. In this paper, by Karamata regular variation theory and the method of lower and upper solutions, we give an exact boundary behavior for the unique solu- tion near the boundary to the singular Dirichlet problem −∆_∞u = b(x)g(u), u > 0, x ∈ Ω, u|_∂Ω = 0, where Ω is a bounded domain with smooth boundary in R^N, g ∈ C¹((0,∞),(0,∞)), g is decreasing on (0,∞) and the function b ∈ C(Ω^¯) which is positive in Ω. We find a new structure condition ong which plays a crucial role in the boundary behavior of the solutions.

Keywords: infinity-Laplacian, singular Dirichlet problem, the exact asymptotic behav- ior, lower and upper solutions.

2010 Mathematics Subject Classification: 35J25, 35B40, 35J67.

1 Introduction and the main results

LetΩbe a bounded domain with smooth boundary inR^N (N≥2). In this paper, we consider the exact asymptotic behavior near the boundary to the following singular Dirichlet problem

−_∆∞u=b(x)g(u)_, u>_0, x∈ _Ω, u|_∂Ω=_{0, (1.1)} where the operator ∆_∞ is the∞-Laplacian, and it is defined as

∆_∞u:=hD²uDu,Dui=

∑

D_iuD_ijuD_ju, (1.2)

bsatisfies

(b₁) b∈C(_Ω^¯)is positive in Ω, andgsatisfies

(_g1) _g∈C¹((_0,∞)_,(_0,∞))_{, lims→0}+ g(_s) =_∞andgis decreasing on(_0,∞)_.

BCorresponding author. Email: mi-ling@163.com

This operator (1.2) is called the infinity Laplacian, which was first introduced in the work of Aronsson [2] in connection with the geometric problem of finding the so-called absolutely minimizing functions inΩ. As a result of the high degeneracy of the∞-Laplacian, the associ- ated Dirichlet problems may not have classical solutions. Therefore solutions are understood in the viscosity sense, a concept introduced by Crandall, Lions [13] and Crandall, Evans, Lions [12], and to be defined in Section 2. By using the viscosity solutions, Jensen [21] proved the existence and uniqueness of the viscosity solutions to the Dirichlet problem to the infinity harmonic equation. Later, Lu and Wang [23] obtained a uniqueness theorem for the Dirichlet problem to the infinity harmonic equation in the perspective of PDE. The infinity Laplace equation in turn is a very topical differential operator that appears in many contexts and has been extensively studied, see, for instance, [3,5,6,11,24–26,29,32,33,40] and the references therein.

Next, let us review the following singular elliptic boundary value problem involving the classical Laplace operator∆, i.e.

−_∆u=b(x)g(u), u>0, x∈_Ω, u|_∂Ω =0. (1.3) Problem (1.3) arises in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrical materials and has been discussed and extended by many authors in many contexts, for instance, the existence, uniqueness, regularity and boundary behavior of solutions, see, [1,4,14–20,22,28,30,34,35,41–43,45,46] and the references therein.

The pioneering work of problem (1.3) is Crandall, Rabinowitz, Tartar [14] and Fulks, May- bee [15]. Forb ≡ 1 in Ωand g satisfying (g₁), [14] and [15] derived that problem (1.3) has a unique solutionu∈C^2+α(_Ω)∩C(_Ω^¯). Moreover, in [14], the following result was established:

ifψ₁ ∈C[0,δ₀]∩C²(0,δ₀]is the local solution to the problem

−ψ⁰⁰₁(t) =g(ψ₁(t)), ψ₁(t)>0, 0<t<δ₀, ψ₁(0) =0, (1.4) then there exist positive constantsc₁andc2such that

c₁ψ₁(d(x))≤u(x)≤c₂ψ₁(d(x)) near∂Ω.

In particular, wheng(u) =u^−γ,γ>1,uhas the property

c₁(d(x))^2/(1+γ) ≤u(x)≤c₂(d(x))^2/(1+γ) near∂Ω. (1.5) By constructing global subsolutions and supersolutions, Lazer and McKenna [22] showed that (1.5) continued to hold on ¯Ω. Then, u ∈ H₀¹(_Ω) if and only if γ < 3. This is a basic characteristic of problem (1.3).

It is very worthwhile to point out that Cîrstea and Rˇadulescu [8–10] introduced the Kara- mata regular variation theory which is a basic tool in stochastic process to study the boundary behavior and uniqueness of solutions to boundary blow-up elliptic problems and obtained a series of rich and significant information about the boundary behavior of solutions. For fur- ther insight on the boundary blow-up elliptic problems, please refer to [36,37,44] and the references therein.

Later, by means of Karamata regular variation theory, Zhang et al. [42,43,45,46] proved the first or second boundary expansion of solutions to problem (1.3). The author et al. [28,30]

further proved the second boundary expansion of solutions to problem (1.3).

Ben Othman, Maagli, Masmoudi, Zribi [4] and Gontara, Maagli, Masmoudi, Turki [19]

introduced a large class of functions b(x) which belong to the Kato class K(_Ω) and proved the boundary behavior of solutions for problem (1.1) when gis normalized regularly varying at zero with index −γ (γ > 0). Later, Zhang et al. [45] extend the previous results on the boundary behavior of the solution u of problem (1.1) to the case where the weight functions b(x)belong to the Kato class K(_Ω)or b(x)lie into a class of functionsΛthat was introduced by Cîrstea and Rˇadulescu in [8–10] for non-decreasing functions and by Mohammed in [31] for nonincreasing functions as the set of positive monotonic functionsC¹(0,δ₀)∩L¹(0,δ₀) (δ₀>0) which satisfy

tlim→0⁺

d dt

K(t) k(t)

=:C_k ∈[0,∞), K(t) =

Z _t

0 k(s)ds. (1.6) Recently, N. Zeddini et al. [41] gave a common proof for theorems in [45] and extended these results.

Now let us return to problem (1.1).

WhenΩis a bounded domain that satisfies both the uniform interior and uniform exterior sphere conditions and b ≡ 1 inΩ, Bhattacharya and Mohammed [5] established that: let g satisfy (g₁) andube a solution of (1.1), then there are positive constantsaandc, with 0<a <c such that

ψ⁻_a¹(√

2d(x))≤u(x)≤ ψ_c⁻¹(√ 2d(x)) whered(x)is the distance ofxfrom∂Ω, and

ψa(t) =

Z _t

0

1

G_a(s)^{14ds, Ga}(t) =

Z _a

t g(s)ds, 0< t<a.

Recently, the author [29] extended the result in [5] to the weight function bwhich belong to the setΛ. Theorem 1.1 in [29] established the following result: letgsatisfy (g₁) and

(g⁰₂) there existsγ>1 such that

slim→0⁺

g⁰(s)s

g(s) =:−γ;

bsatisfy (b₁) and

(b⁰₂) there exist somek∈ _Λand a positive constant b₀ ∈_Rsuch that

d(limx)→0

b(x)

k⁴(d(x)) =b₀.

IfC_k(γ+₃)>4, then for the unique solutionuof problem (1.1), it holds that lim

d(x)→0

u(x) φ K⁴³(d(x))

=ξ₀, (1.7)

whereφis uniquely determined by Z _φ(t)

0

ds g(s)¹³

= t, t>0 (1.8)

and

ξ₀ = ^27b0(3+γ) 64 (3+γ)C_k−4

!₃₊¹

γ

. (1.9)

For convenience, we introduce the following class of functions.

Let Λ1 denote the set of all Karamata functions ˆL, which are normalized slowly varying at zero defined on(0,a)for somea>0 by

Lˆ(t) =c₀exp _{Z a}

1

s

y(ν) ν dν

, s∈(0,a₁),

for somea₁ ∈(0,a), wherec0>0 and the functiony ∈C((0,a₁])with lim_s→0⁺y(s) =0.

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 weight functionblies into Λ1 and the nonlinear term gsatisfies the following structure condition

(g2) there existsCg >0 such that lims→0

1 3g²³(s)^g

0(s)

Z _s

0 g^−1/3(ν)dν=−Cg, i.e.

lims→0(g¹³(s))⁰

Z _s

0 g^−1/3(ν)dν=−C_g. A complete characterization ofgin (g₂) is provided in Lemma3.2.

Note that in this paper we extend the previous results in all two directions. We extend g(u)to a more general class of functions which include the condition (g⁰₂) andb(x)belongs to another class of functionsΛ1.

Our main results are summarized as follows.

Theorem 1.1. Let g satisfy(g₁)–(g₂), b satisfy(b₁)and (b₂) There exists a positive constant b₀ ∈_Rsuch that

lim

d(_x)→0

b(x)

a(d(x)) =b₀, where

a(t) =t^−λL(t), L∈ _Λ1, λ≤4 and Z _η

0 s^1−3λL(s)ds<_∞ for someη>0. (1.10) If C_g <1 and 4C_g+λ(1−C_g)>1,for the unique solution u of problem(1.1), it holds that

d(limx)→0

u(_x)

φ h(d(x)) =ξ0, (1.11)

whereφis uniquely determined by(1.8), h(t) =

Z _t

0 s^1−3λL¹³(s)ds, (1.12) and

ξ₀=

3b₀

(4−λ)Cg+ (λ−1)

1−Cg 3

. (1.13)

Remark 1.2 (Existence and uniqueness [5, Corollary 6.3.]). Let g : (0,∞) → (0,∞) be non- increasing and b ∈ C(_Ω) be a positive function such that sup_x∈Ωb(x) < ∞. The singular boundary value problem (1.1) admits a unique solution.

Remark 1.3. By the following Proposition2.7, one can see that whenλ<4,hin (1.12) satisfies h(t)∼= ³

4−λt^4−3λL¹³(t)_.

Remark 1.4. Some basic examples of the functions which satisfy(g₂)are (i₁) When g(s) =s^−γ, γ>0, C_g = ^γ

γ+3,

φ(t) = ((γ+3)t)/3₃₊³_γ

, ∀ t>0.

(_i2) _When g(s) =s^−γe^(−lns)β, γ>_0, β< _1, β6=0, s∈(_0,s₀]_, s₀∈ (_{0, 1})_, C_g = ^γ

γ+3. (i₃) When g(s) =β⁻³s^3(1+β)e^3s−β, β>0, s ∈(0,

β 1+β

¹

β], C_g =1.

(i₄) When g(s) =β⁻³s^3(1+β)e^−3s−βe^e3s−β, β>0, s ∈(0,s₀], s₀∈(0, 1), C_g =1.

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 Theorem1.1will be given in Section 4.

2 Preparation

Our approach relies on Karamata regular variation theory established by Karamata in 1930 which is a basic tool in the theory of stochastic process (see [7,27,39] and the references therein). In this section, we first give a brief account of the definition and properties of regularly varying functions involved in our paper (see [7,27,39]).

Definition 2.1. A positive measurable function f defined on [a,∞), for some a > 0, is called regularly varying at infinity with index ρ, written as f ∈ RVρ, if for each ξ > 0 and some ρ∈_R,

slim→_∞

f(ξs)

f(s) =ξ^ρ. (2.1)

In particular, whenρ=0, f is calledslowly varying at infinity.

Clearly, if f ∈ RVρ, then L(s):= f(s)/s^ρ is slowly varying at infinity.

Definition 2.2. A positive measurable function f defined on [a,∞), for some a > 0, is called rapidly varying at infinityif for each ρ>1

slim→_∞

f(s)

s^ρ =_∞. (2.2)

We also see that a positive measurable function g defined on (0,a) for some a > 0, is regularly varying at zero with indexσ(written as g ∈ RVZ_σ) ift → g(1/t)belongs to RV−σ. Similarly, gis calledrapidly varying at zeroif t→g(1/t)is rapidly varying at infinity.

Proposition 2.3(Uniform convergence theorem). If f ∈ RV_ρ, then (2.1) holds uniformly forξ ∈ [c₁,c2] with 0 < c₁ < c2. Moreover, if ρ < 0, then uniform convergence holds on intervals of the form(a₁,∞)with a₁ > 0; ifρ >0, then uniform convergence holds on intervals(0,a₁]provided f is bounded on(0,a₁]for all a₁>0.

Proposition 2.4(Representation theorem). A function L is slowly varying at infinity if and only if it may be written in the form

L(s) = ϕ(s)exp _{Z s}

a1

y(τ) τ dτ

, s ≥a₁, (2.3)

for some a1 ≥a, where the functionsϕand y are measurable and for s→_{∞, y}(_s)→0andϕ(_s)→c0, with c₀ >0.

We say that

Lˆ(s) =c₀exp _{Z s}

a1

y(τ) τ dτ

, s ≥a₁, (2.4)

isnormalizedslowly varying at infinity and

f(s) =c₀s^ρLˆ(s), s ≥a₁, (2.5) isnormalizedregularly varying at infinity with indexρ(and written as f ∈ NRV_ρ).

Similarly, g is called normalized regularly varying at zero with index ρ, written as g ∈ NRVZρ ift →g(_1/t)_{belongs to NRV−ρ.}

A function f ∈ RV_ρbelongs to NRV_ρ if and only if f ∈C¹[a₁,∞)for somea₁ >0 and lim

s→_∞

s f⁰(s)

f(s) = ρ. (2.6)

Proposition 2.5. If functions L,L₁are slowly varying at zero, then

(_i) L^ρ (for everyρ ∈ _R), c₁L+c₂L₁ (c₁ ≥ 0, c₂ ≥ 0 with c₁+c₂ > 0), L◦L₁ (if L₁(t) → 0as t→0⁺), are also slowly varying at zero;

(ii) for everyρ >0and t→0⁺,

t^ρL(t)→0, t^−ρL(t)→_∞;

(iii) forρ ∈_Rand t→0⁺,ln(L(t))/lnt →0andln(t^ρL(t))/lnt→ρ.

Proposition 2.6.

(_i) _{If g1}∈ RVZρ₁, g2 ∈RVZρ2 withlim_t→0⁺g2(_t) =_{0,then g1}◦g2 ∈RVZρ₁ρ2. (ii) If g∈RVZ_ρ,then g^α ∈RVZ_ραfor everyα∈_R.

Proposition 2.7(Asymptotic behavior). If a function L is slowly varying at zero, then for a > 0 and t→0⁺,

(i) Rt

0s^ρL(s)ds∼= (ρ+1)⁻¹t^1+ρ L(t), forρ >−1;

(_ii) Ra

t s^ρL(s)ds∼= (−ρ−1)⁻¹t^1+ρ L(t), forρ<−1.

Proposition 2.8. If a function L be defined on (0,η],is slowly varying at zero. Then we have

tlim→0⁺

L(t) Rη

t L(s)

s ds

=0. (2.7)

If furtherRη 0

L(s)

s ds converges, then we have

tlim→0⁺

L(t) Rt

0 L(s)

s ds

=0. (2.8)

Proposition 2.9 ([46, Proposition 2.6]). Let Z ∈ C¹(0,η] be positive and lim_t→0^{+ sZ}_Z⁰₍⁽_s^s₎⁾ = +_∞. Then Z is rapidly varying to zero at zero.

Proposition 2.10 ([46, Proposition 2.7]). Let Z ∈ C¹(0,η)be positive and lim_t→0⁺ sZ⁰(s)

Z(s) = −_∞.

Then Z is rapidly varying to infinity at zero.

Next, we recall here the precise definition of viscosity solutions for problem (1.1).

Definition 2.11. A functionu∈C(_Ω)is a viscosity subsolution of the PDE∆_∞u= −b(x)g(u) in Ω if for every ϕ ∈ C²(_Ω), with the property that u− ϕ has a local maximum at some x0 ∈_{Ω, then}

∆_∞ϕ(x₀)≥ −b(x₀)g(u(x₀)).

Definition 2.12. A function ¯u∈C(_Ω)is a viscosity supersolution of the PDE∆∞u=−b(x)g(u) inΩif for everyϕ∈C²(_Ω), with the property that ¯u−ϕhas a local minimum at somex₀∈ _Ω, then

∆_∞ϕ(x0)≤ −b(x0)g(u¯(x0)).

Definition 2.13. A functionu ∈C(_Ω)is a viscosity solution of the PDE∆_∞u =−b(x)g(u)in Ωif it is both a subsolution and a supersolution.

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. Let

a(t) =t^−λL(t) and

h(t) =

Z _t

0 s^1−3λ(L(s))¹³ds, where t∈ (0,δ₀), λ≤4, Rη

0 s^1−3λ(L(s))¹³ds<_∞for someη>0and L∈ _Λ1.Then (i) lim

t→0⁺ (h⁰(t))⁴

h(t)a(t) = ⁴⁻₃^λ and lim

t→0⁺ th⁰(t)

h(t) = ⁴⁻₃^λ; (ii) lim

t→0⁺ th⁰⁰(t)

h⁰(t) = ¹⁻₃^λ; (iii) lim

t→0⁺

(h⁰(t))²h⁰⁰(t)

a(t) = ¹⁻₃^λ.

Proof. (i) Sinceh⁰(t) =t^1−3λ(L(t))¹³, then (h⁰(t))⁴

h(t)a(t) = ^t

4−4λ 3 L⁴³(t) t^−λL(t)Rt

0s^1−3λ(L(s))¹³ds = ^t

4−λ 3 L¹³(t) Rt

0s^1−3λ(L(s))¹³ds and

th⁰(t)

h(t) = ^t

4−λ 3 L¹³(t) Rt

0s^1−3λ(L(s))¹³ds.

Hence, whenλ<4, by Proposition2.7, we get lim_t→0^{+ (}_h^h_(t⁰⁽₎^t_a⁾⁾_(t⁴₎ =lim_t→0^{+ th}_h(⁰⁽_t^t₎⁾ = ⁴⁻₃^λ; whenλ=4, by Proposition2.8, we get lim_t→0⁺ (h⁰(t))⁴

h(t)a(t) =lim_t→0⁺ th⁰(t) h(t) =0.

(ii) By a direct computation, we get h⁰⁰(t) = ¹−λ

3 t^−2+3λ(L(t))¹³ +¹

3t^1−3λ(L(t))⁻²³L⁰(t) and

th⁰⁰(t) h⁰(t) = ¹

3 tL⁰(t)

L(t) + ¹−λ 3 . It follows byL∈_Λ1 that lim_t→0^{+ tL}_L⁰₍⁽_t^t₎⁾ =0. Hence,

tlim→0⁺

th⁰⁰(t)

h⁰(t) = ¹−λ 3 . (iii) Since

(h⁰(t))²h⁰⁰(t) a(t) = ^th

00(t) h⁰(t)

(h⁰(t))³ ta(t) = ^th

00(t) h⁰(t) ^, by (ii), we get

tlim→0⁺

(h⁰(t))^p−2h⁰⁰(t)

a(t) = ¹−λ 3 .

Lemma 3.2. Let g satisfy(g₁)–(g₂).

(i) If g satisfies(g₂), then C_g≤1;

(_ii) _(g2)holds for C_g ∈(_{0, 1})if and only if g∈ NRV−γ;withγ>_0.In this caseγ=3C_g/(₁−C_g)_; (iii) (g₂)holds for Cg=0if and only if g is normalized slowly varying at zero;

(iv) if(g₂)holds with C_g =1,then g is rapidly varying to infinity at zero;

(v) if

slim→0⁺

g⁰⁰(s)g(s)

(g⁰(s))² =1, (3.1)

then g satisfies(g₂)with Cg=1.

Proof. Sincegsatisfies (g₁) and is strictly decreasing on(0,S₀), we see that 0<

Z _s

0

1

g^1/3(ν)^dν< ^s

g^1/3(s)^, ∀s∈(0,S₀), i.e.,

0<g^1/3(s)

Z _s

0

1

g^1/3(ν)^dν<s, ∀s∈(0,S0), (3.2) and

lims→0g^1/3(s)

Z _s

0

1

g^1/3(ν)^dν=_{0. (3.3)}

(i) Let

I(s) =− ¹ 3g²³(s)^g

0(s)

Z _s

0 g^−1/3(ν)dν, ∀s ∈(0,s₀). Integrate I(t)from 0 tos and integrate by parts, we obtain by (3.3) that

Z _s

0 I(t)dt=−g^1/3(s)

Z _s

0

1

g^1/3(ν)^dν+s, ∀s∈ (0,s₀), i.e.

0< ^g

1/3(s) s

Z _s

0

1

g^1/3(ν)^dν=1− Rs

0 I(t)dt

s , ∀s∈ (0,s₀). It follows from L’Hospital’s rule that

0≤ lim

s→0⁺

g^1/3(s) s

Z _s

0

1

g^1/3(ν)^dν=1− lim

s→0⁺I(s) =1−C_g. (3.4) So (i) holds.

(ii) When (g2) holds withCg ∈(_{0, 1}), it follows by (3.4) that

slim→0⁺

g(s)

sg⁰(s) = lim

s→0⁺

g^1/3(s)Rs

0 1

g^1/3(ν)dν sg⁰(s)Rs

0 1

g^1/3(ν)dνg¹³⁻¹(s) = −¹−Cg

3C_g , (3.5)

i.e.,g ∈ NRV_{−3Cg/(1−Cg)}.

Conversely, when g ∈ NRV−γ with γ > 0, i.e., lim_s→0^{+ sg}_g(⁰⁽_s^s₎⁾ =−γ and there exist pos- itive constant η and ˆL ∈ _Λ1 such that g(s) = c₀s^−γLˆ(s), s ∈ (0,η]. It follows by (2.8) and Proposition2.7(i) that

− lim

s→0⁺

1 3g²³(s)^g

0(s)

Z _s

0 g^−1/3(ν)dν= −¹ 3 lim

s→0⁺

sg⁰(s) g(s) s^lim→0⁺

g^1/3(s) s

Z _s

0 g^−1/3(ν)dν

= ^γ 3 lim

s→0⁺s^−γ3−1(L^ˆ(s))¹³

Z _s

0 ν

γ

3 Lˆ(ν)⁻¹³dν

= ^γ

3+γ = Cg. (iii) ByC_g =0 and the proof of (ii), one can see that

slim→0⁺

sg⁰(s)

g(s) = lim

s→0⁺

sg⁰(s)Rs

0 1

g^1/3(ν)dνg¹³⁻¹(s) g^1/3(s)Rs

0 1

g^1/3(ν)dν

=₃

slim→0⁺

g^1/3(s) s

Z _s

0

1 g^1/3(ν)^dν

⁻1 slim→0⁺

1 3g¹⁻¹³(s)^g

0(s)

Z _s

0

g^−1/3(ν)dν

=0,

i.e.,gis normalized slowly varying at zero.

Conversely, when gis normalized slowly varying at zero, i.e., lim_s→0^{+ sg}_g(⁰⁽_s^s₎⁾ =0, it follows by (3.4) that

slim→0⁺

1 3g¹⁻¹³(_s)^g

0(s)

Z _s

0 g^−1/3(ν)dν= lim

s→0⁺

1 3

sg⁰(s) g(s)

g^1/3(s) s

Z _s

0

1

g^1/3(ν)^dν=0.

(iv) ByCg =1 and the proof of (ii), we see that lim_s→0⁺ _sg^g(0(^s)s) =0, i.e., lim_s→0^{+ sg}_g⁰₍⁽_s^s₎⁾ = −_{∞, we} see by Proposition2.10thatg is rapidly varying to infinity at zero.

(v) By (3.1) and L’Hospital’s rule, we obtain that lims→0

g(s)

sg⁰(s) =lim

s→0 g(s) g⁰(s)

s =lim

s→0

d ds

g(s) g⁰(s)

=1−lim

s→0

g(s)g⁰⁰(s)

(g⁰(s))² =0. (3.6) Hence, by (g₁) and (3.6), we get that

lims→0

g²³(s) g⁰(s) =lim

s→0

g(s) sg⁰(s)

s

g¹³(s) =lim

s→0

g(s) sg⁰(s)^lims→0

s

g¹³(s) =0. (3.7) It follows by the L’Hospital’s rule and (3.7) that

lims→0

1 3g²³(_s)^g

0(s)

Z _s

0

g^−1/3(ν)dν

=lim

s→0

1 3

Rs

0 g^−1/3(ν)dν

g²³(s) g⁰(s)

=lim

s→0

1 3

1

2

3− ^g₍^00(s)g(s)

g⁰(s))²

=−1, i.e.Cg=1.

Lemma 3.3. Let g satisfy(g₁)–(g₂)andφbe the solution to the problem Z _φ(t)

0

ds

(g(s))¹³ =t, ∀t >0.

Then

(i) φ⁰(t) = g(φ(t))¹³, φ(t)>0, t>0, φ(0) =0andφ⁰⁰(t) = ¹₃ g(φ(t))⁻¹³g⁰(φ(t)), t>0;

(ii) φ∈ NRVZ₁−Cg andφ⁰ ∈ NRVZ−Cg;

(iii) lim_t→0⁺ _φ(ξh^t_(t)) =0uniformly forξ ∈[c₁,c₂]with0<c₁<c₂,where h is given as in(1.12).

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

(ii) It follows from (i), (3.5) and (g₂) that

tlim→0⁺

tφ⁰(t)

φ(t) = lim

t→0⁺

t(g(φ(t)))^p−11 φ(t)

=lim

s→0

(g(s))¹³ Rs 0

dν (g(ν))¹3

s =1−Cg,

i.e.,φ∈ NRVZ₁−Cg, and

tlim→0⁺

tφ⁰⁰(t) φ⁰(t) = ¹

3 lim

t→0⁺

g⁰(φ(t))(g(φ(t)))¹³ Rφ(t)

0 (g(ν))⁻¹³dν g(φ(t))

= ¹ 3 lim

s→0⁺

g⁰(s)(g(s))¹³ Rs

0(g(ν))⁻¹³dν g(s)

=−Cg.

(iii) By Lemma 3.1 (i), we see h ∈ NRVZ4−λ

3 . It follows by Proposition 2.4 that φ◦h ∈ NRVZ⁽4−λ)(1−Cg)

3

. Since 4Cg+λ(1−Cg)>1, the result follows by Proposition2.5(ii).

4 Proof of the Theorem

In this section, we prove Theorem1.1. First, we need the following result.

Lemma 4.1(Comparison principle [5, Lemma 4.3]). Suppose that f :Ω×_R→_Ris continuous, f(x,t)is non-decreasing in t.Assume further that f has one sign (either positive or negative) inΩ×_R. If u,v ∈C(_Ω^¯)are such that

∆∞u≥ f(x,u), ∆∞v≤ f(x,v) and u≤v on∂Ω, then u≤v inΩ.

First fix ε > 0. For any δ₀ > 0, we define Ωδ0 = {x ∈ _Ω : 0 < d(x) < δ₀}. Since Ω is C²-smooth, choose δ₁ ∈ (0,δ₀) such that d ∈ C²(_Ωδ1) and |∇d(x)| = 1, ∀ x ∈ _Ωδ1, and consequently∆∞d=0 inΩδ1 in the viscosity sense.

Proof of Theorem1.1. Letv∈C(_Ω^¯)be the unique solution of the problem

−_∆∞v=1, v >0, x∈_Ω, v|_∂Ω=0. (4.1) By Theorem 7.7 in [5], we see that

c₁d(x)≤v(x)≤c₂d(x), ∀x ∈_Ω near∂Ω. (4.2) wherec₁,c2are positive constants.

Now, we define

¯

u_ε = (ξ₀+ε)φ h(d(x)) _{for any} x∈_Ωδ1, where his given as in (1.12).

Let

η(t) = (ξ₀+ε)φ h(t), t∈(0,δ₁).

Note thathandφare all increasing in their respective definition domains. Therefore, whenδ₁ is small enough,ηis increasing in(0,δ₁). Letζ be the inverse ofη. One can easily check that

ζ⁰(t) = ¹

η⁰(ζ(t)) = (ξ₀+ε)φ⁰ h(ζ(t))h⁰(ζ(t))⁻¹ (4.3)

and

ζ⁰⁰(t) = − (ξ0+ε)φ⁰ h(ζ(t))h⁰(ζ(t))⁻³

× (ξ₀+ε)φ⁰⁰ h(ζ(t))(h⁰(ζ(t)))²

+(ξ₀+ε)φ⁰ h(ζ(t))h⁰⁰(ζ(t)). (4.4) Let (x₀,ψ) ∈ _Ωδ1 ∩C²(_Ωδ1) be a pair such that ¯uε ≥ ψ in a neighborhood N of x₀ and

¯

u_ε(x₀) =ψ(x₀)Then ϕ=ζ(ψ)∈C²(_Ωδ1), and

d(x)≥ ϕ(x) in N, d(x0) = ϕ(x0).

Since∆_∞d=0 inΩδ₁, we have∆_∞ϕ(x₀)≤0. A simple computation shows that

∆∞ϕ= ζ⁰⁰(ψ)(ζ⁰(ψ))²|Dψ|⁴+ (ζ⁰(ψ))³_∆∞ψ.

It follows by∆_∞ϕ(x₀)≤0 andζ⁰ >0 that

∆∞ψ(x0)≤ −ζ⁰⁰(ψ(x0))(ζ⁰(ψ(x0)))⁻¹|Dψ(x0)|⁴.

Moreover, since|Dd(x)| = 1 for x ∈ _Ωδ1 andd−ϕattains a local maximum at x₀, it follows that

1=|Dd(x₀)|= |ζ⁰(ψ(x₀))Dψ(x₀)|. Hence

∆_∞ψ(x₀)≤ −ζ⁰⁰(ψ(x₀))(ζ⁰(ψ(x₀)))⁻⁵. Combing with (4.3) and (4.4), we further obtain

∆_∞ψ(x₀)≤ ((ξ₀+ε))³ φ⁰ h(ϕ(x₀))³a(ϕ(x₀))

×

"

φ⁰⁰ h(ϕ(x₀))h(ϕ(x₀)) φ⁰ h(ϕ(x0))

(h⁰(ϕ(x₀)))⁴

h(ϕ(x₀))a(ϕ(x₀))+^h

00(ϕ(x₀))(h⁰(ϕ(x₀)))² a(ϕ(x₀))

# . Hence,

∆_∞ψ(x₀) +b(x₀)g(u¯ε(x₀))

≤((ξ₀+ε))³ φ⁰ h(ϕ(x₀))³a(ϕ(x₀))

×

"

φ⁰⁰ h(ϕ(x0))h(ϕ(x0)) φ⁰ h(ϕ(x₀))

(h⁰(ϕ(x₀)))⁴

h(ϕ(x₀))a(ϕ(x₀))+ ^h

00(ϕ(x₀))(h⁰(ϕ(x₀)))² a(ϕ(x₀)) + ((ξ₀+ε))^{−3 b}(x₀)

a(ϕ(x₀))

g(u¯_ε(x₀)) φ⁰ h(ϕ(x₀))³

#

=:((ξ₀+ε))³ φ⁰ h(d(x₀))³a(d(x₀))I(x₀).

Notice that h(d(x₀)) → 0 as δ₁ → 0 (and thereby x₀ tends to the boundary of Ω). Then, it follows from Lemmas3.1and3.3that

I(x₀)→ (λ−4)C_g+ (1−λ)

3 +b₀(ξ₀+ε)^−3−γ asδ₁ →0.

By the choice ofξ₀, we have I(x₀)<0 providedδ_1ε ∈(0,^δ₂¹)small enough. Thus

∆∞ψ(x₀)≤ −b(x₀)g(u¯_ε(x₀))_,

i.e., ¯u_ε is a supersolution of equation (1.1) in Ωδ_1ε. In a similar way, we can show that

u_ε = (ξ₀−ε)φ h(d(x)) is a subsolution of equation (1.1) inΩδ_1ε.

Letu ∈C(_Ω)be the unique solution to problem (1.1). We assert that there exists Mlarge enough such that

u(_x)≤ Mv(_x) +u¯ε(_x)_{, uε}(_x)≤u(_x) +_Mv(_x)_{, x}∈_Ωδ1ε, (4.5) wherev is the solution of problem (4.1).

In fact, we can chooseM large enough such that

u(x)≤u¯ε(x) +Mv(x) and u_ε(x)≤u(x) +Mv(x) on {x∈_Ω:d(x) =δ_1ε}. We see by (g₁) that ¯u_ε(x) +Mv(x)andu(x) +Mv(x)are also supersolutions of equation (1.1) in Ωδ_1ε. Sinceu = u¯_ε+Mv = u+Mv = u_ε = _{0 on} ∂Ω, (4.5) follows by (g1) and Lemma4.1.

Hence, forx ∈_Ωδ1ε

ξ₀−ε− ^Mv(x)

φ h(d(x)) ≤ ^u(x) φ h(d(x))

and u(x)

φ h(d(x)) ≤ξ₀+ε+ ^Mv(x) φ h(d(x))^. Consequently, by (4.2) and Lemma3.3 (iii),

ξ₀−ε≤lim inf

d(x)→0

u(x) φ(h(d(x)))^; lim sup

d(x)→0

u(x)

φ(h(d(x))) ≤ξ₀+ε.

Thus, lettingε→0, we obtain (1.11).

Thus the proof is finished by lettingε→0.

Acknowledgements

The author is greatly indebted to the anonymous referee for the very valuable suggestions and comments which surely improved the quality of the presentation. This work was partially supported by NSF of China (Grant Nos. 11301250, 11401287 ) and NSF of Shandong Province (Grant No. ZR2013AQ004).

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