A priori bounds and a Liouville theorem on a half-

arXiv:0709.2821v1[math.AP]18Sep2007APRIORIBOUNDSANDALIOUVILLETHEOREMONAHALF-SPACEFORHIGHER-ORDERELLIPTICDIRICHLETPROBLEMSWOFGANGREICHELANDTOBIASWETHAbstract.Weconsiderthe2m-thorderellipticboundaryvalueproblemLu=f(x,u)onaboundedsmoothdomainΩ⊂RNwithDirichletboundaryconditionsu=∂∂νu=...=(∂∂ν)m−1u=0on∂Ω.TheoperatorLisauniformlyellipticoperatoroforder2mgivenbyL=−PNi,j=1aij(x)∂2∂xi∂xj
m+P|α|≤2m−1bα(x)Dα.Forthenonlinearityweassumethatlims→∞f(x,s)sq=h(x),lims→−∞f(x,s)|s|q=k(x)whereh,k∈C(Ω)arepositivefunctionsandq1ifN≤2m,1qN+2mN−2mifN2m.Weproveaprioribounds,i.e,weshowthatkukL∞(Ω)≤Cforeverysolutionu,whereC0isaconstant.Thesolutionsareallowedtobesign-changing.Theproofisdonebyablow-upargumentwhichreliesonthefollowingnewLiouville-typetheoremonahalf-space:ifuisaclassical,bounded,non-negativesolutionof(−Δ)mu=uqinRN+withDirichletboundaryconditionson∂RN+andq1ifN≤2m,1q≤N+2mN−2mifN2mthenu≡0.1.IntroductionAprioriboundsforsolutionsofellipticboundaryvalueproblemshavebeenofmajorimportanceatleastasfarbackasSchauder’sworkinthe1930s.Inthispaperweproveaprioriestimatesonbounded,smoothdomainsΩ⊂RNforsolutionsofhigherorderboundaryvalueproblemsoftheform(1.1)Lu=f(x,u)inΩ,u=∂∂νu=...=∂∂νm−1u=0on∂Ω.Hereνistheunitexteriornormalon∂ΩandL=−NXi,j=1aij(x)∂2∂xi∂xjm+X|α|≤2m−1bα(x)Dαisauniformlyellipticoperatorwithcoefficientsbα∈L∞(Ω)andaij∈C2m−2(Ω)suchthatthereexistsaconstantλ0withλ−1|ξ|2≤PNi,j=1aij(x)ξiξj≤λ|ξ|2forallξ∈RN,x∈Ω.Ourmainresultisthefollowing:Theorem1.SupposeΩ⊂RNisaboundeddomainwith∂Ω∈C2m.Letm∈Nandassumethatq1ifN≤2mand1qN+2mN−2mifN2m.SupposefurtherthatthereexistDate:February1,2008.2000MathematicsSubjectClassification.Primary:35J40;Secondary:35B45.Keywordsandphrases.Higherorderequation,aprioribounds,Liouvilletheorems,movingplanemethod.12WOFGANGREICHELANDTOBIASWETHpositive,continuousfunctionsk,h:Ω→(0,∞)suchthat(1.2)lims→+∞f(x,s)sq=h(x),lims→−∞f(x,s)|s|q=k(x)uniformlywithrespecttox∈Ω.ThenthereexistsaconstantC0dependingonlyonthedataaij,bα,Ω,N,q,h,ksuchthatkuk∞≤Cforeverysolutionuof(1.1).Remark2.Supposethenonlinearitydependsonarealparameterλ,i.e,fλ:Ω×R→Randlims→+∞fλ(x,s)λsq=h(x),lims→−∞fλ(x,s)λ|s|q=k(x)uniformlywithrespecttox∈Ωandλ∈[λ0,∞)whereλ00.ThentheaprioriboundofTheorem1dependsadditionallyonλ0butnotonλ.Thisisimportantinthestudyofglobalsolutionbranchesofaparameterdependentversionof(1.1),whichwewillpursueinfuturework.Wearefocusingonthecaseofsuperlinearnonlinearitiesf(x,u)withsubcriticalgrowth.Amodelnonlinearityisf(x,s)=|s|q.OurresultsholdwithnorestrictionontheshapeofthedomainΩandforgeneral,possiblysign-changingsolutions.Thisisimportantsincethelackofthemaximumprincipleforhigherorderequationsdoesnotallowtorestrictattentiontopositivesolutionsonly.Inthesecond-ordercasem=1aprioriboundsforpositivesolutionshavebeenestablishedforsubcritical,superlinearnonlinearitiesviadifferentmethodsbyBrezis,Turner[7],Gidas,Spruck[13],DeFigueiredo,LionsandNussbaum[10]andrecentlybyQuittner,Souplet[20]andMcKenna,Reichel[19].Inthehigher-ordercasem≥2thetheoryisfarlessdevelopedandstronglydependsonthetypeofboundaryconditionsconsidered.ForDirichletboundaryconditionsweonlyknowofaresultofSoranzo[23],whoprovedaprioriboundsforpositiveradialsolutionsonaballifL=(−Δ)m.ForNavierboundaryconditionsthepictureismorecomplete.LetL=(−L0)mwhereL0=aij∂2∂xi∂xj+bα∂∂xαisasecondorderoperatorandsupposetheboundaryconditionsareofNavier-type:(1.3)u=(−L)u=...=(−L)m−1u=0on∂Ω.Soranzo[23]provedaprioriboundsforpositivesolutionsifL0=ΔandΩisaboundedsmoothconvexdomain.Recently,Sirakov[22]improvedthisresulttogeneraloperatorsL=(−L0)mandgeneralboundedsmoothdomains.Bothauthorsstronglyusethefactthattheboundaryconditions(1.3)allowtowritetheproblemasacoupledsystemofsecondorderequations,whereeachequationiscomplementedwithDirichletboundaryconditions.Inthiscasemaximumprinciplesareavailable.Incontrast,thehigherorderDirichletproblemcannotberewrittenasasystemandthereforerequiresdifferenttechniques.Inourapproachweextendtheso-called“scalingargument”ofGidasandSpruck[13],whichtheyusedtodealwiththesecondordercasem=1andpositivesolutions.Letusgiveabriefsketchoftheirmethod.GidasandSpruckassumethatthereexistsasequenceofpositivesolutionswithL∞-normtendingto+∞.Afterrescalingthesolutionstonorm1andblowing-upthecoordinatesonecantakealimitoftherescaledsolutionsandobtainsAPRIORIBOUNDSFORHIGHER-ORDERELLIPTICPROBLEMS3anontrivialpositivesolutionofalimitboundaryvalueproblem−Δu=uqoneitherthefull-spaceRNorthehalf-spaceRN+={x∈RN:x10}togetherwithDirichletboundaryconditions.ThenacontradictionisreachedprovidedthataLiouville-typeresultisavailable,i.e.,aresultwhichshowsthatthenon-negativesolutionsofthelimitproblemmustbeidenticallyzero.ForsubcriticalqGidasandSpruck[13],[14]provedboththefull-spaceandthehalf-spaceLiouvilletheoremfor−Δu=uqviathemethodofmovingplanes.InordertodealwiththehigherorderDirichletproblem(1.1)andsolutionswhichmaychangesign,theblowupprocedurehastobemodified.Indeed,evenunderassumption(1.2),thereseemstobenodirectargumenttoexcludethecaseofnegativeblowup(i.e.,theexistenceofasequenceofsolutionswhichisnotuniformlybound