Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 971268, 20 pages
doi:10.1155/2010/971268
Research Article
Existence and Asymptotic Behavior of Boundary
Blow-Up Solutions for Weighted px-Laplacian
Equations with Exponential Nonlinearities
Li Yin,1 Yunrui Guo,2 Jing Yang,1 Bibo Lu,3 and Qihu Zhang1, 4
1
Department of Mathematics and Information Science, Zhengzhou University of Light Industry,
Zhengzhou, Henan 450002, China
2
Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China
3
School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, Henan 454000, China
4
School of Mathematics and Statistics, Huazhong Normal University, Wuhan, Hubei 430079, China
Correspondence should be addressed to Qihu Zhang, zhangqh1999@yahoo.com.cn
Received 21 March 2010; Accepted 3 October 2010
Academic Editor: Pavel Drábek
Copyright q 2010 Li Yin et al. 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.
This paper investigates the following px-Laplacian equations with exponential nonlinearities:
−Δpx u ρxefx,u 0 in Ω, ux → ∞ as dx, ∂Ω → 0, where −Δpx u −div|∇u|px−2 ∇u
is called px-Laplacian, ρx ∈ CΩ. The asymptotic behavior of boundary blow-up solutions is
discussed, and the existence of boundary blow-up solutions is given.
1. Introduction
The study of differential equations and variational problems with nonstandard px-growth
conditions is a new and interesting topic. On the background of this class of problems, we
refer to 1–3
. Many results have been obtained on this kind of problems, for example, 4–18
.
On the regularity of weak solutions for differential equations with nonstandard px-growth
conditions, we refer to 4, 5, 8
. On the existence of solutions for px-Laplacian equation
Dirichlet problems in bounded domain, we refer to 7, 9, 15, 18
. In this paper, we consider
the following px-Laplacian equations with exponential nonlinearities
−Δpx u ρxefx,u 0,
ux −→ ∞,
in Ω,
as dx, ∂Ω −→ 0,
P
2
Abstract and Applied Analysis
where −Δpx u − div|∇u|px−2 ∇u and Ω B0, R ⊂ RN is a bounded radial domain
B0, R {x ∈ RN | |x| < R}. Our aim is to give the asymptotic behavior and the existence
of boundary blow-up solutions for problem P.
Throughout the paper, we assume that px, ρx, and fx, u satisfy the following.
H1 px ∈ C1 Ω is radial and satisfies
1 < p− ≤ p < ∞,
x ∈ Ω.
where p− inf px,
p sup px.
Ω
Ω
1.1
H2 fx, u is radial with respect to x, fx, · is increasing, and fx, 0 0 for any
H3 f : Ω × R → R is continuous and satisfies
fx, t ≤ C1 C2 |t|γx ,
∀x, t ∈ Ω × R,
1.2
where C1 , C2 are positive constants and 0 ≤ γ ∈ CΩ.
H4 ρx ∈ CΩ is a radial nonnegative function, and there exists a constant σ ∈
R/2, R such that
ρ0 R − r−βr ≤ ρr ≤ ρ1 R − r−β1 r
for r ∈ σ, R uniformly,
1.3
where ρ0 and ρ1 are positive constants and βr and β1 r are Lipschitz continuous on σ, R
,
which satisfy βr ≤ β1 r < pr for any r ∈ σ, R
.
The operator −Δpx u − div|∇u|px−2 ∇u is called px-Laplacian. Specifically,
if px ≡ p a constant, P is the well-known p-Laplacian problem. If fx, u can be
represented as hxfu, on the boundary blow-up solutions for the following p-Laplacian
equations p is a constant:
−Δp u hxfu 0,
in Ω,
1.4
we refer to 19–26
, and the following generalized Keller-Osserman condition is crucial
∞
1
1
Ft
dt < ∞,
1/p
where Ft t
fsds,
1.5
0
but the typical form of px-Laplacian equation is
−Δpx u |u|qx−2 u 0,
in Ω,
1.6
and there are some differences between the results of 1.4 and 1.6 see 16
.
On the boundary blow-up solutions for the following p-Laplacian equations with
exponential nonlinearities p is a constant:
−Δp u ehxfu 0,
in Ω,
1.7
Abstract and Applied Analysis
3
we refer to 20–22
, but the results on the boundary blow-up solutions for px-Laplacian
equations are rare see 16
.
In 16
, the present author discussed the existence and asymptotic behavior of
boundary blow-up solutions for the following px-Laplacian equations:
−Δpx u fx, u 0,
ux −→ ∞,
in Ω,
as dx, ∂Ω −→ 0,
1.8
on the condition that fx, · satisfies polynomial growth condition.
If px is a function, the typical form of P is the following:
qx−2
−Δpx u ρxe|u|
u
0,
1.9
and the method to construct subsolution and supersolution in 16
cannot give the exact
asymptotic behavior of solutions for P. Our results partially generalized the results of 20–
22
.
Because of the nonhomogeneity of px-Laplacian, px-Laplacian problems are more
complicated than those of p-Laplacian ones see 10
; another difficulty of this paper is that
fx, u cannot be represented as hxfu.
2. Preliminary
In order to deal with px-Laplacian problems, we need some theories on the spaces Lpx Ω,
W 1,px Ω and properties of px-Laplacian, which we will use later see 6, 11
. Let
Lpx Ω u | u is a measurable real-valued function, Ω |ux|px dx < ∞ .
2.1
We can introduce the norm on Lpx Ω by
|u|px
ux px
inf λ > 0 |
λ dx ≤ 1 .
Ω
2.2
The space Lpx Ω, | · |px becomes a Banach space. We call it generalized Lebesgue
space. The space Lpx Ω, | · |px is a separable, reflexive, and uniform convex Banach space
see 6, Theorems 1.10, 1.14
.
The space W 1,px Ω is defined by
W 1,px Ω u ∈ Lpx Ω | |∇u| ∈ Lpx Ω ,
2.3
4
Abstract and Applied Analysis
and it can be equipped with the norm
u
|u|px |∇u|px ,
∀u ∈ W 1,px Ω.
2.4
1,px
1,px
Ω is the closure of C0∞ Ω in W 1,px Ω. W 1,px Ω and W0
Ω are
W0
separable, reflexive, and uniform convex Banach spaces see 6, Theorem 2.1
.
1,px
If u ∈ Wloc Ω ∩ CΩ, u is called a blow-up solution of P when it satisfies
|∇u|px−2 ∇u∇q dx Q
ρxfx, uq dx 0,
Q
1,px
for any domain Q Ω, and maxk − u, 0 ∈ W0
define
Ω
|∇u|px−2 ∇u∇ϕ ρxefx,u ϕ dx,
Q,
1,px
Q Ω such that u ∈ W0
1,px
∀u ∈ Wloc
Q}, and
1,px
Ω ∩ CΩ, ∀ϕ ∈ W0,loc Ω.
2.6
1,px
Lemma 2.1 see 9, Theorem 3.1
. Let h ∈ W 1,px Ω ∩ CΩ, and X h W0
Then, A : X →
2.5
Ω for every positive integer k.
1,px
Let W0,loc Ω {u | there is an open domain
1,px
1,px
A : Wloc Ω ∩ CΩ → W0,loc Ω∗ as
Au, ϕ 1,px
∀q ∈ W0
1,px
W0,loc Ω∗
Ω ∩ CΩ.
is strictly monotone.
Letting g ∈ W0,loc Ω∗ , if g, ϕ ≥ 0,for all ϕ ∈ W0,loc Ω with ϕ ≥ 0 a.e. in Ω, then
1,px
1,px
denote g ≥ 0 in W0,loc Ω∗ ; correspondingly, if −g ≥ 0 in W0,loc Ω∗ , then denote g ≤ 0
1,px
1,px
in W0,loc Ω∗ .
1,px
Definition 2.2. Let u ∈ Wloc Ω ∩ CΩ. If Au ≥ 0 Au ≤ 0 in W0,loc Ω∗ , then u is called
a weak supersolution weak subsolution of P.
1,px
1,px
Copying the proof of 14
, we have the following.
1,px
Lemma 2.3 comparison principle. Let u, v ∈ Wloc
Au − Av ≥ 0,
∗
1,px
in W0
Ω .
1,px
Let ϕx min{ux − vx, 0}. If ϕx ∈ W0
Ω ∩ CΩ satisfy
2.7
Ω (i.e., u ≥ v on ∂Ω), then u ≥ v a.e. in Ω.
Lemma 2.4 see 8, Theorem 1.1
. Under the conditions (H1 ) and (H3 ), if u ∈ W 1,px Ω is a
1,ϑ
bounded weak solution of −Δpx u ρxefx,u 0 in Ω, then u ∈ Cloc
Ω, where ϑ ∈ 0, 1 is a
constant.
Abstract and Applied Analysis
5
3. Asymptotic Behavior of Boundary Blow-Up Solutions
If u is a radial solution for P, then P can be transformed into
pr−2 u r N−1 ρrefr,u ,
r N−1 u u0 u0 ,
u 0 0,
u r ≥ 0,
r ∈ 0, R,
3.1
for 0 < r < R.
It means that ur is increasing.
Theorem 3.1. If fr, u satisfies
fr, u ≥ αus
as u −→ ∞ for r ∈ σ, R uniformly,
3.2
where σ is defined in (H4 ) and α and s are positive constants, then there exists a supersolution Φ1 x
which satisfies Φ1 x → ∞ (as dx, ∂Ω → 0), such that for every solution u of problem P, one
has ux ≤ Φ1 x.
Proof. Define the function gr, a, λ on 0, Rλ as
⎧
1/s
⎪
1
⎪
⎪
k, R0 ≤ r < Rλ ,
a ln
⎪
⎪
⎪
⎪
R − r1−θ − λ
⎪
⎪
⎪
⎪
⎡
⎤
⎪
⎪
1/s−1 pR0 −1/pt−1
⎪
R0
⎪
−θ
1/s
⎪
⎪
1
⎢ a 1 − θR − R0 ⎥
⎪k −
⎪
ln
⎣ ⎦
⎪
⎪
1−θ
1−θ
⎪
R − R0 − λ
r
⎪
s R − R0 − λ
⎪
⎪
⎪
⎪
⎪
⎪
1/pt−1
⎪
N−1
⎪
⎪
R
0
⎪
⎪
×
sin εt − σ
dt
⎪
⎪
⎪
tN−1
⎪
⎪
⎪
⎪
⎪
1/s
⎪
⎨
1
gr, a, λ a ln
, σ < r < R0 ,
⎪
⎪
R − R0 1−θ − λ
⎪
⎪
⎪
⎪
⎪
⎡
⎤
⎪
⎪
1/s−1 pR0 −1/pt−1
⎪
R0
−θ
⎪
1/s
⎪
1
a
−
θR
−
R
1
⎪
⎢
⎥
0
⎪
ln
k−
⎪
⎣ ⎦
⎪
1−θ
1−θ
⎪
⎪
−
λ
−
R
R
σ
s
−
λ
−
R
0
R
⎪
0
⎪
⎪
⎪
⎪
⎪
1/pt−1
⎪
⎪
⎪
R0 N−1
⎪
⎪
⎪
×
sin εt − σ
dt
⎪
⎪
tN−1
⎪
⎪
⎪
⎪
⎪
⎪
1/s
⎪
⎪
⎪
1
⎪
⎪
, r ≤ σ,
⎩ a ln
R − R0 1−θ − λ
3.3
6
Abstract and Applied Analysis
where θ < βR/pR, a > 1/αsup|x|≥R0 px are constants, R0 ∈ σ, R, R − R0 is small
enough, parameter λ ∈ 0, R − R0 1−θ /2
, Rλ satisfies R − Rλ 1−θ − λ 0, ε π/2R0 − σ
1/s
2p 1 s/s 1/1 − θ β /1 − θ
2
k
ln
α
R − R0 1−θ
R0
σ
⎡
1/s
⎣ 2a 1 − θ
sR − R0 ln
2
R − R0 1−θ
R0 N−1
×
sin εt − σ
tN−1
1/s−1 ⎤pR0 −1/pt−1
⎦
3.4
1/pt−1
dt.
Obviously, for any positive constant a, we have gr, a, λ ∈ C1 0, Rλ .
When R0 < r < Rλ < R, we have
1/s−1
1
a1/s
1 − θR − r−θ
g g r, a, λ ln
,
s
R − r1−θ − λ
R − r1−θ − λ
1/s−1pr−1
pr−1 pr−2 1
R − r−θpr−1
1 − θa1/s
g ln
g pr−1 ,
s
R − r1−θ − λ
R − r1−θ − λ
r
g
N−1 pr−2 g
r
N−1
1 − θa1/s
s
pr−1 ln
1
1/s−1pr−1
R − r1−θ − λ
pr − 1 R − r−θpr
× pr 1 − θ Πr
,
R − r1−θ − λ
3.5
where
pr−1 r N−1 1 − θa1/s /s
R − r1−θ − λ
1/s − 11 − θ
Πr R − r N−1 1−θ
pr−1
R − r
pr − 1 r
1 − θa1/s /s
ln 1/ R − r1−θ − λ
R − r1−θ − λ
R − r1−θ
1/s − 1p r
1
ln ln
R − r pr − 1
R − r1−θ − λ
θp r R − r1−θ − λ
1
R − r ln
1−θ
− r
R
pr − 1
R − r
θ
R − r1−θ − λ
R − r1−θ
−p r
R − r1−θ − λ 1−θ
ln
−
λ
.
−
r
R − r
R
pr − 1
R − r1−θ
3.6
Abstract and Applied Analysis
7
If R − R0 is small enough, it is easy to see that
|Πr| ≤ ln
R − R0 1−θ
uniformly,
for λ ∈ 0,
2
1
R − r1−θ − λ
,
3.7
and then
1/s−1pr−11
pr−1 1/s
1
N−1 pr−2 N−1 1 − θa
r
g
g ≤r
ln
s
R − r1−θ − λ
pr − 1 R − r−θpr
× pr ,
R − r1−θ − λ
3.8
∀r ∈ R0 , Rλ .
Thus, when 0 < R − R0 is small enough, from 3.5 and 3.8, for λ ∈ 0, R − R0 1−θ /2
uniformly, we have
pr−2 r N−1 g g
≤ 2r
N−1
1 − θa1/s
s
≤r
N−1
ρr
pr−1 ln
R − r
1/s−1pr−11 pr − 1 R − r−θpr
pr
R − r1−θ − λ
R − r1−θ − λ
αa
1
1−θ
1
s
−λ
r N−1 ρreαg ≤ r N−1 ρrefr,g ,
∀r ∈ R0 , Rλ .
3.9
Thus, when 0 < R − R0 is small enough, the following inequality is valid for λ ∈
0, R − R0 1−θ /2
uniformly:
pr−2 r N−1 g g ≤ r N−1 ρrf r, g ,
∀r ∈ R0 , Rλ .
3.10
8
Abstract and Applied Analysis
Obviously, if R − R0 is small enough, then g ≥ 2p s 1/s 1/1 − θ |β| /1 −
θ/α ln2/R − R0 1−θ 1/s is large enough. Since λ ∈ 0, R − R0 1−θ /2
,
pr−2 r N−1 g g
⎡
−θ
⎢ a 1 − θR − R0 εR0 N−1 ⎣ s R − R0 1−θ − λ
1/s
⎡
a1/s 1 − θR − R0 −θ
≤ εR0 N−1 ⎣
s1/2R − R0 1−θ
⎡
2a1/s 1 − θ
≤ εR0 N−1 ⎣
sR − R0 N−1
≤ εR0 ln
1
R − R0 ln
2
R − R0 1−θ
1/s−1
1−θ
2
R − R0 1−θ
−λ
⎤pR0 −1
⎥
⎦
cosεr − σ
1/s1 ⎤pR0 −1
⎦
1/s1 ⎤pR0 −1
⎦
!
"s1/s1−θ1 p
2a1/s 1 − θ
2
s
R − R0
s
≤ r N−1 ρreαg ≤ r N−1 ρrefr,g ,
σ < r < R0 .
3.11
Thus,
pr−2 r N−1 g g ≤ r N−1 ρrefr,g ,
σ < r < R0 .
3.12
Obviously,
pr−2 r N−1 g g 0 ≤ r N−1 ρrefr,g ,
0 ≤ r < σ.
3.13
Since gx, a, λ g|x|, a, λ is a C1 function on B0, Rλ , if 0 < R − R0 is small enough
R0 depends on R, p, s, α, from 3.10, 3.12, and 3.13, for any λ ∈ 0, R−R0 1−θ /2
, we can
see that g|x|, a, λ is a supersolution for P on B0, Rλ , and then g|x|, a, 0 is a supersolution
for P.
Defining the function gm |x|, a − gr, a − , 1/m on 0, R1/m , where a − >
1/αsup|x|≥R0 px , then gm |x|, a − is a supersolution for P on B0, R − 1/m. If u is a
solution for P, according to the comparison principle, we get that gm |x|, a − ≥ ux for
any x ∈ B0, R1/m . For any x ∈ B0, R \ B0, R0 , we have gm |x|, a − ≥ gm1 |x|, a − ,
when m is large enough. Thus
ux ≤ lim gm |x|, a − ,
m−→∞
∀x ∈ B0, R \ B0, R0 .
3.14
Abstract and Applied Analysis
9
When dx, ∂Ω > 0 is small enough, we have
lim gm |x|, a − <
a ln
m−→∞
1/s
1
k ≤ g|x|, a, 0.
R − r1−θ
3.15
According to the comparison principle, we get that g|x|, a, 0 ≥ ux, for all x ∈
B0, R; then Φ1 x Φ1 |x| g|x|, a, 0 is a radial upper control function of all of
the solutions for P, and Φ1 x Φ1 |x| is a radial supersolution for P. The proof is
completed.
Theorem 3.2. If fr, u satisfies
fr, u −→ −∞ as u −→ −∞ for r ∈ σ, R uniformly,
fr, u ≤ δus
3.16
as u −→ ∞ for r ∈ σ, R uniformly,
where σ is defined in (H4 ) and δ and s are positive constants, then there exists a subsolution Φ2 x
which satisfies Φ2 x → ∞ (as dx, ∂Ω → 0), such that for every solution ux for problem P,
one has ux ≥ Φ2 x.
Proof. We will prove this theorem in the following two cases.
i β1 R > 0.
ii β1 R ≤ 0.
Case 1 β1 R > 0. Let z1 be a radial solution of
−Δpx z1 x −μ,
where μ > 2maxr∈0,R0 ρr 12p
z1 |x|. Then, z1 satisfies
in Ω1 B0, σ,
−1/p− −1
z1 0,
3.17
on ∂Ω1 ,
is a positive constant. We denote z1 z1 r pr−2 − r N−1 z1 z1 −r N−1 μ, z1 σ 0,
z1 0 0,
σ 1/pr−1
rμ 1/pr−1
rμ ,z1 −
dr.
z1 N
r N
3.18
Denote hb r, λ on σ, R0 as
hb r, λ R0 r
⎡
N−1
R0 tN−1
−θ
t − σ ⎢ b1 − θR − R0 ⎣ R0 − σ s R − R 1−θ λ
0
b ln
1/s−1
1
R − R0 1−θ
λ
⎤pR0 −1
⎥
⎦
1/pt−1
σN−1 R0 − t σμ N−1
R0 − σ N
t
dt.
3.19
10
Abstract and Applied Analysis
It is easy to see that
σμ 1/pσ−1
−hb σ, λ z1 σ ,
N
1/s−1
1
b1 − θR − R0 −θ
−hb R0 , λ .
b ln
R − R0 1−θ λ
s R − R0 1−θ λ
3.20
Define the function vr, b, λ on 0, R as
⎧
1/s
⎪
⎪
1
⎪
⎪
− k∗ ,
b ln
⎪
⎪
1−θ
⎪
λ
−
r
R
⎪
⎪
⎪
⎪
⎪
1/s
⎪
⎨
1
vr, b, λ b ln
− k∗ − hb r, λ,
1−θ
⎪
⎪
R − R0 λ
⎪
⎪
⎪
⎪
⎪ 1/s
⎪
σ
⎪
⎪
1
rμ 1/pr−1
⎪
⎪
dr b ln
− k∗ − hb σ, λ,
⎪
⎩− N R − R0 1−θ λ
r
R0 ≤ r < R,
σ < r < R0 ,
r ≤ σ,
3.21
where θ ∈ β1 R/pR, 1, b ∈ 0, 1/δinf|x|≥R0 px are constants, R0 ∈ σ, R,R − R0 is small
enough, parameter λ ∈ 0, R − R0 1−θ /2
, and
∗
k M
b ln
1
R − R0 1−θ
1/s
,
3.22
where M satisfies
σN−1
1
≥ r N−1 ρrefr,y ,
R0 − σ
∀y ≤ −M, ∀r ∈ 0, R0 .
3.23
Obviously, for any positive constant b, vr, b, λ ∈ C1 0, R.
By computation, when r ∈ R0 , R, we have
1/s−1
1
b1/s
1 − θR − r−θ
ln
,
v v r, b, λ s
R − r1−θ λ
R − r1−θ λ
1/s−1pr−1
pr−1 1/s
pr−2 1
−
θb
R − r−θpr−1
1
v ln
v pr−1 ,
s
R − r1−θ λ
R − r1−θ λ
Abstract and Applied Analysis
r
v
N−1 pr−2 v
11
1/s−1pr−1
pr−1 1
1 − θb1/s
ln
r
s
R − r1−θ λ
pr − 1 R − r−θpr−1−1
×
θ Λr,
pr−1
R − r1−θ λ
N−1
3.24
where
pr−1 r N−1 1 − θb1/s /s
1/s − 11 − θ
Λr R − r N−1 pr−1
1/s
pr − 1 r
1 − θb /s
ln 1/ R − r1−θ λ
R − r1−θ λ
θp r
1/s − 1p r
1
1
1−θ
×R − r R − r ln ln
R − r ln
1−θ
R − r
pr − 1
pr − 1
R − r λ
−p r
1 − θ
R − r1−θ R − r ln R − r1−θ λ .
pr − 1
R − r1−θ λ
3.25
By computation, when R − R0 is small enough, for λ ∈ 0, R − R0 1−θ /2
uniformly, we
have
pr−2 v
r N−1 v ≥r
N−1
1 − θb1/s
s
pr−1 ln
1
1/s−1pr−1
R − r1−θ λ
!
"
pr − 1 R − r−θpr−1−1
1
×
θ 1−
pr−1
2
R − r1−θ λ
1/s−1pr−1
pr−1 1
θ N−1 1 − θb1/s
ln
≥ r
2
s
R − r1−θ λ
pr − 1 R − r−θpr−1−1
×
R − r1−θ
pr
1−θ
R − r λ
1/s−1pr−1 pr−1 pr − 1 R − r−θpr
1
θ N−1 1 − θb1/s
ln
≥ r
pr
2
s
R − r1−θ λ
R − r1−θ λ
≥ r N−1 ρ1 R − r−β1 r eδv
≥ r N−1 ρrefr,v ,
s
∀r ∈ R0 , R.
3.26
12
Abstract and Applied Analysis
Then, for λ ∈ 0, R − R0 1−θ /2
uniformly, we have
pr−2 r N−1 v v ≥ r N−1 ρrefr,v ,
∀r ∈ R0 , R.
3.27
When R − R0 is small enough, for all r ∈ σ, R0 , since v ≤ −M, it is easy to see that
pr−2 N−1 pr−2 h r N−1 v v ≥ r
h
⎡
1 ⎢ b1 − θR − R0 −θ
R0 N−1
⎣ R0 − σ s R − R 1−θ λ
0
− σN−1
≥ σN−1
b ln
1
1/s−1
R − R0 1−θ λ
⎤pR0 −1
⎥
⎦
1 σμ R0 − σ N
1
R0 − σ
≥ r N−1 ρrefr,v ,
3.28
Then,
pr−2 r N−1 v v ≥ r N−1 ρrefr,v ,
∀r ∈ σ, R0 .
3.29
Obviously,
pr−2 v r N−1 μ ≥ r N−1 ρrefr,v ,
r N−1 v ∀r ∈ 0, σ.
3.30
Combining 3.27, 3.29, and 3.30, when R − R0 is large enough, for any λ ∈ 0, R −
R0 1−θ /2
, one can see that vr, a, λ is a subsolution for P.
Define the function vm r, b on B0, R as
!
"
1
vm r, b vm r, b ,
,
m
3.31
where is a small enough positive constant such that b < 1/δinf|x|≥R0 px.
For any m 1, 2, . . ., we can see that vm r, b ∈ C1 0, R is a subsolution for P
on BR0 , R. According to the comparison principle, we get that vm r, b ≤ ux for any
x ∈ B0, R. For any x ∈ B0, R \ B0, R0 , we have vm |x|, b ≤ vm1 |x|, b . Thus
ux ≥ lim vm |x|, b ,
m−→∞
∀x ∈ B0, R \ B0, R0 .
When dx, ∂Ω is small enough, we have lim vm |x|, b > v|x|, b, 0.
m−→∞
3.32
Abstract and Applied Analysis
13
According to the comparison principle, we get that v|x|, b, 0 ≤ ux, ∀ x ∈ B0, R;
then Φ2 x Φ2 |x| v|x|, b, 0 is a radial lower control function of all of the solutions for
P, and Φ2 x is a radial subsolution for P.
Case 2 β1 R ≤ 0. Let μ > 2maxr∈0,R0 ρr 12p
b r, λ on σ, R0 as
b r, λ R0 r
−1/p− −1
be a positive constant. Denote
#
1/s−1 $pR0 −1
b
R0 N−1 t − σ
−1
b ln R λ − R0 tN−1 R0 − σ sR λ − R0 3.33
1/pt−1
σN−1 R0 − t σμ N−1
dt.
R0 − σ N
t
It is easy to see that
σμ 1/pσ−1
−b σ, λ z1 σ ,
N
−b R0 , λ 1/s−1
b
b ln R λ − R0 −1
.
sR λ − R0 3.34
Define the function ηr, b, λ on B0, R as
⎧
1/s
⎪
⎪
− k∗ ,
b ln R λ − r−1
⎪
⎪
⎪
⎪
⎪
⎪
1/s
⎨
b ln R λ − R0 −1
− k∗ − b r, λ,
ηr, b, λ ⎪
⎪
⎪
σ 1/pr−1
⎪
⎪
1/s
⎪
rμ ⎪
⎪
−
dr b ln R λ − R0 −1
− k∗ − b σ, λ,
⎩
r N
R0 ≤ r < R,
σ < r < R0 ,
r ≤ σ,
3.35
where b ∈ 0, 1/δinf|x|≥R0 px − β1 x
is a constant, R0 ∈ σ, R,R − R0 is small enough,
parameter λ ∈ 0, R − R0 /2
, and
!
k∗ M b ln
1
R − R0
"1/s
,
where M is defined in 3.23.
Obviously, for any positive constant b, ηr, b, λ ∈ C1 0, R.
3.36
14
Abstract and Applied Analysis
Similar to the proof of Case 1, when R − R0 is small enough, we have
pr−2 r N−1 η η
≥r
N−1
b1/s
s
pr−1
≥ r N−1 ρrefr,η ,
"
1/s−1pr−1 !
1
1−
pr − 1 R λ − r−pr ln R λ − r−1
2
∀r ∈ R0 , R.
3.37
When R − R0 is small enough, for all r ∈ σ, R0 , from the definition of k∗ , it is easy to
see that
pr−2 r N−1 η η ≥ σN−1
1
≥ r N−1 ρrefr,η .
R0 − σ
3.38
Obviously
pr−2 r N−1 η η r N−1 μ ≥ r N−1 ρrefr,η ,
∀r ∈ 0, σ.
3.39
Combining 3.37, 3.38, and 3.39, when R − R0 is large enough, for any λ ∈ 0, R −
R0 /2
, one can see that ηr, a, λ is a subsolution for P.
Define the function ηm r, b ε on B0, R as
"
!
1
,
ηm r, b ε η r, b ε,
m
3.40
where ε is a small enough positive constant such that b ε < 1/δinf|x|≥R0 px .
We can see that ηm r, b ε ∈ C1 0, R is a subsolution for P for any m 1, 2 . . ..
According to the comparison principle, we get that ηm r, b ε ≤ ux for any x ∈ B0, R. For
any x ∈ B0, R \ B0, R0 , we have ηm |x|, b ε ≤ ηm1 |x|, b ε. Then,
ux ≥ lim ηm |x|, b ε,
m → ∞
∀x ∈ B0, R \ B0, R0 .
3.41
When dx, ∂Ω is small enough, we have
lim ηm |x|, b ε > η|x|, b, 0.
m−→∞
3.42
According to the comparison principle, we get that η|x|, b, 0 ≤ ux, ∀ x ∈ B0, R;
then Φ2 x Φ2 |x| η|x|, b, 0 is a radial lower control function of all of the solutions for
P, and Φ2 x Φ2 |x| is a radial subsolution for P.
Abstract and Applied Analysis
15
Theorem 3.3. If fr, u satisfies
fr, u
δ
u−→∞
us
lim
as u −→ ∞ for r ∈ σ, R uniformly,
3.43
where σ is defined in (H4 ), δ and s are positive constants, ρr ρ0 R − r−βr , where βR < pR,
then each solution ux for P satisfies
lim |x|−→R
pR/δ
ux
ln 1/R − |x|
1−θ
1/s 1,
where θ βR
.
pR
3.44
Proof. It is easy to be seen from Theorems 3.1 and 3.2
4. The Existence of Boundary Blow-Up Solutions
Theorem 4.1. If infx∈Ω px > N and fr, u satisfies
fr, u ≥ aus
as u → ∞ for r ∈ σ, R uniformly,
4.1
where σ is defined in (H4 ), a and s are positive constants, then P possesses a boundary blow-up
solution.
Proof. In order to deal with the existence of boundary blow-up solutions, let us consider the
problem
−Δpx u ρrefx,u 0,
ux c,
in Ω0 ,
for x ∈ ∂Ω0 ,
4.2
where c is a positive constant and Ω0 Ω is a radial subdomain of Ω. Since infx∈Ω px > N,
then W1,px Ω0 → Cα Ω0 , where α ∈ 0, 1. The relative functional of 4.2 is
ϕ
Ω0
1
|∇ux|px dx px
Ω0
Fx, udx,
4.3
u
1,px
Ω0 , then ϕ possesses a
where Fx, u 0 efx,t dt. Since ϕ is coercive in X : c W0
nontrivial minimum point u. So, problem 4.2 possesses a weak solution u.
Since aus ≤ fr, u ≤ C1 C2 |u|γx , from Theorems 3.1 and 3.2, we get that P possesses
a supersolution g ∗ x and a subsolution g∗ x, which satisfy g ∗ x ≥ g∗ x, when dx, ∂Ω
the distance from x to ∂Ω is small enough. According to the comparison principle, we get
that g ∗ x ≥ g∗ x for any x ∈ Ω.
16
Abstract and Applied Analysis
Denote Dj {x | |x| < 1 − 1/j 1R} j 1, 2, . . .. Let us consider the problem
−Δpx uj ρxefx,uj 0,
uj x g∗ x,
and the relative functional is
ϕ
Dj
in Dj ,
4.4
for x ∈ ∂Dj ,
px
1 ∇uj x dx px
ρxF x, uj dx.
4.5
Dj
1,px
Let g∗j x g∗ x|Dj . Since the functional ϕ is coercive in Xj g∗j x W0
Dj ,
then ϕ has a nontrivial minimum point uj . Therefore, problem 4.4 has a weak solution uj .
According to the comparison principle, we get that g∗ x ≤ uj x for any x ∈ Dj
j 1, 2, . . . . Since uj x g∗ x for any x ∈ ∂Dj , then uj x ≤ uj1 x for any x ∈ ∂Dj
j 1, 2, . . .. According to the comparison principle, we get that uj x ≤ uj1 x for any
x ∈ Dj j 1, 2, . . ..
Since g ∗ x is a supersolution and g ∗ x ≥ g∗ x for any x ∈ Ω, so we have uj x g∗ x ≤ g ∗ x for any x ∈ ∂Dj j 1, 2, . . .. According to the comparison principle, we get
that uj x ≤ g ∗ x for any x ∈ Dj j 1, 2, . . ..
Since g ∗ x and g∗ x are locally bounded, from Lemma 2.4, each weak solution of
1,α
function. The C1,α interior regularity result implies that the sequences {uj } and
4.4 is a Cloc
{∇uj } are equicontinuous in D2 , and hence we can choose a subsequence, which we denoted
by {u1j }, such that u1j → w1 and ∇u1j → 1 uniformly on D1 for some w1 ∈ CD1 and
1 ∈ CD1 N . In fact, 1 ∇w1 on D1 , and from the interior C1,α estimate, we conclude that
∇w1 ∈ Cα D1 N for some 0 < α < 1. Thus, w1 ∈ W 1,px D1 ∩C1,α D1 . From the C1,α interior
regularity result, we see that |∇uj |p−1 |∇ϕ| ≤ C|∇ϕ| on D1 , and since the function ξ → |ξ|p−2 ξ
is continuous on RN , it follows that |∇u1j x|p−2 ∇u1j x · ∇ϕx → |∇w1 x|p−2 ∇w1 x · ∇ϕx
for x ∈ D1 . Thus, by the dominated convergence theorem, we have
D1
1 p−2 1
∇uj x ∇uj x · ∇ϕxdx −→
|∇w1 x|p−2 ∇w1 x · ∇ϕxdx,
D1
1,px
∀ϕ ∈ W0
D1 .
4.6
Furthermore, since 0 ≤ fu1j ≤ fu1j1 and fu1j x → fw1 x for each x ∈ D1 , by
the monotone convergence theorem, we obtain
ρe
fu1j ρefw1 q dx,
q dx −→
D1
D1
1,px
∀q ∈ W0
4.7
D1 .
Therefore, it follows that
p−2
|∇w1 x|
D1
ρefw1 q dx 0,
∇w1 x · ∇qxdx D1
1,px
∀q ∈ W0
and hence w1 is a weak solution for −Δpx w1 ρefw1 0 on D1 .
D1 ,
4.8
Abstract and Applied Analysis
17
Thus, there exists a subsequence of {uj } which we denote it by{u1j }, such that u1j → w1
in D1 as j → ∞, where w1 ∈ W 1,px D1 ∩ C1,α1 D1 and satisfies
|∇w1 |
px−2
ρxefx,w1 q dx 0,
∇w1 ∇q dx D1
D1
1,px
∀q ∈ W0
D1 .
4.9
Similarly, we can prove that there exists a subsequence of {u1j } which we denote
by{u2j }, such that u2j → w2 in D2 as j → ∞, where w2 ∈ W 1,px D2 ∩ C1,α2 D2 satisfies
w1 w2 |D1 and
px−2
|∇w2 |
ρxefx,w2 q dx 0,
∇w2 ∇q dx D2
D2
1,px
∀q ∈ W0
D2 .
4.10
Repeating the above steps, we can get a subsequence of {uij | j 1, 2, . . .} which we
denote by {ui1
| j 1, 2, . . .} i 1, 2, . . . and satisfies the following.
j
i
10 For any fixed i, {ui1
j } is a subsequence of {uj }.
→ wi1 in Di1 as j → ∞, where wi1 ∈ W 1,px Di1 ∩
20 For any fixed i, ui1
j
C1,αi1 Di1 satisfies wi wi1 |Di .
30 For any fixed i, wi satisfies
ρxefx,wi q dx 0,
|∇wi |px−2 ∇wi ∇q dx Di
Di
1,px
∀q ∈ W0
Di .
4.11
Thus, we can conclude that
j
i {uj } is a subsequence of {uj },
1,px
ii there exists a function w ∈ Wloc
1,α
Ω ∩ Cloc
Ω such that wi w|Di , and for any
j
x ∈ Ω, there exists a constant jx such that when j ≥ jx , uj x is defined at x, and
j
limj → ∞ uj x wx,
iii
Ω
|∇w|px−2 ∇w∇q dx Ω
ρxefx,w q dx 0,
Obviously, w is a boundary blow-up solution for P.
This completes the proof.
1,px
∀q ∈ W0,loc Ω.
4.12
18
Abstract and Applied Analysis
In Theorem 4.1, when infx∈Ω px > N, the existence of solutions for P is given.
In the following, we will consider the existence of solutions for P in the general case
1 < infx∈Ω px ≤ supx∈Ω px < ∞. We need to do some preparation. Let us consider
pr−2 r N−1 u u r N−1 ρrefr,u ,
u 0 0,
r ∈ 0, Rλ ,
I
uRλ d,
where Rλ ∈ 0, R and d is a constant.
Lemma 4.2. If Φ2 Rλ ≤ d ≤ Φ1 Rλ , where Φ1 and Φ2 are defined in Theorems 3.13.2, respectively,
then 4.13 has a solution u satisfying
Φ2 r ≤ ur ≤ Φ1 r,
∀r ∈ 0, Rλ .
4.13
Proof. Denote
⎧
⎪
efr,Φ1 r arctanur − Φ1 r,
⎪
⎪
⎨
hr, u efr,u ,
⎪
⎪
⎪
⎩ fr,Φ2 r
arctanur − Φ2 r,
e
ur > Φ1 r,
Φ2 r ≤ ur ≤ Φ1 r,
4.14
ur < Φ2 r.
Let ρE t ρ|t|, and hE t, u h|t|, u, for all t ∈ −Rλ , Rλ . Let us consider the
even solutions of the following
p|t|−2 u |t|N−1 ρE thE t, u,
|t|N−1 u u−Rλ d,
t ∈ −Rλ , Rλ ,
II
uRλ d.
It is easy to see that u is an even solution for 4.15 if and only if u is even and
ud−
Rλ r
1−N
|t|
t
1/pt−1
N−1
|s|
ρshs, usds
dt,
∀r ∈ 0, Rλ .
4.15
0
R
t
Denote Ψu, μ μd − μ r λ |t|1−N 0 |s|N−1 ρshs, usds
1/pt−1 dt. Similar to the
proof of Lemma 2.3 of 18
, for any μ ∈ 0, 1
, it is easy to see that Ψu, μ is compact
continuous and bounded from CE1 0, Rλ to CE1 0, Rλ , where CE1 0, Rλ {u ∈ C1 0, Rλ |
u is even}. Thus, u Ψu, 1 has a solution u in CE1 0, Rλ and satisfies u 0 limr → 0 u r 0. Then, u|t| is an even solution for 4.15.
Denote Φ1,E t Φ1 |t|,Φ2,E t Φ2 |t|. From the definitions of Φ1 and Φ2 , we can
see that Φ1 0 0 Φ2 0; therefore, Φ1,E t and Φ2,E t are supersolution and subsolution
for 4.15, respectively.
Abstract and Applied Analysis
19
Since Φ2 Rλ ≤ uRλ ≤ Φ1 Rλ and hE t, · is increasing, from the comparison
principle, we have
Φ2,E t ≤ ut ≤ Φ1,E t,
∀t ∈ −Rλ , Rλ .
4.16
It means that u is a solution for 4.13 and u satisfies
Φ2 r ≤ ur ≤ Φ1 r,
∀r ∈ 0, Rλ .
4.17
Thus u is a radial solution for P. This completes the proof.
Theorem 4.3. If fr, u satisfies
fr, u ≥ aus
as u −→ ∞ for r ∈ σ, R uniformly,
4.18
where σ is defined in (H4 ) and a and s are positive constants, then P possesses a boundary blow-up
solution.
Proof. From Lemma 4.2, we have that 4.4 has a weak solution uj x uj |x| uj r. Similar
to the proof of Theorem 4.1, we can obtain the existence of solutions for P.
Acknowledgments
This paper is partly supported by the National Science Foundation of China 10701066&
10926075 & 10971087 China Postdoctoral Science Foundation-funded project 20090460969,
and the Natural Science Foundation of Henan Education Committee 2008-755-65.
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