Zhang, “Existence of solutions for px-Laplacian Dirichlet problem,” Nonlinear Anal-ysis: Theory, Methods & Applications, vol.. Zhang, “A strong maximum principle for differential equation
Trang 1Volume 2008, Article ID 279306, 8 pages
doi:10.1155/2008/279306
Research Article
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 − p x u
e f x,u 0 in Ω, ux → ∞ as dx, ∂Ω → 0, where − p x u −div|∇u| p x−2 ∇u is called
px-Laplacian The singularity of boundary blow-up solutions is discussed, and the existence of bound-ary 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 to1,2 , 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
−Δp x u e f x,u 0 in Ω,
where−Δp x u −div|∇u| p x−2 ∇u, Ω B0, R ⊂ R Nis a bounded radial domainB0, R {x ∈ R N | |x| < R} Our aim is to give the existence and asymptotic behavior of solutions for
problemP
Throughout the paper, we assume that px and fx, u satisfy that
H1 px ∈ C1Ω is radial and satisfies
1 < p−≤ p< ∞, where p− inf
Ω p x, p sup
Trang 2H2 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
f x, t ≤ C1 C2|t| γ x , ∀x, t ∈ Ω × R, 1.2
where C1, C2are positive constants, 0≤ γ ∈ CΩ.
The operator−Δp x u −div|∇u| p x−2 ∇u is called px-Laplacian Especially, if px ≡
pa 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 f x, u cannot be represented as hxfu.
2 Preliminary
In order to deal with px-Laplacian problems, we need some theories on spaces L p xΩ and
L p xΩ
u | u is a measurable real-valued function,
Ω
u xp x dx <∞
. 2.1
We can introduce the norm on L p xΩ by
|u| p x inf
λ > 0|
Ω
u x λ p x dx≤ 1
The space L p xΩ, |·|p x becomes a Banach space We call it generalized Lebesgue space The spaceL p xΩ, |·|p x is a separable, reflexive, and uniform convex Banach space
see 3, Theorems 1.10, 1.14
The space W 1,pxΩ is defined by
and it can be equipped with the norm
u |u| p x∇u p x , ∀u ∈ W 1,px Ω. 2.4
W01,px Ω is the closure of C∞
0 Ω in W 1,px Ω W 1,px Ω and W01,pxΩ are separable, reflexive, and uniform convex Banach spacessee 3, Theorem 2.1
If u ∈ Wloc1,px Ω ∩ CΩ, u is called a solution of P if it satisfies
Q
∇u p x−2 ∇u∇qdx
Q
f x, uqdx 0, ∀q ∈ W01,px Q, 2.5
for any domain QΩ, and max k − u, 0 ∈ W01,px Ω for any k ∈ N
Let W 0,loc 1,px Ω {u| there exists an open domain Q Ω s.t u ∈ W01,px Q} For any
u ∈ Wloc1,px Ω ∩ CΩ and ϕ ∈ W 0,loc 1,px Ω, define A : Wloc1,px Ω∩CΩ → W 0,loc 1,pxΩ∗ as
Au, ϕ Ω|∇u| p x−2 ∇u∇ϕ e f x,u ϕ dx.
Trang 3Lemma 2.1 see 5, Theorem 3.1 Let h ∈ W1,px Ω ∩ CΩ, X h W 0,loc 1,px Ω ∩ CΩ Then,
A : X → W 0,loc 1,pxΩ∗is strictly monotone.
Let g ∈ W 0,loc 1,pxΩ∗, if g, ϕ ≥ 0, for all ϕ ∈ W 0,loc 1,px Ω, ϕ ≥ 0 a.e in Ω, then denote g ≥ 0
in W 0,loc 1,pxΩ∗; correspondingly, if −g ≥ 0 in W 0,loc 1,pxΩ∗, then denote g ≤ 0 in W 0,loc 1,pxΩ∗ Definition 2.2 Let u ∈ Wloc1,px Ω ∩ CΩ If Au ≥ 0 Au ≤ 0 in W 0,loc 1,pxΩ∗, then u is called a
weak supersolutionweak subsolution of P
Copying the proof of9 , we have the following lemma
Lemma 2.3 comparison principle Let u, v ∈ Wloc1,px Ω ∩ CΩ satisfy Au − Av ≥ 0 in
W 0,loc 1,pxΩ∗ Let ϕ x min {ux − vx, 0} If ϕx ∈ W 0,loc 1,px Ω (i.e., u ≥ v on ∂Ω), then
u ≥ v a.e in Ω.
Lemma 2.4 see 4, Theorem 1.1 Under the conditions (H1) and (H3), if u ∈ W 1,px Ω is a
bounded weak solution of−Δp x u e f x,u 0 in Ω, then u ∈ C 1,ϑ
locΩ, where ϑ ∈ 0, 1 is a constant.
3 Main results and proofs
If u is a radial solution ofP, then P can be transformed into
r N−1|u|p r−2 u
r N−1e f r,u , r ∈ 0, R,
u 0 u0, u0 0, ur ≥ 0 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
f r, u ≥ αu s as u −→ ∞ for r ∈ σ, R uniformly, 3.2
where α and s are positive constants, then there exists a continuous function Φ1x which satisfies
Φ1x → ∞ (as dx, ∂Ω → 0), and such that, if u is a weak solution of problem P, then ux ≤
Φ1x.
Proof Let R0 ∈ σ, R Denote
Θr, a, λ
R0
r
⎡
⎣a
a ln
R −R0−λ −1 1/s−1
s
R −R0−λ
⎤
⎦
pR o −1/pt−1
Ro N−1
t N−1 sin εt−σ
1/pt−1
dt.
3.3
Define the function gr, a on 0, R as
g r, a
⎧
⎪
⎪
⎪
⎪
a ln R − r−1 1/s k, R0≤ r < R,
k − Θr, a, 0 a ln R − R0−1 1/s , σ < r < R0,
k − Θσ, a, 0 a ln R − R0−1 1/s , r ≤ σ,
3.4
Trang 4where a > 1/α sup |x|≥R0 p x is a constant, R0 ∈ σ, R, and R − R0 is small enough,
ε π/2R0− σ and k 2p/α ln R − R0−11/s Θσ, 2a, 0.
Obviously, for any positive constant a, gr, a ∈ C10, R.
When R0< r < R, we have
r N−1|g|p r−2 g
r N−1
a 1/s s
p r−1
p r − 1
R − r p r
lnR−r−1 1/s−1pr−1 1Πr , 3.5 where
Πr 1/s − 1
lnR − r−1
r N−1a 1/s /sp r−1
r N−1
a 1/s /s p r−1
pr − 1 R − r
−p r ln R − r
p r − 1 R − r
1/s − 1pr ln ln R − r−1
p r − 1 R − r.
3.6
IfR − R0 is small enough, it is easy to see |Πr| ≤ 1/2; from 3.5, we have
r N−1|g|p r−2 g
≤ 2r N−1
a 1/s
s
p r−1
p r − 1R − r −pr lnR − r−1 1/s−1pr−1
≤ r N−1
1
R − r
αa
r N−1e αg s ≤ r N−1e f r,g, ∀r ∈ R0, R
.
3.7
Obviously, if R − R0is small enough, then g ≥ 2p/α ln R−R0−11/s is large enough,
so we have
r N−1|g|p r−2
g
ε Ro N−1
a
a ln
R − R0
−1 1/s−1
s
R − R0
pR o−1
cos
ε r − σ
≤ r N−1e αg s ≤ r N−1e f r,g , σ < r < R0.
3.8
Obviously,
r N−1|g|p r−2
g
0 ≤ r N−1e f r,g , 0≤ r < σ. 3.9
Since g|x|, a is a C1function on B0, R, if 0 < R − R0is small enoughR0depends 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 g m r, a − on 0, R − 1/m as
gm r, a −
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
a − ln
R− 1
m − r
−1 1/s
m ,
k− Θ
r, a − , 1
m
a − ln
R− 1
m − R0
−1 1/s
, σ < r < R0,
k− Θ
σ, a − , 1
m
a − ln
R− 1
m − R0
−1 1/s
, r ≤ σ,
3.10
Trang 5where 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 p x.
Obviously, g m |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 g m |x|, a− ≥ ux for any x ∈ B0, R−1/m For any x ∈ B0, R − 1/m \ B0, R0, we have g m |x|, a − ≥ g m1|x|, a − Thus,
u x ≤ lim
m→∞g m
|x|, a − , ∀x ∈ B0, R \ B 0, R0
When dx, ∂Ω > 0 is small enough, we have
lim
m→∞gm
|x|, a − <
a ln R − r−1 1/s k ≤ g |x|, a . 3.12
According to the comparison principle, we obtain that g|x|, a ≥ ux, for all x ∈ B0, R,
then Φ1x 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
f r, u ≤ βu s as u −→ ∞ for r ∈ σ, R uniformly, 3.13
where β and s are positive constants, then there exists a continuous function Φ2x which satisfies
Φ2x → ∞ (as dx, ∂Ω → 0), and such that, if ux is a solution of problem P, then ux ≥
Φ2x.
Proof Let z1be a radial solution of
−Δp x z1x −μ in Ω1 B0, σ, z1 0 on ∂Ω1, 3.14
where μ > 2 is a positive constant We denote z1 z1r z1|x|, then z1 satisfies z1σ 0,
z10 0, and
z1
rμ N1/pr−1, z1 −
σ
r
Denote h b r, δ on σ, R0 as
hb r, δ
R0
r
R oN−1
t N−1
t − σ
R0− σ
b
b ln
R δ − R0
−1 1/s−1
s
R δ − R0
p R o−1
σ N−1
t N−1
R0− t
R0− σ
N tμ
1/pt−1
p σ−11/pt−1
dt.
3.16
It is easy to see that
−h
b σ, 0 z
1σ
σμ N1/pσ−1, − h
b
R0, 0
b
b ln
R − R0
−1 1/s−1
s
R − R0
Trang 6Define the function vr, b on B0, R as
v r, b
⎧
⎪
⎪
⎪
⎪
⎩
b ln R − r−1 1/s − k∗, R0≤ r < R,
b ln
R − R0
−1 1/s
− k∗− h b r, 0, σ < r < R0,
−
σ
r
rμ N1/pr−1 dr b ln
R − R0
−1 1/s
− k∗− h b σ, 0, r ≤ σ,
3.18
where b ∈ 0, 1/βinf |x|≥R0 p x is a constant, R0 ∈ σ, R, and R − R0 is small enough, and
k∗ 2p/β ln 2R − R0−11/s
Obviously, for any positive constant b, vr, b ∈ C10, R.
Similar to the proof ofTheorem 3.1, when R − R0is small enough, we have
r N−1|v|p r−2 v
≥ r N−1e f r,v, ∀r ∈ R0, R
When R − R0is small enough, for all r ∈ σ, R0, since fr, v ≤ 0, then
r N−1|v|p r−2 v
≥ 1 2
Ro N−1
R0− σ
b
b ln
R − R0
−1 1/s−1
s
R − R0
p R0−1
Obviously,
r N−1|v|p r−2
v
Combining3.19, 3.20, and 3.21, we can see that vr, a is a subsolution of P
Define the function v m r, b on B0, R as
vm r, b
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
b ln
R 1
m − r
−1 1/s
b lnR 1
m − R0
−1 1/s
− k∗− h b
r, 1 m
, σ < r < R0,
−
σ
r
μr N1/pr−1 dr
b ln
R1
m −R0
−1 1/s
−k∗−h b
σ, 1 m
, r ≤ σ,
3.22
where is a small-enough positive constant such that b < 1/βinf |x|≥R0 p x.
We can see that v m r, b ∈ C10, R is a subsolution of P on BR0, R, according
to the comparison principle, we get that v m |x|, b ≤ ux for any x ∈ B0, R For any
x ∈ B0, R \ B0, R0, we have v m |x|, b ≤ v m1|x|, b Thus,
u x ≥ lim
m→∞vm
|x|, b , ∀x ∈ B0, R \ B 0, R0
When dx, ∂Ω is small enough, we have
lim
m→∞vm
|x|, b > v
From the comparison principle, we obtain v|x|, b ≤ ux, ∀x ∈ B0, R, then Φ2x
v |x|, b is a lower control function of all of the solutions of P
Trang 7Theorem 3.3 If inf x∈Ωp x > N and there exists a σ ∈ R/2, R such that
f r, u ≥ au s as u −→ ∞ for r ∈ σ, R uniformly, 3.25
where a and s are positive constants, thenP possesses a solution.
Proof In order to deal with the existence of boundary blow-up solutions ofP, let us consider the problem
−Δp x u e f x,u 0 in Ω,
where j 1, 2, Since inf x∈Ωp x > N, then W 1,px Ω → C α Ω, where α ∈ 0, 1 The
relative functional of3.26 is
ϕ u
Ω
1
p x ∇ux p x
dx
where Fx, u u
nontrivial minimum point u j, then problem3.26 possesses a weak solution u j According to
the comparison principle, we get u j x ≤ u j1x for any x ∈ Ω and j 1, 2, Since Φ1x
defined in Theorem 3.1 is a supersolution, according to the comparison principle, we have
uj x ≤ Φ1x on Ω for all j 1, 2, Since Φ1x is locally bounded, fromLemma 2.4, every weak solution ofP is a locally C 1,ϑ
loc function Thus,{u j x} possesses a subsequence we still
denote it by{u j 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 Committee2007110037
References
1 Y Chen, S Levine, and M Rao, “Variable exponent, linear growth functionals in image restoration,”
SIAM Journal on Applied Mathematics, vol 66, no 4, pp 1383–1406, 2006.
2 M R ´uzicka, Electrorheological Fluids: Modeling and Mathematical Theory, vol 1748 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.
3 X.-L Fan and D Zhao, “On the spaces L p x Ω and W m,p x Ω,” Journal of Mathematical Analysis and Applications, vol 263, no 2, pp 424–446, 2001.
4 X.-L Fan, “Global C 1,α regularity for variable exponent elliptic equations in divergence form,” Journal
of Differential Equations, vol 235, no 2, pp 397–417, 2007.
5 X.-L Fan and Q Zhang, “Existence of solutions for px-Laplacian Dirichlet problem,” Nonlinear Anal-ysis: Theory, Methods & Applications, vol 52, no 8, pp 1843–1852, 2003.
6 X.-L Fan, Q Zhang, and D Zhao, “Eigenvalues of px-Laplacian Dirichlet problem,” Journal of Math-ematical Analysis and Applications, vol 302, no 2, pp 306–317, 2005.
7 O Kov´aˇcik and J R´akosn´ık, “On spaces L p x Ω and W k,p x Ω,” Czechoslovak Mathematical Journal,
vol 41116, no 4, pp 592–618, 1991.
Trang 88 S G Samko, “Density C∞
0 RN in the generalized Sobolev spaces W m,p xRN ,” Doklady Rossi˘ıskaya Akademiya Nauk, vol 369, no 4, pp 451–454, 1999.
9 Q Zhang, “A strong maximum principle for differential equations with non standard px-growth conditions,” Journal of Mathematical Analysis and Applications, vol 312, no 1, pp 24–32, 2005.
10 Q Zhang, “Existence of positive solutions for elliptic systems with non standard px-growth condi-tions via sub-supersolution method,” Nonlinear Analysis: Theory, Methods & Applicacondi-tions, vol 67, no 4,
pp 1055–1067, 2007.
11 Q Zhang, “Existence and asymptotic behavior of blow-up solutions to a class of px-Laplacian prob-lems,” Journal of Mathematical Analysis and Applications, vol 329, no 1, pp 472–482, 2007.
12 Q Zhang, “Existence of positive solutions for a class of px-Laplacian systems,” Journal of Mathemat-ical Analysis and Applications, vol 333, no 2, pp 591–603, 2007.
13 Q Zhang, “The asymptotic behavior of solutions for px-Laplace equations,” Journal of Zhengzhou University of Light Industry, vol 24, no 5, pp 92–95, 2007.
14 Q Zhang, “Oscillatory property of solutions for pt-Laplacian equations,” Journal of Inequalities and Applications, vol 2007, Article ID 58548, 8 pages, 2007.
15 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.
16 J B Keller, “On solutions of Δu fu,” Communications on Pure and Applied Mathematics, vol 10, pp.
503–510, 1957.
17 A V Lair, “A necessary and sufficient condition for existence of large solutions to semilinear elliptic
equations,” Journal of Mathematical Analysis and Applications, vol 240, no 1, pp 205–218, 1999.
18 A Mohammed, “Existence and asymptotic behavior of blow-up solutions to weighted quasilinear
equations,” Journal of Mathematical Analysis and Applications, vol 298, no 2, pp 621–637, 2004.
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