Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 279306, 8 pages
doi:10.1155/2008/279306
Research Article
Boundary Blow-Up Solutions to px-Laplacian
Equations with Exponential Nonlinearities
Qihu Zhang
Department of Mathematics and Information Science, Zhengzhou University of Light Industry,
Zhengzhou, Henan 450002, China
Correspondence should be addressed to Qihu Zhang, zhangqh1999@yahoo.com.cn
Received 18 August 2007; Accepted 25 November 2007
Recommended by M. Garcia-Huidobro
This paper investigates the px-Laplacian equations with exponential nonlinearities −px u
efx,u 0 in Ω, ux → ∞ as dx, ∂Ω → 0, where −px u −div|∇u|px−2 ∇u is called pxLaplacian. The singularity of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.
Copyright q 2008 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 px-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, 1–15. In this paper,
we consider the px-Laplacian equations with exponential nonlinearities
−Δpx u efx,u 0
ux −→ ∞
in Ω,
P
as dx, ∂Ω −→ 0,
where −Δpx u −div|∇u|px−2 ∇u, Ω B0, R ⊂ RN is a bounded radial domain B0, R {x ∈ RN | |x| < R} . Our aim is to give the existence and asymptotic behavior of solutions for
problem P.
Throughout the paper, we assume that px and fx, u satisfy that
H1 px ∈ C1 Ω is radial and satisfies
1 < p− ≤ p < ∞,
where p− inf px, p sup px;
Ω
Ω
1.1
2
Journal of Inequalities and Applications
H2 fx, u is radial with respect to x, fx, · is increasing and fx, 0 0 for any x ∈ Ω;
H3 f : Ω × R → R is a continuous function and satisfies
fx, t ≤ C1 C2 |t|γx , ∀x, t ∈ Ω × R,
1.2
where C1 , C2 are positive constants, 0 ≤ γ ∈ CΩ.
The operator −Δpx u −div|∇u|px−2 ∇u is called px-Laplacian. Especially, if px ≡
p a constant, P is the well-known p-Laplacian problem see 16–18.
Because of the nonhomogeneity of px-Laplacian, px-Laplacian problems are more
complicated than those of p-Laplacian ones see 6; and 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 spaces Lpx Ω and
W 1,px Ω, and properties of px-Laplacian, which we will use later see 3, 7. Let
px
px
L Ω u | u is a measurable real-valued function,
ux
dx < ∞ .
2.1
Ω
We can introduce the norm on Lpx Ω by
ux px
inf λ > 0 |
λ dx ≤ 1 .
Ω
|u|px
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 3, 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
and it can be equipped with the norm
u |u|px ∇upx ,
∀u ∈ W 1,px Ω.
1,px
2.4
1,px
W0
Ω is the closure of C0∞ Ω in W 1,px Ω. W 1,px Ω and W0
Ω are separable,
reflexive, and uniform convex Banach spaces see 3, Theorem 2.1.
1,px
If u ∈ Wloc Ω ∩ CΩ, u is called a solution of P if it satisfies
px−2
1,px
∇u
∇u∇qdx fx, uqdx 0, ∀q ∈ W0
Q,
2.5
Q
Q
1,px
for any domain Q Ω, and max k − u, 0 ∈ W0
Ω for any k ∈ N .
1,px
1,px
Let W0,loc Ω {u| there exists an open domain Q Ω s.t. u ∈ W0
Q}. For any
1,px
1,px
1,px
1,px
u ∈ Wloc Ω ∩ CΩ and ϕ ∈ W0,loc Ω, define A : Wloc Ω ∩ CΩ → W0,loc Ω∗ as
Au, ϕ Ω |∇u|px−2 ∇u∇ϕ efx,u ϕdx.
Qihu Zhang
3
1,px
Lemma 2.1 see 5, Theorem 3.1. Let h ∈ W 1,px Ω ∩ CΩ, X h W0,loc Ω ∩ CΩ. Then,
A : X → W0,loc Ω∗ is strictly monotone.
1,px
Let g ∈ W0,loc Ω∗ , if g, ϕ ≥ 0, for all ϕ ∈ W0,loc Ω, ϕ ≥ 0 a.e. in Ω, then denote g ≥ 0
1,px
1,px
in W0,loc Ω∗ ; correspondingly, if −g ≥ 0 in W0,loc Ω∗ , then denote g ≤ 0 in W0,loc Ω∗ .
1,px
1,px
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 9, we have the following lemma.
1,px
Lemma 2.3 comparison principle. Let u, v ∈ Wloc
1,px
W0,loc Ω∗ .
u ≥ v a.e. in Ω.
Ω ∩ CΩ satisfy Au − Av ≥ 0 in
1,px
Let ϕx min {ux − vx, 0}. If ϕx ∈ W0,loc Ω (i.e., u ≥ v on ∂Ω), then
Lemma 2.4 see 4, 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 efx,u 0 in Ω, then u ∈ Cloc
Ω, where ϑ ∈ 0, 1 is a constant.
3. Main results and proofs
If u is a radial solution of P, then P can be transformed into
pr−2 r N−1 |u |
u0 u0 ,
u
r N−1 efr,u ,
u 0 0,
u r ≥ 0
r ∈ 0, R,
for 0 < r < R.
3.1
It means that ur is increasing.
Theorem 3.1. If there exists a constant σ ∈ R/2, R such that
fr, u ≥ αus
as u −→ ∞ for r ∈ σ, R uniformly,
3.2
where α and s are positive constants, then there exists a continuous function Φ1 x which satisfies
Φ1 x → ∞ (as dx, ∂Ω → 0), and such that, if u is a weak solution of problem P, then ux ≤
Φ1 x.
Proof. Let R0 ∈ σ, R. Denote
Θr, a, λ R0
r
⎡ 1/pt−1
−11/s−1 ⎤pRo −1/pt−1 N−1
Ro
a a ln R−R0 −λ
⎦
⎣
sin εt−σ
dt.
tN−1
s R−R0 −λ
3.3
Define the function gr, a on 0, R as
⎧
1/s
⎪
k,
R0 ≤ r < R,
a ln R − r−1
⎪
⎪
⎪
⎨
−1 1/s
, σ < r < R0 ,
gr, a k − Θr, a, 0 a ln R − R0 ⎪
⎪
⎪
⎪
1/s
⎩
k − Θσ, a, 0 a ln R − R0 −1
, r ≤ σ,
3.4
4
Journal of Inequalities and Applications
where a > 1/α sup |x|≥R0 px is a constant, R0 ∈ σ, R, and R − R0 is small enough,
ε π/2R0 − σ and k 2p /α ln R − R0 −1 1/s Θσ, 2a, 0.
Obviously, for any positive constant a, gr, a ∈ C1 0, R.
When R0 < r < R, we have
1/s pr−1
1/s−1pr−1 pr − 1 a
r N−1 |g |pr−2 g r N−1
ln R−r−1
1Πr ,
pr
s
R − r
3.5
where
Πr 1/s − 1
ln R − r−1
N−1 1/s
pr−1 a /s
r
R − r
pr−1
r N−1 a1/s /s
pr − 1
1/s − 1p r ln ln R − r−1
−p r ln R − r
R − r.
R − r pr − 1
pr − 1
3.6
If R − R0 is small enough, it is easy to see |Πr| ≤ 1/2; from 3.5, we have
r
N−1
pr−2 |g |
g
≤ 2r
N−1
≤r
N−1
a1/s
s
1
R−r
pr−1
αa
r
pr − 1R − r
N−1 αg s
e
≤r
−pr N−1 fr,g
e
ln R − r−1
1/s−1pr−1
1/s
Obviously, if R − R0 is small enough, then g ≥ 2p /α ln R−R0 −1 so we have
pr−2 r N−1 |g |
g
3.7
, ∀r ∈ R0 , R .
is large enough,
−1 1/s−1 pRo −1
N−1 a a ln R − R0
ε Ro
cos εr − σ
s R − R0
s
≤ r N−1 eαg ≤ r N−1 efr,g ,
3.8
σ < r < R0 .
Obviously,
pr−2 r N−1 |g |
g
0 ≤ r N−1 efr,g ,
0 ≤ r < σ.
3.9
Since g|x|, a is a C1 function on B0, R, if 0 < R − R0 is small enough R0 depends on
R, p, s, α, from 3.7, 3.8, and 3.9, we can see that g|x|, a is a supersolution of P.
Define the function gm r, a − on 0, R − 1/m as
⎧
−1 1/s
1
1
⎪
⎪
⎪
k,
R0 ≤ r < R− ,
−
r
a
−
ln
R
−
⎪
⎪
m
m
⎪
⎪
⎪
⎪
−1 1/s
⎨
1
1
gm r, a − k − Θ r, a − ,
, σ < r < R0 ,
a
−
ln
R
−
−
R
0
⎪
m
m
⎪
⎪
⎪
⎪
−1 1/s
⎪
⎪
⎪
1
1
⎪
⎩k − Θ σ, a − ,
a − ln R −
− R0
, r ≤ σ,
m
m
3.10
Qihu Zhang
5
where m is a big-enough integer such that 0 < 1/m ≤ R − R0 /2, ε π/2R0 − σ, 0 < < 1, is
a positive small constant such that αa − > sup |x|≥R0 px.
Obviously, gm |x|, a− is a supersolution of P on B0, R−1/m. If u is a solution of P,
according to the comparison principle, we get that gm |x|, a− ≥ ux for any x ∈ B0, R−1/m.
For any x ∈ B0, R − 1/m \ B0, R0 , we have gm |x|, a − ≥ gm1 |x|, a − . Thus,
ux ≤ lim gm |x|, a − , ∀x ∈ B0, R \ B 0, R0 .
m→∞
3.11
When dx, ∂Ω > 0 is small enough, we have
1/s
k ≤ g |x|, a .
lim gm |x|, a − < a ln R − r−1
m→∞
3.12
According to the comparison principle, we obtain that g|x|, a ≥ ux, for all x ∈ B0, R,
then Φ1 x g|x|, a is an upper control function of all of the solutions of P. The proof is
completed.
Theorem 3.2. If there exists a σ ∈ R/2, R such that
fr, u ≤ βus
as u −→ ∞ for r ∈ σ, R uniformly,
3.13
where β and s are positive constants, then there exists a continuous function Φ2 x which satisfies
Φ2 x → ∞ (as dx, ∂Ω → 0), and such that, if ux is a solution of problem P, then ux ≥
Φ2 x.
Proof. Let z1 be a radial solution of
−Δpx z1 x −μ
in Ω1 B0, σ, z1 0 on ∂Ω1 ,
3.14
where μ > 2 is a positive constant. We denote z1 z1 r z1 |x|, then z1 satisfies z1 σ 0,
z1 0 0, and
z1
1/pr−1
rμ ,
N
σ 1/pr−1
rμ z1 − dr.
r N
3.15
Denote hb r, δ on σ, R0 as
hb r, δ R0 r
−1 1/s−1 pRo −1
Ro N−1 t − σ b b ln R δ − R0
tN−1 R0 − σ
s R δ − R0
pσ−1 1/pt−1
σN−1 R0 − t tμ 1/pt−1
N−1
dt.
R0 − σ N t
3.16
It is easy to see that
−hb σ, 0
z1 σ
1/pσ−1
σμ ,
N
−
hb
−1 1/s−1
b b ln R − R0
.
R0 , 0 s R − R0
3.17
6
Journal of Inequalities and Applications
Define the function vr, b on B0, R as
⎧
1/s
⎪
− k∗ ,
R0 ≤ r < R,
b ln R − r−1
⎪
⎪
⎪
⎪
−1 1/s
⎨
b ln R − R0
− k ∗ − hb r, 0,
σ < r < R0 ,
vr, b ⎪
⎪
σ
1/pr−1
⎪
rμ −1 1/s
⎪
⎪
⎩− dr b ln R − R0
− k ∗ − hb σ, 0, r ≤ σ,
N
3.18
r
where b ∈ 0, 1/βinf |x|≥R0 px is a constant, R0 ∈ σ, R, and R − R0 is small enough, and
k ∗ 2p /β ln 2R − R0 −1 1/s .
Obviously, for any positive constant b, vr, b ∈ C1 0, R.
Similar to the proof of Theorem 3.1, when R − R0 is small enough, we have
r N−1 |v |
pr−2 v
≥ r N−1 efr,v , ∀r ∈ R0 , R .
3.19
When R − R0 is small enough, for all r ∈ σ, R0 , since fr, v ≤ 0, then
r
N−1
pr−2 |v |
v
N−1 −1 1/s−1 pR0 −1
b b ln R − R0
1 Ro
≥
≥ r N−1 efr,v .
2 R0 − σ
s R − R0
3.20
Obviously,
pr−2 r N−1 |v |
v
r N−1 μ ≥ r N−1 efr,v , ∀r ∈ 0, σ.
3.21
Combining 3.19, 3.20, and 3.21, we can see that vr, a is a subsolution of P.
Define the function vm r, b on B0, R as
⎧
−1 1/s
⎪
1
⎪
⎪
− k∗ ,
R0 ≤ r < R,
−
r
b
ln
R
⎪
⎪
⎪
m
⎪
⎪
−1 1/s
⎪
⎨
1
1
b ln R − R0
,
σ < r < R0 ,
− k ∗ − hb r,
vm r, b m
m
⎪
⎪
⎪
⎪
σ 1/pr−1
⎪
−1 1/s
⎪
μr ⎪
1
1
⎪
∗
⎪
, r ≤ σ,
dr b ln R −R0
−k −hb σ,
⎩− m
m
r N
3.22
where is a small-enough positive constant such that b < 1/βinf |x|≥R0 px.
We can see that vm r, b ∈ C1 0, R is a subsolution of P on BR0 , R, according
to the comparison principle, we get that vm |x|, 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 , ∀x ∈ B0, R \ B 0, R0 .
3.23
m→∞
When dx, ∂Ω is small enough, we have
lim vm |x|, b > v |x|, b .
m→∞
3.24
From the comparison principle, we obtain v|x|, b ≤ ux, ∀x ∈ B0, R, then Φ2 x v|x|, b is a lower control function of all of the solutions of P.
Qihu Zhang
7
Theorem 3.3. If inf x∈Ω px > N and there exists a σ ∈ R/2, R such that
fr, u ≥ aus
as u −→ ∞ for r ∈ σ, R uniformly,
3.25
where a and s are positive constants, then P possesses a solution.
Proof. In order to deal with the existence of boundary blow-up solutions of P, let us consider
the problem
−Δpx u efx,u 0
ux j
in Ω,
for x ∈ ∂Ω,
3.26
where j 1, 2, . . . . Since inf x∈Ω px > N, then W 1,px Ω → Cα Ω, where α ∈ 0, 1. The
relative functional of 3.26 is
ϕu Ω
px
1 ∇ux dx px
Ω
Fx, udx,
3.27
u
1,px
Ω, then ϕ possesses a
where Fx, u 0 efx,t dt. Since ϕ is coercive in Xj : j W0
nontrivial minimum point uj , then problem 3.26 possesses a weak solution uj . According to
the comparison principle, we get uj x ≤ uj1 x for any x ∈ Ω and j 1, 2, . . . . Since Φ1 x
defined in Theorem 3.1 is a supersolution, according to the comparison principle, we have
uj x ≤ Φ1 x on Ω for all j 1, 2, . . . . Since Φ1 x is locally bounded, from Lemma 2.4, every
1,ϑ
weak solution of P is a locally Cloc
function. Thus, {uj x} possesses a subsequence we still
denote it by {uj x}, such that lim j→∞ uj u is a solution of P.
Acknowledgments
This work was supported by the National Science Foundation of China 10701066 &
10671084and China Postdoctoral Science Foundation 20070421107 and the Natural Science
Foundation of Henan Education Committee 2007110037.
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