Part I Elliptic equations and systems

SingularityanddecayestimatesinsuperlinearproblemsviaLiouville-typetheorems.PartI:EllipticequationsandsystemsPeterPol´aˇcik∗SchoolofMathematics,UniversityofMinnesotaMinneapolis,MN55455,USAe-mail:[e-mail]†DepartmentofAppliedMathematicsandStatistics,ComeniusUniversity,Mlynsk´adolina,84248Bratislava,Slovakiae-mail:[e-mail],G´eom´etrieetApplications,InstitutGalil´ee,Universit´eParis-Nord,93430Villetaneuse,Francee-mail:[e-mail],westudysomenewconnectionsbetweenLiouville-typetheo-remsandlocalpropertiesofnonnegativesolutionstosuperlinearellipticprob-lems.Namely,wedevelopageneralmethodforderivationofuniversal,point-wiseaprioriestimatesoflocalsolutionsfromLiouville-typetheorems,whichprovidesasimplerandunifiedtreatmentforsuchquestions.Themethodisbasedonrescalingargumentscombinedwithakey“doubling”property,anditisdifferentfromtheclassicalrescalingmethodofGidasandSpruck.Asanimportantheuristicconsequenceofourapproach,itturnsoutthatuniversalboundednesstheoremsforlocalsolutionsandLiouville-typetheoremsarees-sentiallyequivalent.∗SupportedinpartbyNSFGrantDMS-0400702†SupportedinpartbyVEGAGrant1/3021/0612Pol´aˇcik,QuittnerandSoupletThemethodenablesustoobtainnewresultsonuniversalestimatesofspatialsingularitiesforellipticequationsandsystems,underoptimalgrowthassump-tionswhich,unlikepreviousresults,involveonlythebehaviorofthenonlinearityatinfinity.ForsemilinearsystemsofLane-Emdentype,theseseemtobethefirstresultsonsingularityestimatestocoverthefullsubcriticalrange.Inaddi-tion,wegiveanaffirmativeanswertotheso-calledLane-Emdenconjectureinthreedimensions.1IntroductionTheaimofthispaperistostudysomenewconnectionsbetweenLiouville-typetheo-remsandlocalpropertiesofnonnegativesolutionstosuperlinearellipticproblems.Inallthepaper,theword“solution”alwaysrefersto“nonnegativesolution”,regardlessofwhetheritisspecificallymentioned.Bya(nonlinear)Liouville-typetheorem,weheremeanthestatementofnonexistenceofnontrivialboundedsolutionsonthewholespaceoronahalf-space.Inthelasttwodecades,followingtheseminalpaper[17],Liouville-typetheo-remshavebeenwidelyused,inconjunctionwithrescalingarguments,inderivationofpointwiseaprioriestimatesofsolutionsofboundaryvalueproblems(see[12]forasurvey,formorerecentresultsandreferencesseee.g.[25,15,7,31,14,32]).Inthispaper,wedevelopageneralmethodforderivationofpointwiseaprioriestimatesoflocalsolutions(i.e.,onanarbitrarydomainandwithoutanyboundaryconditions),fromLiouville-typetheorems.Thismethodenablesustoobtainnewresultsonuniversalestimatesofspatialsingularitiesforellipticproblems.Atthesametime,itgivessomewhatsimplerproofsofseveralknownresultsandprovidesaunifiedtreatmentforsuchquestions.Asafurthermotivationtoourapproach,letusmentionthatinanimportantrecentpaper[29]onquasilinearellipticequations,SerrinandZouhaveobservedthatLiouvilletheoremscanbeseenasaconsequenceandalimitingcaseofuniversalboundednesstheorems.Forinstance,forthemodelequation−Δu=up(1.1)withsubcriticalp1,thenonexistenceofnontrivialsolutionsinRnisadirectcorollarytotheuniversalboundednessresult[11,Lemma1],whichstatesthatanyclassicalsolutionof(1.1)inanarbitrarydomainΩ⊂Rnsatisfiesu(x)≤C(n,p)dist−2p−1(x,∂Ω),x∈Ω.(1.2)Inotherwords,citing[29,p.82]:“theybothprovideupperboundsfornonnegativesolutions”,withtheLiouvilletheorem“beingtheextremecasewherethedomainisallofRnandtheupperboundbecomeszero”,anduniversalboundednesstheoremsprovide“acontinuousembeddingoftheLiouvilletheoreminafamilyofresultsforSingularityestimatesviaLiouville-typetheorems3anexpandingsequenceofboundeddomains”.Fromthispointofview,aremarkableconsequenceoftheapproachinthepresentpaperisthattheconverseisalsotrue,sothatLiouvilletheoremsanduniversalboundednesstheoremsareinfactequivalent(forproblemswithhomogeneousnonlinearitiessuchas(1.1),or(3.2)and(4.1)below).Asaconsequenceofourmethod,weimproveanumberofknownresultsonsemilinearandquasilinearellipticequations.Inparticular,althoughitwasnaturaltoexpectthatsingularityestimatesshoulddependonlyonthebehaviorofthenon-linearityatinfinity,allpreviousresults[16,5,29,4]requiredglobalassumptions.Incontrast,ourmethoddoesnotrelyonanyglobalassumptions,thusourresultscon-firmtheaboveexpectation.WealsotreatsemilinearsystemsofLane-Emdentype;thesingularityestimatesthatweobtainseemtobethefirstresultsofthistypetocoverthefullsubcriticalrange(belowtheso-calledSobolevhyperbola).Otherby-productsofthemethodarestrongLiouville-typetheorems,thatisstate-mentsonnonexistenceofnontrivialsolutions(boundedornot)onthewholespaceoronahalf-space.Inparticular,wegiveanaffirmativeanswertotheso-calledLane-Emdenconjectureforellipticsystemsinthreedimensions.Themethodinthispaperisbasedonrescalingargumentscombinedwithakey“doubling”property(seeLemma5.1below).Aheuristicexplanationofourapproach,andofthedifferenceswiththe“classical”rescalingmethod[17](whereglobalaprioriestimatesarederivedfromLiouville-typeresults)isgivenatthebeginningofSection5.Thedoublingpropertyisanextensionofanideaof[18].Inthatwork(seealsothereferencesin[24]),asimila