Volume 2008, Article ID 621621, 19 pagesdoi:10.1155/2008/621621 Research Article The Method of Subsuper Solutions for Weighted Qihu Zhang, 1, 2 Xiaopin Liu, 2 and Zhimei Qiu 2 1 Departme
Trang 1Volume 2008, Article ID 621621, 19 pages
doi:10.1155/2008/621621
Research Article
The Method of Subsuper Solutions for Weighted
Qihu Zhang, 1, 2 Xiaopin Liu, 2 and Zhimei Qiu 2
1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry,
Zhengzhou, Henan 450002, China
2 School of Mathematics Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China
Correspondence should be addressed to Zhimei Qiu,zhimeiqiu@yahoo.com.cn
Received 23 May 2008; Accepted 21 August 2008
Recommended by Marta Garcia-Huidobro
This paper investigates the existence of solutions for weighted pr-Laplacian ordinary boundary
value problems Our method is based on Leray-Schauder degree As an application, we give the
existence of weak solutions for px-Laplacian partial differential equations.
Copyrightq 2008 Qihu Zhang 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
1 Introduction
In this paper, we consider the existence of solutions for the following weighted pr-Laplacian
ordinary equation with right-hand terms depending on the first-order derivative:
−wrup r−2 u
fr, u,
wr1/pr−1 u
0, ∀r ∈T1, T2
, P with one of the following boundary value conditions:
u
T1
c, u
T2
g
u
T1
,
w
T11/pT1−1
u
T1
0, u
T2
d, 1.2
g
u
T1
,
w
T1
1/pT1−1
u
T1
0, h
u
T2
,
w
T2
1/pT2−1
u
T2
0, 1.3
u
T1
uT2
, w
T1u
T1p T1 −2u
T1
wT2u
T2p T2 −2u
T2
, 1.4
where p ∈ CT1, T2, R and pr > 1; w ∈ CT1, T2, R satisfies 0 < wr, ∀r ∈ T1, T2,
andwr −1/pr−1 ∈ L1T1, T2; −wr|u|p r−2 uis called the weighted pr-Laplacian; the
Trang 2notationwT11/pT1 −1uT1 means limr→ T
1wr 1/pr−1 ur exists and
w
T1
1/pT1−1
u
T1
: lim
r → T 1
wr1/pr−1 ur, 1.5
similarly
w
T21/pT2−1
u
T2 : lim
r → T− 2
wr1/pr−1 ur; 1.6
where gx, y and hx, y are continuous and increasing in y for any fixed x, respectively.
The study of differential equations and variational problems with nonstandard pr-growth conditions is a new and interesting topic Many results have been obtained on these kinds of problem, for example,1 18 If wr ≡ pr ≡ p a constant, P is the well-known
p-Laplacian problem Because of the nonhomogeneity of px-Laplacian, px-Laplacian problems are more complicated than those of p-Laplacian, many methods and results for p-Laplacian problems are invalid for px-Laplacian problems For example,
1 if Ω ⊂ Rnis an open bounded domain, then the Rayleigh quotient
λ p x inf
u ∈W01,pxΩ\{0}
Ω
1/px|∇u| p x dx
Ω
1/px|u| p x dx 1.7
is zero in general, and only under some special conditions λ p x > 0see 4, but the fact that
λ p > 0 is very important in the study of p-Laplacian problems In19, the author considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:
−Δpu λfu, v λu α v γ inΩ,
−Δqv λgu, v λu δ v βinΩ,
u v 0 on ∂Ω,
S
the first eigenfunction is used to constructing the subsolution of problem S successfully
On the px-Laplacian problems, maybe px-Laplacian does not have the first eigenvalue and the first eigenfunction Because of the nonhomogeneity of px-Laplacian, the first eigenfunction cannot be used to construct the subsolution of px-Laplacian problems, even
if the first eigenfunction of Laplacian exists.On the existence of solutions for
px-Laplacian equations Dirichlet problems via subsuper solution methods, we refer to13,14;
2 if wr ≡ pr ≡ p a constant and −Δpu > 0, then u is concave, this property is used extensively in the study of one-dimensional p-Laplacian problems, but it is invalid for
−Δpr It is another difference on −Δpand−Δpr : −|u|p r−2 u;
3 on the existence of solutions of the typical pr-Laplacian problem:
−up r−2 u
|u| q r−2 u C, r ∈ 0, 1, 1.8
because of the nonhomogeneity of pt-Laplacian, when we use critical point theory to deal
with the existence of solutions, we usually need the corresponding functional is coercive or satisfy Palais-Smale conditions If 1≤ maxr∈0,1 qr < minr ∈0,1 pr, then the corresponding
functional is coercive, if maxr ∈0,1 pr < minr ∈0,1 qr, then the corresponding functional
Trang 3satisfies Palais-Smale conditions see 3 But if minr∈0,1 pr ≤ qr ≤ maxr ∈0,1 pr,
one can see that the corresponding functional is neither coercive nor satisfying Palais-Smale conditions, the results on this case are rare
There are many papers on the existence of solutions for p-Laplacian boundary value
problems via subsuper solution methodsee 20–24 But results on the sub-super-solution
method for px-Laplacian equations and systems are rare In this paper, when pr is
a general function, we establish several sub-super-solution theorems for the existence of
solutions for weighted pr-Laplacian equation with Dirichlet, Robin, and Periodic boundary
value conditions Moreover, the case of minr ∈0,1 pr ≤ qr ≤ maxr ∈0,1 pr is discussed.
Our results partially generalize the results of13,14,20,25
Let T1 < T2 and I T1, T2, the function f : I × R × R → R is assumed to be
Caratheodory, by this we mean the following:
i for almost every t ∈ I, the function ft, ·, · is continuous;
ii for each x, y ∈ R × R, the function f·, x, y is measurable on I;
iii for each ρ > 0, there is a αρ ∈ L1I, R such that, for almost every t ∈ I and every
x, y ∈ R × R with |x| ≤ ρ, |y| ≤ ρ, one has
ft, x, y ≤ αρt. 1.9
We set C CI, R, C1 {u ∈ C | uis continuous inT1, T2, limr→ T
1wr|u|p r−2 ur
and limr → T−
2wr|u|p r−2 u 0 supr ∈T1,T2 1 0
1/pr−1 u 0 The spaces C and C1 will be equipped with the norm 0 and 1, respectively
We say a function u : I → R is a solution of P, if u ∈ C1 and wr|u|p r−2 ur is
absolutely continuous and satisfiesP almost every on I.
Functions α, β ∈ C1are called subsolution and supersolution ofP, if |α|p r−2 αr and
|β|p r−2 βr are absolutely continuous and satisfy
−wrαp r−2 α
fr, α,
wr1/pr−1 α
≤ 0, a.e on I,
−wrβp r−2 β
fr, β,
wr1/pr−1 β
≥ 0, a.e on I.
1.10
Throughout this paper, we assume that α ≤ β are subsolution and supersolution,
respectively Denote
Ω0t, x | t ∈ I, x ∈αt, βt ,
Ω1t, x, y | t ∈ I, x ∈αt, βt, y∈ R .
1.11
We also assume that
H1 |ft, x, y| ≤ A1t, xK1t, x, y A2t, xK2t, x, y, for all t, x, y ∈ Ω1, where
Ait, x i 1, 2 are positive value and continuous on Ω0, Kit, x, y i 1, 2 are positive
value and continuous onΩ1
H2 There exist positive numbers M1 and M2 such that K1t, x, y ≤ |y|φ|y|,
K2t, x, y ≤ M1φ|y|, for |y| ≥ M2, where φ ∈ C1, ∞, 1, ∞ is increasing and satisfies
∞
1 1/φy 1/p− −1dy ∞, where p− minr∈I pr.
Trang 4Our main results are as the following theorem.
Theorem 1.1 If f is Caratheodory and satisfies (H1) and (H2), α and β satisfy αT1 ≤ c ≤ βT1, αT2 ≤ d ≤ βT2, then P with 1.1 possesses a solution.
Theorem 1.2 If f is Caratheodory and satisfies (H1) and (H2), α and β satisfy αT2 ≤ d ≤ βT2, and
g
α
T1
,
w
T1
1/pT1−1
α
T1
≥ 0 ≥ gβ
T1
,
w
T1
1/pT1−1
β
T1
, 1.12
thenP with 1.2 possesses a solution.
Theorem 1.3 If f is Caratheodory and satisfies (H1) and (H2), α and β satisfy
g
α
T1
,
w
T1
1/pT1−1
α
T1
≥ 0 ≥ gβ
T1
,
w
T1
1/pT1−1
β
T1
,
h
α
T2
,
w
T2
1/pT2−1
α
T2
≤ 0 ≤ hβ
T2
,
w
T2
1/pT2−1
β
T2
,
1.13
thenP with 1.3 possesses a solution.
Theorem 1.4 If f is Caratheodory and satisfies (H1) and (H2), α and β satisfy
α
T1
αT2 < βT1 βT2,
w
T1α
T1p T1 −2α
T1
≥ wT2α
T2p T2 −2α
T2
,
w
T1β
T1p T1 −2β
T1
≤ wT2β
T2p T2 −2β
T2
,
1.14
thenP with 1.4 possesses a solution.
As an application, we consider the existence of weak solutions for the following
px-Laplacian partial differential equation:
−div|∇u| p x−2 ∇u fx, u, |x| n−1/px−1 |∇u| 0, ∀x ∈ Ω, 1.15
whereΩ is a bounded symmetric domain in Rn , p ∈ CΩ; R is radially symmetric We will write px p|x| pr, and pr satisfies 1 < pr ∈ C, f ∈ CΩ × R × R, R is radially symmetric with respect to x, namely, f x, u, v f|x|, u, v fr, u, v, and f satisfies the
Caratheodory condition
2 Preliminary
Denote ϕr, x |x| p r−2 x, ∀r, x ∈ I × R Obviously, ϕ has the following properties.
Lemma 2.1 ϕ is a continuous function and satisfies
i for any r ∈ T1, T2, ϕr, · is strictly increasing;
ii ϕr, · is a homeomorphism from R to R for any fixed r ∈ I.
Trang 5For any fixed r ∈ I, denote ϕ−1r, · as
ϕ−1r, x |x| 2−pr/pr−1 x, for x ∈ R \ {0}, ϕ−1r, 0 0. 2.1
It is clear that ϕ−1r, · is continuous and send bounded sets into bounded sets.
Let us now consider the simple problem
wrϕr, ur fr, 2.2 with boundary value condition1.1, where f ∈ L1 If u is a solution of2.2 with 1.1, by integrating2.2 from T1to r, we find that
wrϕr, ur wT1
ϕ
T1, u
T1
r
T1
ftdt. 2.3 Denote
Ffr r
T1
ftdt, a wT1
ϕ
T1, u
T1
then
ur uT1
r
T1
ϕ−1
r,
wr−1a Ffrdr. 2.5 The boundary conditions imply that
T2
T1
ϕ−1
r,
wr−1a Ffrdr d − c. 2.6
For fixed h ∈ C, we denote
Λha T2
T1
ϕ−1
r,
wr−1a hrdr c − d. 2.7
We have the following lemma
Lemma 2.2 The function Λ h has the following properties i For any fixed h ∈ C, the equation
has a unique solution ah ∈ R.
ii The function a : C → R, defined in (i), is continuous and sends bounded sets to bounded sets.
Proof i Obviously, for any fixed h ∈ C, Λh· is continuous and strictly increasing, then, if
2.8 has a solution, it is unique
Since wr −1/pr−1 ∈ L1T1, T2 and h ∈ C, it is easy to see that
lim
a→ ∞Λha ∞, lim
a→ −∞Λha −∞ 2.9
Trang 6It means the existence of solutions ofΛha 0.
In this way, we define a function ah : CT1, T2 → R, which satisfies
T2
T1
ϕ−1
r,
wr−1 ah hrdr 0. 2.10
ii We claim that
ah ≤ |c − d|
T2
T1ϕ−1
r,
wr−1dr 1
p1
0, ∀h ∈ C. 2.11
If it is false Without loss of generality, we may assume that there are some h ∈ C such
that
ah >
|c − d|
T2
T1ϕ−1
r,
wr−1dr 1
p1
0, 2.12 then
ah h >
|c − d|
T2
T1ϕ−1
r,
wr−1dr 1
p1
,
T2
T1
ϕ−1
r,
wr−1 ah hrdr d − c
>
|c − d|
T2
T1ϕ−1
r,
wr−1dr 1 T2
T1
ϕ−1
r,
wr−1dr d − c
|c − d| T2
T1
ϕ−1
r,
wr−1dr d − c
> 0.
2.13
It is a contradiction Thus,2.11 is valid It mens that a sends bounded sets to bounded
sets
Finally, to show the continuity of a, let {un} be a convergent sequence in C and un →
u, as n → ∞ Obviously, { aun} is a bounded sequence, then it contains a convergent
subsequence{ aun j } Let aun j → a0as j → ∞ Since
T2
T1
ϕ−1
r,
wr−1 aun j
un j rdr 0, 2.14
letting j → ∞, we have
T2
T1
ϕ−1
r,
wr−1a0 urdr 0, 2.15 fromi, we get a0 au, it means a is continuous.
This completes the proof
Trang 7Now, we define a : L1 → R is defined by
ah aFh. 2.16
It is clear that a is a continuous function which send bounded sets of L1into bounded sets ofR, and hence it is a complete continuous mapping
We continue now with our argument previous toLemma 2.2 By solving for uin2.3 and integrating, we find
ur uT1
Fϕ−1
r,
wr−1af Ffr r. 2.17 Let us define
Kht Fϕ−1
r,
wr−1ah Fh t, ∀t ∈T1, T2
. 2.18
We denote by Nf u : C1× T1, T2 → L1, the Nemytsky operator associated to f
defined by
N f ur fr, ur,wr1/pr−1 ur, a.e on I. 2.19
It is easy to see the following lemma
Lemma 2.3 u is a solution of P with boundary value condition 1.1 if and only if u is a solution
of the following abstract equation:
u c KN f u. 2.20
Lemma 2.4 The operator K is continuous and sends equi-integrable sets in L1into relatively compact sets in C1.
Proof It is easy to check that Kht ∈ C1 Sincewr −1/pr−1 ∈ L1, and
wt1/pt−1 Kht ϕ−1
t,
ah Fh, ∀t ∈T1, T2
, 2.21
it is easy to check that K is a continuous operator from L1to C1
Let now U be an equi-integrable set in L1, then there exists ρ ∈ L1, such that
ut ≤ ρt a.e in I, for any u ∈ U. 2.22
We want to show that KU ⊂ C1is a compact set
Let{un} be a sequence in KU, then there exist a sequence {hn} ∈ U such that un
Khn For t1, t2∈ I, we have that
F
hn
t1
− Fhn
t2 ≤ t2
t1
ρtdt
2.23 Hence, the sequence{Fhn} is uniformly bounded and equicontinuous, then there
exists a subsequence of {Fhn} which is convergent in C, and we name the same Since
the operator a is bounded and continuous, we can choose a subsequence of {ahn Fhn}
which we still denote {ahn Fhn} that is convergent in C, then
wtϕt,
Khnt ahn
Fhn
2.24
is convergent in C Since
K
h n
t Fwr−1/pr−1 ϕ−1
r,
a
h n
Fh n
t, ∀t ∈T1, T2
, 2.25
according to the continuous of ϕ−1and the integrability ofwr −1/pr−1 in L1, then Khn
is convergent in C Then, we can conclude that {un} convergent in C1
Trang 8Lemma 2.5 Let α, β ∈ C1 be subsolution and supersolution of P, respectively, which satisfies αt ≤ βt for any t ∈ T1, T2, then there exists a positive constant L such that, for any solution x of
P with 1.1 1/pt−1 x 0≤ L.
Proof We denote
μ0 T2
T1
A1
t, xt A2
t, xtdt, a0 maxwr1/pr−1 | r ∈T1, T2
,
σ maxβs − αt | t, s ∈ T1, T2 ,
γ maxwt1/pt−1 A1t, x | t, x ∈ Ω0
,
2.26
then there exists a t0∈ T1, T2 such that
w
t01/pt0−1
x
t0 ≤ a0x
t0 ≤ a0
σ
T2− T1
2.27 FromH2, there exist positive numbers σ1and N1such that
N1≥ σ1≥ max
r ∈I
M2 a0
σ
T2− T1 1
p r
,
N1
σ1
1
φ
y 1/pr−1 dy > γσ M1μ0, for r∈T1, T2
uniformly
2.28
Assume that our conclusion is not true, combining2.27, then there exists t1, t2 ⊂
T1, T2 such that wr 1/pr−1 xkeeps the same sign ont1, t2, and
w
t1xp t1 −2x
t1
σ1, w
t2xp t2 −2x
t2
N1, 2.29
or inversely
w
t1xp t1 −2x
t1
−σ1, w
t2xp t2 −2x
t2
−N1. 2.30 For simplicity, we assume that the former appears Hence,
γσ M1μ0<
N1
σ1
1
φ
y 1/pr−1 dy
t2
t1
wrxp r−1
φ
wrxp r−11/pr−1 dr
t2
t1
f
r, x,
wr1/pr−1 x
φ
wr1/pr−1x
dr
≤ t2
t1
wr1/pr−1 A1
r, xrxdr M1μ0
≤ γσ M1μ0,
2.31
which is impossible The proof is completed
Trang 9Let us consider the auxiliary SBVP of the form
wrup r−2 u
fr, Rr, u, R1
wr1/pr−1 u
R2r, udef fr, u, r ∈T1, T2
,
2.32 where
Rt, u
⎧
⎪
⎪
⎪
⎪
βt, ut > βt,
u, αt ≤ ut ≤ βt, αt, ut < αt,
R1y
⎧
⎪
⎪
⎪
⎪
L1, y > L1,
y, |y| ≤ L1,
−L1, y < −L1,
2.33
where
L1 1 max
L, sup
r ∈T1,T2
wr1/pr−1
βr, sup
r ∈T1,T2
wr1/pr−1
αr, 2.34
where L is defined inLemma 2.5, and
R2t, u
⎧
⎪
⎪
⎨
⎪
⎪
⎩
et, u u − βt
1 u2 ut > βt,
0, αt ≤ ut ≤ βt, et, u u − αt
1 u2 ut < αt,
2.35
where et, u 1 A1t, Rt, u A2t, Rt, u.
Lemma 2.6 Let the conditions of Lemma 2.5 hold, and let ut be any solution of SBVP with 1.1
satisfies αT1 ≤ c ≤ βT1 and αT2 ≤ d ≤ βT2, then αt ≤ ut ≤ βt, for any t ∈ T1, T2 Proof We will only prove that ut ≤ βt for any t ∈ T1, T2 The argument of the case of
αt ≤ ut for any t ∈ T1, T2 is similar
Assume that ut > βt for some t ∈ T1, T2, then there exist a t0 ∈ T1, T2 and a
positive number δ such that ut0 βt0 δ, ut ≤ βt δ, for any t ∈ T1, T2 Hence,
w
t01/pt0−1
u
t0
w
t01/pt0−1
β
t0
. 2.36
There exists a positive number η such that ut > βt, for any t ∈ J : t0− η, t0 η ⊂
T1, T2 From the definition of β, u, and f we conclude that
wrβp r−2 β
≤ fr, β,
wr1/pr−1 β
fr, β < fr, u ont0− η1, t0 η1
, 2.37
Trang 10where η1∈ 0, η is small enough For any r ∈ t0, t0 η1, we have
r
t0
wrβp r−2 β
dr <
r
t0
f r, udr r
t0
wrup r−2 u
dr. 2.38
From2.36 and 2.38, we have
βp r−2 β<up r−2 u on
t0, t0 η1
it means that
β δ< uon
t0, t0 η1
It is a contradiction to the definition of t0, so ut ≤ βt, for any t ∈ T1, T2
3 Proofs of main results
In this section, we will deal with the proofs of main results
Proof of Theorem 1.1 From Lemmas 2.5 and 2.6, we only need to prove the existence of solutions for SBVP with1.1 Obviously, u is a solution of SBVP with 1.1 if and only if
u is a solution of
u Φf u : c KN f u. 3.1
We set
C1c,du ∈ C1 | uT1
c, uT2
d . 3.2
Obviously, N f u sends C1 into equi-integrable sets in L1 Similar to the proof of
sets in C1, thenΦf u is compact continuous.
Obviously, for any u ∈ C1, we haveΦf u ∈ C1
c,d, andΦf C1 is bounded By virtue
of Schauder fixed point theorem,Φf u has at least one fixed point u in C1
c,d Then, u is a
solution of SBVP with1.1 This completes the proof
Proof of Theorem 1.2 Let d with αT2 ≤ d ≤ βT2 be fixed According toTheorem 1.1,P with the following boundary value condition:
u1
T1
αT1
, u1
T2
possesses a solution u1such that
αt ≤ u1t ≤ βt, ∀t ∈T1, T2
Since limr→ T
1 wru
1p r−2 u1r exists, we have
u1r − u1
T1
r
T1
wt−1/pt−1wt1/pt−1 u1tdt
w
T11/pT1−1
u1
T1 r T
wt−1/pt−11 o1dt.
3.5
... < /p>2.31 < /p>
which is impossible The proof is completed < /p> Trang 9
Let us consider the auxiliary... < /p>
< /p>
< /p>
dr < /p>
≤ t2 < /p>
t1 < /p>
< /p>
w r 1/ p r −1... < /p>
< /p>
f < /p>
r, x, < /p>
w r 1/ p r −1 x < /p>
φ < /p>
w r 1/ p r −1x