Volume 2008, Article ID 621621,19pages doi:10.1155/2008/621621
Research Article
The Method of Subsuper Solutions for Weighted
p
r
-Laplacian Equation Boundary Value Problems
Qihu Zhang,1, 2Xiaopin Liu,2and Zhimei Qiu2
1Department of Mathematics and Information Science, Zhengzhou University of Light Industry,
Zhengzhou, Henan 450002, China
2School of Mathematics Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China
Correspondence should be addressed to Zhimei Qiu,[email protected]
Received 23 May 2008; Accepted 21 August 2008
Recommended by Marta Garcia-Huidobro
This paper investigates the existence of solutions for weightedpr-Laplacian ordinary boundary value problems. Our method is based on Leray-Schauder degree. As an application, we give the existence of weak solutions forpx-Laplacian partial differential equations.
Copyrightq2008 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 weightedpr-Laplacian ordinary equation with right-hand terms depending on the first-order derivative:
−wrupr−2ufr, u,wr1/pr−1u0, ∀r∈T1, T2
, P
with one of the following boundary value conditions:
uT1
c, uT2
d, 1.1
guT1
,wT1
1/pT1−1uT
1
0, uT2
d, 1.2
guT1
,wT1
1/pT1−1uT
1
0, huT2
,wT2
1/pT2−1uT
2
0, 1.3
uT1
uT2
, wT1u
T1pT1−2u
T1
wT2u
T2pT2−2u
T2
, 1.4
wherep ∈ CT1, T2,Randpr > 1;w ∈ CT1, T2,Rsatisfies 0 < wr,∀r ∈ T1, T2,
andwr−1/pr−1∈L1T
1, T2;−wr|u|pr−2u
notationwT11/pT1−1uT1means limr→T1wr1/pr−1urexists and
wT1
1/pT1−1
uT1
: lim r→T1
wr1/pr−1
ur, 1.5
similarly
wT2
1/pT2−1uT
2
: lim r→T2−
wr1/pr−1ur; 1.6
wheregx, yandhx, yare continuous and increasing inyfor any fixedx, respectively. The study of differential equations and variational problems with nonstandardpr -growth conditions is a new and interesting topic. Many results have been obtained on these kinds of problem, for example,1–18. Ifwr≡pr≡pa constant,Pis 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 forpx-Laplacian problems. For example,
1ifΩ⊂Rnis an open bounded domain, then the Rayleigh quotient
λpx inf
u∈W01,pxΩ\{0}
Ω
1/px|∇u|pxdx
Ω
1/px|u|pxdx 1.7
is zero in general, and only under some special conditionsλpx>0see4, but the fact that
λp >0 is very important in the study ofp-Laplacian problems. In19, the author considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:
−Δpuλfu, v λuαvγ inΩ,
−Δqvλgu, v λuδvβinΩ,
uv0 on∂Ω,
S
the first eigenfunction is used to constructing the subsolution of problem Ssuccessfully. On the px-Laplacian problems, maybepx-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 ofpx-Laplacian problems, even if the first eigenfunction of px-Laplacian exists.On the existence of solutions for px -Laplacian equations Dirichlet problems via subsuper solution methods, we refer to13,14;
2ifwr≡ pr ≡pa constantand−Δpu >0, thenuis concave, this property is used extensively in the study of one-dimensionalp-Laplacian problems, but it is invalid for
−Δpr. It is another difference on−Δpand−Δpr:−|u|pr−2u;
3on the existence of solutions of the typicalpr-Laplacian problem:
−upr−2u|u|qr−2uC, r∈0,1, 1.8
because of the nonhomogeneity ofpt-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∈0,1qr<minr∈0,1pr, then the corresponding
satisfies Palais-Smale conditions see 3. But if minr∈0,1pr ≤ qr ≤ maxr∈0,1pr,
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 forp-Laplacian boundary value problems via subsuper solution methodsee20–24. 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 weightedpr-Laplacian equation with Dirichlet, Robin, and Periodic boundary value conditions. Moreover, the case of minr∈0,1pr ≤ qr ≤ maxr∈0,1pris 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:
ifor almost everyt∈I,the functionft,·,·is continuous;
iifor eachx, y∈R×R, the functionf·, x, yis measurable onI;
iiifor eachρ > 0,there is aαρ ∈L1I,Rsuch that, for almost everyt ∈I and every
x, y∈R×Rwith|x| ≤ρ,|y| ≤ρ, one has
ft, x, y≤αρt. 1.9
We setCCI,R,C1{u∈C|uis continuous inT
1, T2, limr→T1wr|u|pr−2ur and limr→T2−wr|u|pr−2ur exist}. Denote u0 supr∈T1,T2|ur| and u1 u0
wr1/pr−1u
0. The spaces C and C1 will be equipped with the norm ·0 and ·1,
respectively.
We say a functionu : I → Ris a solution ofP, ifu ∈ C1 andwr|u|pr−2uris
absolutely continuous and satisfiesPalmost every onI.
Functionsα, β∈C1are called subsolution and supersolution ofP, if|α|pr−2αrand
|β|pr−2βrare absolutely continuous and satisfy
−wrαpr−2αfr, α,wr1/pr−1α≤0, a.e. onI,
−wrβpr−2βfr, β,wr1/pr−1β≥0, a.e. onI.
1.10
Throughout this paper, we assume that α ≤ β are subsolution and supersolution, respectively. Denote
Ω0
t, x|t∈I, x∈αt, βt ,
Ω1
t, 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 1,2are positive value and continuous onΩ0,Kit, x, y i 1,2are 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, whereφ∈C1,∞,1,∞is increasing and satisfies
∞
1 1/φy1/p −−1
Our main results are as the following theorem.
Theorem 1.1. Iff is Caratheodory and satisfies (H1) and (H2),αandβsatisfyαT1≤ c≤ βT1,
αT2≤d≤βT2, thenPwith1.1possesses a solution.
Theorem 1.2. Iff is Caratheodory and satisfies (H1) and (H2),αandβsatisfyαT2≤d≤ βT2,
and
gαT1
,wT1
1/pT1−1αT
1
≥0≥gβT1
,wT1
1/pT1−1βT
1
, 1.12
thenPwith1.2possesses a solution.
Theorem 1.3. Iffis Caratheodory and satisfies (H1) and (H2),αandβsatisfy
gαT1
,wT1
1/pT1−1αT
1
≥0≥gβT1
,wT1
1/pT1−1βT
1
,
hαT2
,wT2
1/pT2−1
αT2
≤0≤hβT2
,wT2
1/pT2−1
βT2
,
1.13
thenPwith1.3possesses a solution.
Theorem 1.4. Iffis Caratheodory and satisfies (H1) and (H2),αandβsatisfy
αT1
αT2< β
T1 β
T2,
wT1α
T1pT1−2α
T1
≥wT2α
T2pT2−2α
T2
,
wT1β
T1pT1−2β
T1
≤wT2β
T2pT2−2β
T2
,
1.14
thenPwith1.4possesses a solution.
As an application, we consider the existence of weak solutions for the followingpx -Laplacian partial differential equation:
−div|∇u|px−2∇ufx, u,|x|n−1/px−1|∇u|0, ∀x∈Ω, 1.15
whereΩ is a bounded symmetric domain inRn,p ∈CΩ;Ris radially symmetric. We will writepx p|x| pr, andprsatisfies 1 < pr ∈ C,f ∈ CΩ×R×R,Ris radially symmetric with respect tox, namely,fx, u, v f|x|, u, v fr, u, v, andf satisfies the Caratheodory condition.
2. Preliminary
Denoteϕr, x |x|pr−2x,∀r, x∈I×R. Obviously,ϕhas the following properties.
Lemma 2.1. ϕis a continuous function and satisfies
ifor anyr∈T1, T2,ϕr,·is strictly increasing;
For any fixedr∈I, denoteϕ−1r,·as
ϕ−1r, x |x|2−pr/pr−1x, forx∈R\ {0}, ϕ−1r,0 0. 2.1
It is clear thatϕ−1r,·is continuous and send bounded sets into bounded sets. Let us now consider the simple problem
wrϕr, urfr, 2.2
with boundary value condition1.1, wheref ∈ L1. Ifuis a solution of2.2with1.1, by
integrating2.2fromT1tor, we find that
wrϕr, urwT1
ϕT1, u
T1
r
T1
ftdt. 2.3
Denote
Ffr r
T1
ftdt, awT1
ϕT1, u
T1
, 2.4
then
ur uT1
r
T1
ϕ−1r,wr−1aFfrdr. 2.5
The boundary conditions imply that
T2
T1
ϕ−1r,wr−1aFfrdr d−c. 2.6
For fixedh∈C, we denote
Λha
T2
T1
ϕ−1r,wr−1ahrdrc−d. 2.7
We have the following lemma.
Lemma 2.2. The functionΛhhas the following properties.iFor any fixedh∈C, the equation
Λha 0 2.8
has a unique solutionah ∈R.
iiThe functiona:C → R, defined in (i), is continuous and sends bounded sets to bounded sets.
Proof. iObviously, for any fixedh∈C,Λh·is continuous and strictly increasing, then, if
2.8has a solution, it is unique. Since wr−1/pr−1 ∈L1T
1, T2andh∈C, it is easy to see that
lim
It means the existence of solutions ofΛha 0.
In this way, we define a functionah :CT1, T2 → R, which satisfies
T2
T1
ϕ−1r,wr−1ah hrdr 0. 2.10
iiWe claim that
ah≤
|c−d| T2
T1ϕ
−1r,wr−1dr 1
p1
h0, ∀h∈C. 2.11
If it is false. Without loss of generality, we may assume that there are someh∈Csuch that
ah>
|c−d| T2
T1ϕ
−1r,wr−1dr 1
p1
h0, 2.12
then
ah h >
|c−d| T2
T1ϕ
−1r,wr−1dr 1
p1
, T2
T1
ϕ−1r,wr−1ah hrdrd−c
>
|c−d| T2
T1ϕ
−1r,wr−1dr 1
T2
T1
ϕ−1r,wr−1drd−c
|c−d| T2
T1
ϕ−1r,wr−1drd−c
>0.
2.13
It is a contradiction. Thus,2.11is valid. It mens thatasends bounded sets to bounded sets.
Finally, to show the continuity ofa, let{un}be a convergent sequence inCandun →
u, as n → ∞. Obviously, {au n} is a bounded sequence, then it contains a convergent subsequence{au nj}. Letau nj → a0asj → ∞. Since
T2
T1
ϕ−1r,wr−1aunj
unjr
dr0, 2.14
lettingj → ∞, we have
T2
T1
ϕ−1r,wr−1a0ur
dr 0, 2.15
fromi, we geta0 au , it meansais continuous.
Now, we define a:L1 → Ris defined by
ah aFh. 2.16
It is clear thatais a continuous function which send bounded sets ofL1into bounded
sets ofR, and hence it is a complete continuous mapping.
We continue now with our argument previous toLemma 2.2. By solving foruin2.3
and integrating, we find
ur uT1
Fϕ−1r,wr−1af Ffr r. 2.17 Let us define
Kht Fϕ−1r,wr−1ah Fh t, ∀t∈T1, T2
. 2.18
We denote byNfu : C1×T1, T2 → L1, the Nemytsky operator associated tof
defined by
Nfur f
r, ur,wr1/pr−1ur, a.e. onI. 2.19
It is easy to see the following lemma.
Lemma 2.3. uis a solution ofPwith boundary value condition1.1if and only ifuis a solution of the following abstract equation:
ucKNfu
. 2.20
Lemma 2.4. The operatorKis continuous and sends equi-integrable sets inL1into relatively compact
sets inC1.
Proof. It is easy to check thatKht∈C1. Sincewr−1/pr−1 ∈L1, and
wt1/pt−1Kht ϕ−1t,ah Fh, ∀t∈T 1, T2
, 2.21
it is easy to check thatKis a continuous operator fromL1toC1.
Let nowUbe an equi-integrable set inL1, then there existsρ∈L1, such that
ut≤ρt a.e. inI, for anyu∈U. 2.22 We want to show thatKU⊂C1is a compact set.
Let{un}be a sequence inKU, then there exist a sequence{hn} ∈ Usuch thatun
Khn. Fort1, 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 operatorais bounded and continuous, we can choose a subsequence of{ahn Fhn}
which we still denote{ahn Fhn}that is convergent inC, then
wtϕt,Khn
tahn
Fhn
2.24 is convergent inC. Since
Khn
t Fwr−1/pr−1ϕ−1r,ahn
Fhn t, ∀t∈
T1, T2
, 2.25
according to the continuous ofϕ−1and the integrability ofwr−1/pr−1inL1, thenKh
Lemma 2.5. Let α, β ∈ C1 be subsolution and supersolution of P, respectively, which satisfies
αt≤βtfor anyt∈T1, T2, then there exists a positive constantLsuch that, for any solutionxof
Pwith1.1whichsatisfiesαt≤xt≤βt,one haswt1/pt−1x0≤L.
Proof. We denote
μ0
T2
T1
A1
t, xtA2
t, xtdt, a0 max
wr1/pr−1|
r ∈T1, T2 ,
σmaxβs−αt|t, s∈T1, T2 ,
γmaxwt1/pt−1A1t, x|t, x∈Ω0 ,
2.26
then there exists at0∈T1, T2such that
wt0
1/pt0−1xt
0≤a0x
t0≤a0
σ T2−T1
. 2.27
FromH2, there exist positive numbersσ1andN1such that
N1≥σ1≥max
r∈I
M2a0
σ T2−T1
1
pr
, N1
σ1
1
φy1/pr−1dy > γσM1μ0, forr∈
T1, T2
uniformly.
2.28
Assume that our conclusion is not true, combining2.27, then there exists t1, t2 ⊂
T1, T2such thatwr1/pr−1xkeeps the same sign ont1, t2,and
wt1xpt1−2x
t1
σ1, w
t2xpt2−2x
t2
N1, 2.29
or inversely
wt1xpt1−2x
t1
−σ1, w
t2xpt2−2x
t2
−N1. 2.30
For simplicity, we assume that the former appears. Hence,
γσM1μ0<
N1
σ1
1
φy1/pr−1dy
t2
t1
wrxpr−1
φwrxpr−11/pr−1 dr
t2
t1
fr, x,wr1/pr−1x
φwr1/pr−1x dr
≤ t2
t1
wr1/pr−1A 1
r, xrxdrM1μ0
≤γσM1μ0,
2.31
Let us consider the auxiliary SBVP of the form
wrupr−2ufr, Rr, u, R1
wr1/pr−1uR
2r, 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
L11max
L, sup
r∈T1,T2
wr1/pr−1βr, sup
r∈T1,T2
wr1/pr−1αr, 2.34
whereLis defined inLemma 2.5, and
R2t, u
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
et, uu−βt
1u2 ut> βt,
0, αt≤ut≤βt,
et, uu−αt
1u2 ut< αt,
2.35
whereet, u 1A1t, Rt, u A2t, Rt, u.
Lemma 2.6. Let the conditions ofLemma 2.5hold, and letutbe any solution of SBVP with1.1
satisfiesαT1≤c≤βT1andαT2≤d≤βT2, thenαt≤ut≤βt,for anyt∈T1, T2.
Proof. We will only prove thatut ≤ βtfor anyt ∈ T1, T2. The argument of the case of
αt≤utfor anyt∈T1, T2is similar.
Assume thatut > βtfor some t ∈ T1, T2, then there exist a t0 ∈ T1, T2 and a
positive numberδsuch thatut0 βt0 δ,ut≤βt δ,for anyt∈T1, T2. Hence,
wt0
1/pt0−1ut
0
wt0
1/pt0−1βt
0
. 2.36
There exists a positive numberηsuch thatut> βt,for anyt∈J: t0−η, t0η⊂
T1, T2. From the definition ofβ, u,andfwe conclude that
wrβpr−2β≤fr, β,wr1/pr−1βfr, β <fr, u ont0−η1, t0η1
whereη1∈0, ηis small enough. For anyr∈t0, t0η1, we have
r
t0
wrβpr−2βdr < r
t0
fr, udr r
t0
wrupr−2udr. 2.38
From2.36and2.38, we have
βpr−2β<upr−2u ont0, t0η1
, 2.39
it means that
βδ< uont
0, t0η1
. 2.40
It is a contradiction to the definition oft0, sout≤βt,for anyt∈T1, T2.
3. Proofs of main results
In this section, we will deal with the proofs of main results.
Proof ofTheorem 1.1. From Lemmas 2.5 and 2.6, we only need to prove the existence of solutions for SBVP with1.1. Obviously, uis a solution of SBVP with1.1 if and only if
uis a solution of
u Φfu:cKNfu. 3.1
We set
C1c,du∈C1 |uT1
c, uT2
d . 3.2
Obviously, Nfu sends C1 into equi-integrable sets in L1. Similar to the proof of Lemma 2.4, we can conclude thatK sends equi-integrable sets inL1 into relatively compact
sets inC1, thenΦfuis compact continuous. Obviously, for anyu ∈C1, we haveΦ
fu ∈ C1c,d, andΦfC1is bounded. By virtue of Schauder fixed point theorem,Φfuhas at least one fixed pointu inC1c,d. Then,uis a solution of SBVP with1.1. This completes the proof.
Proof ofTheorem 1.2. LetdwithαT2≤d≤βT2be fixed. According toTheorem 1.1,Pwith
the following boundary value condition:
u1
T1
αT1
, u1
T2
d, 3.3
possesses a solutionu1such that
αt≤u1t≤βt, ∀t∈
T1, T2
. 3.4
Since limr→T1wru1pr−2u1rexists, we have
u1r−u1
T1
r
T1
wt−1/pt−1wt1/pt−1u
1t
dt
wT1
1/pT1−1u
1
T1
r
T1
wt−1/pt−11o1dt.
Similarly,
αr−αT1
wT1
1/pT1−1αT
1
r
T1
wt−1/pt−11o1dt. 3.6
Obviously
0≤ lim r→T1
u1r−αr
r T1
wt−1/pt−1dt
wT1
1/pT1−1u
1
T1
−wT1
1/pT1−1αT
1
, 3.7
then, we can conclude that
wT1
1/pT1−1u
1
T1
≥wT1
1/pT1−1αT
1
. 3.8
Sinceu1T1 αT1, andgx, yis increasing iny, we have
gu1
T1
,wT1
1/pT1−1u
1
T1
≥gαT1
,wT1
1/pT1−1αT
1
≥0. 3.9
We may assume thatgu1T1,wT11/pT1−1u1T1>0, or we get a solution forP
with1.2.
Sinceu1is a solution ofP, it is also a subsolution ofP. Similarly,Pwith boundary
value condition
v1
T1
βT1
, v1
T2
d, 3.10
possesses a solutionv1such that
u1t≤v1t≤βt, ∀t∈
T1, T2
, 3.11
which satisfies
wT1
1/pT1−1v
1
T1
≤wT1
1/pT1−1βT
1
, 3.12
then
gv1
T1
,wT1
1/pT1−1
v1T1
≤gβT1
,wT1
1/pT1−1
βT1
≤0. 3.13
Obviously, u1t and v1t are subsolution and supersolution of P with 1.2,
respectively. According toTheorem 1.1,Pwith boundary value condition
xT1
u1
T1
v1
T1
2 , x
T2
d, 3.14
possesses a solutionxsuch that
u1t≤xt≤v1t, ∀t∈
T1, T2
. 3.15
We may assume thatgxT1,wT11/pT1−1xT1/0, or we get a solution forP
IfgxT1,wT11/pT1−1xT1 > 0, then denoteu2t xtandv2t v1t; if
gxT1,wT11/pT1−1xT1 < 0, then denotev2t xtandu2t u1t. It is easy to
see thatu2tandv2tboth are solutions ofPand satisfy
gu2
T1
,wT1
1/pT1−1u
2
T1
>0> gv2
T1
,wT1
1/pT1−1v
2
T1
,
u2t≤v2t, ∀t∈
T1, T2
, u2t, v2t
⊆u1t, v1t
, ∀t∈T1, T2
, u2 T2
dv2
T2 , v2 T1 −u2
T1 v1 T1 −u1
T1
2 .
3.16
Repeated the step, we get two sequences{un}and{vn}, all are solutions ofP, and satisfy
gun
T1
,wT1
1/pT1−1u
n
T1
>0> gvn
T1
,wT1
1/pT1−1v
n
T1
, 3.17
unt≤vnt, ∀t∈
T1, T2
, un1t, vn1t
⊆unt, vnt
, ∀t∈T1, T2
, 3.18
un
T2
dvn
T2
, 3.19
vn1
T1
−un1
T1 vn T1 −un
T1
2 . 3.20
According to Lemma 2.5, {unt} and {vnt} both are bounded in C1, then
{wT11/pT1−1unT1} is a bounded set and has a convergent subsequence. Note that
{unt}are solutions ofPand satisfy
wrϕr, unranF
Nf
un
r, 3.21
where
FNf
un r r T1 Nf un
dt, anw
T1
ϕT1, un
T1
. 3.22
Similar to the proof ofLemma 2.4,{unt}possesses a convergent subsequence{unit}
inC1, and then{a
n}is bounded. From2, we can see that{unt}and{vnt}have uniform
C1,αregularity. We may assume thatu
nit → utinC1andvnjt → vtinC1.
It is easy to see thatut≤vtboth are solutions ofP. From the definition of{unt} and{vnt}, we can see that
uT2
dvT2
. 3.23
Combining3.18and3.20, we have
ut≤vt, ∀t∈T1, T2
,
uT1 lim
j→ ∞uni
T1
lim
j→ ∞vni
T1
vT1
. 3.24
Similar to3.7, we have
wT1
1/pT1−1uT
1
≤wT1
1/pT1−1vT
1
From3.17and the continuity ofg, we can see that
guT1
,wT1
1/pT1−1uT
1
≥0≥gvT1
,wT1
1/pT1−1vT
1
. 3.26
From3.25,3.26, and the increasing property ofgx, ywith respect toy, we have
guT1
,wT1
1/pT1−1uT
1
0gvT1
,wT1
1/pT1−1vT
1
. 3.27
Thus,uandvboth are solutions ofPwith1.2. This completes the proof.
Proof ofTheorem 1.3. According toTheorem 1.2,Ppossesses a solutionu1such that
gu1
T1
,wT1
1/pT1−1u 1
T1
0,
u1
T2
αT2
,
αt≤u1t≤βt, ∀t∈
T1, T2
.
3.28
Similar to the proof of3.7, we have
wT2
1/pT2−1u
1
T2
≤wT2
1/pT2−1αT
2
. 3.29
Obviously,hu1T2,wT21/pT2−1u1T2≤0. We may assume that
hu1T2,wT21/pT2−1u1T2<0, 3.30
or we get a solution forPwith1.3, thenu1is a subsolution ofPwith1.3.
According toTheorem 1.2,Ppossesses a solutionv1such that
gv1
T1
,wT1
1/pT1−1v 1
T1
0,
v1
T2
βT2
,
u1t≤v1t≤βt, ∀t∈
T1, T2
.
3.31
Similarly,hv1T2,wT21/pT2−1v1T2≥0. We may assume that
hv1
T2
,wT2
1/pT2−1v
1
T2
>0, 3.32
or we get a solution forPwith1.3, thenv1is a supersolution ofPwith1.3.
According toTheorem 1.2,Ppossesses a solutionxsuch that
gxT1
,wT1
1/pT1−1xT 1
0, xT2
u1
T2
v1
T2
2 ,
u1t≤xt≤v1t, ∀t∈
T1, T2
.
3.33
We may assume thathxT2,wT21/pT2−1xT2/0, or we get a solution forP
ifhxT2,wT21/pT2−1xT2<0, then denotev2t v1tandu2t xt. It is easy to
see thatu2tandv2tboth are solutions ofPand satisfy
hu2
T2
,wT2
1/pT2−1u
2
T2
<0< hv2
T2
,wT2
1/pT2−1v
2
T2
,
u2t≤v2t, ∀t∈
T1, T2
, u2t, v2t
⊆u1t, v1t
, ∀t∈T1, T2
, v2 T2 −u2
T2 v1 T2 −u1
T2
2 .
3.34
Repeating the step, similar to the proof ofTheorem 1.2, we get two sequences{un}and
{vn}, all are solutions ofP, and satisfy
gun
T1
,wT1
1/pT1−1
unT1
0gvn
T1
,wT1
1/pT1−1
vnT1
,
hun
T2
,wT2
1/pT2−1
unT2
<0< hvn
T2
,wT2
1/pT2−1
vnT2
,
unt≤vnt, ∀t∈
T1, T2
, un1t, vn1t
⊆unt, vnt
, ∀t∈T1, T2
.
vn1
T2
−un1
T2 vn T2 −un
T2
2 .
3.35
Similar to the proof of Theorem 1.2, {unt} and {vnt} possess convergent subsequence {unit} and {vnjt} in C1, respectively. We may assume thatunit → ut
inC1, and similarv
njt → vtinC
1. It is easy to see thatut≤ vtboth are solutions of
Pwith1.3. This completes the proof.
Proof ofTheorem 1.4. According toTheorem 1.1,Ppossesses solutionu1which satisfies
u1
T1
αT1
, u1
T2
αT2
, αt≤u1t≤βt, t∈
T1, T2
. 3.36
We may assume thatwT1ϕT1, u1T1/wT2ϕT2, u1T2, or we get a solution for
Pwith1.4, thenwT1ϕT1, u1T1 > wT2ϕT2, u1T2, andu1 is a subsolution ofP.
According toTheorem 1.1,Ppossesses solutionsv1which satisfies
v1
T1
βT1
, v1
T2
βT2
, u1t≤v1t≤βt, t∈
T1, T2
. 3.37
We may assume thatwT1ϕT1, v1T1/wT2ϕT2, v1T2, or we get a solution for
Pwith1.4, thenwT1ϕT1, v1T1< wT2ϕT2, v1T2, andv1is a supersolution ofP.
According toTheorem 1.1,Ppossesses solutionsxand satisfies
xT1
u1 T1 v1 T1
2 x
T2
, u1t≤xt≤v1t, t∈
T1, T2
. 3.38
Similar to the proof ofTheorem 1.2, we obtainuandvthat are solutions ofP, which satisfy
ut≤vt, t∈T1, T2
, 3.39
uT1
uT2
vT1
vT2
, 3.40
wT1
ϕT1, u
T1
≥wT2
ϕT2, u
T2
, 3.41
wT1
ϕT1, v
T1
≤wT2
ϕT2, v
T2
From3.39and3.40, we have
wT1
ϕT1, u
T1
≤wT1
ϕT1, v
T1
,
wT2
ϕT2, u
T2
≥wT2
ϕT2, v
T2
.
3.43
From3.41,3.42, and3.43, we can conclude thatPwith1.4possesses a solution. This completes the proof.
On the case of minr∈−R,Rpr≤qr≤maxr∈−R,Rpr, we consider
−|u|pr−2uC|u|qr−2uer r∈−R, R,
u−R uR 0,
I
whereqr, er ∈ C−R, R,R, minr∈−R,Rpr ≤ qr ≤ maxr∈−R,Rpr,Cis a positive
constant. Denote
p max
r∈−R,Rpr, p
− min
r∈−R,Rpr. 3.44
We have the following corollary.
Corollary 3.1. Ifp∈CR,1,∞is even,Rsatisfies
R≤1C max r∈−R,Rer
−p−1/pp−−1
, 3.45
thenIpossesses at least a nontrivial solution.
Proof. It is easy to see thatα≡0 is a subsolution ofI. Denote
βr 1−
r
0
|μs|1/ps−1−1μs ds, 3.46
whereμis a positive constant satisfyingβR 0. Sincepis even, thenβ−R 0. It is easy to see that 0≤βr≤1,∀r∈−R, R, and
−βpr−2βμ R
0
|s|1/ps−1ds1−pξ≥R 0
|s|1/p−1ds1−pξ
≥ R
0
|s|1/p−1ds1−p
−
≥1C max
r∈−R,Rer≥C|β|
qr−2βer,
3.47
whereξ ∈ −R, R. Then,βis a supersolution ofI. FromTheorem 1.1, one can see thatI
4. Applications in PDE
LetΩ⊂Rnbe an open bounded domain. In this section, we always denote
p max x∈Ω
px, p− min
x∈Ω px. 4.1
Let us now consider1.15with one of the following boundary value conditions:
u|∂Ω0, 4.2
∇u0, ∀x∈∂Ω. 4.3
Ifuis a radial solution of1.15, then it can be transformed into
−rn−1upr−2urn−1fr, u,|r|n−1/pr−1u0, r∈T1, T2
,whereT1≥0, 4.4
and the boundary value condition will be transformed into1.1,1.2, or1.3, respectively.
Theorem 4.1. If4.4has subsolution and supersolutionαandβ,respectively, satisfyingαt≤βt for anyt∈T1, T2, andfis continuous and satisfies (H1)-(H2), in each of the following cases:
i0< T1< T2,Ω {x∈Rn|T1<|x|< T2},αT1≤0≤βT1, andαT2≤0≤βT2;
ii0 T1 < T2,Ω {x∈Rn |T1 < |x| < T2} B0;T2\ {0}, andp− > n;αT1≤ 0 ≤
βT1,αT2≤0≤βT2;
iii0T1 < T2,Ω {x∈Rn| |x|< T2}B0;T2, andp− > n;wT11/pT1−1αT1≥
0≥wT11/pT1−1βT1,αT2≤0≤βT2;
then1.15with4.2has at least one weak radially symmetric solutionu.
Proof. Notice that rn−1−1/pr−1 ∈ L10, T
2 and satisfies 0 < rn−1,∀r ∈ 0, T2. We can
conclude the existence of solutions for 4.4 with 1.1,1.2, or 1.3, from Theorems 1.1,
1.2, and1.3. If limr→0rn−1|u|pr−2ur 0, notice that
upr−2
ur≤r1−n r
0
tn−1ft, u,|t|n−1/pt−1udt
≤ r
0
ft, u,|t|n−1/pt−1udt−→0 asr −→0,
4.5
then we haveu0 0. This completes the proof.
Similarly, we have the following theorem.
Theorem 4.2. If4.4has subsolution and supersolutionαandβ,respectively, satisfyingαt≤βt for anyt∈T1, T2, and
wT1
1/pT1−1αT
1
≥0≥wT1
1/pT1−1βT
1
,
wT2
1/pT2−1αT
2
≤0≤wT2
1/pT2−1βT
2
,
andfis continuous and satisfies (H1)-(H2), in each of the following cases:
i0< T1< T2;Ω {x∈Rn|T1<|x|< T2};
ii0T1< T2;Ω {x∈Rn|T1<|x|< T2}B0;T2\{0}orΩ B0;T2;p∈C1Ω;R
andp−> n;
then1.15with4.3has at least one weak radially symmetric solutionu.
On the case ofp−≤qx≤p, we consider
−div|∇u|px−2∇uC|u−1|qx−2uex, x∈Ω,
ux 0, x∈∂Ω, II
whereΩ {x ∈ Rn | 0 < |x| < R},qx, ex ∈ CΩ,R, 2 ≤ n < p− ≤ qx ≤ p,Cis a positive constant.
We have the following corollary.
Corollary 4.3. Ifp∈CRn,1,∞is radial, andRsatisfies
R≤min
1,
1− 1 2p−−1
p−1 n−3/2
1Cmaxx∈Ωex
1/p−−3/2
, 4.7
thenIIpossesses at least a nontrivial solution.
Proof. It is easy to see thatα≡0 is a subsolution ofII. Denote
βr 1−
r
0
μs−1/21/ps−1ds, 4.8
whereμis a positive constant satisfyingβR 0. It is easy to see that 0≤βr≤1,∀r∈0, R, and
−rn−1βpr−2βμ
n−3
2
rn−5/2 R
0
|s|−1/2ps−1ds1−pξn−3
2
rn−5/2
≥ R
0
|s|−1/2p−−1ds1−pξn−3
2
rn−1
≥R−1/2p−−111−p−
1− 1 2p−−1
p−1
n−3
2
rn−1
≥rn−11Cmax x∈Ω ex
≥rn−1C|β−1|qx−2βex,
4.9
where ξ ∈ Ω. Then, β is a supersolution of II. From Theorem 4.1, one can see that II
Acknowledgments
This work is partly supported by the National Science Foundation of China10701066 and 10671084, China Postdoctoral Science Foundation20070421107, and the Natural Science Foundation of Henan Education Committee2007110037.
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