Concentration-compactness at the mountain pass level in semilinear elliptic problems

KyrilTintarev∗

DepartmentofMathematics

UppsalaUniversitySE-75106Uppsala,Swedenkyril.tintarev@math.uu.se

February2,2008

Abstract

Theconcentrationcompactnessframeworkforsemilinearellipticequationswithoutcompactness,setoriginallybyP.-L.Lionsforcon-strainedminimizationinthecaseofhomogeneousnonlinearity,isextendedheretothecaseofgeneralnonlinearitiesinthestandardmountainpasssettingofAmbrosetti–Rabinowitz.Inthesesetting,existenceofsolutionsatthemountainpasslevelcisverifiedunderasingleassumptioncnonlinearitiesthatoscillateaboutthe“criticalstem”F(u)=|u|2∗

.2000MathematicsSubjectClassification:35J20,35J60,49J35

Keywords:Semilinearellipticequations,concentrationcompactness,mountainpass,positivesolutions,variationalproblems.

1Introduction

InthispaperwestudyexistencefortheclassicalsemilinearellipticprobleminadomainΩ⊂RN,

−∆u+λu=f(x,u),(1.1)withtheDirichletboundarycondition.Thenumberλisassumedtobe

greaterthanthebottomofthespectrumfortheDirichletLaplacianinΩ.WeconsiderherecaseswherethecorrespondentSobolevimbeddinglackscompactness,namelywhenΩ=RNorwhenthenonlinearityf(x,s)hasthe

∗

growthofthecriticalmagnitude|s|2−1,where2∗=2N

HomogeneityofthenonlinearitymeansthatonecanrelplacetheLagrangemultiplierfortheminimizerwith1bymultiplyingtheminimizerof(1.2)byanappropriatepositiveconstant.Sincethelossofcompactnessintheseproblemsisduetonon-compacttransformations(shiftsordilations),discretesequencesofthesetransformationdefineasymptoticproblems(orproblemsatinfinity).Applyingsuchsequencesonecanimmediatelyseethatthein-equalityc≤c∞,wherec∞istheconstrainedminimumforanasymptoticcounterpartof(1.2),isalwaystrue.Inthefamousfour-paperseriesofP.L.-Lions([21,22,23,24],thestrictinequalityc2,p=2∗onboundeddomains,[9]).

Intheautonomous(f(x,s)=f(s))subcriticalcaseonRN,theimbed-dingofthesubspaceofH1-radiallysymmetricfunctionsintoLpiscompact([16]),sothatexistenceofminimizersin(1.2)followsfromastandardweak

2

󰀄󰀂

󰀅2/p.

a(x)|u|pdxΩ

(1.2)

continuity/lowersemicontinuityargument.Lagrangemultipliersherecanbesetto1evenforgeneralnonlinearitiesf(s)ofsubcriticalgrowth,sinceautonomousproblemsonRNpossessadditionalhomogeneity,namelytheonewithrespecttothetransformations→u(·/s).Existenceresultsforthesub-criticalautonomouscaseareduetoBerestyckiandLions[5].IncasetheimbeddingofthesubspaceofradialfunctionsintoL2∗

thecritical

isnolongercompact,andconcentrationcompactnessargumentwasusedbyFlucherandM¨uller[18]toprovedexistenceofminimizersforgeneralautonomousnon-linearityfunderthepenaltyconditionc󰀇

t

=

f(s)ds,

0

asF(t)>F∞(t),whereF∞istheappropriateasymptoticcounterpartofF.Inthecaseofsubcriticalnonlinearity,morerefinedrealizationsofthepenaltyconditioncF∞doesnotyieldc3givesanunredeemableerror,indeed,[9]providesanon-existencecounterexample.

1.2Problemswithouthomogeneity

Inproblemswithouthomogeneity,Lagrangemultipliersfordifferentlevelsoftheconstraintfunctionalhavetobeevaluated,whichremainsingeneralanopenproblem.Onecanstillconsiderinthiscaseminimizationofthefunctional

G(u)=

1

InsofarasonecanverifythePalais–Smaleconditionforboundedcriti-calsequences,whenthenonlinearitylackshomogeneity,onehastoconsiderunconstrainedminimaxstatementsofthemountainpasstype.

Belowweusethefollowingterminology.Anumberc∈RiscalledacriticalvalueofaC1-functionalGinaBanachspace,ifitisthevalueG(u)atacriticalpointu(G′(u)=0),anditiscalledacriticallevelofafunctionalifthereisasequenceuk,calledcriticalsequence,suchthatG(uk)→candG′(uk)→0.ThePalais–Smalecondition(PS)csaysthateverycriticalsequenceatthelevelchasaconvergentsubsequence,inwhichcasethecriticallevelbecomesacriticalvalue.

Thereisageneralresultonweakconvergenceofcriticalsequencesforsemilinearellipticequationstoanon-zerosolutionduetoRabinowitz[28],buthisproofdoesnotverify(PS)ccondition.

Thereareseveralimportantreasonstoverify(PS)c,suchas:utilityofminimaxstatementstocalculateorestimatecriticalvalues;sortingobtainedbydifferentmethodsbythevaluesofthefunctional;estimatingtheMorseindexofacriticalpoint;andexistenceofmultiplecriticalpointsviaminimaxprinciplesthatrequirethe(PS)c.Ontheotherhand,the(PS)c-conditioninnon-compactproblemsiswellknowntofailoncriticallevelsproducedbydivergentboundedsequencesoftheform

uk=w+

(n)

m󰀆n=1

gkw∞,

(n)

(1.4)

wheregkarepairwiseasymptoticallyorthogonalsequencesoftransforma-tionsresponsibleforthelossofcompactness(actionsoftranslationsordila-tions),wisacriticalpointofthefunctionalandw∞isacriticalpointofthe

asymptoticproblem.

Ourapproachisbasedontheobservationthat(1.4)isessentiallytheonlywayaboundedcriticalsequencecandiverge,whichleadstoaconclusionthat(PS)cintheAmbrosetti-Rabinowitzsettingsholdswheneverthemountainpasslevelsatisfiesc4

decompositionfrom[29].

VerificationofPalais–Smaleconditionatthemountainpasslevelistrivialwhentheproblemhashomogeneityandthushasanequivalentconstrainedminimizationstatement.ThishasbeenalreadyobservedbyCerami,Fortu-natoandStruwe[11]whoconsideredautonomousproblemwiththecriticalstemnonlinearityonboundeddomainsinthemountainpasssetting,not-ingthatthesolvabilityconditionc0,namelysuchthatforeveryM>0,thesetV−1(0,M)hasafinitemeasure.Wedonotsurveytheliteraturehereforthecriticalcase,referringthereadertothebibliographyinthebooksofChabrowski,[7],Flucher[19]andWillem[35]andtotherecentsurveyofBartsch,WangandWillem[2].Asarule,the

∗

nonlinearityconsideredinliteratureisoftheforma(x)|u|2plusasubcriticalterm,andthispaperconsidersamoregeneralcase.

Ourpaperisorganizedasfollows.Forthesakeofsimplicity,weassumethatthenonlinearityF(x,s)iscontinuouslydifferentiablewithrespecttosandthatitsderivativeadmitsrequiredasymptoticfunctionsasuniformlimits,sotheasymptoticfunctionalsandtheasymptoticequationsarewelldefined.Section2givesageneralizationoftheexistenceresultofFlucherandM¨ullerinthesensethatthenonlinearityatinfinityisdefinedduetodiscretedilations,whichgenerallygivesasmallerF∞(involvedinthepenaltycondi-tionF>F∞)thantheupperlimitdefinedin[18].Acaseinpointhereisthe

∗

nonlinearitysuchthatt→F(et)e−2tisaperiodicfunction.Thisperiodicity

∗

impliesthatF(s)oscillatesaboutthecritical“stem”s2.Somesortofoscil-5

lationsaboutthecritical“stem”arenecessaryforexistenceofsolutionsinthezeromasscase(λ=0).Specifically,underthegrowthbounds(2.18)inthezeromasscase,themountainpasssolutionsoftheautonomousequation(1.1)onRN(whichareequivalentlyprovidedasconstrainedminima)satisfythewell-knownPohoˇzaevidentity(see[27]forboundeddomain,[5]forRN):

󰀇

|∇u|2dx=2∗

󰀇(F(u)−λu2)dx.(1.5)RN

RN

Validityofthisidentityrequiresthatuanditsgradientaredecayingsuffi-cientlyfastatinfinity.Thesedecayratesareverifiedinthecaseλ>0with

subcriticalnonlinearityin[5]andforλ=0andthenonlinearityFbounded

bythecriticalstemC|u|2∗

in[17].Forpositivesolutions,Pohoˇzaevidentityisequivalentto

󰀇∞

s2∗

+1d0s2∗

d|{u≤s}|=0,whichimpliesthat

F(s)−λs2

2Autonomouscriticalproblem

∞

WeconsiderthespaceD1,2(RN),N>2,acompletionofC0(Ω)withrespecttothegradientnorm

󰀁󰀇󰀃1/2

󰀔u󰀔=|∇u|2dx,

RN

andweequipthespaceD1,2(RN)withthegroupD(N,Z,γ)ofunitaryoper-atorsgeneratedbytheshifts

DRN={u→u(·−y),y∈RN}

andbytheactionofdiscretedilationswithafixedscalingfactorγ>1,

δZ,γ={u→γ

def

N−2

def

2

andnotethatG∈C(D1,2(RN)).Let

󰀔u2󰀔−ψ(u),

(2.3)

ΦG={ϕ∈C([0,∞)→H1(RN)):ϕ0=0,limG(ϕt)=−∞}.

t→∞

(2.4)

IfΦG=∅,wesetc(G)=+∞,otherwise

c(G)=inf

.

Proposition2.1.ThesetΦGisnonemptyifandonlyifsupF>0,inwhich

∞

caseΦGcontainsapathut(x)=u(x/t)withu∈C0andψ(u)>0.

7

def

ϕ∈ΦGt∈[0,∞)

supG(ϕt).(2.5)

Proof.Notfirstthatthepathusisacontinuousmapfrom(0,∞)toD1,2(RN)thatextendsbycontinuityasu0=0:󰀔us󰀔2=sN−2󰀔u󰀔2,andinparticularlims→0󰀔us󰀔=0.Sincethenormiscontinuous,itsufficestoprovecontinuity

∞

ofusinD′(RN).Indeed,withϕ∈C0(Ω)ands,s0>0,byLebesgueconvergencetheorem,

󰀇󰀇󰀇

uϕ(s0·)ass→s0.usϕ=sNuϕ(s·)→sN0IfsupF>0,thensupψ>0andΦG=∅,sinceitcontainsapathus(x)=

u(x/s)withψ(u)>0.Indeed,bythechangeofvariablesinrespectiveintegrals,

1

G(us)=

2

jj

f(γ−f(γ−

N−2

N−2

2

2

j

s),j∈Z,s∈R.(2.6)

Itisuniquelydefinedbyitsvaluesontheintervals(1,γ)and(−γ,−1).IfFisdifferentiable,oneobviouslyhas

f(s)=γ−

N+2

2

j

s),j∈Z,s∈R.(2.7)

NotethatforanygivenFthatadmitsasymptoticfunctionsF±,theyareselfsimilar.

ThefollowingstatementgeneralizesTheorem5.2from[34].Proposition2.2.Assume(2.1)andassume,foreachofthesigns“+”and“−”,thateithersupF±>0,orF±=0withsupF>0.Let

κ(t)=supψ(u),t>0

󰀋u󰀋2=tdef

(2.8)

8

andLetκ±(t)bethevalue(2.8)correspondingtothefunctionalsψ±.IfFsatisfies(2.6),orifκ(1)>max{κ−(1),κ+(1)},thenthemaximumin(2.8)isattained.

Furthemore,theinequalityκ(t)>κ±(t)holdswheneverF≥F±withthestrictinequalityinaneighborhoundofzero.Proof.1.Bysubstitutionu(s)=v(s/t

1

2

,(2.9)

soitsufficestoprovethelemmafort=1.

2.AssumenowthatFsatisfies(2.6).IfF=0,thenκ=0andanyfunctionwiththegivennormisamaximizerforψ.AssumenowthatsupF>0.Letukbeaminimizingsequencein(2.8),thatis,󰀔uk󰀔2=1

(n)(n)

andψ(uk)→κ(1).Letyk∈RN,jk∈Z,w(n)∈D1,2(RN)andtheindexsetsN+∞,N−∞,N0⊂NbeasinTheorem6.1.NotethatbyLemma6.2,

󰀆

κ(t)=limψ(uk)=ψ(w(n)).(2.10)

n∈N

Atthesametimefrom(6.3)

1=󰀔uk󰀔≥

2

󰀆

n∈N

󰀔w(n)󰀔2.

(2.11)

Letv(n)(x)=w(n)(snx)withsn=󰀔w(n)󰀔

2

Relations(2.13)and(2.14)canholdsimultaneouslyifandonlyifthereisan0∈Nsuchthatsn0=1,whilesn=0whenevern=n0.Consequently,

N−2

wehavefrom(6.4)uk−γ

uk(γ·+yk0),wehaveuˆk→w(n0)inL2(RN).Atthe

sametime,bytheweaklowersemicontinuityofnorms,󰀔w(n0)󰀔2≤1,while

∗

bycontinuityofψinL2(RN)wehaveψ(w(n0))=κ(1).Sincethelattercanholdonlywhen󰀔w(n0)󰀔2=1,w(n0)isthedesiredmaximizer.

3.Considernowthegeneralcasewithκ(1)>κ±(1)andnotethat,sinceF±satisfies(2.6),themaximumforκ±(1)isattainedduetotheprevious

nn

step.Letukbeaminimizingsequencein(2.8)andletyk∈RN,jk∈Z,wn∈D1,2(RN)andtheindexsetsN+∞,N−∞,N0⊂NbeasinTheorem6.1.NotethatbyLemma6.2,

󰀆󰀆󰀆

(n)(n)

ψ+(w(n)),(2.15)ψ−(w)+ψ(w)+κ(t)=limψ(uk)=

2

jk

(n0)

−jk

(n0)

(n)

∗

n∈N0n∈N−∞n∈N+∞

Let,asinthestep2,v(n)(x)=w(n)(snx)withsn=󰀔w(n)󰀔

󰀆

sNn

κ(1)n∈N

−∞

2

(2.16)

+

κ+(1)

)<1,relations(2.16)and(2.14)canhold

simultaneouslyifandonlyifthereisan0∈N0suchthatsn0=1,whilesn=0

(n)

whenevern=n0.Consequently,wehavefrom(6.4)uk−w(n0)(·−yk0)→0

∗

inL2(RN).Similarlytothestep2weconcludethatw(n0)isthedesiredmaximizer.

4.ThelastassertionofthepropositionisobviousifwetakeintoaccountthattherangeofanyfunctioninD1,2(RN)isaconnectedsetwhoseclosurecontainszero.

κ(1)

Remark2.3.Notethatonealwayshasκ(t)≥κ±(t),sinceifwisamaxi-mizerforκ±(1),thenψ(γN−2

N−2

x)

isamaximizerfort>0.

10

Wenowconnectthemaximizers(2.8)withthemountainpassvaluesforG.

Proposition2.4.AssumethatsupF>0andthat

|f(s)|≤C|s|2

∗−1

.(2.18)

Thenanymaximizerwfor(2.8)correspondingto

t=t0def

=(2∗

κ(1))

−

2

2

sN−2󰀔v󰀔2−sNψ(v)hasa

singlecriticalpoint,amaximum,whichisnecessarilyattainedats=1sinceG′(v)=0.Then

c(G)≤maxs≥0

G(vs)=G(v),

whichverifies(ii).

Assumenowthat(2.8)hasamaximizerw.ByRemark2.3,wisacriticalpointofG.Bytheargumentabove,c(G)≤G(w).Ontheotherhand,sinceanypathus∈ΦGstartsattheoriginandisunbounded,wehave,usingthe

notationuˆ=u(󰀔u󰀔−

2

2N

2r2

s−r2N

2r2s

−r

2

r2−r

2N

3

Non-autonomouscriticalprobleminRN.

Letnowf(x,s)∈C(RN×R),N>2.

Assumethatforsomeγ>1thefollowinglimitsexistandthatthecon-vergenceisuniform:

f0(s)def

=|xlim|→∞

f(x,s),

f−(s)def

=

j∈Zlim,j→−∞

γN+2

2

j

s),f+(s)def

=

j∈Zlim

γ

N+2

2

j

,j→+∞

s).

Let

F(x,s)=

󰀇

s

f(x,σ)dσ,

0

andassumetheinequality

|f(x,s)|≤C|s|2

∗−1

,s∈R,x∈RN.

Let

ψ(u)=󰀇

F(x,u)dx,

RN

G(u)=

1(3.1)

casetheinequalityc(G)w2

t;kdef

=γ

N−2

jk

s),

itiseasytoseethatforanyǫ>0thereisakǫ∈Nsothatforallk>kǫandt≥ǫ,

ψ(wt;k)=󰀇

F(x,wN

t;k)=tN

󰀇Fk(tx,w)→tNψ#(w)=ψ#(wt).RRN

Letusredefinethepathwt;kfort∈[0,ǫ],by

t

Proof.Sincec(G)<∞,supF>0andfromthecondition(R)triviallyfollowsthemountainpassgeometry,namely,thenon-emptyΨGandc(G)>0.Notealsothatonecanpasstothelimitin(R),sotheconditionholdsalsoforF#.Bythestandardmountainpassreasoning,thefunctionalGpossessesacriticalsequenceuk∈D1,2(RN),thatis,G(uk)→c(G)andG′(uk)→0.Alsobyastandardargument,itfollowsfrom(R)thatthesequenceukisboundedinD1,2(RN).ConsidernowtherenamedsubsequenceofukgivenbyTheorem6.1.Ifw(n)=0in(6.4)foralln≥2,thenuk→w(1)in∗

L2,ψ′(uk)convergesinD1,2(RN),andfromG′(uk)→0itfollowsthatukconvergesinD1,2(RN)toacriticalpointofGatthelevelc(G).

Letusassumenowthatforsomem≥2,w(m)=0.Dueto(6.3)andLemma6.5,wehavethefollowingestimateofc(G)frombelow:

󰀆󰀆󰀆

(n)(n)(1)

G−(w(n)).G+(w)+G0(w)+c(G)=limG(uk)≥G(w)+

n∈N0

n∈N+∞

n∈N−∞

(3.6)

Notethatwithnecessity,w(1)isacriticalpointofG,andw(n),n≥2,arecriticalpointsofcorrespondentG#.LetGm=G0ifm∈N0,Gm=G+ifm∈N+andGm=G−ifm∈N−.Thecorrespondentasymptoticnonlinearity󰀂1

(1)

wewilldenoteasFm.Duetocondition(R),G(w)=[

4Criticalcase,problemsindomains

LetnowΩ⊂RN,N>2,beadomainwith

Remark4.3.RepetitionoftheproofofTheorem3.2alsogivesthatifFislikeinTheorem3.2,Ω⊂RNisadomainandinequalities(3.4)aresatisfied,

1,2

(Ω)satisfies(PS)c-thenthefunctionalGunderstoodinrestrictiontoD0

conditionatthemountainpasslevelc(G;Ω)andthushasacriticalpointatthislevel.

Remark4.4.Theorem4.2andRemark4.3,unliketheircounterpartonRN,cannotclaimthatc(G;Ω)F#(s),sincetheproblematinfinityissupportedonadifferentdomain.Furthermore,[9]

∗

offersacounterexampleforN=3andF(s)=|s|2+λu2withsufficientlysmallpositiveλ.

Ontheotherhand,ifΩisaboundeddomain,N>3andF(x,s)≥F+(s)+ǫs2withsomeǫ>0,thenc(G;Ω)Proposition4.5.UnderconditionsofTheorem4.2,c(G;Ω)≤c(G+).

∞

(Ω)beasequenceconvergentinD1,2(RN)toacriticalProof.Letwk∈C0

pointwforG+satisfyingG+(w)=c(G+).

ws;j,k(x)=γ

N−2

ǫ

wǫ;j,k.Then,with

Ft(x,u)=tNF(tx,t−

N−2

2

1

sN−2󰀔wk󰀔2−sNψ+(wk).

Itiseasytoseethatask→∞,therighthandsideconvergesto

max

s≥0

5Subcriticalcase

Considernowthe“positivemass”caseλ>0withthefunctionalψdefinedbyexpression(2.2)ontheSobolevspaceH1(RN),N≥1,equippedwiththeequivalentnormLet

󰀔u󰀔2=󰀇

(∇|u|+λ|u|2)dx.

RN

G(u)=

1

Proof.ThebeginningoftheproofiscompletelyanalogoustothatforThe-orem3.2andcanbeabbreviated.WeapplyTheorem6.1totheboundedcriticalsequence,noting,similarlytoRemark4.1,thatN−∞=∅and,more-over,N+∞=∅sinceF+=0.With(6.6)takingtheroleof(6.5)wearriveatanimmediateanalogof(3.6):

󰀆(1)

G(w)+G∞(w(n))≤c(G)n≥2

AsintheproofofTheorem3.2,allthetermsinthelefthandsidearenon-negativedueto(R).Assumethatthereexistsm≥2suchthatw(m)=0.

Observethepatht→w(m)(·/t)isoftheclassΦG∞.Indeed,itscontinuityinD1,2(RN)-normwasshowninProposition2.1,andtheproofofremainingcontinuity,inL2normisanalogous.Thenonlinearitysatisfiestherequire-mentsof[5]forPohoˇzaevidentity(1.5).Thus,since

󰀇

1

(F∞(w(m))−λ|w(m)|2)dx=

RN

2

N

tN−2󰀔w(m)󰀔2D1,2(RN)−t

󰀇

RN

(F∞(w(m))−λ|w(m)|2)dx,

convergesto−∞whent→∞.Furthermore,themaximumofthisexpression

overt≥0isisclearlyattainedatasinglepoint,whichisnecessarilyt=1,sincesince(G′∞(w(m)),w(m))=0.Thus

c(G∞)≤maxG∞(w(m)(·/t))=G∞(w(m))dx.

t≥0

(5.6)

Ontheotherhand,since,G∞(w(n))≥0foralln>1andG(w(1))≥0,wehavefrom(5.5),

G∞(w(m))≤c(G)1

(Ω)isconnectedandcontainszero.anyfunctioninH0

18

Remark5.3.Theorem5.2canbetriviallygeneralizedtothecaseofperiodiccoefficients.LetV∈L∞(RN),V>0,beZN-periodic,thatissatisfy

V(x+y)=V(x)forallx∈RN,y∈ZN

WemayequipH1(RN)withanequivalentnorm

󰀇

󰀔u󰀔2=(|∇u|2+V(x)u2)dx,

RN

(5.7)

Assumethatthereisafunctionf∞∈C(RN×R),ZN-periodicinthefirst

argument,suchthatf(x+yk,s)→f∞(x,s)foranysequenceyk∈ZN,|yk|→∞.Theorem5.2remainstruealsounderthesemodifications.

6

6.1

Appendix

WeakconvergencedecompositioninD1,2(RN)

ThefollowingtheoremisTheoremfrom[34],withthedilationfactor2replacedbygeneralγ.

Theorem6.1.Letuk∈D1,2(RN),N>2,beaboundedsequence.Let

(n)(n)

γ>1.Thereexistw(n)∈D1,2(RN),yk∈RN,jk∈Zwithk,n∈N,anddisjointsetsN0,N+∞,N−∞⊂N,suchthat,forarenumberedsubsequenceofuk,

w

(n)

=w-limγ

−N−2

2

jk

(n)

w(n)(γjk(·−yk))→0inL2(RN),

(n)

(n)

∗

(6.4)

andtheseriesaboveconvergesuniformlyink.

(1)(n)(n)

Furthermore,1∈N0,yk=0;jk=0whenevern∈N0;jk→−∞

(n)(n)

(resp.jk→+∞)whenevern∈N−∞(resp.n∈N+∞);andyk=0

(n)(n)

whenever|γjkyk|isbounded.

19

Thefollowingstatementsare,respectively,Lemma5.6(anelementarymodification)andRemark3.4from[34].

Lemma6.2.LetF∈C(RN×R)satisfy|F(x,s|≤c|s|2,letγ>0,N>2,andassumethatthefollowinglimitsexistandareuniforminx∈RN:

F+(s)=

def

j∈Z,j→+∞

∗

lim

γ−NJF(γ−jx,γ

N−2

2

j),

F0(s)=limF(x,s).

|x|→∞

(n)

def

Letuk∈D1,2(RN),w(n),yk∈RNandletjk∈Z,N0,N+∞,N−∞⊂NbeasprovidedbyTheorem6.1.Then

󰀇

F(uk)(6.5)lim

k→∞RN

󰀆󰀇󰀆󰀇󰀆󰀇

F−(w(n)).F+(w(n))+F0(w(n))+=

n∈N0

RN

n∈N+∞

RN

n∈N−∞

RN

(n)

Lemma6.3.LetF∈C(RN×R)satisfy5.2,5.3,andassumethatthe

followinguniformlimitexists:

F∞(s)=limF(x,s).

|x|→∞def

Letuk∈H1(RN),w(n),yk∈RNbeasprovidedbyTheorem6.1withN±=∅.Then󰀇󰀆󰀇

F∞(w(n)).(6.6)F(uk)=lim

k→∞

RN

n∈N

RN

(n)

Acknowledgments

TheauthorthanksMosheMarcusandIanSchindlerfortheirencouragingre-marks.ThispaperwaswrittenasVisitingProfessoratUniversityofToulouse

1andtheauthorexpresseshisgratitudetoJ.FleckingerandtherestofthefacultyatCeremathfortheirwarmhospitality.

20

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