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The existence of positive solutions is given, and the asymptotic behavior of solutions near boundary is discussed.. In [4,7], Fan and Zhao give the regularity of weak solutions for differ

Volume 2007, Article ID 19349, 9 pages

doi:10.1155/2007/19349

Research Article

Existence and Asymptotic Behavior of Positive Solutions to

p(x)-Laplacian Equations with Singular Nonlinearities

Qihu Zhang

Received 17 July 2007; Accepted 27 August 2007

Recommended by M Garc´ıa-Huidobro

This paper investigates thep(x)-Laplacian equations with singular nonlinearities −Δp(x) u

= λ/u γ(x) in Ω, u(x) =0 on ∂Ω, where−Δp(x) u = −div(|∇ u | p(x) −2 ∇ u) is called

p(x)-Laplacian The existence of positive solutions is given, and the asymptotic behavior of solutions near boundary is discussed

Copyright © 2007 Qihu Zhang This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

The study of differential equations and variational problems with nonstandard p(x)-growth conditions is a new and interesting topic We refer to [1,2], the background of these problems Many results have been obtained on this kind of problems, for exam-ple, [2–13] In [4,7], Fan and Zhao give the regularity of weak solutions for differential equations with nonstandard p(x)-growth conditions On the existence of solutions for p(x)-Laplacian problems in bounded domain, we refer to [5,11,12]

In this paper, we consider thep(x)-Laplacian equations with singular nonlinearities:

− p(x) u = λ

u γ(x) inΩ,

u(x) =0 on∂Ω,

(P)

where− p(x) u = −div(|∇ u | p(x) −2∇ u) is called p(x)-Laplacian,Ω⊂ R N is a bounded domain with C2 boundary ∂Ω If p(x) ≡ p (a constant), then ( P) is the well-known

p-Laplacian problem There are many results on the existence of positive solutions for p-Laplacian problems with singular nonlinearities (see [14–18]), but the results on the existence of positive solutions for p(x)-Laplacian problems with singular nonlinearities

2 Journal of Inequalities and Applications

are rare Our aim is to give the existence of positive solutions for problem (P), and give the asymptotic behavior of positive solutions near boundary

Throughout the paper, we assume that 0< γ(x) ∈ C( Ω) and p(x) satisfy

(H1) p(x) ∈ C1(Ω), 1 < p − ≤ p+< + ∞, wherep − =infΩp(x), p+=supΩp(x).

Because of the nonhomogeneity ofp(x)-Laplacian, p(x)-Laplacian problems are more

complicated than those ofp-Laplacian ones, many results and methods for p-Laplacian

problems are invalid forp(x)-Laplacian problems (see [6]), and another difficulty of this paper is that f (x, u) =1/uγ(x)cannot be represented as h(x) f (u) Our results partially

generalized the results of [18]

2 Preliminary

In order to deal with p(x)-Laplacian problems, we need some theories on the spaces

L p(x)(Ω), W1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see [3,8]) Let

L p(x)(Ω)=



u | u is a measurable real-valued function,



Ω

u(x)p(x)

dx < ∞



,

C+

0(Ω)=u ∈ C(Ω)| u > 0 in Ω, u =0 on∂Ω.

(2.1)

We can introduce the norm onL p(x)(Ω) by

| u | p(x) =inf



μ > 0 |



Ω



u(x) μ p(x) dx ≤1

The space (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 [3, Theorems 1.10, Theorem 1.14])

The spaceW1,p(x)(Ω) is defined by

W1,p(x)(Ω)=u ∈ L p(x)(Ω)| ∇ u  ∈ L p(x)(Ω), (2.3) and it can be equipped with the norm

| u | = | u | p(x)+|∇ u | p(x), ∀ u ∈ W1,p(x)(Ω) (2.4)

W01,p(x)(Ω) is the closure of C∞

0(Ω) in W1,p(x)(Ω), W1,p(x)(Ω) and W1,p(x)

0 (Ω) are sep-arable, reflexive, and uniform convex Banach spaces (see [3, Theorem 2.1])

Ifu ∈ Wloc1,p(x)(Ω)∩ C0+(Ω), u is called a positive solution of (P) ifu(x) satisfies



Q |∇ u | p(x) −2 ∇ u ∇ qdx −



Q

λ

u γ(x) qdx =0, ∀ q ∈ W01,p(x)(Q), (2.5) for any domainQΩ

LetW0,loc1,p(x)(Ω)= { u | there is an open domainQΩ s.t u ∈ W01,p(x)(Q)}, and define

A : Wloc1,p(x)(Ω)∩ C+

0(Ω)→(W0,loc1,p(x)(Ω))∗as

Au, ϕ  =



Ω |∇ u | p(x) −2 ∇ u ∇ ϕ − λ

u γ(x) ϕ

whereu ∈ Wloc1,p(x)(Ω)∩ C+

0(Ω), ϕ∈ W0,loc1,p(x)(Ω); then we have the following lemma Lemma 2.1 (see [5, Theorem 3.1]) A : Wloc1,p(x)(Ω)∩ C0+(Ω)→(W0,loc1,p(x)(Ω))∗ is strictly monotone.

Let g ∈(W0,loc1,p(x)(Ω))∗ , if g, ϕ  ≥ 0, for all ϕ ∈ W0,loc1,p(x)(Ω), ϕ≥ 0 a.e in Ω, then denote

g ≥ 0 in (W0,loc1,p(x)(Ω))∗ ; correspondingly, if − g ≥ 0 in (W0,loc1,p(x)(Ω))∗ , then denote g ≤ 0 in

(W0,loc1,p(x)(Ω))∗

Definition 2.2 Let u ∈ Wloc1,p(x)(Ω)∩ C+

0(Ω) If Au ≥0(Au≤0) in (W0,loc1,p(x)(Ω))∗, thenu

is called a weak supersolution (weak subsolution) of (P)

Copying the proof of [10], we have the following lemma

Lemma 2.3 (comparison principle) Let u, v ∈ Wloc1,p(x)(Ω)∩ C( Ω) be positive and satisfy

Au − Av ≥ 0 in (W0,loc1,p(x)(Ω))∗ Let ϕ(x) =min{ u(x) − v(x), 0 } If ϕ(x) ∈ W0,loc1,p(x)(Ω) (i.e.,

u ≥ v on ∂Ω), then u ≥ v a.e in Ω.

Lemma 2.4 (see [7]) If g(x, u) is continuous onΩ× R , u ∈ W1,p(x)(Ω) is a bounded weak

solution of − p(x) u + g(x, u) = 0 in Ω, u = w0 on ∂ Ω, where w0∈ W1,p(x)(Ω), then u ∈

C1,locα(Ω), where α∈ (0, 1) is a constant.

3 Existence of positive solutions

In order to deal with the existence of positive solutions, let us consider the problem

− p(x) u = λ

| u |+a n γ(x)

inΩ,

u(x) =0 forx ∈ ∂Ω,

(3.1)

where{ a n } is a positive strictly decreasing sequence and limn →+∞ a n =0 We have the following lemma

Lemma 3.1 For any n =1, 2, , problem ( 3.1 ) possesses a weak positive solution  n ∈

C( Ω).

Proof The relative functional of (3.1) is

ϕ =



Ω

1

p(x) ∇ u(x)p(x)

dx −



ΩF n(x, u)dx, (3.2) whereF n(x, u)=u

0λ/(( | t |+a n)γ(x))dt Since ϕ is coercive in W01,p(x)(Ω), then ϕ possesses

a nontrivial minimum point n, then|  n |is also a nontrivial minimum point of problem (3.1), then (3.1) possesses a weak positive solution The proof is completed 

4 Journal of Inequalities and Applications

Here and hereafter, we will use the notationd(x, ∂Ω) to denote the distance of x ∈Ω

to the boundary ofΩ Denote d(x) = d(x, ∂Ω) and ∂Ω  = { x ∈Ω| d(x) < } Since∂Ω

isC2 regularly, then there exists a positive constant σ such that d(x) ∈ C2(∂Ω2σ), and

|∇ d(x) | ≡1 Letδ ∈(0, (1/3)σ) be a small enough constant Denote

v1(x)=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

δ +

d(x)

δ

2 − t δ

2/(p − −1)

dt, δ ≤ d(x) < 2δ,

δ +

2δ

δ

(2δ− t) δ

2/(p − −1)

dt, 2 ≤ d(x).

(3.3)

Obviously,v1(x)∈ C1(Ω)∩ C+

0(Ω)

Lemma 3.2 If λ > 0 is large enough, then v1(x) is a subsolution of (P ).

Proof Since |∇ d(x) | ≡1, whenλ > 0 is large enough, we have

− p(x) v1= − d(x) ≤ λ

v1(x)γ(x), ∀ x ∈ Ω, d(x) < δ. (3.4)

By computation, whenδ < d(x) < 2δ, we have

− p(x) v1= −div 2 − d(x)

δ

2/(p − −1)p(x) −1

∇ d(x)



δ

2/(p − −1) p(x) −1

 d(x)

δ

2/(p − −1) p(x) −1 

∇ d(x) ∇ p(x)

ln 2 − d(x) δ

2/(p − −1)

+2

δ

p(x) −1

p − −1

δ

 (2(p(x) −1)/(p − −1))−1

.

(3.5) Whenλ > 0 is large enough, it is easy to see that

− p(x) v1≤ λ

v1(x)γ(x), ∀ x ∈ Ω, δ < d(x) < 2δ,

− p(x) v1=0≤ λ

v1(x)γ(x), ∀ x ∈ Ω, 2δ < d(x).

(3.6)

From (3.4) and (3.6), we can conclude thatv(x) is a subsolution of (P) 

Theorem 3.3 If λ > 0 is a large enough constant, then problem ( P ) possesses only one posi-tive solution u λ , and u λ is increasing with respect to λ.

Proof Denote u n =  n+a n, where nis a solution of (3.1) Since{ u n }is a sequence of positive solutions of

− p(x) u = λ

u γ(x) inΩ,

u(x) = a n forx ∈ ∂Ω,

(II)

then everyu nis subsolution and supersolution of− p(x) u = λ/u γ(x)inΩ According to comparison principle, we have thatu n ≥ u n+1forn =1, 2, Since v1(x) is a subsolution

of (P) andv1(x)=0 on∂ Ω, then u n ≥ u n+1 ≥ v1forn =1, 2, According toLemma 2.4,

we have that{ u n }has uniformC1,αlocal regularity property, and hence we can choose

a subsequence, which we denoted by{ u1

n }, such thatu1

n → w and ∇ u1

n → h inΩ In fact,

h = ∇ w inΩ

For any domainDΩ, for any ϕ ∈ W01,p(x)(D) The C1,αregularity result implies that the sequences{ u n }and{∇ u n }are equicontinuous inD; from the C1,αestimate we con-clude that∇ w ∈ C α(D) for some 0 < α < 1 Thus w∈ W1,p(x)(D)∩ C1,α(D) From the

C1,αregularity result, we see that|∇ u1

n | p −1|∇ ϕ | ≤ C |∇ ϕ |onD, and since the function

ξ → | ξ | p −2 ξ is continuous on Rn, it follows that |∇ u1

n(x)| p −2 ∇ u1

n(x)· ∇ ϕ(x) →

|∇ w(x) | p −2 ∇ w(x) · ∇ ϕ(x) for x ∈ D Thus, by the dominated convergence theorem, for

anyϕ ∈ W01,p(x)(D), we can see that



D ∇ u1

n(x)p −2

∇ u1

n(x)· ∇ ϕ(x)dx −→



D ∇ w(x)p −2

∇ w(x) · ∇ ϕ(x)dx. (3.7)

Furthermore, since 0≤ λ/([u1

n(x)]γ(x))≤ λ/([u1n+1(x)]γ(x)), and λ/([u1

n(x)]γ(x))→

λ/([w(x)] γ(x)) for eachx ∈ D, by the monotone convergence theorem we obtain



D

λ



u1

n(x)γ(x) ϕdx −→



D

λ



w(x)γ(x) ϕdx, ∀ ϕ ∈ W01,p(x)(D) (3.8) Therefore, it follows that



D ∇ w(x)p −2

∇ w(x) · ∇ ϕ(x)dx −



D

λ



w(x)γ(x) ϕdx =0, ∀ ϕ ∈ W01,p(x)(D), (3.9)

and hencew is a weak solution of −Δp(x) w = λ/([w(x)] γ(x)) onD.

Obviously,w is a solution of ( P), and satisfiesw ≥ v1 According to comparison prin-ciple, it is easy to see that (P) possesses only one positive solution, andu λ is increasing

6 Journal of Inequalities and Applications

4 Asymptotic behavior of positive solutions

In the following, we will useC ito denote positive constants

Theorem 4.1 If u is a positive weak solution of problem ( P ), then C2d(x) ≤ u(x) as x → ∂ Ω Proof Similar to the proof ofLemma 3.2, there exists a positive constantC2 such that whenδ > 0 is small enough, then v2(x)= C2d(x) is a subsolution of ( P) on∂Ω δ Thus

u(x) ≥ v2(x)= C2d(x) on ∂Ω δ The proof is completed  Denoteγ ∗ =maxx ∈ ∂Ω2σγ(x) and γ ∗ =minx ∈ ∂Ω2σγ(x).

Theorem 4.2 If 1 ≤ γ ∗ < γ ∗ , for any weak solution u of problem ( P ), we have

C3 

d(x)θ1

≤ u(x) ≤ C4 

d(x)θ2

where θ1=maxd(x) ≤ σ(p(x)/(p(x)−1 +γ(x))), θ2=mind(x) ≤ σ(p(x)/(p(x)−1 +γ(x))) Proof FromTheorem 4.1we only consider (P) in the case of 1< γ ∗ < γ ∗ Denote

v3(x)=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

a

aδ θ+

d(x)

δ aθδ θ −1 2 − t

δ

2/(p − −1)

dt, δ ≤ d(x) < 2δ,

aδ θ+

 2δ

δ aθδ θ −1 2 − t

δ

2/(p − −1)

dt, 2 ≤ d(x),

(4.2)

wherea and θ are positive constants and satisfy θ ∈(0, 1), 0< δ is small enough.

Obviously,v3(x)∈ C1(Ω)∩ C+

0(Ω) By computation,

− p(x) v3(x)= −(aθ)p(x) −1(θ−1)

p(x) −1 d(x) (θ −1)( p(x) −1)−1

1 +Π(x) , d(x) < δ,

(4.3) where

Π(x) = d (∇ p ∇ d) ln aθ

(θ−1)

p(x) −1 +d

(∇ p ∇ d) ln d

p(x) −1 +d

 d

(θ−1)

p(x) −1 . (4.4)

Obviously| Π(x) | ≤1/2, when δ > 0 is small enough Let θ= θ1anda ∈(0, 1) is small enough, whenδ ∈(0,a) is small enough, we can conclude that

− p(x) v3(x)≤ λ

v (x)γ(x), d(x) < δ. (4.5)

By computation, whenδ < d(x) < 2δ, we have

− p(x) v3= −div



aθδ θ −1 2 − d(x)

δ

2/(p − −1)p(x) −1

∇ d(x)

= −



aθδ θ −1 2 − d(x)

δ

2/(p − −1)p(x) −1

×∇ d(x) ∇ p(x)

lnaθδθ −1 2 − d(x)

δ

2/(p − −1)

−



aθδ θ −1 2 − d(x)

δ

2/(p − −1) p(x) −1

 d(x)

+2

δ

p(x) −1

p − −1

aθδ θ −1 p(x) −1

δ

 (2(p(x) −1) /(p − −1))−1

.

(4.6)

Thus, there exists a positive constantC ∗such that

− p(x) v3 ≤ C ∗ δ(θ −1)( p(x) −1)−1, δ < d(x) < 2δ. (4.7) Obviously,

v3(x)≤ a(θ + 1)δ θ, δ < d(x) < 2δ. (4.8) Letθ = θ1, whena ∈(0, 1) is small enough,δ ∈(0,a) is small enough, then

− p(x) v3(x)≤ λ

v2(x)γ(x), δ < d(x) < 2δ. (4.9)

It is easy to see that

− p(x) v3(x)=0≤ λ

v2(x)γ(x), 2δ < d(x) (4.10) Combining (4.5), (4.9), and (4.10), it is easy to see that whenθ = θ1,a ∈(0, 1) is small enough andδ ∈(0,a) is small enough, then v(x) is a subsolution of (P), thenu(x) ≥

C3[d(x)]θ1on∂Ωδ

Similarly, when δ > 0 is small enough, θ = θ2, and a ≥maxx ∈ ∂Ωδ(u(x)/δθ) is large enough, we can see thatv(x) is a supersolution of ( P) on∂Ωδ, andu(x) ≤ a[d(x)] θ2 on

Theorem 4.3 If lim d(x) →0 p(x) = p0and lim d(x) →0 p(x)/(p(x) −1 +γ(x)) = s, where s ≤1

is a positive constant, u is a solution of ( P ), then

lim

d(x) →0

u(x)

C

d(x) s =1, C = lim

d(x) →0

θ p(x) −1(1− θ)

p(x) −1

 1/(p(x) −1+ γ(x))

. (4.11)

8 Journal of Inequalities and Applications

Theorem 4.4 If 1 ≥ γ ∗ , for any positive constant θ ∈ (0, 1), u is a weak solution of problem ( P ), then there exists a positive constant C5such that C1d(x) ≤ u(x) ≤ C5(d(x))θ as x → ∂Ω Proof According to Theorem 4.1, it only needs to proveu(x) ≤ C5(d(x))θ asx → ∂Ω Define a function on∂Ω δasv4(x)= C5(d(x))θ, whereC5≥(1/δθ) maxx ∈ ∂Ωδ u(x) Similar

to the proof ofTheorem 4.2, whenδ > 0 is small enough, then v4(x) is a supersolution of (P) on∂Ω δ, thenu(x) ≤ v4(x)= C5(d(x))θon∂Ω δ The proof is completed 

Theorem 4.5 If γ ∗ < 1 < γ ∗ , u is a weak solution of problem ( P ), then there exists a positive constant C6 such that C1d(x) ≤ u(x) ≤ C6(d(x))θ as x → ∂ Ω, where θ =mind(x) ≤ δ(p(x)/ (p(x)−1 +γ(x))).

Proof According to Theorem 4.1, it only needs to proveu(x) ≤ C6(d(x))θ asx → ∂Ω Similar to the proof ofTheorem 4.2, whenδ > 0 is small enough, then v5(x)= C6(d(x))θ

is a supersolution of (P) on∂Ωδ, then u(x) ≤ v5(x)= C6(d(x))θ on ∂Ωδ The proof is

Acknowledgments

This work was partially supported by the National Science Foundation of China (no

10701066 and no 10671084) and the Natural Science Foundation of Henan Education Committee (no 2007110037)

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Qihu Zhang: Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, Henan, China

Email address:zhangqh1999@yahoo.com.cn

6 Journal of Inequalities and Applications

4 Asymptotic behavior of positive solutions< /b>...