A priori bounds for superlinear elliptic equations with semidefinite nonlinearity Yūki Naito.pdf

该【A priori bounds for superlinear elliptic equations with semidefinite nonlinearity Yūki Naito 】是由【金钏】上传分享，文档一共【23】页，该文档可以免费在线阅读，需要了解更多关于【A priori bounds for superlinear elliptic equations with semidefinite nonlinearity Yūki Naito 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..NonlinearAnalysis151(2017)18–40ContentslistsavailableatScienceDirectNonlinearAnalysisate/naAprioriboundsforsuperlinearellipticequationswithsemidefinitenonlinearityYˉukiNaitoa,?,TakashiSuzukib,YoheiToyotabaDepartmentofMathematics,GraduateSchoolofScienceandEngineering,EhimeUniversity,Matsuyama790-8577,JapanbDivisionofMathematicalScience,DepartmentofSystemsInnovation,GraduateSchoolofEngineeringScience,OsakaUniversity,Osaka560-8531,JapanarticleinfoabstractArticlehistory:WederiveaprioriboundsforpositivesolutionsofthesuperlinearellipticproblemsReceived30April2016??u=a(x)uponaboundeddomain?inRN,wherea(x)isH¨oldercontinuousAccepted23November2016in?.Ourmainmotivationistostudythecasewherea(x)≥0,a(x)?≡0anda(x)CommunicatedbyEnzoMitidierihassomezerosetsin?.Weshowthat,inthiscase,thescalingargumentsreduceMSC:theproblemofaprioriboundstotheLiouville-typeresultsfortheequation??u=A(x′)upinRN,whereAisthecontinuousfunctiondefinedonthesubspaceRkwithprimary35B451≤k≤Nandx′∈–?2016TheAuthor(s).-NC-NDlicenseKeywords:(/licenses/by-nc-nd//).AprioriestimatesLiouville-???u=a(x)upin?,()u(x)=φ(x)on??,where?isaboundeddomaininRN(N≥2)withsmoothboundary??,p>1,a(x)isH¨oldercontinuousandsatisfiesa(x)≥0,a(x)?≡0in?,andφ∈C(??)satisfiesφ≥0on??.Inordertoproveexistenceandmultiplicityofpositivesolutionsto(),itisimportanttoobtainaprioriboundsforitspositivesolutions(see,.[4]).ItiswellknownthatLiouville-typeresultsenableone?-mailaddresses:******@ehime-(),******@-(),y-******@-().http://dx./.-546X/?2016TheAuthor(s).-NC-NDlicense(/licenses/by-nc-nd//).:.../NonlinearAnalysis151(2017)18–(x)iscontinuousandstrictlypositiveon?,aprioriboundswasobtainedbyGidasandSpruck[7].Themethodin[7]isbasedonrescalingandLiouvilletheoremsfornonnegativesolutionstotheequation??u=upeitherinRNorinahalf-[7]forthesemidefinitefunctiona(x),namely,a(x)≥0,a(x)?≡0in?anda(x)hassomezerosetsin?.In[7],theuniformpositivityofa(x)playsacrucialroletoderivetheentirespaceproblem??u=upinRNorinahalf-,inthesemidefinitecase,thescalingargumentsreducetheproblemofaprioriboundstotheLiouville-typeresultsfornonnegativesolutionstotheequation??u=A(x′)upinRN,whereAisthecontinuousfunctiondefinedonRkwith1≤k≤Nandx′∈={x∈?:a(x)=0}.FirstweconsiderthecaseZ=?\ω,whereω?RNisanopensetwithC1-boundary?ωsatisfyingω??.Forx∈?ω,?a(x)standsforlim?a(x).Wedenotebyξ(x)00x∈ω,x→x00theouterunitnormalvectoron?:(A)In(),a(x)isH¨oldercontinuousandsatisfiesa(x)≥0,a(x)?≡0in?.ThesetZisgivenbyZ=?\ω,whereω?RNisanopensetwithC1smoothboundary?ωsatisfyingω??.Thefollowingeither(i)or(ii)musthold.(i)a∈C1(ω)and?a(x)?=0forallx∈?ω.(ii)a∈C2(ω)and?a(x)=0forallx∈?∈?ω,onehas0d2lim2a(x0?sξ(x0))?=0.()s→0+(A)(x)≥0in?,ifa∈C1(?),then?a(x)=0forallx∈?()requiresthattheHessianmatrixofa(x)isnot0atx=,weseethatd2?lima(x?sξ(x))=?αa(x)(?ξ(x))α,s→0+ds200x00|α|=2whereαisthemulti-index(seeSection3).Ifa(x)isgivenbyanonnegativepartofsomefunction,a(x)maysatisfy(i)of(A).<p<(N+2)/(N?2)and(A)>0,andletφsatisfy∥φ∥L∞(??)≤()satisfy∥u∥L∞(?)≤C,wheretheconstantC>0dependsonlyon?,a,.(i),wecanreplacethecondition(A)bythatasatisfies,foreachx0∈?ω,???1a(x)=χω(x)·??αa(x0)(x?x0)α+o(|x?x0|m0)?asx→x0m!x|α|=m00with?αα?xa(x0)(?ξ(x0))?=0,|α|=m0whereχAisthecharacteristicfunctionofA?RNandm0=1or2.(SeeSection3.):.../NonlinearAnalysis151(2017)18–40(ii)Itisanopenquestionwhetheraprioriboundsholdinthecase?ω∩???=?.Inparticular,itisnaturalandinterestingtoconsiderthecasewhere?ωcrosses??=?butZ?=?andZ??.In[7,Section4]aprioriboundswereestablishedinthecasewherea(x)hasafinitenumberofisolatedzerointheinteriorof?.Seealso[13,].Wehereconsiderthefollowingcases.(B)In(),a∈C2(?)satisfiesa(x)≥0,a(x)?≡0in?.Letk∈{2,3,...,N},andletω?RkbeanopensetwithC1smoothboundary?ωsatisfying?ω×{0}N?k={(x′,x′′)∈Rk×RN?k:x′∈?ω,x′′=0}??.ThesetZ??isgivenbyeitherZ=?ωifk=NandZ=?ω×{0}N?kifk∈{2,3,...,N?1}.Forx0∈?ω,denotebyξ(x0)∈Rktheouterunitnormalvectoron?ωatx0,andbyν1(x0)=ξ(x0)×{0}N?kifk∈{2,3,...,N?1}.Inthecasek=N,foreachx0∈Z,onehasd2lim2a(x0?sξ(x0))?=→0dsInthecasek∈{2,3,...,N?1},foreachx0∈Z,thereexistsν2=ν2(x0)∈{0}k×SN?k?1satisfyingd2lim2a(x0?s(σ1ν1+σ2ν2))?=0()s→0dsforallσ1,σ2∈Rwithσ2+σ2=(B)isZ=SN?1ifk=NandZ=Sk?1×{0}N?kifk∈{2,3,...,N?1}.<p<(N+2)/(N?2)and(B)>0,andletφsatisfy∥φ∥L∞(??)≤()satisfy∥u∥L∞(?)≤C,wheretheconstantC>0dependsonlyon?,a,,wewillemploytheargumentbasedonrescalingandLiouville-≤k≤=(x′,x′′)∈RN,wherex′=(x1,x2,...,xk)andx′′=(xk+1,xk+2,...,xN).Weconsidertheproblem??u=A(x′)up,u≥0inRN,()whereA∈C(Rk)satisfiesA(x′)≥0andA(x′)?≡0forx′∈(r)=min{A(y′):|y′|=r}forr>≤k≤+2?p(N?2)A(r2y′)isstrictlyincreasinginr>0foreachfixedy′∈RkwithA(r2y′)>∈C2(RN)beasolutionoftheproblem().(i)Letk=1ork=≡0.(ii)Let3≤k≤??α(r)>0()r→∞forsome?>0with(k+?)/(k?2)>p,thenu≡0.:.../NonlinearAnalysis151(2017)18–=1,<p<(N+2)/(N?2)andh(t)=|t|αorh(t)=(t+)αforsomeα>0,wheret+=max{t,0}.Thenanysolutionu=u(x)∈C2(RN)to??u=h(x)up,u≥0inRN()-typeresultsfor()werealsoobtainedbyDuandLi[5,],typically,forthecasesh(t)=(t+)αor|t|αtwithsomeα≥[5],Liouville-typeresultsarevalidforp>1,andderivedbythemonotonicitypropertyofsolutionsto().pletelydifferentapproachtoshowtheLiouville-typeresultsfor().Assumethat()holdswith?=(N+2)/(N?2)≤(k+2)/(k?2)for3≤k≤,weobtainthefollowing.′?≤k≤(x)=i=1σixi,whereσi>0isaconstantfor1≤i≤<p<(N+2)/(N?2).Thenanysolutionu=u(x)∈C2(RN)to??u=Q(x′)up,u≥=N,werefertoPhanandSouplet[13]fortheLiouville-typetheoremsandaprioriboundsofsolutionstotheHardy–H′enonequation?u+|x|?up=0withp>1and?∈(x)in()changessign,aprioriboundsofpositivesolutionswereobtainedby,.,[1–3,5].Let?+={x∈?|a(x)>0},??={x∈?|a(x)<0},and?0={x∈?|a(x)=0}.Berestycki,Capuzzo-DucettaandNirenberg[2]andChenandLi[3]consideredthecasewhereasatisfiesa∈C2(?),?0=?+∩??,|?a(x)|?=0forallx∈??0isaC2manifoldofdimensionN?1intheinteriorof?.Underthiscondition,aprioriboundswereobtainedbyemployingtherescalingandLiouville-typetheoremsfortheproblem??u=x1up,u≥,AmannandL′opez-G′omez[1]andDuandLi[5]consideredthecasewhere?0mayhavenonemptyinteriorin?,namely,?0??,?(int?0),??+,and???aresmooth,andthereexisttheexponentγ>0andthecontinuousfunctionα(x)whichispositiveandboundedawayfrom0near??+suchthata(x)=α(x)[dist(x,??+)]γin?+.Inthiscase,Liouville-typetheoremfortheproblem()withh(t)=(t+)γhasbeenusedtoobtainaprioriboundsofpositivesolutions(see[5,]).Wealsoconsiderthecorrespondingparabolicinitial–boundaryvalueproblem??u=?u+a(x)up,x∈?,t>0,?tu=0,x∈??,t>0,()????u(x,0)=u0(x),x∈?,where?isasmoothboundeddomaininRNandinitialdatau0∈C(?)satisfiesu0(x)≥0forx∈?.Wewillshowaprioriboundsofglobalsolutionsto()when(A)or(B)(x)≡1,apriori:.../NonlinearAnalysis151(2017)18–40boundofglobalsolutionsto()wasprovenbyGiga[9]fornonnegativesolutions,andbyQuittner[17]forsign-(x)changessign,aprioriboundofglobalpositivesolutionswasshownbyQuittnerandSimondon[18].Itshouldbementionedthatuniversalandaprioriboundofglobalsolutionsaswellastheblow-upestimateswereshownfromparabolicLiouvilletheoremsbyPol′aˇcikandetal.[15]fora(x)≡1,byPhan[12]fora(x)=|x|αwithα>?2,andbyPol′aˇcikandQuittner[14],Xing[21]andF¨oldes[6],wereferto[19,ChapterI].,,,inSection5,weshowanalogousresultsforthecorrespondingparabolicinitial–().+2?p(N?2)A(r2y′)isstrictlyincreasinginr>0foreachfixedy′∈RkwithA(r2y′)>,foranynonnegativesolutionu∈C2(RN)of()hastheformu=u(x′).Furthermore,ifk=1ork=2,thenu≡≤k≤??α(r)>0()r→∞forsome?>0with(k+?)/(k?2)>∈C2(RN)of()hastheformu=u(x′),thenu≡,,().Bythestrongmaximumprinciple,umustsatisfyu>0inRNoru≡,weassumethatu>0inRN,andshowthatuhastheformu=u(x′).≥,ifN=2weregardu=u(x)asafunctionofx=(x1,x2,x3)?2y=2,v(y)=|x|u(x),()|x|thefunctionvsatisfies??v=φ(|y|,y′)vp,v>0inRN\{0},()whereA(y′/|y|2)φ(|y|,y′)=()|y|N+2?p(N?2)withy′=(y1,...,yk).BytheassumptionofA,φ(r,y′)isstrictlydecreasinginr>0foreachfixedy′∈Rkwithφ(r,y′)>(0)=liminfr→0v(r).(),weobtain?2?N?v(0)>0andv(y)=O|y|as|y|→∞.():.../NonlinearAnalysis151(2017)18–=(0,...,0,ν,...,ν)∈RNwith|ν|=1.()k+1NForλ≥0,putTλ={y∈RN|y·ν=λ}andΣλ={y∈RN|y·ν<λ}.Fory∈Σλ,defineyλ=y?2(y·ν)ν+2λν,()thatis,yλisthereflectionofy∈>|yλ|>|y|fory∈Σ.()λInfact,putzλ=(yλ+y)/(),wehaveyλ?y=2λν?2(y·ν)νandzλ=y?(y·ν)ν+·ν=λand|yλ|2?|y|2=(yλ+y)·(yλ?y)=4zλ·(λν?(y·ν)ν)=4λ(λ?(y·ν)).Thus,fory∈Σλwithλ>0,wehave|yλ|2?|y|2>0,andhence()(r)byV(r)=sup{v(y):|y|≥r}forr>-handsideof(),wehaveV(r)=O(r2?N)asr→∞.()SinceV(r)isnonincreasingand()holds,fory∈Σλwithλ>0,wehavev(y)≤V(|y|)andv(yλ)≤V(|yλ|)≤V(|y|).()Forλ≥0,definew(y)=v(y)?v(yλ)fory∈Σ\{0}.()>0,wλsatisfies?wλ+cλ(y)wλ≤0inΣλ\{0},()wherecλsatisfies0≤cλ(y)≤pφ(|y|,y′)V(|y|)p?1fory∈Σλ\{0}.()|yλ|>|y|andφ(r,y′)isdecreasinginr>0,wehave0=?v(y)+φ(|y|,y′)v(y)p??v(yλ)?φ(|yλ|,y′)v(yλ)p≥?(v(y)?v(yλ))+φ(|y|,y′)(v(y)p?v(yλ)p)=?wλ(y)+cλ(y)wλ(y),fory∈Σλ\{0},where?1c(y)=φ(|y|,y′)p(sv(y)+(1?s)v(yλ))p?()thatp(sv(y)+(1?s)v(yλ))p?1≤pV(|y|)p?()holds.:.../NonlinearAnalysis151(2017)18–=W(y),definedon|y|≥R0withsomeR0≥1,suchthat,forλ>0,?W+cλ(y)W≤0for|y|>R0andliminfW(y)>0.()|y|→∞=max{A(y′):|y′|≤1}.Since|y′/|y|2|≤1/|y|≤1for|y|≥1,wehaveA(y′/|y|2)≤C1for|y|≥(),itfollowsthat′C1|φ(|y|,y)|≤N+2?p(N?2)for|y|≥1.()|y|PutpC1V(|y|)p?1Φ(y)=fory∈RN\{0}.|y|N+2?p(N?2)ThenΦ∈C(RN\{0}).From()and(),wehavecλ(y)≤Φ(y)for|y|≥1.??4?SinceV(r)satisfies(),weobtainΦ(y)=O|y|as|y|→∞.By[10,],thereexistsafunctionW=W(y)>0,definedon|y|≥R0withsomeR0≥1,satisfying?W+Φ(|y|)W=0for|y|>R0andliminfW(y)>0.|y|→∞Thus()holds.D