Research

# Existence of positive solutions for variable exponent elliptic systems

Samira Ala1*, Ghasem Alizadeh Afrouzi2, Qihu Zhang3 and Asadollah Niknam4

Author Affiliations

1 Department of Mathematics, Sciences and Research, Islamic Azad University (IAU) Tehran, Iran

2 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

3 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China

4 Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

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Boundary Value Problems 2012, 2012:37  doi:10.1186/1687-2770-2012-37

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/37

 Received: 30 October 2011 Accepted: 3 April 2012 Published: 3 April 2012

© 2012 Ala et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We consider the system of differential equations

- Δ p ( x ) u = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] in  Ω , - Δ p ( x ) v = λ p ( x ) [ g ( x ) b ( v ) + h ( u ) ] in  Ω , u = v = 0 on  Ω ,

where Ω ⊂ ℝN is a bounded domain with C2 boundary ∂Ω, 1 < p(x) ∈C1 ( Ω ̄ ) is a function. Δ p ( x ) u  = div  ( | u | p ( x ) - 2 u ) is called p(x)-Laplacian. We discuss the existence of positive solution via sub-super solutions without assuming sign conditions on f(0), h(0).

MSC: 35J60; 35B30; 35B40.

##### Keywords:
positive solutions; p(x)-Laplacian problems; sub-supersolution

### 1. Introduction

The study of diferential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc., (see[1-3]). Many results have been obtained on this kind of problems, for example [1,3-8]. In [7], Fan gives the regularity of weak solutions for differential equations with variable exponent. On the existence of solutions for elliptic systems with variable exponent, we refer to [8,9]. In this article, we mainly consider the existence of positive weak solutions for the system

( P ) - Δ p ( x ) u = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] in  Ω , - Δ p ( x ) v = λ p ( x ) [ g ( x ) b ( v ) + h ( u ) ] in  Ω , u = v = 0 on  Ω ,

where Ω ⊂ ℝN is a bounded domain with C2 boundary Ω, 1 < p(x) ∈ C1 ( Ω ̄ ) is a function. The operator Δ p ( x ) u  = div  ( | u | p ( x ) - 2 u ) is called p(x)-Laplacian. Especially, if p(x) ≡ p (a constant), (P) is the well-known p-Laplacian system. There are many articles on the existence of solutions for p-Laplacian elliptic systems, for example [5,10]. Owing to the nonhomogeneity of p(x)-Laplacian problems are more complicated than those of p-Laplacian, many results and methods for p-Laplacian are invalid for p(x)-Laplacian; for example, if Ω is bounded, then the Rayleigh quotient

λ p ( x ) = inf u W 0 1 , p ( x ) ( Ω ) \ { 0 } Ω 1 p ( x ) | u | p ( x ) d x Ω 1 p ( x ) | u | p ( x ) d x

is zero in general, and only under some special conditions λp(x) > 0 (see [11]), and maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist, but the fact that the first eigenvalue λp > 0 and the existence of the first eigenfunction are very important in the study of p-Laplacian problems. There are more difficulties in discussing the existence of solutions of variable exponent problems.

Hai and Shivaji [10], consider the existence of positive weak solutions for the following p-Laplacian problems

( I ) - Δ p u = λ f ( v ) in  Ω , - Δ p v = λ g ( u ) in  Ω , u = v = 0 on  Ω

the first eigenfunction is used to construct the subsolution of p-Laplacian problems success-fully. On the condition that λ is large enough and

lim u + f M ( g ( u ) ) 1 ( p - 1 ) u p - 1 = 0 , for every M > 0 ,

the authors give the existence of positive solutions for problem (I).

Chen [5], considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:

( II ) - Δ p u = λ f ( u , v ) = λ u α v γ in  Ω , - Δ q v = λ g ( u , v ) = λ u δ v β in  Ω , u = v = 0 on  Ω

the first eigenfunction is used to construct the subsolution of problem(II), the main results are as following

(i) If α, β ≥ 0, γ, δ > 0, θ = (p - 1 - α)(q - 1 - β) - γδ > 0, then problem (II) has a positive weak solution for each λ > 0;

(ii) If θ = 0 and pγ = q(p - 1 - α), then there exists λ0 > 0 such that for 0 < λ < λ0, then problem (II) has no nontrivial nonnegative weak solution.

On the p(x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist. Even if the first eigenfunction of p(x)-Laplacian exist, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of p(x)-Laplacian problems. Zhang [12] investigated the existence of positive solutions of the system

- Δ p ( x ) u = λ p ( x ) f ( v ) in  Ω , - Δ p ( x ) v = λ p ( x ) g ( u ) in  Ω , u = v = 0 on  Ω ,

In this article, we consider the existence of positive solutions of the system

- Δ p ( x ) u = λ p ( x ) F ( x , u , v ) in  Ω , - Δ p ( x ) v = λ p ( x ) G ( x , u , v ) in  Ω , u = v = 0 on  Ω ,

where p(x) ∈ C1 ( Ω ̄ ) is a function, F(x, u, v) = [g(x)a(u) + f(v)], G(x, u, v) = [g(x)b(v) +h(u)], λ is a positive parameter and Ω ⊂ ℝN is a bounded domain.

To study p(x)-Laplacian problems, we need some theory on the spaces Lp(x)(Ω), W1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see [6,13]). If Ω ⊂ ℝN is an open domain, write

C + ( Ω ) = { h : h C ( Ω ) , h ( x ) > 1 f o r x Ω } , h + = sup  x Ω h ( x ) , h - = inf x Ω h ( x ) , for any  h C ( Ω ) .

Throughout the article, we will assume that:

(H1) Ω ⊂ ℝN is an open bounded domain with C2 boundary Ω.

(H2) p(x) ∈ C1 ( Ω ̄ ) and 1 < p- ≤ p+.

(H3) a, b C1([0, )) are nonnegative, nondecreasing functions such that

lim u + a ( u ) u p - - 1 = 0 , lim u + b ( u ) u p - - 1 = 0 .

(H4) f, h : [0, +) → R are C1, monotone functions, limu→+∞ f(u) = +, limu→+∞ h(u) = +, and

lim u + f M ( h ( u ) ) 1 ( p - - 1 ) u p - - 1 = 0 , M > 0 .

(H5) g : [0, +) (0, +) is a continuous function such that L 1 = min x Ω ̄ g ( x ) , and L 2 = max x Ω ̄ g ( x ) .

Denote

L p ( x ) ( Ω ) = u | u is a measurable real - valued function , Ω | u ( x ) | p ( x ) d x < .

We introduce the norm on Lp(x)(Ω) by

| u | p ( x ) = inf λ > 0 : Ω u ( x ) λ p ( x ) d x 1 ,

and (Lp(x)(Ω), |.|p(x)) becomes a Banach space, we call it generalized Lebesgue space. The space (Lp(x)(Ω), |.|p(x)) is a separable, reflexive, and uniform convex Banach space (see [[6], Theorems 1.10 and 1.14]).

The space W1,p(x)(Ω) is defined by W1,p(x)(Ω) = {u Lp(x) : | u| Lp(x)}, and it is equipped with the norm

u = | u | p ( x ) + | u | p ( x ) , u W 1 , p ( x ) ( Ω ) .

We denote by W01,p(x)(Ω) is the closure of C 0 Ω in W1,p(x)(Ω). W1,p(x)(Ω) and W 0 1 , p ( x ) ( Ω ) are separable, reflexive, and uniform convex Banach space (see [[6], Theorem 2.1] We define

( L ( u ) , v ) = Ω | u | p ( x ) - 2 u v d x , v , u W 0 1 , p ( x ) ( Ω ) ,

then L : W 0 1 , p ( x ) ( Ω ) ( W 0 1 , p ( x ) ( Ω ) ) * is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [[14], Theorem 3.1]).

If u , v W 0 1 , p ( x ) ( Ω ) , ( u , v ) is called a weak solution of (P) if it satisfies

Ω | u | p ( x ) - 2 u q d x = Ω λ p ( x ) F ( x , u , v ) q d x , q W 0 1 , p ( x ) ( Ω ) , Ω | v | p ( x ) - 2 v q d x = Ω λ p ( x ) G ( x , u , v ) q d x , q W 0 1 , p ( x ) ( Ω ) .

Define A : W 1 , p ( x ) ( Ω ) ( W 0 1 , p ( x ) ( Ω ) ) * as

< A u , φ > = Ω ( | u | p ( x ) - 2 u φ + l ( x , u ) φ ) d x , u W 1 , p ( x ) ( Ω ) , φ W 0 1 , p ( x ) ( Ω ) ,

where l(x, u) is continuous on Ω ̄ × , and l(x, .) is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [15], we have the following lemma.

Lemma 1.1. (Comparison Principle). Let u, v W1,p(x)(Ω) satisfying Au - Av ≥ 0 in ( W 0 1 , p ( x ) ( Ω ) ) * , φ ( x ) = min { u ( x ) - v ( x ) , 0 } . If φ ( x ) W 0 1 , p ( x ) ( Ω ) (i.e., u ≥ v on ∂Ω ), then u ≥ v a.e. in Ω.

Here and hereafter, we will use the notation d(x, Ω) to denote the distance of x ∈ Ω to the boundary of Ω.

Denote d(x) = d(x, Ω) and Ω ϵ = { x Ω | d ( x , Ω ) < ϵ } . Since Ω is C2 regularly, then there exists a constant δ ∈ (0, 1) such that d ( x ) C 2 ( Ω 3 δ ¯ ) , and |d(x)| ≡ 1.

Denote

v 1 ( x ) = γ d ( x ) , d ( x ) < δ , γ δ + δ d ( x ) γ 2 δ - t δ 2 p - - 1 ( L 1 + 1 ) 2 p - - 1 d t , δ d ( x ) < 2 δ , γ δ + δ 2 δ γ 2 δ - t δ 2 p - - 1 ( L 1 + 1 ) 2 p - - 1 d t , 2 δ d ( x ) .

Obviously, 0 v 1 ( x ) C 1 ( Ω ̄ ) . Considering

- Δ p ( x ) w ( x ) = η in  Ω , w = 0 on  Ω , (1)

we have the following result

Lemma 1.2. (see [16]). If positive parameter η is large enough and w is the unique solution of (1), then we have

(i) For any θ∈ (0, 1) there exists a positive constant C1 such that

C 1 η 1 p + - 1 + θ max  x Ω ̄ w ( x ) ;

(ii) There exists a positive constant C2 such that

max  x Ω ̄ w ( x ) C 2 η 1 p - - 1 .

### 2. Existence results

In the following, when there be no misunderstanding, we always use Ci to denote positive constants.

Theorem 2.1. On the conditions of (H1) - (H5), then (P) has a positive solution when λ is large enough.

Proof. We shall establish Theorem 2.1 by constructing a positive subsolution (Φ1, Φ2) and supersolution (z1, z2) of (P), such that Φ1 ≤ z1 and Φ2 ≤ z2. That is (Φ1, Φ2) and (z1, z2) satisfies

Ω | Φ 1 | p ( x ) - 2 Φ 1 q d x Ω λ p ( x ) g ( x ) a ( Φ 1 ) q d x + Ω λ p ( x ) f ( Φ 2 ) q d x , Ω | Φ 2 | p ( x ) - 2 Φ 2 q d x Ω λ p ( x ) g ( x ) b ( Φ 2 ) q d x + Ω λ p ( x ) h ( Φ 1 ) q d x ,

and

Ω | z 1 | p ( x ) - 2 z 1 q d x Ω λ p ( x ) g ( x ) a ( z 1 ) q d x + Ω λ p ( x ) f ( z 2 ) q d x , Ω | z 2 | p ( x ) - 2 z 2 q d x Ω λ p ( x ) g ( x ) b ( z 2 ) q d x + Ω λ p ( x ) h ( z 1 ) q d x ,

for all q W 0 1 , p ( x ) ( Ω ) with q ≥ 0. According to the sub-supersolution method for p(x)-Laplacian equations (see [16]), then (P) has a positive solution.

Step 1. We construct a subsolution of (P).

Let σ ∈ (0, δ) is small enough. Denote

ϕ ( x ) = e k d ( x ) - 1 , d ( x ) < σ , e k σ - 1 + σ d ( x ) k e k σ 2 δ - t 2 δ - σ 2 p - - 1 d t , σ d ( x ) < 2 δ , e k σ - 1 + σ 2 δ k e k σ 2 δ - t 2 δ - σ 2 p - - 1 d t , 2 δ d ( x ) .

It is easy to see that ϕ C 1 ( Ω ̄ ) . Denote

α = min inf  p ( x ) - 1 4 ( sup | p ( x ) | + 1 ) , 1 , ζ = min { a ( 0 ) L 1 + f ( 0 ) , b ( 0 ) L 1 + h ( 0 ) , - 1 } .

By computation

- Δ p ( x ) ϕ = - k ( k e k d ( x ) ) p ( x ) - 1 ( p ( x ) - 1 ) + ( d ( x ) + ln k k ) p d + d k , d ( x ) < σ , 1 2 δ - σ 2 ( p ( x ) - 1 ) p - - 1 - 2 δ - d 2 δ - σ ln  k e k σ 2 δ - d 2 δ - σ 2 p - - 1 p d + Δ d × ( k e k σ ) p ( x ) - 1 2 δ - d 2 δ - σ 2 ( p ( x ) - 1 ) p - - 1 - 1 ( L 1 + 1 ) , σ < d ( x ) < 2 δ , 0 , 2 δ < d ( x ) .

From (H3) and (H4), there exists a positive constant M > 1 such that

f ( M - 1 ) 1 , h ( M - 1 ) 1 .

Let σ = 1 k ln M , then

σ k = ln  M . (2)

If k is sufficiently large, from (2), we have

- Δ p ( x ) ϕ - k p ( x ) α , d ( x ) < σ . (3)

Let -λζ = , then

k p ( x ) α - λ p ( x ) ζ ,

from (3), then we have

- Δ p ( x ) ϕ λ p ( x ) ( a ( 0 ) L 1 + f ( 0 ) ) λ p ( x ) ( g ( x ) a ( ϕ ) + f ( ϕ ) ) , d ( x ) < σ . (4)

Since d ( x ) C 2 ( Ω 3 δ ¯ ) , then there exists a positive constant C3 such that

- Δ p ( x ) ϕ ( k e k σ ) p ( x ) - 1 2 δ - d 2 δ - σ 2 ( p ( x ) - 1 ) p - - 1 - 1 . 2 ( p ( x ) - 1 ) ( 2 δ - σ ) ( p - - 1 ) - 2 δ - d 2 δ - σ ln  k e k σ 2 δ - d 2 δ - σ 2 p - - 1 p d + Δ d C 3 ( k e k σ ) p ( x ) - 1 In  k , σ < d ( x ) < 2 δ .

If k is sufficiently large, let -λζ = , we have

C 3 ( k e k σ ) p ( x ) - 1 ln  k = C 3 ( k M ) p ( x ) - 1 ln  k λ p ( x ) ,

then

- Δ p ( x ) ϕ λ p ( x ) ( L 1 + 1 ) , σ < d ( x ) < 2 δ .

Since ϕ (x) 0 and a, f are monotone, when λ is large enough, then we have

- Δ p ( x ) ϕ λ p ( x ) ( g ( x ) a ( ϕ ) + f ( ϕ ) ) , σ < d ( x ) < 2 δ . (5)

Obviously

- Δ p ( x ) ϕ = 0 λ p ( x ) ( L 1 + 1 ) λ p ( x ) ( g ( x ) a ( ϕ ) + f ( ϕ ) ) , 2 δ < d ( x ) . (6)

Combining (4), (5), and (6), we can conclude that

- Δ p ( x ) ϕ λ p ( x ) ( g ( x ) a ( ϕ ) + f ( ϕ ) ) , a . e .  on  Ω . (7)

Similarly

- Δ p ( x ) ϕ λ p ( x ) ( g ( x ) b ( ϕ ) + h ( ϕ ) ) , a . e . on  Ω . (8)

From (7) and (8), we can see that (ϕ1, ϕ2) = (ϕ, ϕ) is a subsolution of (P).

Step 2. We construct a supersolution of (P).

We consider

- Δ p ( x ) z 1 = λ p + μ ( L 2 + 1 ) in  Ω , - Δ p ( x ) z 2 = λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) in  Ω , z 1 = z 2 = 0 on  Ω ,

where β = β ( λ p + ( L 2 + 1 ) μ ) = max x Ω ̄ z 1 ( x ) . We shall prove that (z1, z2) is a supersolution for (p).

For q W 0 1 , p ( x ) ( Ω ) with q ≥ 0, it is easy to see that

Ω | z 2 | p ( x ) - 2 z 2 q d x = Ω λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) q d x Ω λ p + L 2 h ( β ( λ p + ( L 2 + 1 ) μ ) ) q d x + Ω λ p + h ( z 1 ) q d x . (9)

Since lim u + f M ( h ( u ) ) 1 ( p - - 1 ) u p - - 1 = 0 ,when μ is sufficiently large, combining Lemma 1.2 and (H3), then we have

h ( β ( λ p + ( L 2 + 1 ) μ ) ) b C 2 λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) 1 p - - 1 b ( z 2 ) (10)

Hence

Ω | z 2 | p ( x ) - 2 z 2 q d x Ω λ p + g ( x ) b ( z 2 ) q d x + Ω λ p + h ( z 1 ) q d x . (11)

Also

Ω | z 1 | p ( x ) - 2 z 1 q d x = Ω λ p + ( L 2 + 1 ) μ q d x

By (H3), (H4), when μ is sufficiently large, combining Lemma 1.2 and (H3), we have

( L 2 + 1 ) μ 1 λ p + 1 C 2 β ( λ p + ( L 2 + 1 ) μ ) p - - 1 L 2 a ( β ( λ p + ( L 2 + 1 ) μ ) ) + f C 2 λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) 1 p - - 1 .

Then

Ω | z 1 | p ( x ) - 2 z 1 q d x Ω λ p + g ( x ) a ( z 1 ) q d x + Ω λ p + f ( z 2 ) q d x . (12)

According to (11) and (12), we can conclude that (z1, z2) is a supersolution for (P).

It only remains to prove that ϕ1 ≤ z1 and ϕ2 ≤ z2.

In the definition of v1(x), let γ = δ 2 ( max x Ω ̄ ϕ ( x ) + max x Ω ̄ | ϕ ( x ) | ) . We claim that

ϕ ( x ) v 1 ( x ) , x Ω . (13)

From the definition of v1, it is easy to see that

ϕ ( x ) 2 max  x Ω ̄ ϕ ( x ) v 1 ( x ) , when  d ( x ) = δ ,

and

ϕ ( x ) 2 max  x Ω ̄ ϕ ( x ) v 1 ( x ) , when  d ( x ) δ .

It only remains to prove that

ϕ ( x ) v 1 ( x ) , when  d ( x ) < δ .

Since v 1 - ϕ C 1 ( Ω δ ¯ ) , then there exists a point x 0 Ω δ ¯ such that

v 1 ( x 0 ) - ϕ ( x 0 ) = min x 0 Ω δ ¯ [ v 1 ( x ) - ϕ ( x ) ] .

If v1(x0) - ϕ(x0) < 0, it is easy to see that 0 < d(x0) < δ, and then

v 1 ( x 0 ) - ϕ ( x 0 ) = 0 .

From the definition of v1, we have

| v 1 ( x 0 ) | = γ = 2 δ max  x Ω ̄ ϕ ( x ) + max  x Ω ̄ | ϕ ( x ) | > | ϕ ( x 0 ) | .

It is a contradiction to ∇v1(x0) - ϕ(x0) = 0. Thus (13) is valid.

Obviously, there exists a positive constant C3 such that

γ   C 3 λ .

Since d ( x ) C 2 ( Ω 3 δ ¯ ) , according to the proof of Lemma 1.2, then there exists a positive constant C4 such that

- Δ p ( x ) v 1 ( x ) C * γ p ( x ) - 1 + θ C 4 λ p ( x ) - 1 + θ , a . e . in  Ω , where  θ ( 0 , 1 ) .

When η λ p + is large enough, we have

- Δ p ( x ) v 1 ( x ) η .

According to the comparison principle, we have

v 1 ( x ) w ( x ) , x Ω . (14)

From (13) and (14), when η λ p + and λ ≥ 1 is sufficiently large, we have

ϕ ( x ) v 1 ( x ) w ( x ) , x Ω . (15)

According to the comparison principle, when μ is large enough, we have

v 1 ( x ) w ( x ) z 1 ( x ) , x Ω .

Combining the definition of v1(x) and (15), it is easy to see that

ϕ 1 ( x ) = ϕ ( x ) v 1 ( x ) w ( x ) z 1 ( x ) , x Ω .

When μ ≥ 1 and λ is large enough, from Lemma 1.2, we can see that β ( λ p + ( L 2 + 1 ) μ ) is large enough, then λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) is large enough. Similarly, we have ϕ2 ≤ z2. This completes the proof. □

### 3. Asymptotic behavior of positive solutions

In this section, when parameter λ → +, we will discuss the asymptotic behavior of maximum of solutions about parameter λ, and the asymptotic behavior of solutions near boundary about parameter λ.

Theorem 3.1. On the conditions of (H1)-(H5), if (u, v) is a solution of (P) which has been given in Theorem 2.1, then

(i) There exist positive constants C1 and C2 such that

C 1 λ max  x Ω ̄ u ( x ) C 2 λ p + ( L 2 + 1 ) μ 1 p - - 1 (16)

C 1 λ max  x Ω ̄ v ( x ) C 2 λ p + ( L 2 + 1 ) h C 2 ( λ p + ( L 2 + 1 ) μ ) 1 p - - 1 1 p - - 1 (17)

(ii) for any θ ∈ (0, 1), there exist positive constants C3 and C4 such that

C 3 λ d ( x ) u ( x ) C 4 ( λ p + ( L 2 + 1 ) μ ) 1 / ( p - - 1 ) ( d ( x ) ) θ , a s d ( x ) 0 , (18)

C 3 λ d ( x ) v ( x ) C 4 λ p + ( L 2 + 1 ) h C 2 ( λ p + ( L 2 + 1 ) μ ) 1 p - - 1 1 p - - 1 ( d ( x ) ) θ , a s d ( x ) 0 (19)

where μ satisfies (10).

Proof. (i) Obviously, when 2δ ≤ d(x), we have

u ( x ) , v ( x ) ϕ ( x ) = e k σ - 1 + σ 2 δ k e k σ 2 δ - t 2 δ - σ 2 p - - 1 d t - λ ζ α σ 2 δ M 2 δ - t 2 δ - σ 2 p - - 1 d t ,

then there exists a positive constant C1 such that

C 1 λ max  x Ω ̄ u ( x ) and C 1 λ max  x Ω ̄ v ( x ) .

It is easy to see

u ( x ) z 1 ( x ) max  x Ω ̄ z 1 ( x ) C 2 ( λ p + ( L 2 + 1 ) μ ) 1 p - - 1 ,

then

max  x Ω ̄ u ( x ) C 2 λ p + ( L 2 + 1 ) μ 1 p - - 1 .

Similarly

max  x Ω ̄ v ( x ) C 2 λ p + ( L 2 + 1 ) h C 2 λ p + ( L 2 + 1 ) μ 1 p - - 1 1 p - - 1

Thus (16) and (17) are valid.

(ii) Denote

v 3 ( x ) = α ( d ( x ) ) θ , d ( x ) ρ ,

where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, δ) is small enough.

Obviously, v3(x) ∈ C1ρ), By computation

- Δ p ( x ) v 3 ( x ) = - ( α θ ) p ( x ) - 1 ( θ - 1 ) ( p ( x ) - 1 ) ( d ( x ) ) ( θ - 1 ) ( p ( x ) - 1 ) - 1 ( 1 + Π ( x ) ) , d ( x ) < ρ ,

where

Π x = d p d In α θ θ - 1 p x - 1 + d p d In d p x - 1 + d Δ d θ - 1 p x - 1 .

Let α = 1 ρ C 2 λ p + L 2 + 1 μ 1 / p - - 1 , where ρ > 0 is small enough, it is easy to see that

α p x - 1 λ p + μ L 2 + 1  and  Π x 1 2 .

where ρ > 0 is small enough, then we have

- Δ p x v 3 x λ p + μ L 2 + 1 .

Obviously v3(x) ≥ z1(x) on Ωρ. According to the comparison principle, we have v3(x) ≥ z1 (x) on Ωρ. Thus

u x C 4 λ p + L 2 + 1 μ 1 / p - - 1 d x θ , a s d x 0 .

Let α = 1 ρ C 2 λ p + L 2 + 1 h C 2 λ p + L 2 + 1 μ 1 p - - 1 1 p - - 1 , when ρ > 0 is small enough, it is easy to see that

α p x - 1 λ p + L 2 + 1 h C 2 λ p + L 2 + 1 μ 1 p - - 1 .

Similarly, when ρ > 0 is small enough, we have

v x C 4 λ p + L 2 + 1 h C 2 λ p + L 2 + 1 μ 1 p - - 1 1 p - - 1 d x θ a s d x 0

Obviously, when d(x) < σ, we have

u x , v x ϕ x = e k d x - 1 C 3 λ d x .

Thus (18) and (19) are valid. This completes the proof. □

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors typed, read and approved the final manuscript.

### Acknowledgements

The authors would like to appreciate the referees for their helpful comments and suggestions. The third author partly supported by the National Science Foundation of China (10701066 & 10971087).

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