Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 609047, 14 pages
doi:10.1155/2008/609047
Research Article
Boundary Asymptotic and Uniqueness of
Solution for a Problem with px-Laplacian
Ionel Rovenţa
Department of Mathematics, University of Craiova, 200585 Craiova, Romania
Correspondence should be addressed to Ionel Rovenţa, roventaionel@yahoo.com
Received 14 April 2008; Revised 10 June 2008; Accepted 30 August 2008
Recommended by Alexander Domoshnitsky
The goal of this paper is to prove the boundary asymptotic behavior of solutions for weighted
px-Laplacian equations that take infinite value on a bounded domain.
Copyright q 2008 Ionel Rovenţa. 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
Let Ω be a bounded domain in Rn , and let f be a nonnegative, nondecreasing, and C1 function
on R .
We study here the asymptotic boundary behaviour of solutions for the problem
Δpx u gxfu,
x ∈ Ω,
ux −→ ∞, as dist x, ∂Ω −→ 0,
1.1
where Ω ⊆ Rn is a bounded domain with C2 boundary. In additional conditions, the
uniqueness of solutions is also discussed. For further details, see 1, 2.
Here, Δpx u : div|∇ux|px−2 ∇ux is the px-Laplacian, a function defined on
Rn with 1 < px < ∞.
First, Bieberbach in 3, considered the problem
Δu gxfu,
x ∈ Ω,
ux −→ ∞, as dist x, ∂Ω −→ 0.
1.2
He shows that 1.2 admits a unique solution if Ω is a smooth planar domain, f is the
exponential function, and g is constant 1. Rademacher 4 extended the results to the threedimensional domains; Keller 5 and Osserman 6 gave a necessary and sufficient conditions
to solve the problem 1.2 in the n-dimensional case if the domain satisfies inner and outer
sphere conditions.
2
Journal of Inequalities and Applications
Bandle and Marcus 7 found the boundary asymptotic of solutions for 1.2, when g
is continuous and positive, and f satisfies fmt ≤ m1α ft for all 0 < m < 1 and all t ≥ t0 /m.
Cı̂rstea and Rădulescu 8–10 prove the uniqueness and asymptotic behavior of solutions for
problem 1.2, when f is regularly varying, and g ∈ C0,α Ω is a nonnegative function which
is allowed to vanish on the boundary.
In the study of p-Laplacian equations, the main difficulty arises from the lack of
compactness, but here the px-Laplacian possesses more complicated inhomogeneous
nonlinearities, so we need some special techniques. Problems of this type arise in many
areas of applied physics including nuclear physics, field theory, solid waves, and problems of
false vacuum. Zhang in 11 gives the existence and singularity of blowup solutions for the
problem
−Δpx u hx, u 0,
x ∈ Ω,
ux −→ ∞, as dist x, ∂Ω −→ 0,
1.3
where Ω B0, R ⊂ RN , px and fx, u satisfy the following.
H1 px ∈ C1 Ω is a radial symmetric function and satisfies
1 < p− ≤ p < N,
where p− inf px,
Ω
p sup px.
Ω
1.4
H2 hx, u is radial with respect to x, hx, · is increasing, and hx, 0 0 for any x ∈ Ω.
H3 h : Ω × R → R is a continuous function and satisfies
|fx, t| ≤ C1 C2 |t|αx−1 ,
for every x, t ∈ Ω × R, C1 ≥ 0, C2 ≥ 0, α ∈ CΩ,
1 ≤ αx < p∗ x :
1.5
Npx
.
N − px
2. Preliminaries
We recall in what follows some definitions and basic properties of the generalized Lebesgue1,px
Sobolev spaces Lpx Ω, W 1,px Ω, and W0
Ω, where Ω is an open subset of Rn . In that
context, we refer to the book of Musielak 12 and the papers of Kováčik and Rákosnı́k 13
and Fan et al. 14–16.
Set
∞
2.1
L∞
Ω h; h ∈ L Ω, ess inf hx > 1 ∀x ∈ Ω .
x∈Ω
For any h ∈ L∞
Ω, we define
h ess sup hx,
x∈Ω
h− ess inf hx.
2.2
x∈Ω
For any px ∈ L∞
Ω, we define the variable exponent Lebesgue space
Lpx Ω u; u is a measurable real-valued function such that
|ux|px dx < ∞ .
Ω
2.3
Ionel Rovenţa
3
We define a norm, the so-called Luxemburg norm, on this space by the formula
ux px
dx
≤
1
.
|u|px inf μ > 0;
μ Ω
2.4
Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many
respects: they are Banach spaces 13, Theorem 2.5, the Hölder inequality holds 13, Theorem
2.1, they are reflexive if and only if 1 < p− ≤ p < ∞ 13, Corollary 2.7, and continuous
functions are dense if p < ∞ 13, Theorem 2.11. The inclusion between Lebesgue spaces
also generalizes naturally 13, Theorem 2.8. If 0 < |Ω| < ∞ and r1 , r2 are variable exponents
so that r1 x ≤ r2 x almost everywhere in Ω, then there exists the continuous embedding
Lr2 x Ω → Lr1 x Ω, whose norm does not exceed |Ω| 1.
We denote by Lp x Ω the conjugate space of Lpx Ω, where 1/px 1/p x 1.
For any u ∈ Lpx Ω and v ∈ Lp x Ω, the Hölder type inequality
1
1
uv dx ≤
|u|px |v|p x
p− p−
Ω
2.5
holds true.
An important role in manipulating the generalized Lebesgue-Sobolev spaces is played
by the modular of the Lpx Ω space, which is the mapping ρpx : Lpx Ω → R defined by
ρpx u Ω
|u|px dx.
2.6
If u ∈ Lpx Ω and p < ∞, then the following relations hold true:
p−
p
p
p−
|u|px > 1 ⇒ |u|px ≤ ρpx u ≤ |u|px ,
|u|px < 1 ⇒ |u|px ≤ ρpx u ≤ |u|px ,
2.7
|un − u|px −→ 0 ⇐⇒ ρpx un − u −→ 0.
Next, we define the variable Sobolev space:
W 1,px Ω {u ∈ Lpx Ω; |∇u| ∈ Lpx Ω}.
2.8
On W 1,px Ω, we may consider one of the following equivalent norms:
upx |u|px |∇u|px ,
2.9
∇ux px ux px
|u| inf μ > 0;
dx ≤ 1 .
μ μ 2.10
or
Ω
1,px
We also define W0
W 1,px Ω and
Ω as the closure of C0∞ Ω in W 1,px Ω. Assuming p− > 1, the spaces
1,px
W0
Ω
are separable and reflexive Banach spaces.
4
Journal of Inequalities and Applications
Set
Ipx u 1,px
For all u ∈ W0
Ω
|∇u|px |u|px dx.
2.11
Ω, the following relations hold true:
−
−
u > 1 ⇒ up ≤ Ipx u ≤ up ,
2.12
u < 1 ⇒ up ≤ Ipx u ≤ up .
Finally, we remember some embedding results regarding variable exponent LebesgueSobolev spaces. For the continuous embedding between variable exponent Lebesgue-Sobolev
spaces we refer to 15, Theorem 1.1. If p : Ω → R is Lipschitz continuous and p < N, then
for any q ∈ L∞
Ω with px ≤ qx ≤ Npx/N − px, there is a continuous embedding
W 1,px Ω → Lqx Ω. In what concerns the compact embedding, we refer to 15, Theorem
1.3. If Ω is a bounded domain in Rn , px ∈ CΩ, p > N, then for any qx ∈ L∞
Ω
with ess infx∈Ω Npx/N − px − qx > 0 there is a compact embedding W 1,px Ω →
Lqx Ω.
Our aim is to find the asymptotic boundary behaviour to the problem
Δpx u gxfu,
x ∈ Ω,
ux −→ ∞, as dist x, ∂Ω −→ 0.
2.13
#
Suppose p ∈ L∞
and limdistx,∂Ω→0 px p . Consider g ∈ CΩ nonnegative, and f
satisfies
f ∈ C1 0, ∞,
f0 0, ft > 0 , and f is nondecreasing on 0, ∞,
t
∞
1
dt
<
∞,
where
Ft
:
fsds.
1/p#
1 Ft
0
2.14
2.15
We will refer to condition 2.15 as the generalized Keller-Osserman condition. Allow
g to be unbounded on Ω or to vanish on ∂Ω.
3. Boundary asymptotic and uniqueness
Let f satisfy the Keller-Osserman condition. Consider ψ as a function defined on 0, ∞ such
that
∞
1
ds,
3.1
ψt 1/p#
#
t q Fs
where q# is the Holder conjugate of p# , that is, 1/q# 1/p# 1. The function ψ is decreasing
so ψ has an inverse φ : 0, ψ0 → 0, ∞. Moreover, φ is a decreasing function with
limt→0 φt ∞, and some computations give us
1/p#
φ t ψ −1 t −q# Fφt
,
|φ t|p
#
−2 φ t q#
fφt.
p#
3.2
Ionel Rovenţa
5
Now we will see that
1/q#
φ t p# q# Fφt
− φ t q#
fφt
3.3
.
Definition 3.1. A positive measurable function defined on a, ∞, for some a > 0 is said to be
regularly varying at infinity of index q ∈ R, written f ∈ RVq , provided that
ftz
tq
z→∞ fz
∀t > 0.
lim
3.4
Remark 3.2. Let f ∈ RVσ1 σ 2 > p be a nondecreasing and continuous function. Then,
F ∈ RVσ2 and therefore F −1/p ∈ RV−σ−2/p . Since −σ − 2/p < −1, the observation made
in 1 allows us to conclude that F −1/p ∈ L1 1, ∞ and hence f satisfies the generalized
Keller-Osserman condition.
Remark 3.3. Let f ∈ RVσ1 . By the change of variable s tz, we infer that
z
Fz fsds 1
3.5
zftzdt.
0
0
Because f ∈ RVσ1 is continuous, there exists an > 0 such that for every z > we
have ftz/fz ≤ tσ1 1. Hence, using Lebesgue’s dominated convergence theorem, the
limit shifts with the integral and so
Fz
z→∞ zfz
1
ftz
dt z→∞ fz
1
tσ1 dt lim
lim
0
0
1
.
σ2
3.6
Lemma 3.4. Consider f satisfies the Keller-Osserman condition, then one has
#
Fs1/q
0.
s→∞
fs
3.7
lim
Proof. We sketch the proof only in the context of regularly varying functions. For
further details regarding general case, see 17. Let f ∈ RVσ1 σ > p# − 2. From
fz/Fz 1/zzfz/Fz and using limz→∞ Fz/zfz 1/σ 2, then there
exists a constant C > 0 such that limz→∞ Fz/zσ2 C. Finally, limz→∞ fz/zσ1 limz→∞ Fz/zσ2 zfz/Fz Cσ 2. It remains to consider the limit
1/q#
#
Fz1/q
Fz/zσ2 lim
lim
z→∞ fz
z→∞
fz/zσ1
z−1/p
#
σ21
0,
3.8
because −1/p# σ 2 1 < 0.
Another way to prove the lemma is to consider the derivative
#
Fs1/p #
#
1
fzFz−1/q > 0.
p#
3.9
#
If Fs1/p is bounded then there exists M > 0 such that 0 < Fs1/p <
#
M, ∀t > 1. By integrating from 1 to t, it follows that 0 < Fs1/p < Mt and also
6
Journal of Inequalities and Applications
#
1/Mt < 1/Fs1/p < ∞, ∀t > 1. If we integrate again from 1 to ∞, we obtain a contradiction
with the Keller-Osserman condition.
#
In conclusion, the derivative of Fs1/q is unbounded.
Using L’Hospitals rule, we should obtain much more
#
Fz1/q
∞.
z→∞
z
3.10
lim
The following lemma will be useful later; see 1.
Lemma 3.5. Let q# ∈ R be the Holder conjugate of p# > 1. If f ∈ RVσ1 σ > p# − 2, then
∞
#
#
limz→∞ Fz1/q /fz z Fs−1/p ds σ 2 − p# /p# 2 σ.
Proof. By applying L’Hospitals rule, we obtain
#
σ 2 − p#
zFz−1/p
1 zfz
lim
−
1
lim ∞
.
#
#
z→∞
p#
Fs−1/p ds z→∞ p Fz
z
3.11
In conclusion,
σ 2 − p#
Fz
Fz1/q
zFz−1/p
lim
.
lim
z→∞ fz ∞ Fs−1/p# ds
z→∞ ∞ Fs−1/p# ds zfz
p# 2 σ
z
z
#
#
3.12
Corollary 3.6. Let f ∈ RVσ1 σ > p# be a continuous function. Then,
q# σ p# − 2
|φ t|p −2 φ t
− #
.
lim
t→0
tfφt
σ2
p
#
3.13
Proof. See 1, Corollary 3.2.
1,px
We say that u ∈ Wloc Ω is a continuous local weak solution to the equation Δpx u gxfu on the domain Ω if and only if
1,px
|∇u|px−2 ∇u∇ϕdx −
gxfuxϕdx, ϕ ∈ W0
D,
3.14
D
D
for every subdomain D Ω.
1,px
∗
Ω as
Define L : W 1,px Ω → W0
Lu, ϕ |∇u|px−2 ∇u∇ϕdx gxfuxϕdx,
Ω
3.15
Ω
1,px
for every u ∈ W 1,px Ω, v ∈ W0
Ω.
1,px
Lemma 3.7 Comparison principle. Let u, v ∈ W 1,px Ω satisfy Lu − Lv ≥ 0 in W0
1,px
and consider ϕx min{ux − vx, 0}. If ϕx ∈ W0
in Ω.
Proof. See 18.
∗
Ω
Ω (i.e., u ≥ v on ∂Ω), then u ≥ v a.e.
Ionel Rovenţa
7
Remark 3.8. L defined below satisfy the condition from 18. Here, we use Lu ≥ Lv in the form
px−2
px−2
|∇v|
∇v∇ϕdx gxfvxϕdx ≤
|∇u|
∇u∇ϕdx gxfuxϕdx.
Ω
Ω
Ω
Ω
3.16
Given a real number b, let Λb be the class of all positive monotonic functions k ∈
L1 0, ν ∩ C1 0, ν that satisfy
t
lim
t→0
0
ks
ds b.
kt
3.17
We will let Λ stand for the union of all Λb as b ranges over 0, ∞. For k ∈ Λ, we use the
t
notation λt 0 ksds.
t
Remark 3.9. Observe that limt→0 0 ks/ktds 0 for any k ∈ Λ. Moreover, in 3.17,
0 ≤ b ≤ 1 if k is nondecreasing and b ≥ 1 if k is nonincreasing. This is true because
t
t
ks/ktds 1/kt 0 ksds tkξ/kt, for an ξ ∈ 0, t.
0
Let z be a C2 function on a domain Ω in Rn , f satisfies the Keller-Osserman condition
and v φz.
A direct computation show that
Δpx v px − 1|φ z|px−2 φ z|∇z|px |φ z|px−2 φ zΔpx z
|φ z|px−1 ln|φ z||∇z|px−2 ∇px∇zx.
3.18
Theorem 3.10. Suppose f ∈ RVσ1 , σ ≥ p# − 2 satisfies 2.14. Let Ω ⊆ Rn be a bounded C2
domain and let g ∈ CΩ be a nonnegative function such that
lim
gx
dx→0 kdxpx
A
3.19
for some positive constant A and some k ∈ Λb . Then, any local weak solution u of 2.13 satisfies
#
1/2σ−p# p b2 σ − p# ux
lim
.
dx→0 φλdx
A2 σ
3.20
Proof. Consider ρ > 0 and
Ωρ : {x ∈ Ω : 0 < dx < ρ}.
3.21
We know that Ω is a C2 bounded domain, so there exists positive constant μ,
depending only on Ω, such that
d ∈ C2 Ωμ ,
|∇d| ≡ 1 on Ωμ ,
3.22
where dx distx, ∂Ω. The existence of such a constant is given by the smoothness of the
domain Ω. For ρ ∈ Γ : 0, μ/2, let
Ω−ρ : Ωμ \ Ωρ ,
Ωρ : Ωμ−ρ .
3.23
8
Journal of Inequalities and Applications
The proof concerns two cases which discuss the monotonicity of the function k.
In the first case, we take k ∈ Λb for some b and k nonincreasing on 0, ν for some
ν > 0. Without loss of generality, we can consider ν > μ.
Let
z± x : λdx ± λρ,
for x ∈ Ω±ρ .
3.24
Some computations give us
|∇z± |px kpx d|∇d|px ,
Δpx z± px − 1kpx−2 dk d|∇d|px kpx−1 dΔpx d
3.25
kpx−1 d lnkd|∇d|px−2 ∇px∇dx.
Using these computations in 3.18 with v± φz± , we find that
Δpx v± px − 1|φ z± |px−2 φ z± |∇z± |px |φ z± |px−2 φ z± Δpx z±
|φ z± |px−1 ln|φ z± ||∇z± |px−2 ∇px∇z± x
px − 1kpx d|φ z± |px−2 φ z± |∇d|px
px − 1kpx−2 dk d|φ z± |px−2 φ z± |∇d|px
3.26
kpx−1 d|φ z± |px−2 φ z± Δpx d
kpx−1 d lnkd|∇d|px−2 ∇px∇dx|φ z± |px−2 φ z± kpx−1 d|∇d|px−2 ∇px∇dx|φ z± |px−1 ln|φ z± |.
Given 0 < < A/2, we define two numbers ϑ± by
ϑ± :
p# b2 σ − p# A ± 22 σ
1/2σ−p# ,
3.27
where A is the limit in 3.19.
Let w± x ϑ± φz± x, x ∈ Ω±ρ . For simplicity, let Lz : −Δpx z gxfz.
In order to apply the comparison principle, we have to prove that for ρ ∈ Γ and μ ≥ 0
sufficiently small, we have
Lw− ≥ 0 on Ω−ρ ,
Lw ≤ 0 on Ωρ .
3.28
From 3.26, recalling that px − 1 px/qx and |∇d| 1 on Ωμ , we find
Lw± gxfw± − Δpx w± Lw± ϑ± px−1 kpx dfφz± gxfϑ± φz± ±
±
±
±
x
D
x
D
x
D
x
,
×
D
2
3
1
4
ϑ± px−1 kpx dfφz± 3.29
Ionel Rovenţa
9
where
D1± x −
|φ z± |px−2 φ z± Δpx d,
kdfφz± D2± x −
px k d |φ z± |px−2 φ z± |∇d|px
,
qx k2 d
fφz± px |φ z± |px−2 φ z± ,
qx
fφz± |φ z± |px−2 φ z± ln kdϑ± φ z± ln φ z± ±
D4 x −
|∇d|px−2 ∇px∇dx.
kdf φ z±
3.30
D3± x −
Recalling that φ < 0 and z± x ≤ 2λdx, we infer
|D1± x| −
|φ z± |px−2 φ z± |Δpx d|
kdfφz± 3.31
px−2 ±
φ z λdx |φ z± |
≤ −2
|Δpx d|.
kdx z± xfφz± Since d ∈ C2 Ωμ , limdistx,∂Ω→0 px p# , by Lemma 3.5 and Remark 3.9 from below,
we obtain
lim
Ω±ρ ×Γdx,ρ→0 ,0 D1± x 0.
3.32
Using limdistx,∂Ω→0 px p# and Corollary 3.2, we have
q# σ 2 − p#
|φ z± |px−2 φ z± −
.
p# σ 2
Ω±ρ ×Γdx,ρ→0 ,0 z± xfφz± lim
3.33
With the same argument, we obtain
|D4± x| ≤ 2
px−2 ±
φ z λdx kdϑ± |φ z± |
ln ±
.
kdx
φ z z± xfφz± 3.34
Next, we prove
lim
Ω±ρ ×Γdx,ρ→0
λdx kdϑ±
ln ± 0.
|φ z |
,0 kdx
3.35
Recall that
1/p#
φ z −q# Fφz
#
where Az Fφz/φzσ2 1/p .
−q# 1/p#
Azφzσ2/p ,
#
3.36
10
Journal of Inequalities and Applications
If we integrate from 0 to t, we obtain
φt−α1
1/p#
−q# −α 1
t
3.37
Asds,
0
where α σ 2/p# > 1. Here, when x → 0, we have dx → ρ and z± x → 0.
Dividing by t and taking to the limit, we have
φt−α1
C,
t→0
t
3.38
lim
where C is a suitable positive constant.
From 3.36, it follows that limt→0 φt/tβ −C1 , where β 1/−α 1σ 2/p# <
0, and C1 is a positive suitable constant.
Now, it will be easy to infer that
kdϑ±
λdx kdϑ±
λdx
ln
lim
ln
|φ z± | Ω±ρ ×Γdx,ρ→0 ,0 kdx |λβ z± |
Ω±ρ ×Γdx,ρ→0 ,0 kdx
lim
lnkt/λβ t
lim
0.
t→0
kt/λt
3.39
Finally, limΩ±ρ ×Γdx,ρ→0 ,0 D4± x 0.
Since k is nonincreasing, we have k dz− ≥ k dλd and k dz ≤ k dλd.
Therefore,
D2− D3− −
px k d |φ z± |px−2 φ z± px |φ z− |px−2 φ z− −
qx k2 d
fφz± qx
fφz− px k dλd |φ z± |px−2 φ z± px |φ z− |px−2 φ z− −
−
D2 D3− ,
≥−
qx k2 d
z− fφz± qx
fφz− 3.40
D .
and similarly D2 D3 ≤ D
2
3
From 3.19, we see that corresponding to , there is 0 < δ < A/2 such that
A − kpx dx ≤ gx ≤ A k p dx,
0 < dx < δ .
3.41
Suppose that the constant μ in 3.22 is chosen such that μ < δ . Using 3.40 in the left
side of the inequality 3.29, we get
Lw− ≥ ϑ− px−1 px
k
dfφz− A − fϑ− φz− ϑ− px−1 fφz− −
−
−
x
D
x
D
x
.
D1− x D
3
4
2
3.42
Using the same method, we obtain
px−1 px
Lw ≤ ϑ k
dfφz A fϑ φz ϑ px−1 fφz D1 x
D2 x D3 x D4 x .
3.43
Ionel Rovenţa
11
From 3.2, we infer
lim
Ω±ρ ×Γdx,ρ→0 ,0 −
px |φ z± |px−2 φ z± −1.
qx
fφz± 3.44
By Lemma 3.5, we have
lim
Ω±ρ ×Γdx,ρ→0
p# b2 σ − p# ±
±
D
.
x
D
x
−
3
2
2σ
,0 3.45
Thus, as Ω±ρ × Γ dx, ρ → 0 , 0 , the expression in the bracket of 3.42 converges
to
#
A − ϑ− 2σ−p −
#
p b2 σ − p# p# b2 σ − p# A−
−1
> 0,
2σ
A − 2
2σ
3.46
and the expression in the bracket of 3.43 converges to
#
A ϑ 2σ−p −
#
p b2 σ − p# p# b2 σ − p# A
−1
< 0.
2σ
A 2
2σ
3.47
Therefore, for ρ ∈ Γ and μ ≥ 0 sufficiently small, we conclude that
Lw− ≥ 0 on Ω−ρ ,
Lw ≤ 0 on Ωρ .
3.48
Thus,
−Δp w− ≥ −gxfw− on Ω−ρ ,
−Δp w ≤ −gxfw on Ωρ .
3.49
Suppose now that k is nondecreasing. Given 0 < < A/2, from 3.19 we deduce
A − kpx dx − ρ ≤ A − kpx dx ≤ gx ≤ A k px dx
≤ A kpx dx ρ.
3.50
Consider
w± x ϑ± φλdx ± ρ ϑ± φy± for x ∈ Ω±ρ .
3.51
Here, y± x : λdx ± ρ. For simplicity, we will use d± for dx ± ρ. Similarly, we
obtain
Lw± ϑ± px−1 kpx d±fφy± gxfϑ± φy± ±
±
±
±
x
T
x
T
x
T
x
,
×
T
2
3
1
4
ϑ± px−1 kpx d±fφy± 3.52
12
Journal of Inequalities and Applications
where
T1± x −
|φ y± |px−2 φ y± Δpx d,
kd±fφy± T2± x −
px k d± |φ y± |px−2 φ y± |∇d|px
,
qx k2 d±
fφy± T3± x −
px |φ y± |px−2 φ y± ,
qx
fφy± T4± x −
|φ y± |px−2 φ y± lnkd±ϑ± |φ y± | ln|φ y± |
|∇d|px−2 ∇px∇dx.
kd±fφy± 3.53
By Lemma 3.5 and Remark 3.9, we obtain a similar inequality by replacing z± with y± :
lim
T1± x 0,
lim
T4± x 0,
Ω±ρ ×Γdx,ρ→0 ,0 Ω±ρ ×Γdx,ρ→0 ,0 lim
Ω±ρ ×Γdx,ρ→0 ,0 T2± x T3± x −
3.54
p# b2 σ − p# .
2σ
Now, suppose u is a nonnegative solution of 2.13. We note that
u ≤ w− Cu μ
on ∂Ω−ρ ,
3.55
where Cu μ : max{ux | dx ≥ μ}. Since φ and k are nonincreasing follows w x ≤
ϑ φλμ for dx μ − ρ. Therefore,
w ≤ u Cw μ on ∂Ωρ ,
3.56
where Cw μ : ϑ φλμ. Furthermore, u and w satisfy the hypothesis from comparison
principle on Ωρ for Gx, t −gxft. Moreover, since f is nondecreasing, w− Cu μ and
u Cw μ satisfy the hypothesis from comparison principle on Ω−ρ and Ωρ , respectively, for
Gx, t −gxft. Therefore by comparison principle, we infer
ux ≤ w− x Cu μ x ∈ ∂Ω−ρ ,
w x ≤ u Cw xμ x ∈ ∂Ωρ .
3.57
Hence for x ∈ Ωρ ∩ Ω−ρ , we have
ϑ −
Cw μ
Cu μ
ux
≤
≤ ϑ− ,
φλdx ± λρ φλdx ± λρ
φλdx ± λρ
Cw μ
Cu μ
ux
≤
≤ ϑ− .
ϑ −
φλdx ± ρ φλdx ± ρ
φλdx ± ρ
3.58
Ionel Rovenţa
13
Letting ρ → 0, we see that
ϑ −
Cw μ
Cu μ
ux
≤
≤ ϑ− ,
φλdx φλdx
φλdx
3.59
for all x ∈ Ωμ . On recalling that φt → ∞ as t → 0, we obtain
ϑ ≤ lim inf
dx→0
ux
ux
≤ lim sup
≤ ϑ− .
φλdx
φλdx
dx→0
3.60
The claimed result follows if we take → 0 . If we want to prove the uniqueness of
solutions for 2.13, we need additional condition on f; see 1. The proof is similar to the
proof from 1.
Acknowledgment
The author would like to express his gratitude to the referee and to the editor for their
detailed and helpful comments. The author has been partially supported by CNCSIS Grant
no. 589/2007.
References
1 A. Mohammed, “Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite
boundary values,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 480–489, 2007.
2 Q. Zhang, “Boundary blow-up solutions to px-Laplacian equations with exponential nonlinearities,” Journal of Inequalities and Applications, vol. 2008, Article ID 279306, 8 pages, 2008.
3 L. Bieberbach, “Δu eu und die automorphen Funktionen,” Mathematische Annalen, vol. 77, no. 2, pp.
173–212, 1916.
4 H. Rademacher, “Einige besondere probleme der partiellen Differentialgleichungen,” in Die
Differential und Integralgleichungen der Mechanik und Physik I, P. Frank and R. von Mises, Eds., pp.
838–845, Rosenborg, New York, NY, USA, 2nd edition, 1943.
5 J. B. Keller, “On solution of Δu fu,” Communications on Pure and Applied Mathematics, vol. 10, no.
4, pp. 503–510, 1957.
6 R. Osserman, “On the inequality Δu ≥ fu,” Pacific Journal of Mathematics, vol. 7, pp. 1641–1647, 1957.
7 C. Bandle and M. Marcus, “‘Large’ solutions of semilinear elliptic equations: existence, uniqueness
and asymptotic behaviour,” Journal d’Analyse Mathématique, vol. 58, no. 1, pp. 9–24, 1992.
8 F.-C. Cı̂rstea and V. Rădulescu, “Uniqueness of the blow-up boundary solution of logistic equations
with absorbtion,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 335, no. 5, pp. 447–
452, 2002.
9 F.-C. Cı̂rstea and V. Rădulescu, “Nonlinear problems with boundary blow-up: a Karamata regular
variation theory approach,” Asymptotic Analysis, vol. 46, no. 3-4, pp. 275–298, 2006.
10 F.-C. Cı̂rstea and V. Rădulescu, “Boundary blow-up in nonlinear elliptic equations of Bieberbach—
Rademacher type,” Transactions of the American Mathematical Society, vol. 359, no. 7, pp. 3275–3286,
2007.
11 Q. Zhang, “Existence and asymptotic behavior of positive solutions to px-Laplacian equations with
singular nonlinearities,” Journal of Inequalities and Applications, vol. 2007, Article ID 19349, 9 pages,
2007.
12 J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin,
Germany, 1983.
13 O. Kováčik and J. Rákosnı́k, “On spaces Lpx and W 1,px ,” Czechoslovak Mathematical Journal, vol.
41116, no. 4, pp. 592–618, 1991.
14 X. Fan and X. Han, “Existence and multiplicity of solutions for px-Laplacian equations in RN ,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 1-2, pp. 173–188, 2004.
15 X. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces W k,px Ω,” Journal of
Mathematical Analysis and Applications, vol. 262, no. 2, pp. 749–760, 2001.
14
Journal of Inequalities and Applications
16 X. Fan and D. Zhao, “On the spaces Lpx Ω and W m,px Ω,” Journal of Mathematical Analysis and
Applications, vol. 263, no. 2, pp. 424–446, 2001.
17 F. Gladiali and G. Porru, “Estimates for explosive solutions to p-Laplace equations,” in Progress
in Partial Differential Equations, Vol. 1 (Pont-à-Mousson, 1997), vol. 383 of Pitman Research Notes in
Mathematics Series, pp. 117–127, Longman, Harlow, UK, 1998.
18 Q. Zhang, “A strong maximum principle for differential equations with nonstandard px-growth
conditions,” Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 24–32, 2005.