A compactness result for a Gelfand-Liouville system with Lipschitz condition

0.07 MB·English

by  Samy Skander Bahoura

A COMPACTNESS RESULT FOR A GELFAND-LIOUVILLE SYSTEM WITH LIPSCHITZ
CONDITION.
SAMYSKANDERBAHOURA
5 ABSTRACT. Wegiveaquantizationanalysistoanellipticsystem(Gelfand-Liouvilletypesystem)withDirich-
1 letcondition.Anapplication,wehaveacompactnessresultforanellipticsystemwithLipschitzcondition.
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MathematicsSubjectClassification: 35J6035B4535B50

y
a
Keywords:quantization,blow-up,boundary,Gelfand-Liouvillesystem,Dirichletcondition,aprioriestimate,Lips-
M
chitzcondition.

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2

] 1. INTRODUCTION AND MAIN RESULTS
P
A
Weset∆ = ∂11 +∂22 onopensetΩofR2 withasmoothboundary.
.
h
t
a
m Weconsiderthefollowingequation:
[

2 −∆u= Vev in Ω ⊂ R2,
v
1 −∆v = Weu in Ω ⊂ R2,
4 (P)

4  u = 0 in ∂Ω,

0 
v = 0 in ∂Ω.
0
. 
1 

0 Here:
5
1
:
v 0 ∈ ∂Ω
i
X
r When u = v, the above system is reduced to an equation which was studied by many authors, with or
a
without the boundary condition, also for Riemann surfaces, see [1-16], one can find some existence and
compactness results, alsoforasystem.
Amongotherresults, wecanseein[6]thefollowingimportant Theorem,
TheoremA.(Brezis-Merle[6]).Considerthecaseofoneequation;if(u ) = (v ) and(V ) = (W ) are
i i i i i i i i
twosequencesoffunctionsrelativelytotheproblem(P)with,0< a ≤V ≤ b < +∞,then,forallcompact
i
setK ofΩ,
1
supu ≤c = c(a,b,K,Ω).
i
K
Theorem B (Brezis-Merle [6]).Consider the case of one equation and assume that (u ) and (V ) are
i i i i
twosequences offunctions relatively tothepreviousproblem (P)with,0 ≤ V ≤ b < +∞,and,
i
euidy ≤ C,
Ω
Z
then,forallcompactsetK ofΩ,
supu ≤ c = c(b,C,K,Ω).
i
K
Next,wecallenergythefollowingquantity:
E = euidy.
Ω
Z
Theboundedness oftheenergy isanecessary condition toworkontheproblem (P)asshowedin[6],by
thefollowingcounterexample.
TheoremC(Brezis-Merle[6]).Considerthecaseofoneequation,thentherearetwosequences(u ) and
i i
(V ) oftheproblem (P)with,0 ≤ V ≤ b < +∞,and,
i i i
euidy ≤ C,
Ω
Z
and
supu → +∞.
i
Ω
Note that in [11], Dupaigne-Farina-Sirakov proved (by an existence result of Montenegro, see [14]) that
thesolutionsoftheabovesystemwhenV andW areconstantscanbeextremalandthisconditionimplythe
boundedness oftheenergyanddirectlythecompactness. Notethatin[10],ifweassume(inparticular) that
′
∇logV and ∇logW and V > a > 0 or W > a > 0 and V,W are nonegative and uniformly bounded
thentheenergy isbounded andwehaveacompactness result.
Notethatinthecaseofoneequation,wecanprovebyusingthePohozaevidentitythatif+∞ > b≥ V ≥
a > 0, ∇V is uniformely Lipschitzian that the energy is bounded when Ω is starshaped. In [13] Ma-Wei,
usingthemoving-planemethodshowedthatthisfactistrueforalldomainΩwiththesameassumptions on
V. In [10] De Figueiredo-do O-Ruf extend this fact to a system by using the moving-plane method for a
system.
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Theorem C, shows that we have not a global compactness to the previous problem with one equation,
perhaps we need more information on V to conclude to the boundedness of the solutions. When ∇logV
isLipschitz function, Chen-Liand Ma-Weisee[7] and[13], showed thatwehave acompactness onallthe
open set. The proof is via the moving plane-Method of Serrin and Gidas-Ni-Nirenberg. Note that in [10],
we have the same result for this system when ∇logV and ∇logW are uniformly bounded. We will see
belowthatforasystemwealsohaveacompactness resultwhenV andW areLipschitzian.
Nowconsider thecaseofoneequation. Inthiscaseourequation haveniceproperties.
IfweassumeV withmoreregularity,wecanhaveanothertypeofestimates,asup+inf typeinequalities.
ItwasprovedbyShafrirsee[16],that,if(u ) ,(V ) aretwosequencesoffunctionssolutionsoftheprevious
i i i i
equation without assumption on the boundary and, 0 < a ≤ V ≤ b < +∞, then we have the following
i
interiorestimate:
a
C supu +infu ≤ c = c(a,b,K,Ω).
i i
b K Ω
(cid:16) (cid:17)
Now, if we suppose (V ) uniformly Lipschitzian with A the Lipschitz constant, then, C(a/b) = 1 and
i i
c = c(a,b,A,K,Ω), see[5].
Hereweareinterested bythecaseofasystem ofthistypeofequation. First,wegivethebehavior ofthe
blow-up points on theboundary and inthe second timewehave aproof ofcompactness of thesolutions to
Gelfand-Liouville typesystemwithLipschitzcondition.
Here,wewriteanextention ofBrezis-MerleProblem(see[6])is:
Problem. Suppose that Vi → V andWi → W inC0(Ω¯),with, 0 ≤ Vi ≤ b1 and 0 ≤ Wi ≤ b2 forsome
positiveconstantsb1,b2. Also,weconsiderasequenceofsolutions(ui),(vi)of(P)relativelyto(Vi),(Wi)
suchthat,
euidx ≤ C1, evidx ≤ C2,
Ω Ω
Z Z
isitpossibletohave:
||ui||L∞ ≤C3 = C3(b1,b2,C1,C2,Ω)?
and,
||vi||L∞ ≤C4 = C4(b1,b2,C1,C2,Ω)?
In this paper we give a caracterization of the behavior of the blow-up points on the boundary and also
a proof of the compactness theorem when V and W are uniformly Lipschitzian. For the behavior of the
i i
blow-uppointsontheboundary, thefollowingcondition areenough,
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0 ≤ Vi ≤ b1, 0 ≤ Wi ≤ b2,
Theconditions V → V andW → W inC0(Ω¯)arenotnecessary.
i i
Butfortheproof ofthecompactness forthe Gelfand-Liouville type system (Brezis-Merle type problem)
weassumethat:
||∇Vi||L∞ ≤ A1, ||∇Wi||L∞ ≤ A2.
Ourmainresultare:
Theorem 1.1. Assumethat maxΩui → +∞and maxΩvi → +∞Where(ui)and (vi)are solutions of
theprobleme(P)with:
0 ≤ Vi ≤ b1, and euidx ≤ C1, ∀ i,
Ω
Z
and,
0 ≤ Wi ≤ b2, and evidx ≤ C2, ∀ i,
Ω
Z
then; after passing to a subsequence, there is a finction u, there is a number N ∈ N and N points
x1,x2,...,xN ∈ ∂Ω,suchthat,
N
∂ u ϕ → ∂ uϕ+ α ϕ(x ), α ≥ 4π,
ν i ν j j j
Z∂Ω Z∂Ω j=1
X
for any ϕ∈ C0(∂Ω),and,
ui → u in Cl1oc(Ω¯ −{x1,...,xN}).
N
∂ u ϕ→ ∂ uϕ+ β ϕ(x ), β ≥ 4π,
ν i ν j j j
Z∂Ω Z∂Ω j=1
X
for any ϕ∈ C0(∂Ω),and,
vi → v in Cl1oc(Ω¯ −{x1,...,xN}).
Inthefollowingtheorem,wehaveaprooffortheglobalaprioriestimatewhichconcerntheproblem(P).
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Theorem 1.2. Assume that (u ),(v ) are solutions of (P) relatively to (V ),(W ) with the following
i i i i
conditions:
x1 = 0 ∈∂Ω,
and,
0 ≤ Vi ≤ b1, ||∇Vi||L∞ ≤ A1, and eui ≤ C1,
Ω
Z
0≤ Wi ≤ b2, ||∇Wi||L∞ ≤ A2, and evi ≤ C2,
Ω
Z
Wehave,
||ui||L∞ ≤ C3(b1,b2,A1,A2,C1,C2,Ω),
and,
||vi||L∞ ≤ C4(b1,b2,A1,A2,C1,C2,Ω),
2. PROOF OF THE THEOREMS
Proofoftheorem1.1:
SinceVievi andWieui areboundedinL1(Ω),wecanextractfromthosetwosequencestwosubsequences
whichconverge totwononegativemeasuresµ1 andµ2.
Ifµ1(x0) < 4π, byaBrezis-Merle estimate forthe firstequation, wehave eui ∈ L1+ǫ around x0,bythe
elliptic estimates, for the second equation, we have vi ∈ W2,1+ǫ ⊂ L∞ around x0, and , returning to the
∞
firstequation, wehaveui ∈ L aroundx0.
Ifµ2(x0) < 4π,thenui andvi arealsolocallybounded aroundx0.
Thus, we take a look to the case when, µ1(x0) ≥ 4π and µ2(x0) ≥ 4π. Byour hypothesis, those points
x0 arefinite.
Wewillseethatinside Ωnosuchpoints exist. Bycontradiction, assumethat, wehaveµ1(x0) ≥ 4π. Let
usconsideraballBR(x0)whichcontainonlyx0 asnonregular point. Thus,on∂BR(x0),thetwosequence
u andv areuniformlybounded. Letusconsider:
i i
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−∆zi = Vievi in BR(x0) ⊂R2,
( zi = 0 in ∂BR(x0).
Bythemaximumprinciple wehave:
z ≤ u
i i
andz → z almosteverywhereonthisball,andthus,
i
ezi ≤ eui ≤ C,
Z Z
and,
ez ≤ C.
Z
but,z isasolution tothefollowingequation:
−∆z = µ1 in BR(x0)⊂ R2,
( z = 0 in ∂BR(x0).
with,µ1 ≥ 4π andthus,µ1 ≥ 4πδx0 andthen,bythemaximumprinciple:
z ≥ −2log|x−x0|+C
thus,
ez = +∞,
Z
whichisacontradiction. Thus,thereisnononregular pointsinsideΩ
Thus,weconsider thecasewherewehavenonregular pointsontheboundary, weusetwoestimates:
∂νuidσ ≤ C1, ∂νvidσ ≤ C2,
Z∂Ω Z∂Ω
and,
′
||∇ui||Lq ≤ Cq, ||∇vi||Lq ≤Cq, ∀ i and 1 < q < 2.
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Wehavethesamecomputations, asinthecaseofoneequation.
Weconsiderapointsx0 ∈ ∂Ωsuchthat:
µ1(x0) < 4π.
We consider a test function on the boundary η we extend η by a harmonic function on Ω, we write the
equation:
−∆((u −u)η) = (V evi −Vev)η+ < ∇(u −u)|∇η >=f
i i i i
with,
|f |≤ 4π−ǫ+o(1) < 4π−2ǫ <4π,
i
Z
−∆((v −v)η) = (W eui −Weu)η+ < ∇(v −v)|∇η >= g ,
i i i i
with,
|g | ≤ 4π−ǫ+o(1) < 4π−2ǫ < 4π,
i
Z
By the Brezis-Merle estimate, we have uniformly, eui ∈ L1+ǫ around x0, by the elliptic estimates, for
the second equation, wehave vi ∈ W2,1+ǫ ⊂ L∞ around x0, and , returning to the first equation, we have
∞
ui ∈ L aroundx0.
Wehavethesamethingifweassume:
µ2(x0) < 4π.
Thus,ifµ1(x0)< 4π orµ2(x0)< 4π,wehaveforR > 0smallenough:
(ui,vi)∈ L∞(BR(x0)∩Ω¯).
Byourhypothesis thesetofthepointssuchthat:
µ1(x0) ≥ 4π, µ2(x0) ≥ 4π,
isfinite, and, outside this setu andv are locally uniformly bounded. Bythe elliptic estimates, wehave
i i
theC1 convergence touandv oneachcompactsetofΩ¯ −{x1,...xN}.
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Proofoftheorem1.2:
Without loss of generality, we can assume that 0 is a blow-up point (either, we use a translation). Also,
byaconformal transformation, wecanassumethatΩ = B+,thehalfball,and∂+B+ istheexterior part,a
1 1
part which not contain 0 and onwhich u and v converge in the C1 norm touand v. Letus consider B+,
i i ǫ
thehalfballwithradiusǫ > 0.
ThePohozaevidentity gives:
∆u < x|∇v > dx = − ∆v <x|∇u > dx+ g(∂ u ,∂ v )dσ, (1)
i i i i ν i ν i
ZBǫ+ ZBǫ+ Z∂+Bǫ+
Thus,
V evi < x|∇v > dx = − W eui < x|∇u > dx+ g(∂ u ,∂ v )dσ, (2)
i i i i ν i ν i
ZBǫ+ ZBǫ+ Z∂+Bǫ+
Afterintegration byparts, weobtain:
V evidx+ < x|∇V > evidx+ < ν|∇V >dσ+
i i i
ZBǫ+ ZBǫ+ Z∂Bǫ+
+ W euidx+ < x|∇W > euidx+ < ν|∇W > dσ =
i i i
ZBǫ+ ZBǫ+ Z∂Bǫ+
= g(∂ u ,∂ v )dσ,
ν i ν i
Z∂+Bǫ+
Also,foruandv,wehave:
Vevdx+ < x|∇V > evdx+ < ν|∇V > dσ+
ZBǫ+ ZBǫ+ Z∂Bǫ+
+ Weudx+ < x|∇W > eudx+ < ν|∇W > dσ =
ZBǫ+ ZBǫ+ Z∂Bǫ+
= g(∂ u,∂ v)dσ,
ν ν
Z∂+Bǫ+
If,wetakethedifference, weobtain:
(1+o(ǫ))( V evidx− Vevdx)+
i
ZBǫ+ ZBǫ+
8
+(1+o(ǫ))( W euidx− Weudx) =
i
ZBǫ+ ZBǫ+
= α1+β1 +o(ǫ)+o(1) = o(1),
acontradiction.
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DEPARTEMENTDEMATHEMATIQUES,UNIVERSITEPIERREETMARIECURIE,2PLACEJUSSIEU,75005,PARIS,FRANCE.
E-mailaddress: [email protected]
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