Hindawi Publishing Corporation
Journal of Inequalities and Applications
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. Garcı́a-Huidobro
This paper investigates the p(x)-Laplacian equations with singular nonlinearities −Δ p(x) u
= λ/uγ(x) in Ω, u(x) = 0 on ∂Ω, where −Δ p(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 Creative 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 example, [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 the p(x)-Laplacian equations with singular nonlinearities:
− p(x) u =
u(x) = 0
λ
uγ(x)
in Ω,
(P)
on ∂Ω,
where − p(x) u = − div(|∇u| p(x)−2 ∇u) is called p(x)-Laplacian, Ω ⊂ RN is a bounded
domain with C 2 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
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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) ∈ C 1 (Ω), 1 < p− ≤ p+ < +∞, where p− = inf Ω p(x), p+ = supΩ p(x).
Because of the nonhomogeneity of p(x)-Laplacian, p(x)-Laplacian problems are more
complicated than those of p-Laplacian ones, many results and methods for p-Laplacian
problems are invalid for p(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), W 1,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,
C0+ (Ω) =
Ω
u(x) p(x) dx < ∞ ,
(2.1)
u ∈ C(Ω) | u > 0 in Ω, u = 0 on ∂Ω .
We can introduce the norm on L p(x) (Ω) by
u(x) p(x)
|u| p(x) = inf μ > 0 |
μ dx ≤ 1. .
(2.2)
Ω
The space (L p(x) (Ω), | · | p(x) ) becomes a Banach space. We call it generalized Lebesgue
space. The space (L p(x) (Ω), | · | p(x) ) is a separable, reflexive, and uniform convex Banach
space (see [3, Theorems 1.10, Theorem 1.14]).
The space W 1,p(x) (Ω) is defined by
W 1,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 ∈ W 1,p(x) (Ω).
(2.4)
1,p(x)
1,p(x)
W0
(Ω) is the closure of C0∞ (Ω) in W 1,p(x) (Ω), W 1,p(x) (Ω) and W0
(Ω) are separable, reflexive, and uniform convex Banach spaces (see [3, Theorem 2.1]).
1,p(x)
If u ∈ Wloc (Ω) ∩ C0+ (Ω), u is called a positive solution of (P) if u(x) satisfies
Q
|∇u| p(x)−2 ∇u∇qdx −
for any domain Q Ω.
λ
Q
qdx
uγ(x)
= 0,
1,p(x)
∀ q ∈ W0
(Q),
(2.5)
Qihu Zhang 3
1,p(x)
1,p(x)
Let W0,loc (Ω) = {u | there is an open domain Q Ω s.t. u ∈ W0
1,p(x)
1,p(x)
A : Wloc (Ω) ∩ C0+ (Ω) → (W0,loc (Ω))∗
Au,ϕ =
1,p(x)
where u ∈ Wloc
Ω
(Q)}, and define
as
|∇u| p(x)−2 ∇u∇ϕ −
λ
u
γ(x) ϕ dx,
(2.6)
1,p(x)
(Ω) ∩ C0+ (Ω), ϕ ∈ W0,loc (Ω); then we have the following lemma.
1,p(x)
1,p(x)
Lemma 2.1 (see [5, Theorem 3.1]). A : Wloc (Ω) ∩ C0+ (Ω) → (W0,loc (Ω))∗ is strictly
monotone.
1,p(x)
1,p(x)
Let g ∈ (W0,loc (Ω))∗ , if g,ϕ ≥ 0, for all ϕ ∈ W0,loc (Ω), ϕ ≥ 0 a.e. in Ω, then denote
1,p(x)
1,p(x)
g ≥ 0 in (W0,loc (Ω))∗ ; correspondingly, if −g ≥ 0 in (W0,loc (Ω))∗ , then denote g ≤ 0 in
1,p(x)
(W0,loc (Ω))∗ .
1,p(x)
1,p(x)
Definition 2.2. Let u ∈ Wloc (Ω) ∩ C0+ (Ω). If Au ≥ 0(Au ≤ 0) in (W0,loc (Ω))∗ , then u
is called a weak supersolution (weak subsolution) of (P).
Copying the proof of [10], we have the following lemma.
1,p(x)
Lemma 2.3 (comparison principle). Let u,v ∈ Wloc (Ω) ∩ C(Ω) be positive and satisfy
1,p(x)
1,p(x)
Au − Av ≥ 0 in (W0,loc (Ω))∗ . Let ϕ(x) = min{u(x) − v(x),0}. If ϕ(x) ∈ W0,loc (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 on Ω × R, u ∈ W 1,p(x) (Ω) is a bounded weak
solution of − p(x) u + g(x,u) = 0 in Ω, u = w0 on ∂Ω, where w0 ∈ W 1,p(x) (Ω), then u ∈
1,α
Cloc
(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
λ
in Ω,
− p(x) u = γ(x)
|u| + an
(3.1)
u(x) = 0 for x ∈ ∂Ω,
where {an } is a positive strictly decreasing sequence and limn→+∞ an = 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=
u
Ω
1 ∇u(x) p(x) dx −
p(x)
Ω
Fn (x,u)dx,
1,p(x)
(3.2)
where Fn (x,u) = 0 λ/((|t | + an )γ(x) )dt. Since ϕ is coercive in W0
(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.
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Journal of Inequalities and Applications
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) < }. Since ∂Ω
is C 2 regularly, then there exists a positive constant σ such that d(x) ∈ C 2 (∂Ω2σ ), and
|∇d(x)| ≡ 1. Let δ ∈ (0,(1/3)σ) be a small enough constant. Denote
⎧
⎪
d(x),
⎪
⎪
⎪
⎪
⎪
d(x) −
⎪
⎪
⎨
2δ − t 2/(p −1)
v1 (x) = ⎪δ +
d(x) < δ,
dt,
δ
δ
⎪
⎪
⎪
2δ −
⎪
⎪
(2δ − t) 2/(p −1)
⎪
⎪
⎩δ +
dt,
δ
δ
δ ≤ d(x) < 2δ,
(3.3)
2δ ≤ d(x).
Obviously, v1 (x) ∈ C 1 (Ω) ∩ C0+ (Ω).
Lemma 3.2. If λ > 0 is large enough, then v1 (x) is a subsolution of (P).
Proof. Since |∇d(x)| ≡ 1, when λ > 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, when δ < d(x) < 2δ, we have
− p(x) v1 = − div
=−
2δ − d(x)
δ
−
+
2δ − d(x)
δ
∇d(x)
2/(p− −1) p(x)−1
2δ − d(x)
δ
2 p(x) − 1
δ p− − 1
2/(p− −1) p(x)−1
d(x)
2/(p− −1) p(x)−1
−
2δ − d(x) 2/(p −1)
∇d(x)∇ p(x) ln
δ
2δ − d(x)
δ
(2(p(x)−1)/(p− −1))−1
.
(3.5)
When λ > 0 is large enough, it is easy to see that
− p(x) v1 ≤ λ
v1 (x)
− p(x) v1 = 0 ≤ γ(x) ,
λ
v1 (x)
∀x ∈ Ω, δ < d(x) < 2δ,
(3.6)
γ(x) ,
∀x ∈ Ω, 2δ < d(x).
From (3.4) and (3.6), we can conclude that v1 (x) is a subsolution of (P).
Qihu Zhang 5
Theorem 3.3. If λ > 0 is a large enough constant, then problem (P) possesses only one positive solution uλ , and uλ is increasing with respect to λ.
Proof. Denote un = n + an , where n is a solution of (3.1). Since {un } is a sequence of
positive solutions of
λ
uγ(x)
− p(x) u =
u(x) = an
in Ω,
(II)
for x ∈ ∂Ω,
then every un is subsolution and supersolution of − p(x) u = λ/uγ(x) in Ω. According to
comparison principle, we have that un ≥ un+1 for n = 1,2,.... Since v1 (x) is a subsolution
of (P) and v1 (x) = 0 on ∂Ω, then un ≥ un+1 ≥ v1 for n = 1,2,.... According to Lemma 2.4,
we have that {un } has uniform C 1,α local regularity property, and hence we can choose
a subsequence, which we denoted by {u1n }, such that u1n → w and ∇u1n → h in Ω. In fact,
h = ∇w in Ω.
1,p(x)
(D). The C 1,α regularity result implies that
For any domain D Ω, for any ϕ ∈ W0
the sequences {un } and {∇un } are equicontinuous in D; from the C 1,α estimate we conclude that ∇w ∈ C α (D) for some 0 < α < 1. Thus w ∈ W 1,p(x) (D) ∩ C 1,α (D). From the
C 1,α regularity result, we see that |∇u1n | p−1 |∇ϕ| ≤ C |∇ϕ| on D, and since the function
ξ → |ξ | p−2 ξ is continuous on Rn , it follows that |∇u1n (x)| p−2 ∇u1n (x) · ∇ϕ(x) →
|∇w(x)| p−2 ∇w(x) · ∇ϕ(x) for x ∈ D. Thus, by the dominated convergence theorem, for
1,p(x)
(D), we can see that
any ϕ ∈ W0
D
1 p −2
∇u (x) ∇u1 (x) · ∇ϕ(x)dx −→
n
n
D
∇w(x) p−2 ∇w(x) · ∇ϕ(x)dx.
(3.7)
Furthermore, since 0 ≤ λ/([u1n (x)]γ(x) ) ≤ λ/([u1n+1 (x)]γ(x) ), and λ/([u1n (x)]γ(x) ) →
λ/([w(x)]γ(x) ) for each x ∈ D, by the monotone convergence theorem we obtain
D
λ
u1n (x)
γ(x) ϕdx −→
D
λ
w(x)
γ(x) ϕdx,
1,p(x)
∀ϕ ∈ W0
(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,
1,p(x)
∀ϕ ∈ W0
(D), (3.9)
and hence w is a weak solution of −Δ p(x) w = λ/([w(x)]γ(x) ) on D.
Obviously, w is a solution of (P), and satisfies w ≥ v1 . According to comparison principle, it is easy to see that (P) possesses only one positive solution, and uλ is increasing
with respect to λ.
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Journal of Inequalities and Applications
4. Asymptotic behavior of positive solutions
In the following, we will use Ci to denote positive constants.
Theorem 4.1. If u is a positive weak solution of problem (P), then C2 d(x) ≤ u(x) as x → ∂Ω.
Proof. Similar to the proof of Lemma 3.2, there exists a positive constant C2 such that
when δ > 0 is small enough, then v2 (x) = C2 d(x) is a subsolution of (P) on ∂Ωδ . Thus
u(x) ≥ v2 (x) = C2 d(x) on ∂Ωδ . The proof is completed.
Denote γ∗ = maxx∈∂Ω2σ γ(x) and γ∗ = minx∈∂Ω2σ γ(x).
Theorem 4.2. If 1 ≤ γ∗ < γ∗ , for any weak solution u of problem (P), we have
C3 d(x)
θ1
θ2
≤ u(x) ≤ C4 d(x)
as x −→ ∂Ω,
(4.1)
where θ1 = maxd(x)≤σ (p(x)/(p(x) − 1 + γ(x))), θ2 = mind(x)≤σ (p(x)/(p(x) − 1 + γ(x))).
Proof. From Theorem 4.1 we only consider (P) in the case of 1 < γ∗ < γ∗ . Denote
⎧ θ
⎪
⎪
⎪a d(x) ,
⎪
⎪
⎪
d(x)
⎪
⎪
⎨
θ
v3 (x) = ⎪aδ +
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩aδ θ +
δ
2δ
δ
d(x) < δ,
aθδ θ−1
aθδ θ−1
2δ − t
δ
2δ − t
δ
2/(p− −1)
dt,
δ ≤ d(x) < 2δ,
(4.2)
2/(p− −1)
dt,
2δ ≤ d(x),
where a and θ are positive constants and satisfy θ ∈ (0,1), 0 < δ is small enough.
Obviously, v3 (x) ∈ C 1 (Ω) ∩ C0+ (Ω). By computation,
(θ−1)(p(x)−1)−1 − p(x) v3 (x) = −(aθ) p(x)−1 (θ − 1) p(x) − 1 d(x)
1 + Π(x) ,
d(x) < δ,
(4.3)
where
Π(x) = d
(∇ p∇d)ln aθ
(∇ p∇d)lnd
d
+d +d
.
(θ − 1) p(x) − 1
p(x) − 1
(θ − 1) p(x) − 1
(4.4)
Obviously |Π(x)| ≤ 1/2, when δ > 0 is small enough. Let θ = θ1 and a ∈ (0,1) is small
enough, when δ ∈ (0,a) is small enough, we can conclude that
− p(x) v3 (x) ≤ λ
v2 (x)
γ(x) ,
d(x) < δ.
(4.5)
Qihu Zhang 7
By computation, when δ < d(x) < 2δ, we have
− p(x) v3 = − div
aθδ
= − aθδ θ−1
θ −1
2δ − d(x)
δ
2δ − d(x)
δ
2/(p− −1) p(x)−1
∇d(x)
2/(p− −1) p(x)−1
−
2δ − d(x) 2/(p −1)
× ∇d(x)∇ p(x) lnaθδ θ−1
(4.6)
δ
− aθδ
+
θ −1
2δ − d(x)
δ
2/(p− −1) p(x)−1
d(x)
(2(p(x)−1)/(p
2 p(x) − 1 θ −1 p(x)−1 2δ − d(x)
aθδ
δ p− − 1
δ
−
−1))−1
.
Thus, there exists a positive constant C ∗ 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)
Let θ = θ1 , when a ∈ (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)
2δ < d(x).
(4.10)
It is easy to see that
− p(x) v3 (x) = 0 ≤ λ
v2 (x)
γ(x) ,
Combining (4.5), (4.9), and (4.10), it is easy to see that when θ = θ1 , a ∈ (0,1) is small
enough and δ ∈ (0,a) is small enough, then v(x) is a subsolution of (P), then u(x) ≥
C3 [d(x)]θ1 on ∂Ωδ .
Similarly, when δ > 0 is small enough, θ = θ2 , and a ≥ maxx∈∂Ωδ (u(x)/δ θ ) is large
enough, we can see that v(x) is a supersolution of (P) on ∂Ωδ , and u(x) ≤ a[d(x)]θ2 on
∂Ωδ . The proof is completed.
Theorem 4.3. If limd(x)→0 p(x) = p0 and limd(x)→0 p(x)/(p(x) − 1 + γ(x)) = s, where s ≤ 1
is a positive constant, u is a solution of (P), then
u(x)
s = 1,
d(x)→0 C d(x)
lim
C = lim
d(x)→0
λ
θ p(x)−1 (1 − θ) p(x) − 1
1/(p(x)−1+γ(x))
.
(4.11)
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Journal of Inequalities and Applications
Proof. It can be obtained easily from Theorem 4.2.
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 C5 such that C1 d(x) ≤ u(x) ≤ C5 (d(x))θ as x → ∂Ω.
Proof. According to Theorem 4.1, it only needs to prove u(x) ≤ C5 (d(x))θ as x → ∂Ω.
Define a function on ∂Ωδ as v4 (x) = C5 (d(x))θ , where C5 ≥ (1/δ θ )maxx∈∂Ωδ u(x). Similar
to the proof of Theorem 4.2, when δ > 0 is small enough, then v4 (x) is a supersolution of
(P) on ∂Ωδ , then u(x) ≤ v4 (x) = C5 (d(x))θ on ∂Ωδ . 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 C1 d(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 prove u(x) ≤ C6 (d(x))θ as x → ∂Ω.
Similar to the proof of Theorem 4.2, when δ > 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
completed.
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