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1、NonlinearDiffer.Equ.Appl.17(2010),391–409c2010Birkh¨auserVerlagBasel/Switzerland1021-9722/10/040391-19NonlinearDifferentialEquationspublishedonlineMarch2,2010DOI10.1007/s00030-010-0059-0andApplicationsNoDEAPrincipaleigenvalueinanunboundeddomainwithindefinitepotentialLiamidiLead
2、iandAkilaYechouiAbstract.Inthiswork,wediscusstheexistenceandthenon-existenceNofprincipaleigenvalueinanunboundeddomainofRforsomepoten-tialswhichchangesign.Wealsogivecertainpropertiesofthisprincipaleigenvalue.MathematicsSubjectClassification(2000).35J20,35J70,35P05,35P30.Keywords
3、.Principaleigenvalue,Ellipticproblems,p-Laplacian,Indefiniteweight,Unboundeddomain,Coercivity.1.IntroductionThisworkismainlyconcernedwiththeexistenceofapositiveprincipaleigen-valueforthefollowingproblem−Δu+V(x)
4、u
5、p−2u=λm(x)
6、u
7、p−2uinΩ,(1.1)pwhereΩisanunboundedsmoothdomaininRN,Δ
8、u:=div(
9、∇u
10、p−2∇u),p10suchthat(1.1)admitsanontrivialu,withu≥0,inasuitableweaksense.Manyworkshavebeendevotedtotheexistenceof
11、aprincipaleigenvalueinthelastyearsduetotheimportanceofthevalidityoftheweightedPoin-car´einequality.ThecaseV≡0hasbeenstudied,amongothers,by[3,8,11,12]underdifferenthypothesisonm.Theyprovethatthereexistsafirstpositiveprincipaleigenvalue,denotedbyλ1(m)anddefinedbyλ(m):=infE(u)=
12、
13、∇u
14、p,u∈Wandm
15、u
16、p=1,1ΩΩinthecaseofboundedandunboundeddomains,whereWisasuitableSobolevspace.ThecaseV≥0istechnicallyverysimilartothecaseV≡0,asthe392L.LeadiandA.YechouiNoDEAenergyassociatedE(u):=(
17、∇u
18、p+V(x)
19、u
20、p)VΩhassimilarpropertieswiththeenergyEdefinedabove(seeforexample[17]).P
21、roblems(1.1)withVchangingsignandsatisfyingcertainconditionswererecentlyconsideredin[18](withm≡1)[13,10](withmindefinite,p=2andadditionalhypothesisonVandm)[5,6,9,11,16](withmindefinite).In[9,16],theuseofthevalueα(V,m):=infE(u);u∈W1,p(Ω),
22、u
23、p=1andm
24、u
25、p=0V0ΩΩplaysanimportantr
26、oleintheproofofexistenceofaprincipaleigenvalue,whereΩisabound
