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Existence and asymptotic behavior of positive solutions for a variable exponent elliptic system without variational structure
Nonlinear Analysis 72 (2010) 354–363
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Existence and asymptotic behavior of positive solutions for a variable exponent elliptic system without variational structureI Qihu Zhang a,b,c,∗ , Zhimei Qiu b , Rong Dong b a
Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China
b
School of Mathematics Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China
c
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, China
article
info
Article history: Received 25 August 2008 Accepted 15 June 2009 MSC: 35J25 35J60 Keywords: Variable exponent elliptic system Subsolution Supersolution
abstract We mainly consider the existence and asymptotic behavior of positive solutions of the following system
−∆p(x) u = λp(x) (uα(x) v γ (x) + h1 (x)) in Ω , −∆q(x) v = λq(x) (uδ(x) v β(x) + h2 (x)) in Ω , u = v = 0 on ∂ Ω , where Ω ⊂ RN is a bounded domain with C 2 boundary ∂ Ω , 1 < p(x), q(x) ∈ C 1 (Ω ) are functions, and −∆p(x) u = −div(|∇ u|p(x)−2 ∇ u) is called p(x)-Laplacian. When α, β, γ , δ satisfy some conditions and λ is large enough, we proved the existence of a positive solution. In particular, we do not assume any symmetric condition, and we do not assume any sign condition on h1 (0) and h2 (0). © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The study of differential equations and variational problems with variable exponents is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, image processing, etc. (see [1–4]). Many results have been obtained on these kinds of problems, for example [5–25]. In [9], Fan give the regularity of weak solutions for differential equations with variable exponents. For the case of the existence of solutions for elliptic systems with variable exponents, we refer to [13,17,23,24]. In this paper, we mainly consider the existence and asymptotic behavior of positive weak solutions for the following system
(P1 )
−∆p(x) u = λp(x) (uα(x) v γ (x) + h1 (x)) in Ω , −∆q(x) v = λq(x) (uδ(x) v β(x) + h2 (x)) in Ω , u = v = 0 on ∂ Ω ,
where p(x), q(x) ∈ C 1 (Ω ) are functions and Ω ⊂ RN is an open bounded domain with C 2 boundary ∂ Ω . The operator −∆p(x) u = −div(|∇ u|p(x)−2 ∇ u) is called p(x)-Laplacian and the corresponding equation is called a variable exponent I Foundation item: Partly supported by the National Science Foundation of China (10701066 & 10671084), China Postdoctoral Science Foundation (20070421107), the Natural Science Foundation of Henan Education Committee (2008-755-65) and the Natural Science Foundation of Jiangsu Education Committee (08KJD110007). ∗ Corresponding author at: Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China. E-mail address: [email protected] (Q. Zhang).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.06.069
Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
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equation. Especially, if p(x) ≡ p (a constant), −∆p(x) is the well-known p-Laplacian and the corresponding equation is called a constant exponent equation. There are many papers on the existence of solutions for constant exponent elliptic systems without variational structure, for example [26–28]. Because of the nonhomogeneity of variable exponent problems, variable exponent problems are more complicated than constant exponent problems, and many results and methods for constant exponent problems are invalid for variable exponent problems; for example, if Ω is bounded, then the Rayleigh quotient
R λp(x) =
inf
1,p(x)
u∈W0
(Ω )\{0}
1
Ω p(x)
R
|∇ u|p(x) dx
1
Ω p(x)
|u|p(x) dx
is zero in general, and only under some special conditions λp(x) > 0 (see [15]), the p(x)-Laplacian does not have the first eigenvalue and the first eigenfunction in general. However, 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 to be discussed in the existence and asymptotic behavior of solutions of variable exponent problems. In [28], the authors consider the existence of positive weak solutions for the following constant exponent problems
−∆p u = λf (v) in Ω , (I ) −∆p v = λg (u) in Ω , u = v = 0 on ∂ Ω , (
the first eigenfunction is used to construct the subsolution of constant exponent problems successfully. Under the condition that λ is large enough and lim
u→+∞
f [M (g (u))1/(p−1) ] up−1
= 0,
for every M > 0,
the authors give the existence of positive solutions for problem (I ). In [27], the author considers the existence and nonexistence of positive weak solution to the following constant exponent elliptic system
−∆p u = λf (u, v) = λuα v γ in Ω , (II ) −∆q v = λg (u, v) = λuδ v β in Ω , u = v = 0 on ∂ Ω , the first eigenfunction is used to construct the subsolution of it, the main results are as following (i) If α, β ≥ 0, γ , δ > 0, $ = (p − 1 − α)(q − 1 − β) − γ δ > 0, then problem (II ) has a positive weak solution for each λ > 0; (ii) If $ = 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 variable exponent 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 exists, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of variable exponent problems. In many cases, the radial symmetric conditions are effective to deal with variable exponent problems. There are many results about the radial variable exponent problems (see [12,13,20–23]). In [23,24], the present author discussed the existence of positive solutions of the following
−∆p(x) u = λf (v) in Ω , (III ) −∆p(x) v = λg (u) in Ω , u = v = 0 on ∂ Ω , (
when it possesses some symmetric conditions. But the asymptotic behavior of positive weak solutions have not been discussed in [23,24,26–28]. Our aim is to give the existence and asymptotic behavior of positive weak solutions for problem (P1 ) without any symmetric conditions. Through a new method to construct sub-supersolution, this paper gives the existence of positive weak solutions for problem (P1 ) via sub-supersolution method. Similarly to (II ), we consider the following
(P2 )
−∆p(x) u = λ(uα(x) v γ (x) + h1 (x)) in Ω , −∆q(x) v = λ(uδ(x) v β(x) + h2 (x)) in Ω , u = v = 0 on ∂ Ω .
It is well known that (P1 ) is equal to (P2 ) if p(x) ≡ p ≡ q(x) (a constant), but for general functions p(x) and q(x), (P1 ) is not equal to (P2 ) even if p(x) = q(x) (a function). Moreover, we discuss the existence of positive weak solutions for problem (P2 ) without any symmetric conditions. Since there exist a p(x)-Laplacian and a q(x)-Laplacian in (P1 ) and (P2 ), we call (P1 ) and (P2 ) are (p(x), q(x))-type, similarly, we call (III ) is (p(x), p(x))-type. There are some differences between the existence of positive solutions of (P1 )
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Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
and (P2 ), and there are some differences between the existence of positive solutions of (p(x), q(x))-type and (p(x), p(x))type. Notice that in problems (I ) and (II ), f and g are independent on the variable x, and the systems are homogeneity. Our results partially generalized the results of [23,24,26–28]. This paper is divided into four sections. In the second section, we do some preparation. In the third section, we will discuss the existence of positive solutions of (P1 ) and (P2 ). In the fourth section, we will discuss the asymptotic behavior of positive solutions of (P1 ) and (P2 ). 2. Preliminaries In order to deal with variable exponent problems, we need some theories on spaces Lp(x) (Ω ), W 1,p(x) (Ω ) and properties of p(x)-Laplacian which we will use later (see [8,19]). Write C+ (Ω ) = {h|h ∈ C (Ω ), h(x) > 1 for x ∈ Ω } , h+ = max h(x), h− = min h(x), for any h ∈ C (Ω ). x∈Ω
x∈Ω
Throughout the paper, we will assume that: (H1 ) Ω ⊂ RN is an open bounded domain with C 2 boundary ∂ Ω . (H2 ) p, q ∈ C 1 (Ω ) satisfy 1 < p− ≤ p+ and 1 < q− ≤ q+ . (H3 ) h1 , h2 , α, β, γ , δ ∈ C (Ω ) satisfy α, β ≥ 0 on Ω and γ , δ > 0 on Ω . (H4 ) 0 ≤ α + < p− − 1, 0 ≤ β + < q− − 1, and
$ := (p− − 1 − α + )(q− − 1 − β + ) − γ + δ + > 0. Denote Lp(x) (Ω ) =
Z
u | u is a measurable real-valued function, Ω
|u(x)|p(x) dx < ∞ .
p(x)
(Ω ) by Z ) u(x) p(x) dx ≤ 1 , = inf λ > 0 Ω λ
We can introduce a norm on L
(
|u|p(x)
and (Lp(x) (Ω ), |·|p(x) ) becomes a Banach space, and we call it variable exponent Lebesgue space. The space (Lp(x) (Ω ), |·|p(x) ) is a separable, reflexive and uniform convex Banach space (see [8, Theorem 1.10, Theorem 1.14]). The space W 1,p(x) (Ω ) is defined by W 1,p(x) (Ω ) = u ∈ Lp(x) (Ω ) | |∇ u| ∈ Lp(x) (Ω ) ,
and it can be equipped with the norm
kuk = |u|p(x) + |∇ u|p(x) ,
∀u ∈ W 1,p(x) (Ω ) . 1,p(x)
1,p(x)
(Ω ) are separable, reflexive and We denote by W0 (Ω ) the closure of C0∞ (Ω ) in W 1,p(x) (Ω ). W 1,p(x) (Ω ) and W0 uniform convex Banach spaces (see [8, Theorem 2.1]), and we call it variable exponent Sobolev space. We define (L(u), v) =
Z Ω
|∇ u|p(x)−2 ∇ u∇v dx,
∀u, v ∈ W01,p(x) (Ω ),
1,p(x)
1,p(x)
(Ω ) → (W0 (Ω ))∗ is a continuous, bounded and strictly monotone operator, and it is a homeomorphism then L : W0 (see [14, Theorem 3.1]). 1,p(x) If (u, v) ∈ (W0 (Ω ), W01,q(x) (Ω )), (u,v ) is called a weak solution of (P1 ) if it satisfies Z Z p(x)−2 |∇ | u ∇ u ∇ϕ dx = λp(x) (uα(x) v γ (x) + h1 (x))ϕ dx, ∀ϕ ∈ W01,p(x) (Ω ) , Ω Ω Z Z λq(x) (uδ(x) v β(x) + h2 (x))ψ dx, ∀ψ ∈ W01,q(x) (Ω ) . |∇v|q(x)−2 ∇v∇ψ dx = Ω
Ω
1,p(x) W0
Define A : W 1,p(x) (Ω ) → (
hAu, ϕi =
Z Ω
(Ω ))∗ as
(|∇ u|p(x)−2 ∇ u∇ϕ + h(x, u)ϕ)dx,
∀u ∈ W 1,p(x) (Ω ), ∀ϕ ∈ W01,p(x) (Ω ),
Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
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where h(x, u) is continuous on Ω × R, h(x, ·) is increasing and satisfies ∗ (x)−1
|h(x, t )| ≤ C1 + C2 |t |p
,
where p∗ (x) =
Np(x)/(N − p(x)), ∞, p(x) ≥ N .
p(x) < N
It is easy to check that A is a continuous bounded mapping. Copying the proof of [25], we have the following lemma. 1,p(x)
Lemma 2.1 (Comparison Principle). . Let u, v ∈ W 1,p(x) (Ω ) satisfying Au − Av ≥ 0 in (W0
v(x), 0}. If ϕ(x) ∈
1,p(x) W0
(Ω ))∗ , ϕ(x) = min{u(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 = {x ∈ Ω | d(x, ∂ Ω ) < }. Since ∂ Ω is C 2 regularly, there exists a constant ` ∈ (0, 1) such that d(x) ∈ C 2 (∂ Ω3` ), and |∇ d(x)| ≡ 1. Denote
ζ d(x), d(x) < `, 2 Z d(x) 2` − t p− −1 ζ ` + dt , ` ≤ d(x) < 2`, ζ v1 (x) = ` ` 2 Z 2` 2` − t p− −1 ζ ` + ζ dt , 2` ≤ d(x). ` ` Obviously, 0 ≤ v1 (x) ∈ C 1 (Ω ). Let us consider the following problem
− ∆p(x) w(x) = η in Ω , w = 0 on ∂ Ω ,
(1)
where η is a positive parameter. Lemma 2.2 (See [11]). If the 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 1
C1 η p+ −1+θ ≤ max w(x); x∈Ω
(ii) There exists a positive constant C2 such that 1
max w(x) ≤ C2 η p− −1 . x∈Ω
Proof. (i) By computation
p(x)−1 −ζ [(∇ p∇ d) ln ζ + ∆d("x)], d(x) < `, #) 2 ! ( 2 ( p ( x ) − 1) 2` − d 2` − d p− −1 − ln ζ ∇ p ∇ d + ∆d `(p− − 1) ` ` −∆p(x) v1 (x) = 2(p−(x)−1) −1 p −1 2` − d p(x)−1 , ` < d(x) < 2`, ×ζ ` 0, 2` < d(x), then −∆p(x) v1 (x) ≤ C∗ ζ p(x)−1+θ a.e. on Ω , for any θ ∈ (0, 1), where C∗ = C∗ (`, θ , p, Ω ) is a positive constant and independent on ζ . + When C∗ ζ p −1+θ = 12 η, we can see that v1 (x) is a subsolution of (1). According to the comparison principle, we can see that v1 (x) ≤ w(x) on Ω . Obviously, ζ ` ≤ maxx∈Ω v1 (x) ≤ 2ζ `, there exists a positive constant C1 such that 1
max w(x) ≥ max v1 (x) ≥ C1 η p+ −1+θ . x∈Ω
x∈Ω
(ii) It is easy to see from Lemma 2.1 of [11]. This completes the proof.
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3. Existence of positive solutions In the following, if it cannot leading to confusion, we always use Ci to denote positive constants. Theorem 3.1. Under the conditions of (H1 )–(H4 ), then (P1 ) has a positive solution when λ is large enough. Proof. We shall establish Theorem 3.1 by constructing a positive subsolution (φ1 , φ2 ) and a supersolution (z1 , z2 ) of (P1 ), such that φ1 ≤ z1 and φ2 ≤ z2 , i.e., (φ1 , φ2 ) and (z1 , z2 ) satisfy
Z Z γ (x) p(x)−2 λp(x) (φ1α(x) φ2 + h1 (x))ϕ dx, ∇φ1 · ∇ϕ dx ≤ |∇φ1 | Ω Ω Z Z β(x) λq(x) (φ1δ(x) φ2 + h2 (x))ψ dx, |∇φ2 |q(x)−2 ∇φ2 · ∇ψ dx ≤ Ω
Ω
and
Z Z γ (x) p(x)−2 ∇ z1 · ∇ϕ dx ≥ λp(x) (z1α(x) z2 + h1 (x))ϕ dx, |∇ z1 | Ω Ω Z Z β(x) λq(x) (z1δ(x) z2 + h2 (x))ψ dx, |∇ z2 |q(x)−2 ∇ z2 · ∇ψ dx ≥ Ω
Ω
1,p(x) W0
1,q(x) W0
for all (ϕ, ψ) ∈ ( (Ω ), (Ω )) with ϕ ≥ 0 and ψ ≥ 0. According to the sub-supersolution method for p(x)Laplacian equations (see [11]), (P1 ) has a positive solution. Step 1. We construct a subsolution of (P1 ). Let σ ∈ (0, `) is small enough. Denote
kd(x) e − 1, d(x) < σ , 2 Z d(x) 2` − t p− −1 kσ dt , σ ≤ d(x) < 2`, e −1+ kekσ φ1 (x) = 2` − σ σ 2 Z 2` 2` − t p− −1 ekσ − 1 + kekσ dt , 2` ≤ d(x), 2` − σ σ kd(x) e − 1, d(x) < σ , 2 Z d(x) 2` − t q− −1 kσ dt , σ ≤ d(x) < 2`, e −1+ kekσ φ2 (x) = 2` − σ σ 2 Z 2` 2` − t q− −1 ekσ − 1 + kekσ dt , 2` ≤ d(x). 2` − σ σ It is easy to see that φ1 , φ2 ∈ C 1 (Ω ). Denote inf p(x) − 1
a = min
,
inf q(x) − 1
4(sup |∇ p(x)| + 1) 4(sup |∇ q(x)| + 1)
,1 ,
b = |h1 (0)| + |h2 (0)| + 1.
By computation
ln k ∆d kd(x) p(x)−1 − k ( ke ) ( p ( x ) − 1 ) + d ( x ) + ∇ p ∇ d + , d(x) < σ , k k ( " ! #) 2 1 2(p(x) − 1) 2` − d 2` − d p− −1 kσ ∇ p ∇ d + ∆d − ln ke 2` − σ p− − 1 2` − σ 2` − σ −∆p(x) φ1 = 2(p−(x)−1) −1 p −1 2` − d kσ p(x)−1 × ( ke ) , σ < d(x) < 2`, 2` − σ 0, 2` < d(x). From (H3 ), there exists a positive constant M > 2 such that γ (x)
(φ1α(x) φ2 Let σ =
1 k
+ h1 (x)) ≥ 1 and
β(x)
(φ1δ(x) φ2
+ h2 (x)) ≥ 1,
∀x ∈ Ω , when φ1 , φ2 ≥ M − 1.
(2)
ln M, we have
σ k = ln M .
(3)
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359
If k is sufficiently large, from (3), we have
− ∆p(x) φ1 ≤ −kp(x) a,
d(x) < σ .
(4)
Let λ = b+1 k, we have a
kp(x) a ≥ λp(x) b, from (2) and (4) and the definition of b, we have γ (x)
− ∆p(x) φ1 ≤ −λp(x) b ≤ λp(x) (φ1α(x) φ2
+ h1 (x)),
d(x) < σ .
(5)
Since d(x) ∈ C 2 (Ω 3` ), there exists a positive constant C3 such that
2(p−(x)−1) −1 p −1 2` − d −∆p(x) φ1 ≤ (kekσ )p(x)−1 2` − σ ( #) " 2 ! 2` − d 2` − d p− −1 2(p(x) − 1) kσ × − ln ke ∇ p ∇ d + ∆ d (2` − σ )(p− − 1) 2` − σ 2` − σ ≤ C3 (kekσ )p(x)−1 ln k, σ < d(x) < 2`. Let λ = b+a 1 k. If k is sufficiently large, we have C3 (kekσ )p(x)−1 ln k = C3 (kM )p(x)−1 ln k ≤ λp(x) , then
− ∆p(x) φ1 ≤ λp(x) ,
σ < d(x) < 2`.
(6)
+ h1 (x)), σ < d(x) < 2`.
(7)
Obviously γ (x)
λp(x) ≤ λp(x) (φ1α(x) φ2
Combining (6) and (7), we have γ (x)
− ∆p(x) φ1 ≤ λp(x) ≤ λp(x) (φ1α(x) φ2
+ h1 (x)),
σ < d(x) < 2`,
when λ is large enough.
(8)
Obviously γ (x)
− ∆p(x) φ1 = 0 ≤ λp(x) (φ1α(x) φ2
+ h1 (x)),
2` < d(x).
(9)
Combining (5), (8) and (9), we can conclude that γ (x)
+ h1 (x)),
a.e. on Ω .
(10)
β(x)
+ h2 (x)),
a.e. on Ω .
(11)
− ∆p(x) φ1 ≤ λp(x) (φ1α(x) φ2 Similarly
− ∆q(x) φ2 ≤ λq(x) (φ1δ(x) φ2
From (10) and (11), we can see that (φ1 , φ2 ) is a subsolution of (P1 ). Step 2. We construct a supersolution of (P1 ) We consider
+ −∆p(x) z1 = λp µ1 in Ω , + q −∆q(x) z2 = λ µ2 in Ω , z1 = z2 = 0 on ∂ Ω , when µ1 and µ2 satisfy some conditions, we shall prove that (z1 , z2 ) is a supersolution for (P1 ). If we could prove that
µ1 ≥ [max z1 (x)]α [max z2 (x)]γ + max |h1 (x)| , +
x∈Ω
+
x∈Ω
x∈Ω
and
µ2 ≥ [max z1 (x)]δ [max z2 (x)]β + max |h2 (x)| , +
x∈Ω
+
x∈Ω
x∈Ω
we would see that (z1 , z2 ) is a supersolution for (P1 ).
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Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
From Lemma 2.2, we have 1
+
+
1
max z1 (x) ≤ C2 (λp µ1 ) p− −1 and max z2 (x) ≤ C2 (λq µ2 ) q− −1 . x∈Ω
x∈Ω
Let 1
1
µ1 = 2[C2 (λp µ1 ) p− −1 ]α [C2 (λq µ2 ) q− −1 ]γ , +
+
+
+
(12)
we only need 1
1
µ2 ≥ 2[C2 (λp µ1 ) p− −1 ]δ [C2 (λq µ2 ) q− −1 ]β , +
+
+
+
when µ1 and µ2 are large enough,
(13)
then, it is easy to see that (z1 , z2 ) is a supersolution for (P1 ). Since 0 ≤ α + < p− − 1 and 0 ≤ β + < q− − 1, from (12), we can see that µ2 is large enough, when µ1 is large enough. From (H3 ) and (H4 ), we can see that (13) is satisfied. Thus (z1 , z2 ) is a supersolution for (P1 ). Step 3. We prove that φ1 ≤ z1 and φ2 ≤ z2 . In the definition of v1 (x), let ζ = 2` (max φ1 (x) + max |∇φ1 (x)|). We claim that x∈Ω
φ1 (x) ≤ v1 (x),
x∈Ω
∀x ∈ Ω .
(14)
From the definition of v1 , it is easy to see that
φ1 (x) ≤ 2 max φ1 (x) ≤ v1 (x),
when d(x) = `,
φ1 (x) ≤ 2 max φ1 (x) ≤ v1 (x),
when d(x) ≥ `.
x∈Ω
and x∈Ω
It only remains to prove that
φ1 (x) ≤ v1 (x),
when d(x) < `.
Since v1 − φ1 ∈ C 1 (∂ Ω` ), then there exists a point x0 ∈ ∂ Ω` such that
v1 (x0 ) − φ1 (x0 ) = min [v1 (x) − φ1 (x)]. x∈∂ Ω`
If v1 (x0 ) − φ1 (x0 ) < 0, it is easy to see that 0 < d(x0 ) < `, and then
∇v1 (x0 ) − ∇φ1 (x0 ) = 0. From the definition of v1 , we have
|∇v1 (x0 )| = ζ =
2
`
(max φ1 (x) + max |∇φ1 (x)|) > |∇φ1 (x0 )| . x∈Ω
x∈Ω
It is a contradiction to ∇v1 (x0 ) − ∇φ1 (x0 ) = 0. Thus (14) is valid. Obviously, there exists a positive constant C3 such that
ζ ≤ C3 λ. Since d(x) ∈ C 2 (∂ Ω3` ), according to the proof of Lemma 2.2, there exists a positive constant C4 such that
−∆p(x) v1 (x) ≤ C∗ ζ p(x)−1+θ ≤ C4 λp(x)−1+θ , p+
When η = λ
a.e. in Ω , where θ ∈ (0, 1).
and λ ≥ 1 is large enough, we have
−∆p(x) v1 (x) ≤ η. According to the comparison principle, we have
v1 (x) ≤ w(x),
∀x ∈ Ω .
(15) p+
From (14) and (15), when η = λ
φ1 (x) ≤ v1 (x) ≤ w(x),
and λ ≥ 1 is sufficiently large, we have
∀x ∈ Ω .
According to the comparison principle, when µ1 is large enough, we have
v1 (x) ≤ w(x) ≤ z1 (x),
∀x ∈ Ω .
(16)
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361
Combining (16) and the definition of v1 (x), we easily see that
φ1 (x) ≤ v1 (x) ≤ w(x) ≤ z1 (x),
∀x ∈ Ω .
From (12), we can see that µ2 is large enough, when µ1 ≥ 1 and λ is large enough. Similarly, we have φ2 ≤ z2 . This completes the proof. Theorem 3.2. Under the conditions of (H1 )–(H4 ), if osc q(x), osc p(x) < 1, and (supx∈∂ Ω (p(x) − 1), infx∈∂ Ω p(x)) ∩ x∈∂ Ω
x∈∂ Ω
(supx∈∂ Ω (q(x) − 1), infx∈∂ Ω q(x)) is nonempty, then (P2 ) has a positive solution when λ is large enough. Proof. Since osc q(x), osc p(x) < 1, from the continuity of p(x) and q(x), without loss of generality, we may assume that x∈∂ Ω
x∈∂ Ω
osc q(x), osc p(x) < 1, then
x∈∂ Ω3`
x∈∂ Ω3`
sup (p(x) − 1) < x∈∂ Ω3`
inf p(x)
and
x∈∂ Ω3`
sup (q(x) − 1) < x∈∂ Ω3`
inf q(x).
x∈∂ Ω3`
We may assume that ` is small enough such that ( sup (p(x)−1), inf p(x))∩( sup (q(x)−1), inf q(x)) is nonempty, x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
then we can find a real number s ∈ ( sup (p(x) − 1), inf p(x)) ∩ ( sup (q(x) − 1), inf q(x)). For any λ > 0, there x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
exists a k such that such that ks = λ. Since s ∈ ( sup (p(x) − 1), inf p(x)) and ks = λ, we can see that kp(x) a ≥ λb and C5 (kekσ )p(x)−1 ln k ≤ λ can be x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
satisfied simultaneously, when λ > 0 is large enough. Since s ∈ ( sup (q(x) − 1), inf q(x)) and ks = λ, we can see that kq(x) a ≥ λb and C5 (kekσ )q(x)−1 ln k ≤ λ can be satisfied simultaneously, when λ > 0 is large enough. Similarly to the proof of Theorem 3.1, we can conclude that (φ1 , φ2 ) is a subsolution of (P2 ). We consider
( −∆p(x) z1 = λµ1 in Ω , −∆q(x) z2 = λµ2 in Ω , z1 = z2 = 0 on ∂ Ω . Similarly to the proof of Theorem 3.1, let 1
1
µ1 = 2[C2 (λµ1 ) p− −1 ]α [C2 (λµ2 ) q− −1 ]γ , +
+
(17)
we only need 1
1
µ2 ≥ 2[C2 (λµ1 ) p− −1 ]δ [C2 (λµ2 ) q− −1 ]β , +
+
when µ1 is large enough,
(18)
then, it is easy to see that (z1 , z2 ) is a supersolution for (P2 ). Similarly to the proof of Theorem 3.1, we can conclude that (P2 ) has a positive solution. 4. Asymptotic behavior of positive solutions In this section, we will discuss the asymptotic behavior of maxima of solutions, and the asymptotic behavior of solutions near boundary, when the parameter goes to ∞. Theorem 4.1. Under the conditions of (H1 )–(H4 ), if (u, v) is a solution of (P1 ) which has been given in Theorem 3.1, then (i) There exist positive constants C1 and C2 such that + − C1 λ ≤ max u(x) ≤ C2 (λp µ1 )1/(p −1) ,
(19)
+ − C1 λ ≤ max v(x) ≤ C2 (λq µ2 )1/(q −1) ,
(20)
x∈Ω
x∈Ω
(ii) For any θ ∈ (0, 1), there exist positive constants C3 and C4 such that + − C3 λd(x) ≤ u(x) ≤ C4 (λp µ1 )1/(p −1) (d(x))θ ,
C3 λd(x) ≤ v(x) ≤ C4 (λ
q+
1/(q− −1)
µ2 )
θ
(d(x)) ,
as d(x) → 0,
(21)
as d(x) → 0,
(22)
where the positive parameters µ1 and µ2 are large enough, and satisfies (12) and (13).
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Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
Proof. (i) Obviously, when 2` ≤ d(x), we have u(x) ≥ φ1 (x) = e
kσ
2`
Z −1+
ke
kσ
Z
2`
kekσ
2 p− −1
2` − σ
σ
v(x) ≥ φ2 (x) = ekσ − 1 +
2` − t
2` − t
2 q− −1
2` − σ
σ
dt ≥ λ
b+1
dt ≥ λ
b+1
a
a
2`
Z
M
2`
2 p− −1
2 q− −1
2` − σ
σ
Z
2` − t
M
σ
2` − t 2` − σ
dt , dt ,
then there exists a positive constant C1 such that C1 λ ≤ max u(x)and C1 λ ≤ max v(x). x∈Ω
x∈Ω
It is easy to see + − u(x) ≤ z1 (x) ≤ max z1 (x) ≤ C2 (λp µ1 )1/(p −1) ,
x∈Ω
then + − max u(x) ≤ C2 (λp µ1 )1/(p −1) .
x∈Ω
Similarly + − max v(x) ≤ C2 (λq µ2 )1/(q −1) .
x∈Ω
Thus (19) and (20) are valid. (ii) Denote
v3 (x) = a(d(x))θ ,
d(x) ≤ ρ,
where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, `) is small enough. Obviously, v3 (x) ∈ C 1 (∂ Ωρ ). 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) < ρ,
where
Π (x) = d
(∇ p∇ d) ln d ∆d (∇ p∇ d) ln aθ +d +d . (θ − 1)(p(x) − 1) (p(x) − 1) (θ − 1)(p(x) − 1)
+ − Let a = ρ1 C2 (λp µ1 )1/(p −1) , when ρ > 0 is small enough, it is easy to see that
(a)p(x)−1 ≥ λp µ1 and +
|Π (x)| ≤
1 2
.
When ρ > 0 is small enough, then we have +
−∆p(x) v3 (x) ≥ λp µ1 . Obviously v3 (x) ≥ z1 (x), when d(x) = ρ or d(x) = 0. When (1 − θ )p+ < 1, it is easy to see that v3 (·) ∈ W 1,p(x) (∂ Ωρ ). According to the comparison principle, we have v3 (x) ≥ z1 (x) on ∂ Ωρ . Thus + − u(x) ≤ C4 (λp µ1 )1/(p −1) (d(x))θ ,
Let a = ρ C2 (λ 1
q+
µ2 )
1/(q− −1)
(a)q(x)−1 ≥ λq µ2 , +
as d(x) → 0.
, it is easy to that
when ρ > 0 is small enough.
Similarly, we have
v(x) ≤ C4 (λq µ2 )1/(q +
− −1)
(d(x))θ ,
as d(x) → 0,
when ρ > 0 is small enough.
Obviously, we have u(x) ≥ φ1 (x) = ekd(x) − 1 ≥ C3 λd(x),
v(x) ≥ φ2 (x) = ekd(x) − 1 ≥ C3 λd(x),
when d(x) < σ is small enough, when d(x) < σ is small enough.
Thus (21) and (22) are valid. This completes the proof.
Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
363
Similarly, we have: Theorem 4.2. Under the conditions of Theorem 3.2, if (u, v) is a solution of (P2 ) which has been given in Theorem 3.2, then: (i) There exist positive constants C1 and C2 such that 1
− C1 λ p+ ≤ max u(x) ≤ C2 (λµ1 )1/(p −1) ,
x∈Ω
C1 λ
1 q+
≤ max v(x) ≤ C2 (λµ2 )1/(q
− −1)
x∈Ω
.
(ii) For any θ ∈ (0, 1), there exist positive constants C3 and C4 such that 1
− C3 λ p+ d(x) ≤ u(x) ≤ C4 (λµ1 )1/(p −1) (d(x))θ ,
C3 λ
1 q+
1/(q− −1)
d(x) ≤ v(x) ≤ C4 (λµ2 )
θ
(d(x)) ,
as d(x) → 0, as d(x) → 0,
where the positive parameters µ1 and µ2 are large enough, and satisfy (17) and (18). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002) 213–259. Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (4) (2006) 1383–1406. M. Růžička, Electrorheological fluids: Modeling and mathematical theory, in: Lecture Notes in Math., vol. 1784, Springer-Verlag, Berlin, 2000. V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv 29 (1987) 33–36. E. Acerbi, G. Mingione, Gradient estimates for the p(x)-Laplacian system, J. Reine Angew. Mat. (Crelles J.) 584 (2005) 117–148. E. Acerbi, G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 (2001) 121–140. A. Coscia, G. Mingione, Hölder continuity of the gradient of p(x)-harmonic mappings, C.R. Acad. Sci. Ser. I Math. 328 (1999) 363–368. X.L. Fan, D. Zhao, On the spaces Lp(x) (Ω ) and W m,p(x) (Ω ), J. Math. Anal. Appl. 263 (2001) 424–446. X.L. Fan, Global C 1,α regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2007) 397–417. X.L. Fan, D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear Anal. 39 (2000) 807–816. X.L. Fan, On the sub-supersolution method for p(x)-Laplacian equations, J. Math. Anal. Appl. 330 (2007) 665–682. X.L. Fan, Y.Z. Zhao, D. Zhao, Compact imbedding theorems with symmetry of Strauss-Lions type for the spaces W 1,p(x) , J. Math. Anal. Appl. 255 (2001) 333–348. X.L. Fan, H.Q. Wu, F.Z. Wang, Hartman-type results for p(t )-Laplacian systems, Nonlinear Anal. 52 (2003) 585–594. X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843–1852. X.L. Fan, Q.H. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005) 306–317. X.L. Fan, D. Zhao, Regularity of minimizers of variational integrals with continuous p(x)-growth conditions, Chinese Ann. Math. 17A (5) (1996) 557–564. A. El Hamidi, Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl. 300 (2004) 30–42. P. Marcellini, Regularity and existence of solutions of elliptic equations with (p.q)-growth conditions, J. Differential Equations 90 (1991) 1–30. S.G. Samko, Denseness of C0∞ (RN ) in the generalized Sobolev spaces W m,p(x) (RN ), Dokl. Ross. Akad. Nauk 369 (4) (1999) 451–454. Q.H. Zhang, Existence of solutions for weighted p(r )-Laplacian system boundary value problems, J. Math. Anal. Appl. 327 (2007) 127–141. Q.H. Zhang, Existence of radial solutions for p(x)-Laplacian equations in RN , J. Math. Anal. Appl. 315 (2) (2006) 506–516. Q.H. Zhang, The asymptotic behavior of solutions for p(x)-Laplace equations, J. Zhengzhou Univ. Light Industry 24 (5) (2007) 92–95. Q.H. Zhang, Existence of positive solutions for elliptic systems with nonstandard p(x)-growth conditions via sub-supersolution method, Nonlinear Anal. 67 (2007) 1055–1067. Q.H. Zhang, Existence of positive solutions for a class of p(x)-Laplacian systems, J. Math. Anal. Appl. 333 (2007) 591–603. Q.H. Zhang, A strong maximum principle for differential equations with nonstandard p(x)-growth conditions, J. Math. Anal. Appl. 312 (1) (2005) 24–32. M. Chhetri, D.D. Hai, R. Shivaji, On positive solutions for classes of p-Laplacian semipositone systems, Discrete Contin. Dyn. Syst. 9 (4) (2003) 1063–1071. C.H. Chen, On positive weak solutions for a class of quasilinear elliptic systems, Nonlinear Anal. 62 (2005) 751–756. D.D. Hai, R. Shivaji, An existence result on positive solutions of p-Laplacian systems, Nonlinear Anal. 56 (2004) 1007–1010.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Existence and asymptotic behavior of positive solutions for a variable exponent elliptic system without variational structureI Qihu Zhang a,b,c,∗ , Zhimei Qiu b , Rong Dong b a
Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China
b
School of Mathematics Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China
c
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, China
article
info
Article history: Received 25 August 2008 Accepted 15 June 2009 MSC: 35J25 35J60 Keywords: Variable exponent elliptic system Subsolution Supersolution
abstract We mainly consider the existence and asymptotic behavior of positive solutions of the following system
−∆p(x) u = λp(x) (uα(x) v γ (x) + h1 (x)) in Ω , −∆q(x) v = λq(x) (uδ(x) v β(x) + h2 (x)) in Ω , u = v = 0 on ∂ Ω , where Ω ⊂ RN is a bounded domain with C 2 boundary ∂ Ω , 1 < p(x), q(x) ∈ C 1 (Ω ) are functions, and −∆p(x) u = −div(|∇ u|p(x)−2 ∇ u) is called p(x)-Laplacian. When α, β, γ , δ satisfy some conditions and λ is large enough, we proved the existence of a positive solution. In particular, we do not assume any symmetric condition, and we do not assume any sign condition on h1 (0) and h2 (0). © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The study of differential equations and variational problems with variable exponents is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, image processing, etc. (see [1–4]). Many results have been obtained on these kinds of problems, for example [5–25]. In [9], Fan give the regularity of weak solutions for differential equations with variable exponents. For the case of the existence of solutions for elliptic systems with variable exponents, we refer to [13,17,23,24]. In this paper, we mainly consider the existence and asymptotic behavior of positive weak solutions for the following system
(P1 )
−∆p(x) u = λp(x) (uα(x) v γ (x) + h1 (x)) in Ω , −∆q(x) v = λq(x) (uδ(x) v β(x) + h2 (x)) in Ω , u = v = 0 on ∂ Ω ,
where p(x), q(x) ∈ C 1 (Ω ) are functions and Ω ⊂ RN is an open bounded domain with C 2 boundary ∂ Ω . The operator −∆p(x) u = −div(|∇ u|p(x)−2 ∇ u) is called p(x)-Laplacian and the corresponding equation is called a variable exponent I Foundation item: Partly supported by the National Science Foundation of China (10701066 & 10671084), China Postdoctoral Science Foundation (20070421107), the Natural Science Foundation of Henan Education Committee (2008-755-65) and the Natural Science Foundation of Jiangsu Education Committee (08KJD110007). ∗ Corresponding author at: Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China. E-mail address: [email protected] (Q. Zhang).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.06.069
Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
355
equation. Especially, if p(x) ≡ p (a constant), −∆p(x) is the well-known p-Laplacian and the corresponding equation is called a constant exponent equation. There are many papers on the existence of solutions for constant exponent elliptic systems without variational structure, for example [26–28]. Because of the nonhomogeneity of variable exponent problems, variable exponent problems are more complicated than constant exponent problems, and many results and methods for constant exponent problems are invalid for variable exponent problems; for example, if Ω is bounded, then the Rayleigh quotient
R λp(x) =
inf
1,p(x)
u∈W0
(Ω )\{0}
1
Ω p(x)
R
|∇ u|p(x) dx
1
Ω p(x)
|u|p(x) dx
is zero in general, and only under some special conditions λp(x) > 0 (see [15]), the p(x)-Laplacian does not have the first eigenvalue and the first eigenfunction in general. However, 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 to be discussed in the existence and asymptotic behavior of solutions of variable exponent problems. In [28], the authors consider the existence of positive weak solutions for the following constant exponent problems
−∆p u = λf (v) in Ω , (I ) −∆p v = λg (u) in Ω , u = v = 0 on ∂ Ω , (
the first eigenfunction is used to construct the subsolution of constant exponent problems successfully. Under the condition that λ is large enough and lim
u→+∞
f [M (g (u))1/(p−1) ] up−1
= 0,
for every M > 0,
the authors give the existence of positive solutions for problem (I ). In [27], the author considers the existence and nonexistence of positive weak solution to the following constant exponent elliptic system
−∆p u = λf (u, v) = λuα v γ in Ω , (II ) −∆q v = λg (u, v) = λuδ v β in Ω , u = v = 0 on ∂ Ω , the first eigenfunction is used to construct the subsolution of it, the main results are as following (i) If α, β ≥ 0, γ , δ > 0, $ = (p − 1 − α)(q − 1 − β) − γ δ > 0, then problem (II ) has a positive weak solution for each λ > 0; (ii) If $ = 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 variable exponent 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 exists, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of variable exponent problems. In many cases, the radial symmetric conditions are effective to deal with variable exponent problems. There are many results about the radial variable exponent problems (see [12,13,20–23]). In [23,24], the present author discussed the existence of positive solutions of the following
−∆p(x) u = λf (v) in Ω , (III ) −∆p(x) v = λg (u) in Ω , u = v = 0 on ∂ Ω , (
when it possesses some symmetric conditions. But the asymptotic behavior of positive weak solutions have not been discussed in [23,24,26–28]. Our aim is to give the existence and asymptotic behavior of positive weak solutions for problem (P1 ) without any symmetric conditions. Through a new method to construct sub-supersolution, this paper gives the existence of positive weak solutions for problem (P1 ) via sub-supersolution method. Similarly to (II ), we consider the following
(P2 )
−∆p(x) u = λ(uα(x) v γ (x) + h1 (x)) in Ω , −∆q(x) v = λ(uδ(x) v β(x) + h2 (x)) in Ω , u = v = 0 on ∂ Ω .
It is well known that (P1 ) is equal to (P2 ) if p(x) ≡ p ≡ q(x) (a constant), but for general functions p(x) and q(x), (P1 ) is not equal to (P2 ) even if p(x) = q(x) (a function). Moreover, we discuss the existence of positive weak solutions for problem (P2 ) without any symmetric conditions. Since there exist a p(x)-Laplacian and a q(x)-Laplacian in (P1 ) and (P2 ), we call (P1 ) and (P2 ) are (p(x), q(x))-type, similarly, we call (III ) is (p(x), p(x))-type. There are some differences between the existence of positive solutions of (P1 )
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Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
and (P2 ), and there are some differences between the existence of positive solutions of (p(x), q(x))-type and (p(x), p(x))type. Notice that in problems (I ) and (II ), f and g are independent on the variable x, and the systems are homogeneity. Our results partially generalized the results of [23,24,26–28]. This paper is divided into four sections. In the second section, we do some preparation. In the third section, we will discuss the existence of positive solutions of (P1 ) and (P2 ). In the fourth section, we will discuss the asymptotic behavior of positive solutions of (P1 ) and (P2 ). 2. Preliminaries In order to deal with variable exponent problems, we need some theories on spaces Lp(x) (Ω ), W 1,p(x) (Ω ) and properties of p(x)-Laplacian which we will use later (see [8,19]). Write C+ (Ω ) = {h|h ∈ C (Ω ), h(x) > 1 for x ∈ Ω } , h+ = max h(x), h− = min h(x), for any h ∈ C (Ω ). x∈Ω
x∈Ω
Throughout the paper, we will assume that: (H1 ) Ω ⊂ RN is an open bounded domain with C 2 boundary ∂ Ω . (H2 ) p, q ∈ C 1 (Ω ) satisfy 1 < p− ≤ p+ and 1 < q− ≤ q+ . (H3 ) h1 , h2 , α, β, γ , δ ∈ C (Ω ) satisfy α, β ≥ 0 on Ω and γ , δ > 0 on Ω . (H4 ) 0 ≤ α + < p− − 1, 0 ≤ β + < q− − 1, and
$ := (p− − 1 − α + )(q− − 1 − β + ) − γ + δ + > 0. Denote Lp(x) (Ω ) =
Z
u | u is a measurable real-valued function, Ω
|u(x)|p(x) dx < ∞ .
p(x)
(Ω ) by Z ) u(x) p(x) dx ≤ 1 , = inf λ > 0 Ω λ
We can introduce a norm on L
(
|u|p(x)
and (Lp(x) (Ω ), |·|p(x) ) becomes a Banach space, and we call it variable exponent Lebesgue space. The space (Lp(x) (Ω ), |·|p(x) ) is a separable, reflexive and uniform convex Banach space (see [8, Theorem 1.10, Theorem 1.14]). The space W 1,p(x) (Ω ) is defined by W 1,p(x) (Ω ) = u ∈ Lp(x) (Ω ) | |∇ u| ∈ Lp(x) (Ω ) ,
and it can be equipped with the norm
kuk = |u|p(x) + |∇ u|p(x) ,
∀u ∈ W 1,p(x) (Ω ) . 1,p(x)
1,p(x)
(Ω ) are separable, reflexive and We denote by W0 (Ω ) the closure of C0∞ (Ω ) in W 1,p(x) (Ω ). W 1,p(x) (Ω ) and W0 uniform convex Banach spaces (see [8, Theorem 2.1]), and we call it variable exponent Sobolev space. We define (L(u), v) =
Z Ω
|∇ u|p(x)−2 ∇ u∇v dx,
∀u, v ∈ W01,p(x) (Ω ),
1,p(x)
1,p(x)
(Ω ) → (W0 (Ω ))∗ is a continuous, bounded and strictly monotone operator, and it is a homeomorphism then L : W0 (see [14, Theorem 3.1]). 1,p(x) If (u, v) ∈ (W0 (Ω ), W01,q(x) (Ω )), (u,v ) is called a weak solution of (P1 ) if it satisfies Z Z p(x)−2 |∇ | u ∇ u ∇ϕ dx = λp(x) (uα(x) v γ (x) + h1 (x))ϕ dx, ∀ϕ ∈ W01,p(x) (Ω ) , Ω Ω Z Z λq(x) (uδ(x) v β(x) + h2 (x))ψ dx, ∀ψ ∈ W01,q(x) (Ω ) . |∇v|q(x)−2 ∇v∇ψ dx = Ω
Ω
1,p(x) W0
Define A : W 1,p(x) (Ω ) → (
hAu, ϕi =
Z Ω
(Ω ))∗ as
(|∇ u|p(x)−2 ∇ u∇ϕ + h(x, u)ϕ)dx,
∀u ∈ W 1,p(x) (Ω ), ∀ϕ ∈ W01,p(x) (Ω ),
Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
357
where h(x, u) is continuous on Ω × R, h(x, ·) is increasing and satisfies ∗ (x)−1
|h(x, t )| ≤ C1 + C2 |t |p
,
where p∗ (x) =
Np(x)/(N − p(x)), ∞, p(x) ≥ N .
p(x) < N
It is easy to check that A is a continuous bounded mapping. Copying the proof of [25], we have the following lemma. 1,p(x)
Lemma 2.1 (Comparison Principle). . Let u, v ∈ W 1,p(x) (Ω ) satisfying Au − Av ≥ 0 in (W0
v(x), 0}. If ϕ(x) ∈
1,p(x) W0
(Ω ))∗ , ϕ(x) = min{u(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 = {x ∈ Ω | d(x, ∂ Ω ) < }. Since ∂ Ω is C 2 regularly, there exists a constant ` ∈ (0, 1) such that d(x) ∈ C 2 (∂ Ω3` ), and |∇ d(x)| ≡ 1. Denote
ζ d(x), d(x) < `, 2 Z d(x) 2` − t p− −1 ζ ` + dt , ` ≤ d(x) < 2`, ζ v1 (x) = ` ` 2 Z 2` 2` − t p− −1 ζ ` + ζ dt , 2` ≤ d(x). ` ` Obviously, 0 ≤ v1 (x) ∈ C 1 (Ω ). Let us consider the following problem
− ∆p(x) w(x) = η in Ω , w = 0 on ∂ Ω ,
(1)
where η is a positive parameter. Lemma 2.2 (See [11]). If the 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 1
C1 η p+ −1+θ ≤ max w(x); x∈Ω
(ii) There exists a positive constant C2 such that 1
max w(x) ≤ C2 η p− −1 . x∈Ω
Proof. (i) By computation
p(x)−1 −ζ [(∇ p∇ d) ln ζ + ∆d("x)], d(x) < `, #) 2 ! ( 2 ( p ( x ) − 1) 2` − d 2` − d p− −1 − ln ζ ∇ p ∇ d + ∆d `(p− − 1) ` ` −∆p(x) v1 (x) = 2(p−(x)−1) −1 p −1 2` − d p(x)−1 , ` < d(x) < 2`, ×ζ ` 0, 2` < d(x), then −∆p(x) v1 (x) ≤ C∗ ζ p(x)−1+θ a.e. on Ω , for any θ ∈ (0, 1), where C∗ = C∗ (`, θ , p, Ω ) is a positive constant and independent on ζ . + When C∗ ζ p −1+θ = 12 η, we can see that v1 (x) is a subsolution of (1). According to the comparison principle, we can see that v1 (x) ≤ w(x) on Ω . Obviously, ζ ` ≤ maxx∈Ω v1 (x) ≤ 2ζ `, there exists a positive constant C1 such that 1
max w(x) ≥ max v1 (x) ≥ C1 η p+ −1+θ . x∈Ω
x∈Ω
(ii) It is easy to see from Lemma 2.1 of [11]. This completes the proof.
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3. Existence of positive solutions In the following, if it cannot leading to confusion, we always use Ci to denote positive constants. Theorem 3.1. Under the conditions of (H1 )–(H4 ), then (P1 ) has a positive solution when λ is large enough. Proof. We shall establish Theorem 3.1 by constructing a positive subsolution (φ1 , φ2 ) and a supersolution (z1 , z2 ) of (P1 ), such that φ1 ≤ z1 and φ2 ≤ z2 , i.e., (φ1 , φ2 ) and (z1 , z2 ) satisfy
Z Z γ (x) p(x)−2 λp(x) (φ1α(x) φ2 + h1 (x))ϕ dx, ∇φ1 · ∇ϕ dx ≤ |∇φ1 | Ω Ω Z Z β(x) λq(x) (φ1δ(x) φ2 + h2 (x))ψ dx, |∇φ2 |q(x)−2 ∇φ2 · ∇ψ dx ≤ Ω
Ω
and
Z Z γ (x) p(x)−2 ∇ z1 · ∇ϕ dx ≥ λp(x) (z1α(x) z2 + h1 (x))ϕ dx, |∇ z1 | Ω Ω Z Z β(x) λq(x) (z1δ(x) z2 + h2 (x))ψ dx, |∇ z2 |q(x)−2 ∇ z2 · ∇ψ dx ≥ Ω
Ω
1,p(x) W0
1,q(x) W0
for all (ϕ, ψ) ∈ ( (Ω ), (Ω )) with ϕ ≥ 0 and ψ ≥ 0. According to the sub-supersolution method for p(x)Laplacian equations (see [11]), (P1 ) has a positive solution. Step 1. We construct a subsolution of (P1 ). Let σ ∈ (0, `) is small enough. Denote
kd(x) e − 1, d(x) < σ , 2 Z d(x) 2` − t p− −1 kσ dt , σ ≤ d(x) < 2`, e −1+ kekσ φ1 (x) = 2` − σ σ 2 Z 2` 2` − t p− −1 ekσ − 1 + kekσ dt , 2` ≤ d(x), 2` − σ σ kd(x) e − 1, d(x) < σ , 2 Z d(x) 2` − t q− −1 kσ dt , σ ≤ d(x) < 2`, e −1+ kekσ φ2 (x) = 2` − σ σ 2 Z 2` 2` − t q− −1 ekσ − 1 + kekσ dt , 2` ≤ d(x). 2` − σ σ It is easy to see that φ1 , φ2 ∈ C 1 (Ω ). Denote inf p(x) − 1
a = min
,
inf q(x) − 1
4(sup |∇ p(x)| + 1) 4(sup |∇ q(x)| + 1)
,1 ,
b = |h1 (0)| + |h2 (0)| + 1.
By computation
ln k ∆d kd(x) p(x)−1 − k ( ke ) ( p ( x ) − 1 ) + d ( x ) + ∇ p ∇ d + , d(x) < σ , k k ( " ! #) 2 1 2(p(x) − 1) 2` − d 2` − d p− −1 kσ ∇ p ∇ d + ∆d − ln ke 2` − σ p− − 1 2` − σ 2` − σ −∆p(x) φ1 = 2(p−(x)−1) −1 p −1 2` − d kσ p(x)−1 × ( ke ) , σ < d(x) < 2`, 2` − σ 0, 2` < d(x). From (H3 ), there exists a positive constant M > 2 such that γ (x)
(φ1α(x) φ2 Let σ =
1 k
+ h1 (x)) ≥ 1 and
β(x)
(φ1δ(x) φ2
+ h2 (x)) ≥ 1,
∀x ∈ Ω , when φ1 , φ2 ≥ M − 1.
(2)
ln M, we have
σ k = ln M .
(3)
Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
359
If k is sufficiently large, from (3), we have
− ∆p(x) φ1 ≤ −kp(x) a,
d(x) < σ .
(4)
Let λ = b+1 k, we have a
kp(x) a ≥ λp(x) b, from (2) and (4) and the definition of b, we have γ (x)
− ∆p(x) φ1 ≤ −λp(x) b ≤ λp(x) (φ1α(x) φ2
+ h1 (x)),
d(x) < σ .
(5)
Since d(x) ∈ C 2 (Ω 3` ), there exists a positive constant C3 such that
2(p−(x)−1) −1 p −1 2` − d −∆p(x) φ1 ≤ (kekσ )p(x)−1 2` − σ ( #) " 2 ! 2` − d 2` − d p− −1 2(p(x) − 1) kσ × − ln ke ∇ p ∇ d + ∆ d (2` − σ )(p− − 1) 2` − σ 2` − σ ≤ C3 (kekσ )p(x)−1 ln k, σ < d(x) < 2`. Let λ = b+a 1 k. If k is sufficiently large, we have C3 (kekσ )p(x)−1 ln k = C3 (kM )p(x)−1 ln k ≤ λp(x) , then
− ∆p(x) φ1 ≤ λp(x) ,
σ < d(x) < 2`.
(6)
+ h1 (x)), σ < d(x) < 2`.
(7)
Obviously γ (x)
λp(x) ≤ λp(x) (φ1α(x) φ2
Combining (6) and (7), we have γ (x)
− ∆p(x) φ1 ≤ λp(x) ≤ λp(x) (φ1α(x) φ2
+ h1 (x)),
σ < d(x) < 2`,
when λ is large enough.
(8)
Obviously γ (x)
− ∆p(x) φ1 = 0 ≤ λp(x) (φ1α(x) φ2
+ h1 (x)),
2` < d(x).
(9)
Combining (5), (8) and (9), we can conclude that γ (x)
+ h1 (x)),
a.e. on Ω .
(10)
β(x)
+ h2 (x)),
a.e. on Ω .
(11)
− ∆p(x) φ1 ≤ λp(x) (φ1α(x) φ2 Similarly
− ∆q(x) φ2 ≤ λq(x) (φ1δ(x) φ2
From (10) and (11), we can see that (φ1 , φ2 ) is a subsolution of (P1 ). Step 2. We construct a supersolution of (P1 ) We consider
+ −∆p(x) z1 = λp µ1 in Ω , + q −∆q(x) z2 = λ µ2 in Ω , z1 = z2 = 0 on ∂ Ω , when µ1 and µ2 satisfy some conditions, we shall prove that (z1 , z2 ) is a supersolution for (P1 ). If we could prove that
µ1 ≥ [max z1 (x)]α [max z2 (x)]γ + max |h1 (x)| , +
x∈Ω
+
x∈Ω
x∈Ω
and
µ2 ≥ [max z1 (x)]δ [max z2 (x)]β + max |h2 (x)| , +
x∈Ω
+
x∈Ω
x∈Ω
we would see that (z1 , z2 ) is a supersolution for (P1 ).
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Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
From Lemma 2.2, we have 1
+
+
1
max z1 (x) ≤ C2 (λp µ1 ) p− −1 and max z2 (x) ≤ C2 (λq µ2 ) q− −1 . x∈Ω
x∈Ω
Let 1
1
µ1 = 2[C2 (λp µ1 ) p− −1 ]α [C2 (λq µ2 ) q− −1 ]γ , +
+
+
+
(12)
we only need 1
1
µ2 ≥ 2[C2 (λp µ1 ) p− −1 ]δ [C2 (λq µ2 ) q− −1 ]β , +
+
+
+
when µ1 and µ2 are large enough,
(13)
then, it is easy to see that (z1 , z2 ) is a supersolution for (P1 ). Since 0 ≤ α + < p− − 1 and 0 ≤ β + < q− − 1, from (12), we can see that µ2 is large enough, when µ1 is large enough. From (H3 ) and (H4 ), we can see that (13) is satisfied. Thus (z1 , z2 ) is a supersolution for (P1 ). Step 3. We prove that φ1 ≤ z1 and φ2 ≤ z2 . In the definition of v1 (x), let ζ = 2` (max φ1 (x) + max |∇φ1 (x)|). We claim that x∈Ω
φ1 (x) ≤ v1 (x),
x∈Ω
∀x ∈ Ω .
(14)
From the definition of v1 , it is easy to see that
φ1 (x) ≤ 2 max φ1 (x) ≤ v1 (x),
when d(x) = `,
φ1 (x) ≤ 2 max φ1 (x) ≤ v1 (x),
when d(x) ≥ `.
x∈Ω
and x∈Ω
It only remains to prove that
φ1 (x) ≤ v1 (x),
when d(x) < `.
Since v1 − φ1 ∈ C 1 (∂ Ω` ), then there exists a point x0 ∈ ∂ Ω` such that
v1 (x0 ) − φ1 (x0 ) = min [v1 (x) − φ1 (x)]. x∈∂ Ω`
If v1 (x0 ) − φ1 (x0 ) < 0, it is easy to see that 0 < d(x0 ) < `, and then
∇v1 (x0 ) − ∇φ1 (x0 ) = 0. From the definition of v1 , we have
|∇v1 (x0 )| = ζ =
2
`
(max φ1 (x) + max |∇φ1 (x)|) > |∇φ1 (x0 )| . x∈Ω
x∈Ω
It is a contradiction to ∇v1 (x0 ) − ∇φ1 (x0 ) = 0. Thus (14) is valid. Obviously, there exists a positive constant C3 such that
ζ ≤ C3 λ. Since d(x) ∈ C 2 (∂ Ω3` ), according to the proof of Lemma 2.2, there exists a positive constant C4 such that
−∆p(x) v1 (x) ≤ C∗ ζ p(x)−1+θ ≤ C4 λp(x)−1+θ , p+
When η = λ
a.e. in Ω , where θ ∈ (0, 1).
and λ ≥ 1 is large enough, we have
−∆p(x) v1 (x) ≤ η. According to the comparison principle, we have
v1 (x) ≤ w(x),
∀x ∈ Ω .
(15) p+
From (14) and (15), when η = λ
φ1 (x) ≤ v1 (x) ≤ w(x),
and λ ≥ 1 is sufficiently large, we have
∀x ∈ Ω .
According to the comparison principle, when µ1 is large enough, we have
v1 (x) ≤ w(x) ≤ z1 (x),
∀x ∈ Ω .
(16)
Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
361
Combining (16) and the definition of v1 (x), we easily see that
φ1 (x) ≤ v1 (x) ≤ w(x) ≤ z1 (x),
∀x ∈ Ω .
From (12), we can see that µ2 is large enough, when µ1 ≥ 1 and λ is large enough. Similarly, we have φ2 ≤ z2 . This completes the proof. Theorem 3.2. Under the conditions of (H1 )–(H4 ), if osc q(x), osc p(x) < 1, and (supx∈∂ Ω (p(x) − 1), infx∈∂ Ω p(x)) ∩ x∈∂ Ω
x∈∂ Ω
(supx∈∂ Ω (q(x) − 1), infx∈∂ Ω q(x)) is nonempty, then (P2 ) has a positive solution when λ is large enough. Proof. Since osc q(x), osc p(x) < 1, from the continuity of p(x) and q(x), without loss of generality, we may assume that x∈∂ Ω
x∈∂ Ω
osc q(x), osc p(x) < 1, then
x∈∂ Ω3`
x∈∂ Ω3`
sup (p(x) − 1) < x∈∂ Ω3`
inf p(x)
and
x∈∂ Ω3`
sup (q(x) − 1) < x∈∂ Ω3`
inf q(x).
x∈∂ Ω3`
We may assume that ` is small enough such that ( sup (p(x)−1), inf p(x))∩( sup (q(x)−1), inf q(x)) is nonempty, x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
then we can find a real number s ∈ ( sup (p(x) − 1), inf p(x)) ∩ ( sup (q(x) − 1), inf q(x)). For any λ > 0, there x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
exists a k such that such that ks = λ. Since s ∈ ( sup (p(x) − 1), inf p(x)) and ks = λ, we can see that kp(x) a ≥ λb and C5 (kekσ )p(x)−1 ln k ≤ λ can be x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
x∈∂ Ω3`
satisfied simultaneously, when λ > 0 is large enough. Since s ∈ ( sup (q(x) − 1), inf q(x)) and ks = λ, we can see that kq(x) a ≥ λb and C5 (kekσ )q(x)−1 ln k ≤ λ can be satisfied simultaneously, when λ > 0 is large enough. Similarly to the proof of Theorem 3.1, we can conclude that (φ1 , φ2 ) is a subsolution of (P2 ). We consider
( −∆p(x) z1 = λµ1 in Ω , −∆q(x) z2 = λµ2 in Ω , z1 = z2 = 0 on ∂ Ω . Similarly to the proof of Theorem 3.1, let 1
1
µ1 = 2[C2 (λµ1 ) p− −1 ]α [C2 (λµ2 ) q− −1 ]γ , +
+
(17)
we only need 1
1
µ2 ≥ 2[C2 (λµ1 ) p− −1 ]δ [C2 (λµ2 ) q− −1 ]β , +
+
when µ1 is large enough,
(18)
then, it is easy to see that (z1 , z2 ) is a supersolution for (P2 ). Similarly to the proof of Theorem 3.1, we can conclude that (P2 ) has a positive solution. 4. Asymptotic behavior of positive solutions In this section, we will discuss the asymptotic behavior of maxima of solutions, and the asymptotic behavior of solutions near boundary, when the parameter goes to ∞. Theorem 4.1. Under the conditions of (H1 )–(H4 ), if (u, v) is a solution of (P1 ) which has been given in Theorem 3.1, then (i) There exist positive constants C1 and C2 such that + − C1 λ ≤ max u(x) ≤ C2 (λp µ1 )1/(p −1) ,
(19)
+ − C1 λ ≤ max v(x) ≤ C2 (λq µ2 )1/(q −1) ,
(20)
x∈Ω
x∈Ω
(ii) For any θ ∈ (0, 1), there exist positive constants C3 and C4 such that + − C3 λd(x) ≤ u(x) ≤ C4 (λp µ1 )1/(p −1) (d(x))θ ,
C3 λd(x) ≤ v(x) ≤ C4 (λ
q+
1/(q− −1)
µ2 )
θ
(d(x)) ,
as d(x) → 0,
(21)
as d(x) → 0,
(22)
where the positive parameters µ1 and µ2 are large enough, and satisfies (12) and (13).
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Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
Proof. (i) Obviously, when 2` ≤ d(x), we have u(x) ≥ φ1 (x) = e
kσ
2`
Z −1+
ke
kσ
Z
2`
kekσ
2 p− −1
2` − σ
σ
v(x) ≥ φ2 (x) = ekσ − 1 +
2` − t
2` − t
2 q− −1
2` − σ
σ
dt ≥ λ
b+1
dt ≥ λ
b+1
a
a
2`
Z
M
2`
2 p− −1
2 q− −1
2` − σ
σ
Z
2` − t
M
σ
2` − t 2` − σ
dt , dt ,
then there exists a positive constant C1 such that C1 λ ≤ max u(x)and C1 λ ≤ max v(x). x∈Ω
x∈Ω
It is easy to see + − u(x) ≤ z1 (x) ≤ max z1 (x) ≤ C2 (λp µ1 )1/(p −1) ,
x∈Ω
then + − max u(x) ≤ C2 (λp µ1 )1/(p −1) .
x∈Ω
Similarly + − max v(x) ≤ C2 (λq µ2 )1/(q −1) .
x∈Ω
Thus (19) and (20) are valid. (ii) Denote
v3 (x) = a(d(x))θ ,
d(x) ≤ ρ,
where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, `) is small enough. Obviously, v3 (x) ∈ C 1 (∂ Ωρ ). 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) < ρ,
where
Π (x) = d
(∇ p∇ d) ln d ∆d (∇ p∇ d) ln aθ +d +d . (θ − 1)(p(x) − 1) (p(x) − 1) (θ − 1)(p(x) − 1)
+ − Let a = ρ1 C2 (λp µ1 )1/(p −1) , when ρ > 0 is small enough, it is easy to see that
(a)p(x)−1 ≥ λp µ1 and +
|Π (x)| ≤
1 2
.
When ρ > 0 is small enough, then we have +
−∆p(x) v3 (x) ≥ λp µ1 . Obviously v3 (x) ≥ z1 (x), when d(x) = ρ or d(x) = 0. When (1 − θ )p+ < 1, it is easy to see that v3 (·) ∈ W 1,p(x) (∂ Ωρ ). According to the comparison principle, we have v3 (x) ≥ z1 (x) on ∂ Ωρ . Thus + − u(x) ≤ C4 (λp µ1 )1/(p −1) (d(x))θ ,
Let a = ρ C2 (λ 1
q+
µ2 )
1/(q− −1)
(a)q(x)−1 ≥ λq µ2 , +
as d(x) → 0.
, it is easy to that
when ρ > 0 is small enough.
Similarly, we have
v(x) ≤ C4 (λq µ2 )1/(q +
− −1)
(d(x))θ ,
as d(x) → 0,
when ρ > 0 is small enough.
Obviously, we have u(x) ≥ φ1 (x) = ekd(x) − 1 ≥ C3 λd(x),
v(x) ≥ φ2 (x) = ekd(x) − 1 ≥ C3 λd(x),
when d(x) < σ is small enough, when d(x) < σ is small enough.
Thus (21) and (22) are valid. This completes the proof.
Q. Zhang et al. / Nonlinear Analysis 72 (2010) 354–363
363
Similarly, we have: Theorem 4.2. Under the conditions of Theorem 3.2, if (u, v) is a solution of (P2 ) which has been given in Theorem 3.2, then: (i) There exist positive constants C1 and C2 such that 1
− C1 λ p+ ≤ max u(x) ≤ C2 (λµ1 )1/(p −1) ,
x∈Ω
C1 λ
1 q+
≤ max v(x) ≤ C2 (λµ2 )1/(q
− −1)
x∈Ω
.
(ii) For any θ ∈ (0, 1), there exist positive constants C3 and C4 such that 1
− C3 λ p+ d(x) ≤ u(x) ≤ C4 (λµ1 )1/(p −1) (d(x))θ ,
C3 λ
1 q+
1/(q− −1)
d(x) ≤ v(x) ≤ C4 (λµ2 )
θ
(d(x)) ,
as d(x) → 0, as d(x) → 0,
where the positive parameters µ1 and µ2 are large enough, and satisfy (17) and (18). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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