Existence and uniqueness of solutions on bounded domains to a FitzHugh-Nagumo type elliptic

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

PACIFICJOURNALOFMATHEMATICSVol.197,No.1,2001

EXISTENCEANDUNIQUENESSOFSOLUTIONSONBOUNDEDDOMAINSTOAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM

CarolusReineckeandGuidoSweers

Inthispaperweprovetheexistenceanduniquenessoftheboundarylayersolutiontoasemilineareigenvalueprob-lemconsistingofacoupledsystemoftwoellipticpartialdif-ferentialequations.Althoughthesystemisnotquasimono-tone,thereexistsatransformationtoaquasimonotonesys-tem.Forthetransformedsystemwemayandwillusemax-imum(sweeping)principleargumentstoderivepointwisees-timates.Adegreeargumentcompletestheuniquenessproof.

1.Introduction.

Weconsiderthefollowingnonlineareigenvalueproblem: in , u=λ(f(u) v) v=λ(δu γv)in ,(Pλ) u=v=0onΓ= ,

withλ,δ,γ>0andwhere RNisasmoothboundeddomain.Asusual,adomainisanopenconnectedset.Thenonlinearityfisassumedtobesmoothandlikeathirdorderpolynomial.Weprovetheexistenceofacurveofpositivesolutions(uλ,vλ)to(Pλ)forλlargeenough.Thesesolutionsareshowntobe,exceptforaboundarylayerofwidthO(λ 1/2),closeto(ρ,(δ/γ)ρ)whereρapositivezerooff(s) (δ/γ)sandf (ρ)<0.Thestabilityofthesesolutionsasequilibriaoftheparabolicsystem inR+× , ut= u+λ(f(u) v)

inR+× ,vt= v+λ(δu γv)(1) u=v=0onR+×Γ,

withappropriateinitialconditionsisalsoproven.Finallyitisshownthatthesesolutionsareuniqueinanappropriateorderinterval.

Thequestionofexistenceofsolutionsto(Pλ)withλ=1andwithdi er-entkindsofnonlinearitieswasstudiedbyKlaasenandMitidieri[9]andDeFigueiredoandMitidieri[7],seealsoRothe[21]andLazerandMcKenna[12].Thefactthatthesecondequationcanbeinvertedtosolvevinterms

183

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

184C.REINECKEANDG.SWEERS

ingthisapproachitwasshownforexamplein[9]withf(u)=u(u 1)(a u),0<a<1/2andin[7]formoregeneralfofthesametype,thatthereexistatleasttwonontrivialsolutions,undertheassumptionsthatδ/γissmallenoughand containsalargeenoughball.Byrescaling,thisimpliesthatthereexistnontrivialsolutionsto(Pλ)ifλislargeandδ/γsmall.Ourtreatmentoftheproblemdi ersfromthevariationalapproachmen-tionedabove.Byimposingsomenaturalrestrictionsontheparameters,whicharesatis edifδ/γissmall,itispossibletomakeatransformationof(Pλ)andamodi cationofftoobtainaquasimonotonesystem.Solutionstothequasimonotonesysteminacertainrangecorrespondtosolutionstotheoriginalproblem.Thisapproachwasalsousedin[19]aswellasin[14]forothersystemsofequations.Theadvantageofworkingwithaquasimono-tonesystemisthatforsuchsystemsacomparisonprincipleholds.Fromthisfollowstheexistenceofsolutionsbetweenanorderedpairofsub-andsuper-solutions.ForsuchsystemsonealsohasananalogueofMcNabb’ssweepingprinciple,see[15],[2],[4]and[22]ingthisquasimonotoneapproachweareabletogiveacompletequali-tativedescriptionofaspeci csolutionto(Pλ).Thisqualitativedescriptionallowsustoproveuniquenessandstabilityresults.ResultsinthisdirectionwereobtainedbyLazerandMcKenna[12]forasystemwithδ=γandfsuchthatf(s)/sisdecreasingonR+.Existenceandpositivityofsolutionswereconsideredin[9]and[7].Ifwesetδ=0in(Pλ)thentheproblemreducestothewellstudiedscalarproblem

(Sλ) u=λf(u)u=0in ,onΓ.

Thereisanextensiveliteratureonsuchkindofproblems.Wejustmention[2],[4],[13]andmorerecently[5].Wenotethatourtreatmentofthequasimonotonesystemissimilartothetreatmentofproblem(Sλ)aswasdonein[2]and[4].Theresultsofthepresentpaperwereannouncedin[20].Thestructureofthepaperisasfollows.Inthenextsectionthepreciseassumptionsonthenonlinearityfarestated,aswellastheconditionswhichweimposeontheparametersγandδ.Itisthenshownhow(Pλ)canbetransformedtoaquasimonotonesystem.Themainresultsarealsostatedinthissection.InSection3weproveseveralauxiliaryresults.TheproofsofthemaintheoremsaregiveninSection4.InAppendixAwede neournotionofsub-andsupersolutionsforquasimonotonesystemsandgivesome

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM185

relatedresults.Inparticularwestateaversionofthesweepingprincipleforquasimonotonesystems.Thisprincipleisusedrepeatedlyintheproofs.

2.Assumptionsandmainresults.

Theassumptionsonfarethefollowing.

ConditionA.Thefunctionf∈C1,1(R),f(0)≥0andthereexistsσ0>0++suchthatforevery0≤σ<σ0thereexistρ σ<ρσwithρσ>0suchthat

± +(1)f(ρ±σ)=σρσandf(s)>σsforρσ<s<ρσ;

+(2)f (s)<0foralls∈(ρ+σ0,ρ0);

(3)Jσ(ρ)>0on(0,ρ+σ)forall0≤σ<σ0where ρ+σ(2)(f(s) σs)ds.Jσ(ρ):=ρ

SeeFigure1.f(t)

s0t

st

-s-0s0++ss+00

Figure1.

Example1.Thefunctionf(u)=au u3witha>0,see[12]and[7],satis esConditionAabovewithσ0=2a/3.

Example2.Considerthefunctionf(u)=u(a u)2(u 1)with a>0.ConditionAholdsifa<1/2.Inthiscaseσ0=2a 5a+2/9.Withthisnonlinearityproblem(Pλ)isanextensionoftheFitzHugh-Nagumoequations,see[9]and[10].

Aswassaidintheintroduction,animportantstepinouranalysisistotransform(Pλ)andtomodifyfinordertoobtainaquasimonotonesystem.Forthede nitionofaquasimonotonesystemandsomeresultsforsuch

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

186C.REINECKEANDG.SWEERS

systemswerefertoAppendixA.Inordertotransformsystem(Pλ)weneedthefollowingassumptionontheparametersδandγ: andsupposethatConditionB1.LetM:=max f (s);0≤s≤ρ+0√γ 2>M.

Wede neβandαby

(3)

(4)β:=(γ M) α=γ β.

IfConditionB1holdsthenβ∈Randα,β>0.Notethat β(β+M)=δ γβandthat

(5)

Onemayverifythat(u,w) u=

w=(Qλ) u= :=1 δ>0.in ,in ,onΓ,isasolutiontoλ(f(u) βu+βw)λ(f(u)+Mu αw)w=0

ifandonlyif(u,βu βw)isasolutionto(Pλ). (s)=f(s)foralls∈[0,ρ+] ∈C1,1(R)beafunctionsatisfyingfLetf0 withf,fboundedonRandwithf(s)+M≥0foralls∈R.Ifwereplace thesystembecomesquasimonotone.Sinceweareinterestedfin(Qλ)byf+insolutions(u,v)to(Pλ)withupositiveandmaxu<ρ+δ/γ≤ρ0wecanassumewithoutlossofgeneralitythefollowing:

ConditionA*.Thefunctionfsatis esConditionAwithfandf boundedandf (s)+M≥0foralls∈Randf(s)≤0fors≥ρ+0.

Anotherconditionwhichweimposeis:

ConditionB2.Theconstantβde nedin(3)satis esβ<σ0.

Underthisconditiononehasforλlargeenoughapositivenontrivialsolutiontothescalarproblem u=λ(f(u) βu)in ,u=0onΓ,

+whichhasitsmaximumintheinterval(ρ ,ρββ),see[4].Thissolutionwill

beusedtoobtainanontrivialsubsolutionto(Qλ)forλlargeenough.Thede nitionofsub-andsupersolutionsisgiveninAppendixA.

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM187

WemakesomeremarksonConditionsB1andB2.Bothconditionsaresatis edifδ/γissmallenough.Moreprecisely,for xedδ>0,B1andB2aresatis edif √2;M+2if0≤δ<σ0γ>2.M+σ0+δ/σ0ifδ≥σ0

Inthe rsttheoremweprovetheexistenceofacurveofpositivesolutionsto(Pλ).

Theorem2.1(Existenceofacurveofsolutions).LetfsatisfyConditionA,letγ,δbesuchthatConditionsB1andB2holdandassumethatΓisC3.¯×C2( ))¯Thenthereexistλ >0andafunctionΛ∈C1([λ ,+∞),C2( )suchthat(uλ,vλ):=Λ(λ)isapositivesolution,i.e.,uλ,vλ≥0,to(Pλ)forallλ≥λ .Furthermoreδ +(1)maxuλ∈(ρ ,ρ)andmaxv∈(ρδ/γ,ρ+λδ/γδ/γδ/γ); δ+(2)limλ→∞Λ(λ)=ρ+uniformlyoncompactsubsetsof .δ/γ,ρδ/γ

Thestabilityofthesolutionsobtainedabovewillbecon- inthetheorem¯×C ¯.Forλ>λ wede nethelinearsideredinthespaceX:=C operatorAλ:D(Aλ) X→Xby

(6)

and

(7)Aλ uv :=D(Aλ):={(u,v)∈X;( u, v)∈X} 00 uv λ f (uλ)1δ γ uv for(u,v)∈D(Aλ).Hereuλisthe rstcomponentofΛ(λ).Inthede nitionofD(Aλ), uand waretobeunderstoodindistributionalsense.

Theorem2.2(Stability).AssumethattheconditionsofTheorem2.1holdandletλ andΛbeasinthattheorem.Foreveryλ≥λ thesolutionΛ(λ)=(uλ,vλ)to(Pλ)isanexponentiallystableequilibriumsolutiontotheinitialvalueproblem(1)i.e.,foreveryλ≥λ thereexistsνλ>0suchthatthespectrumσ(Aλ)iscontainedin{ν∈C;Reν>νλ}.

Ourlasttheoremisaresultontheuniqueness,inarestrictedsense,ofsolutionsto(Pλ).

Theorem2.3(Uniquenessinorderinterval).AssumethattheconditionsofTheorem2.1holdandletλ andΛbeasinthattheorem.Forevery+ functionz∈C0( )withz≥0andmaxz∈(ρ β,ρδ/γ)thereexistsλz>λsuchthatif(u,v)isasolutionto(Pλ)withλ>λzandu∈[z,ρ+δ/γ]then(u,v)=Λ(λ).

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

188C.REINECKEANDG.SWEERS

Ingeneralonecannotexpectuniquenessofsolutions.Indeeditmayforexamplebethecasethatthetrivialsolutionisastablesolutiontotheproblem.Thentherewillexistathird,unstablesolutionin[0,Λ(λ)].ThisisthecasewhenfisasinExample2andConditionsB1andB2hold,see[19].

Weendthissectionwithasummaryofthenotationthatwillbeused.Notation:.¯Wewrite Letu1,u2∈C( ).

u1≥u2ifu1(x)≥u2(x)forallx∈ ;

u1 u2ifu1≥u2andu1=u2;

u1>u2ifu1(x)>u2(x)forallx∈ .¯×C( )¯weshalluse(u,w)(x)=(u(x),w(x)). For(u,w)∈C( )¯×C( ),¯i=1,2.Wewrite Let(ui,wi)∈C( )

(u1,w1)≥(u2,w2)ifu1≥u2andw1≥w2;

(u1,w1) (u2,w2)if(u1,w1)≥(u2,w2)and(u1,w1)=(u2,w2);

(u1,w1)>(u2,w2)ifu1>u2andw1>w2.

If(u1,w1)≥(u2,w2)wedenoteby[(u1,w1),(u2,w2)]theorderinter-val ¯×C( )¯;(u1,w1)≤(u,w)≤(u2,w2).(u,w)∈C( )

∞( )withz≥0andD ( ) ByD+( )wedenotethesetofz∈C0denotestheusualspaceofdistributions.¯wesay u1≤u2inD ( )-senseif Foru1,u2∈C( )

u1( z)dx≤u2zdx

forallz∈D+( ). ForaBanachspaceXwedenotetheboundedlinearoperatorsfromXintoXbyL(X).

3.Preliminaryresults.

3.1.Estimatesforpositivesolutions.

Proposition3.1.LetBbetheunitballinRN.Supposethatfsatis esConditionA*.ThenthereexistsλB>0suchthattheproblem inB, u=λB(f(u) βu+βw)

inB, w=λB(f(u)+Mu αw)(8) u=w=0on B,

hasasolution(UB,WB)withthefollowingproperties:

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM

+(1)0≤(UB,WB)<(ρ+δ/γ, ρδ/γ),with =1 δ/(γβ).

(2)UBandWBareradiallysymmetricwith

(0)=WB(0)=0UB189and UB(r),WB(r)<0on(0,1].

(3)(UB(0),WB(0))>(ρ δ/γ, ρδ/γ)andWB(0)≥ τwhereτ:=UB(0).

Proof.SinceConditionB2holds,Jβ(ρ)>0forall0≤ρ<ρ+β.Thisimpliesthatforλlargeenough,sayλ=λB,thereexistsapositivesolutionto u=λ(f(u) βu)inB,u=0on B,

+withmax∈(ρ β,ρβ),see[4].Then(0)isasubsolutionto(8).Since+++(ρ+, ρ)isasupersolutionwith(0)<(ρ, ρδ/γδ/γδ/γδ/γ)thereexistsa

+solution(UB,WB)with<UB<ρ+δ/γand0<WB< ρδ/γto(8),see

inganextensionduetoTroy,[24],ofresultsofGidas,NiandNirenberg,[8],toquasimonotonesystems,wehavethatUBandWB (0)=W (0)=0andU (r),W (r)<0areradiallysymmetricwithUBBBBontheinterval(0,1).Also( +λBα)WB=λB(f(UB)+MUB)≥0 (1)<0.Letτ:=U(0).WithandbythestrongmaximumprincipleWBBVB=β(UB WB)italsofollowsfromthemaximumprinciplethat

(9)maxVB<(δ/γ)τ.

Indeed,( +λBγ)(VB δτ/γ)=λB(UB τ)≤0inB,withVB=0on Band(9)follows.SinceVBisalsoradiallysymmetricanddecreasing,VB(0)=β(τ WB(0))<(δ/γ)τandhence

WB(0)>(1 δ/(γβ))τ= τ> ρ δ/γ.

(1)=β(U (1) W (1))<0andhenceU (1)<W (1)<0. AlsoVBBBBB

Nextweconstructafamilyofsubsolutionsto(Qλ)usingthefunctionsUBandWB.Thesesubsolutionswillbeusedtodeterminebysweepingtheshapeofthesolutionsto(Qλ)inacertainorderinterval.We xz ∈ andlet

(10)λ :=λBdist(z ,Γ) 2.

|x z |≤(λB/λ)1/2;

|x z |>(λB/λ)1/2,Lemma3.2.Forallλ≥λ weset 1/2 for(UB,WB)(λ/λB)(x z)Zλ(x):=0for

(11)+Y:=(ρ+δ/γ, ρδ/γ)with(UB,WB)asinProposition3.1.ThenZλisasubsolutionto(Qλ)andisasupersolutionto(Qλ)withZλ<Y.

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

190C.REINECKEANDG.SWEERS

Proof.ItfollowsdirectlythatYisasupersolution.ThefunctionZλiscontinuousandZλ(x)=0forx∈Γ.DenotebyZλ,i,i=1,2,thetwocomponentsofZλ.Letz∈D+( ).Then,withBλ=B(z ,(λB/λ)1/2)andndenotingtheoutwardnormal,weobtainbytheGreenidentity:

Zλ,1( z)dx=Zλ,1( z)dx Bλ Zλ,1 z zdSZλ,1( Zλ,1)zdx = Bλ Bλ

≤λ(f(Zλ,1) βZλ,1+βZλ,2)zdx.

AsimilarresultholdsforZλ,2.FinallymaxZλ,1=Zλ,1(z )=τ<ρ+δ/γandmaxZλ,2=Zλ,2(z )=WB(0)< ρ+ δ/γ.

SinceZλisasubsolutionto(Qλ)andYisasupersolutionto(Qλ)withZλ<Ythereexistsatleastonesolutionintheorderinterval[Zλ,Y].Forevery xedλ≥λ wede neforally∈ satisfyingdist(y,Γ)>(λB/λ) 1/2thefunctions

(12)y(x):=Zλ(x+z y).Zλ

RepeatingtheproofofLemma3.2oneseesthatforλ≥λ , y1/2Sλ:=Zλ;y∈ suchthatdist(y,Γ)>(λB/λ)

isafamilyofsubsolutions.Weshallusethesweepingprinciplewithfunc-tionsinSλtoobtain,atleastforλlargeenough,estimatesofsolutionsto(Qλ)intheorderinterval[Zλ,Y].Inordertoestimateasolutionin[Zλ,Y]inallof aswellasontheboundarywemakethefollowingassumptiononΓwhichholdsifΓ∈C3:

satis esauniforminteriorspherecondition,thatis,thereexistsε >0suchthat =∪{B(y,ε);y∈ anddist(y,Γ)>ε }.

Wemaysupposethat ε:={y∈ ;dist(y,Γ)>ε}isconnectedforallε≤ε .

Lemma3.3.Thereexistsλ×>λ andb>0suchthatforallλ>λ×wehavethefollowingestimateforeverysolution(u,w)∈[Zλ,Y]to(Qλ):

(13)(u(x),w(x))>min{bλ1/2dist(x,Γ),τ}(1, ),

with =1 δ/(γβ)andτasinProposition3.1. 2Proof.Letελ:=(λB/λ)1/2andλ×:=maxλ ,λBε .Supposethat(u,w)∈[Zλ,Y]isasolutionto(Qλ)withλ>λ×.Asin[4]thereexistsforeveryy∈ ελacurvein ελconnectingywithz .Usingthesweeping

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM191

yprinciple,PropositionA.6,itfollowsthat(u,w)>Zλforally∈ ελ.Usingy(u(x),w(x))≥supy∈ εZλ(x)one nds(13). λ

Thenextlemmaimprovestheestimatewefoundinthepreviousone.Lemma3.4.Foreveryε>0andλ>λ×thereexistsaconstantb(ε)>0,independentofλ,suchthatforeverysolution(u,w)∈[Zλ,Y]to(Qλ)itholdsthat +1/2(u(x),w(x))>minb(ε)λdist(x,Γ),ρδ/γ ε(1, ),(14)

with =1 δ/(γβ).Inparticularthereexistsb0>0suchthat 1/2+(u(x),w(x))>minb0λdist(x,Γ),ρσ0(1, ).(15)

Proof.Letλ>λ×be xedandsuppose(u,w)∈[Zλ,Y]issolutionto(Qλ).Ifρ+δ/γ ε≤τthen(13)holdswithb(ε)=bandbasinthepreviouslemma.

+Supposeρ+δ/γ ε>τ.Sincef(s) (δ/γ)s>0foralls∈(ρδ/γ,ρδ/γ)

thereexists ε>0suchthat

f(s) (δ/γ)s> ε(s τ)foralls∈[τ,ρ+δ/γ ε].

FromLemma3.3itfollowsthat(u(x),w(x))>(τ, τ)forallx∈ suchthatdist(x,Γ)>λ 1/2τ/b.Forsubsolutionsweneedthefunctione≥0satisfying e=µeinB1(16)e=0on B1

whereµistheprincipaleigenvalueandB1theunitballinRN.Wenormalizeesuchthate(0)=1.Letµε=µ/ εand √ 1/2 :=y∈ ;dist(y,Γ)>(ε+τ/b)λ.

We xy∈ andletB:=B(y,(µε/λ)1/2).Foreveryt∈[0,1]wede nethefunctions(Ut,Wt)onby 1/2(y x),Ut(x):=τ+t(ρ+ ε τ)e(λ/µ)εδ/γ

Wt(x):= Ut(x).

ThenT:={(Ut,Wt);t∈[0,1]}isafamilyofsubsolutionstotheproblem p=λ(f(p) βp+βq)inB, q=λ(f(p)+Mp αq)inB,(17) p=uon B, q=won B.

Usingthesweepingprincipleitfollowsthat

+(u(y),w(y))>(U1(y),W1(y))=(ρ+δ/γ ε, (ρδ/γ ε)).

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

192C.REINECKEANDG.SWEERS

Sincey∈ wasarbitrarywehavethat

+ 1/2(18)(u(x),w(x))>(ρ+.δ/γ ε, (ρδ/γ ε))ifdist(x,Γ)>(µε+τ/b)λ

√ 1De neb(ε):=ττ/b+ε,andnotethatmin{bλ1/2dist(x,Γ),τ}≥√b(ε)λ1/2dist(x,Γ)ifdist(x,Γ)≤(ε+τ/b)λ 1/2.HencebyLemma3.3

(u(x),w(x))>b(ε)λ1/2dist(x,Γ)(1, )√forallxwithdist(x,Γ)≤(ε+τ/b)λ 1/2.Thisproves(14)while(15)+followsbychoosingb0=b(ε)withε=ρ+ δ/γ ρσ0.

ThenextlemmawillbeusedintheproofofTheorem2.3.

+Lemma3.5.Letz0∈C0( )benonnegativewithmaxz0∈(ρ β,ρδ/γ).

Thereexistsλz0>0suchthatif(u,w)isasolutionto(Qλ)withu∈[z0,ρ+δ/γ]andλ>λz0then(u,w)∈[Zλ,Y].

Proof.Firstnotethatifu∈[z0,ρ+δ/γ]then(u,w)∈[(z0,0),Y].Letx0∈ besuchthatz0(x0)=maxz0.Chooseρ∈(ρ β,τ),whereτisasinProposition3.1,andr0>0suchthatρ<z0(x)≤u(x)forallx∈B(x0,r0).+Sincef(s) βs>0foralls∈(ρ β,ρβ)thereexists >0suchthat

f(s) βs> (s ρ)

(19)foralls∈[ρ,τ].Leteandµbeas(16)withe(0)=1.Supposethat r >0.Lety∈B(x0,rλ)be xedandde neonThenrλ:= 0B=B(y, B(x0,r0), (y x).Ut(x)=ρ+t(τ ρ)e

ItholdsthatT:={(Ut,0);t∈[0,1]}isafamilyofsubsolutionsto(17)withuλ,wλ,insteadofu,w.Byasweepingargument,startingwith(U0,0)oneconcludesthat(u(y),w(y))>(U1(y),W1(y))=(τ,0).Sincey∈B(x0,rλ)wasarbitrarywehavethat(u,w)≥(τ,0)onB(x0,rλ).x0x0LetZλ,i,i=1,2,denotethetwocomponentsofZλde nedin(12). x0ThefunctionZλ,BHence,if(19)isreplaced1hassupport(x0, √ 2,thenrλ>Bandbythestrongerconditionλ>(B+2r0x0(u,w)(x)>(Zλ,1,0)forallx∈ . x0Fromthisitfollowsthat(u,w)∈Zλ,Y.Indeed,usingthefactthatx0x0x0 Zλisasubsolutiononehasthat (w Zλ,2)+α(w Zλ,2)≥0inD( )-sense.AsintheproofofLemma3.3itnowfollowsthat(u,w)∈[Zλ,Y]. 2λ>(µ/ )r0.

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM193

3.2.Thesemilinearproblemonthehalfspace.Inthissectionweconsiderthefollowingproblem

U

W U=f(U) βU+βW=f(U)+MU αW=W=0inRN+,inRN+,

on RN+.(20)

Themainresultwhichweproveisthatthereexistsapositivesolution(U,W)to(20)suchthat

(21)x1→∞ + N 1, ρ)uniformlyinx∈R,lim(U,W)x1,x =(ρ+δ/γδ/γ

with =1 δ/(γβ).Moreoverthereexistsonlyonesuchsolutionand(U,W)(x1,x )=(u,w)(x1)where(u,w)asolutiontotheproblem

u w

u(0) u(0)====f(u) βu+βwf(u)+Mu αw0,w(0)=0,κ,w (0)=ν,inR+,inR+,(22)

forsomeappropriateinitialdataκandν.Itisstandardthatwehaveforeverypair(κ,ν)∈R2atleastlocallyauniquesolutionto(22)whichcanbecontinuedtosomemaximuminterval.Wedenotesuchasolutionby(u,w)κ,ν=(uκ,ν,wκ,ν).Firstweshowthatthereexistsauniquepair(κ,ν)suchthatthecorrespondingsolutionexistsforallr∈R+,ispositiveand+tendsto(ρ+δ/γ, ρδ/γ)atin nity.Somepropertiesofthissolutionthatareneededlater,arealsoproven.

Proposition3.6.Assumethatfsatis esConditionA*.Thenthereexistsauniquepair(¯κ,ν¯)suchthatthesolution(u,w)κ¯,ν¯to(22)ispositiveandsatis es

(23)r→∞++lim(u,w)κ¯,ν¯(r)=(ρδ/γ, ρδ/γ),

with =1 δ/(βγ).Moreoverκ¯>ν¯>0and(u,w)κ¯,ν¯hasthefollowingproperties:

++(1)0<(u,w)κ¯,ν¯(r)<(ρδ/γ, ρδ/γ)forallr>0;

(2)uκ¯,ν¯(r)>wκ¯,ν¯(r)forallr>0;

+and(u ,w )(3)(u ,w )κ¯,ν¯(r)>(0,0)forallr∈Rκ¯,ν¯(r)→(0,0)asr→∞.

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

194C.REINECKEANDG.SWEERS

Theproofofthispropositionconsistsofanumberoflemmas.Wealsoneedtoconsiderthefollowingsystem u =f(u) vinR+, inR+, v =δu γv(24) (0)=0, u (0)=0,v u(0)=κ,v(0)=η.

Againwedenotesolutionsto(24)by(u,v)κ,ηwiththeunderstandingthatthesolutionsarede nedonamaximuminterval.Wepointoutthefactthatforκ,ν∈Ritholdsthatthesolution(u,v)κ,β(κ ν)to(24)isgivenby(uκ,ν,β(uκ,ν wκ,ν))where(u,w)κ,νisthesolutionto(22).Forasolution(u,v)=(u,v)κ,ηto(24)wehavethefollowingidentityforallr≥0: u 2 1γ(u) κ2 ((v )2 η2)= 2(25)f(s)ds+2uv v2.0Indeed,di erentiating u(r)1 2γ2 f(s)ds 2u(r)v(r)+v(r)2,H(r):=u(r) v(r)+20

(24)impliesthatH (r)=0forallr≥0.HenceH(r)=H(0)forallr≥0,whichgives(25).Weshalloftenusethefollowingonedimensionalmaximumprinciple,seee.g.,[11,Theorem2.9.2].

Lemma3.7.Ifg∈C2[0,+∞)isbounded,g(0)≥0and g +cg 0withc>0,theng(r)>0forallr>0.Moreover,ifg(0)=0theng (0)>0.Our rstlemmaisonthederivativesofsolutionsto(22).

Lemma3.8.Supposethat(u,w)κ,νisasolutionto(22)withκ,ν>0and(u,w)κ,ν(r)>(0,0)forallr>0.Then(u ,w )κ,ν(r)>(0,0)forallr≥0.Proof.Sincethesystemisquasimonotonethisfollowsfromamovingplaneargument,similartothemethodusedbyGidas,NiandNirenberg[8].Seealso[2]whereasimilarargumentisusedforascalarequation. Lemma3.9.Foraboundedsolution(u,w)κ,νto(22)withuκ,ν 0itholdsthat0<ν<κand0<wκ,ν(r)<uκ,ν(r)forallr>0.

Proof.Denoteby(u,w)thesolution(u,w)κ,ν.Sincewisboundedandsatis es w +αw=f(u)+Mu 0withw(0)=0wehavebyLemma3.7thatν>0andw(r)>0forallr>0.Letη=β(κ ν).Asobservedearlier,thesolution(u,v)=(u,v)κ,ηto(24)isgivenby(u,β(u w)).Sincevisboundedwithv(0)=0and v +γv=δu 0itholdsagainbyLemma3.7thatη>0andv(r)>0forr>0.Henceκ>νandw(r)<u(r)forr>0.

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM195

Lemma3.10.If(u,w)κ,νisapositivesolutionto(22)suchthat(23)holdsthen

r→∞lim(u ,w )κ,ν(r)=0.

+Proof.De neuK(r):=ρ+δ/γ uκ,ν(r+K)andwK(r):= ρδ/γ wκ,ν(r+K).

ItholdsthatuKandwKconvergeuniformlytozeroon[0,1]asK→∞.From(22)wehavethattheyremainboundedinC2[0,1].Byinterpo-lationuK,wKconvergetozeroinC1[0,1].Therefore(u ,w )κ,ν(K)= (u K,wK)(0)→(0,0),asK→∞.

Let(u,w)κ,νbeasolutionto(22)forwhich(23)holds.Then(u,v)κ,ηwithη=β(κ ν)isasolutionto(24)andvκ,η=β(uκ,ν wκ,ν)→(δ/γ)ρ+δ/γasr→∞.UsingLemma3.10andlettingr→∞in(25)weobtainthefollowingrelationshipbetweenκandν: ρ+ 2δ/γδβ(26)κ2 f(s) sds.(κ ν)2=20

Thiswillbeusedtoprovetheuniquenessofsuchsolutions.Nextweshowthatthereexistsinitialdata(¯κ,ν¯)forwhichthecorrespondingsolutionto(22)ispositiveandsatis es(23).

Lemma3.11.Thereexistsκ¯,ν¯∈Rsuchthatthesolution(u,w)κ¯,ν¯to(22)satis es(23).Moreoverthissolutionispositiveand0<ν¯<κ¯.

Proof.Weshallusesuper-andsubsolutionsandLemmaA.4to ndaposi-tivesolutionto inR+, u =f(u) βu+βw w =f(u)+Mu αwinR+,(27) u(0)=w(0)=0,

+satisfying(23).Asasupersolutionwetake(ρ+δ/γ, ρδ/γ).Wehavetocon-

structanonzerosubsolution.Fromaphaseplaneanalysisoneseesthattheinitialvalueproblem =f(u) βuinR+, uu(0)=0, u(0)=(2Jβ(0))1/2,

withJβ(0)>0de nedin(2),hasasolutionu withlimr→∞u (r)=ρ+βand

u,0)isasubsolution.ByLemmaA.4thereu (r)>0forallr≥0.Then( +existsasolution(u,w)to(27)suchthat(0,u )<(u,w)<(ρ+δ/γ, ρδ/γ).At

thisstagewemaychooseeitherthemaximalorminimalsolution.Inthenextlemmaweshallprovethattheyareequal.Let(¯κ,ν¯):=(u (0),w (0)).Then(u,w)isthesolutionto(22)with(κ,ν)=(¯κ,ν¯).

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

196C.REINECKEANDG.SWEERS

Itholdsthatu(r),w(r)>0forallr>0.Indeed,u(r)≥u (r)>0andsincewisboundedwith w +αw=f(u)+Mu 0andw(0)=0wehavebyLemma3.7thatw(r)>0forr>0andthatw (0)=ν>0.ByLemma3.9,κ¯>ν¯>0.Lemma3.8showsthatu (r),w (r)>0forallr>0.Inparticularlimr→∞u(r)=ρandlimr→∞w(r)=ρ exist.Fromtheequationswe ndthat

f(ρ)+βρ βρ = f(ρ) Mρ+αρ =0.

Fromthisonegetsthatρ = ρandthatf(ρ)=(δ/γ)ρ.Sinceρ>ρ+βwe+havethat(ρ,ρ )=(ρ+ δ/γ, ρδ/γ).

OurlastlemmaconcernstheuniquenesspartofProposition3.6.

Lemma3.12.Let(u,w)κ,νbeapositivesolutionto(22)suchthat(23)holds.Then(κ,ν)=(¯κ,ν¯)with(¯κ,ν¯)asinLemma3.11.

Proof.Let( u,0)bethesubsolutionofthepreviouslemma.Firstweshowthattheminimumandmaximumsolutionsto(27)intheorderinterval

+[( u,0),(ρ+δ/γ, ρδ/γ)]

areequal.Let(u,w)κ,νbetheminimalsolutionand(u,w)κ¯,ν¯themaximalsolution.Itmustholdthatκ≤κ¯andν≤ν¯.Ifthesolutionsarenotequalatleastoneoftheseinequalitiesmustbestrict.Supposeν<ν¯.ByLemma3.9wealsohavethatκ>νandκ¯>ν¯andby(26)that

δ β222β2β22δ β222β2β22(28)κ+κν ν=κ¯+κ¯ν¯ ν¯.2 (β2/δ)ν2isstrictlyincreasingThefunctionx→(1 (β2/δ))x2+2(β /δ)xν on[κ,κ¯]becauseithasderivative2δ β2x/δ+2β2ν/δwhichisstrictlypositiveforx∈[κ,κ¯]sinceδ>β2.Hence

β22δ β222β2β22δ β222β2κ+κν ν≤κ¯+κ¯ν ν.κ+2(β2/δ)¯κx (β2/δ)x2hasderivativeThefunctionx→(1 (β2/δ))¯2β2κ¯δ 2β2x/δ.Sincethederivativeisstrictlypositiveon[ν,ν¯]itfollowsthat

δ β222β2β22β22δ β222β2κ¯+κ¯ν¯ ν¯,κ+κν ν<contradicting(28).Ifκ<κ¯we ndacontradictionbythesameargument.Weconcludethatκ=κ¯andν=ν¯andthat(u,w)κ,ν=(u,w)κ¯,ν¯.Itremainstoshowthatanypositivesolution(u,w)κ,νforwhich(23)holds+isintheorderinterval[( u,0),(ρ+ .δ/γ, ρδ/γ)].Firstweshowthatuκ,ν>u

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM197

Supposethatuκ,ν(r)>ρ+βforallr≥R.Wede neutfor0≤t≤Ron

[0,R]by u (r t)fort≤r≤R;(r):=u t0for0≤r<t,

withu asinLemma3.11.Applyingthesweepingprinciplewiththesubsolu-tions{(u u(r),0)=(u t,0);0≤t≤R}one ndsthat(u,w)κ,ν(r)>( 0(r),0)forr∈(0,R).Henceuκ,ν(r)>u (r)forr>0and(u,w)κ,ν≥( u,0).Ontheotherhand,sinceuκ,ν,wκ,νareincreasingbyLemma3.8,itholds+that(u,w)κ,ν<(ρ+, ρδ/γδ/γ).Sincethereisonlyonesolutionintheorder

+interval[( u,0),(ρ+ δ/γ, ρδ/γ)]theuniquenessisproved.

Ourmainresultconcerningsystem(20)isthefollowing.

Proposition3.13.Assumethatfsatis esConditionA*.Thenthereex-istsauniquepositivesolution(U,W)to(20)satisfying(21).Thissolutionisgivenby

(29)(U,W)(x1,x ):=(u,w)κ¯,ν¯(x1)for(x1,x )∈RN 1,

with(u,w)κ¯,ν¯asinLemma3.11.

Proof.Clearly(29)de nesapositivesolutionto(20)satisfying(21).Sup-posethat(U,W)isanypositivesolutionsatisfying(21).De nethefunctions

()(x1):=(supx ∈RN 1U(x1,x ),supx ∈RN 1W(x1,x ));()(x1):=(infx ∈RN 1U(x1,x ),infx ∈RN 1W(x1,x )).

ByLemmaA.5(isasubsolutionand()isasupersolutionto(27).+Moreover()<(ρ+δ/γ, ρδ/γ).Thisfollowsbysweepingusingthefamilyofsupersolutions{(t, t):t≥ρ+δ/γ}.

Since(isasubsolutionthereexistsapositivesolution(u,w) to+(27)with(≤(u,w) <(ρ+, ρδ/γδ/γ).ByLemma3.12wehavethat(u,w) =(u,w)κ¯,ν¯.UsingasweepingargumentasintheproofofLemma3.12itfollowsthat()>( u,0)withu asintheproofofLemma3.11.Hencethereexistsapositivesolution(u,w) to(27)with( u,0)<(u,w) ≤()andbyLemma3.12,(u,w) =(u,w)κ¯,ν¯.Hence()=()whichprovestheuniquenessclaimintheproposition.

3.3.Thelinearizedproblemonthehalfspace.Let(u,w)κ¯,ν¯beasinProposition3.6.Inthisparagraphweconsiderthefollowinglinearsystem: (uN, r¯ Φ=(f) β+ω)Φ+βΨ r¯ωΦinRκ¯,ν¯ + (30) r¯ Ψ=(f (uκ¯ωΨinRN¯,ν¯)+M)Φ+(ω α)Ψ r+, Φ=Ψ=0on RN+.

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

198C.REINECKEANDG.SWEERS

Hereα,β,Mareasin(4),(3)andB1respectively,ω>max{α,M}andr¯∈R.ForthisproblemwehavethefollowingresultofLiouvilletype.

Proposition3.14.Supposethatr¯≥1andthat(Φ,Ψ)isaboundedpositivesolutionto(30).Then(Φ,Ψ)≡(0,0).

Thispropositionwillbeaconsequenceofthefollowinglemma.

Lemma3.15.Suppose ,ψ∈C[0,+∞)areboundedwith ,ψ≥0, (0)=ψ(0)=0anditholdsthat

(31)

(32) ≤f (uκ¯,ν¯) β +βψ,

ψ ≤f (uκ¯,ν¯) +M αψ,

inD (0,∞)-sense.Then (x1)=ψ(x1)=0forallx1≥0.

Proof.Weset(p,q):=(u ,w )κ¯,ν¯andrecallthatp,q>0on[0,∞).Withoutlossofgeneralityweassumethat ,ψ≤1.Wearguebycontradictionandsupposethat( ,ψ)=(0,0).FirstweobservethatifthereexistsK>0suchthat (x1)=ψ(x1)=0forallx1≥Kthenbyasweepingargumenton[0,K]withthefamily{(tp,tq);t≥0}ofsupersolutionsitfollowsthat (x1)=ψ(x1)=0forallx1∈[0,K].Thisisincontradictionwithourassumption.NowletK>0andε>0besuchthatthat

f (uκ¯,ν¯(x1))< ε

andnotethatalso

f (uκ¯,ν¯(x1))+M α< ε

Byour rstobservationwemayassumethat

R(K):=max{ (K)/p(K),ψ(K)/q(K)}>0.

Wede nethefollowingfunctionson[K,∞):

St(x1)= (x1) e√x1 t)forallx1≥K,forallx1≥K.

Tt(x1)=ψ(x1) e(x1 t),

Rt(x1)=max{St(x1)/p(x1),Tt(x1)/q(x1)}

fort≥K.Itholdsthat

≤(f (uκ St¯,ν¯) β)St+βTt,√,

and Tt ≤(f (uκ¯,ν¯)+M)St αTt,

inD (K,∞)-sense.Fort>Kletmt=supx1∈[K,t]Rt(x1).Bythemaximumprincipleonehasthatmt=Rt(K)fortlargeenough.Indeed,sinceforωlargeenough,itholdsinD (K,t)-sensethat

(St mtp) +ω(St mtp)≤0,

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM199

and

(Tt mtq) +α(Tt mtq)≤0,

weseethatmtmustbeattainedinKorint.SinceRt(t)≤0andRt(K)>0,iftislargeenough,weconcludethatmt=Rt(K).Nowletx1∈[K,∞)be xed.Thenforallt>x1largewehave

R(x1)=max{ (x1)/p(x1),ψ(x1)/q(x1)} √ √(x1 t)x1 t)≤Rt(K)+maxe/p(x1),e/q(x1).

Lettingt→∞wededucethatR(x1)≤R(K)andhenceRattainsitsmaximumon[K,∞)inx1=K.Consequentlysupx1∈[0,∞)R(x1)isattainedinsomepointr0∈(0,K].Butthisisincontradictiontothemaximumprinciple.Indeed,inasimilarwayasabove,oneseesthatR(x1)mustattainitsmaximumon[0,K+1]eitherin0orinK+1andnotinK. ToseehowProposition3.14followsfromthislemma,wede ne (x1):=supΦ(x1,x );x ∈RN 1, N 1ψ(x1):=supΨ(x1,x);x∈R.

ThenbyLemmaA.5, ,ψ∈C[0,+∞)with (0)=ψ(0)=0andinD (RN+)-sense

1 1βψ ω ) β+ω) +(f(uκ¯,ν¯1 1 ψ ≤(f(uκ(ω α)ψ ωψ.)+M) +¯,ν¯Sincer¯≥1wededucethat ≤ ≤(f (uκ¯,ν¯) β+ω) +βψ ω

=(f (uκ¯,ν¯) β+ω) +βψ

and

ψ ≤(f (uκ¯,ν¯)+M) +(ω α)ψ ωψ

=(f (uκ¯,ν¯)+M) αψ.

Bythelemma( ,ψ)(x1)=0forx1≥0andhencealso(Φ,Ψ)(x)=(0,0)onRN+.

4.Proofsofthemainresults.

4.1.ProofofTheorem2.1.FromnowonweassumethatΓisC3We . ¯×beginRecallthatXdenotesthespaceC byde ning1some operators. ¯andletC ¯=u∈C1 ¯;u(x)=0forx∈Γ.Fork,λ>0C 0

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

200C.REINECKEANDG.SWEERS

1 1 1 11¯¯de ne λ +k0:C →C0 byu= λ +k0gwith 1 ¯theuniquefunctionsatisfyingu∈C0 λ 1 u+ku=ginD ( )-sense,u=0onΓ. 1 ¯×C1 ¯inXandde netheoperatorLetjbetheembeddingofC00Kk,λ:X→Xby 1 1 λ +k00gg 1 1Kk,λ=j .hh0 λ +k0 1Sincejiscompactand λ 1 +k0iscontinuous,Kk,λisacompactlinearmaponX.Weshallalsousethefactthat Kk,λ L(X)isuniformly

boundedinλ.Thisfollowsfromthefactthat 1 1 1(33) λ +k0g ≤ g ∞∞ ¯¯andforeveryg∈C .We xω>max{α,M}.Forafunctionu∈C λ>0thewede netheoperatorsMu,Tu,λ∈L(X)by f(u) βhhβ(34),=Muggf (u)+M α

and

(35)Tu,λ:=Kω,λ(Mu+ωI).

Operatorsofthiskindwerestudiedextensivelyin[23].Ifu∈[0,ρ+δ/γ]thenTu,λisapositiveirreduciblecompactoperatoronX,see[23,Lemma1.3].Moreover,Tu,λhasapositivespectralradius(seee.g.,[18])whichwedenotebyr(Tu,λ).BytheKrein-RutmanTheorem(seee.g.,[1,Theorem3.1]),r(Tu,λ)isaneigenvalueofTu,λtowhichapositiveeigenfunctionpertains.Inthenextlemmaweprovethatforλlargeenoughitholdsforeverysolution(u,w)∈[Zλ,Y]thatr(Tu,λ)<1.

Lemma4.1.Thereexistsλ >λ×suchthatforallλ>λ andeverysolution(u,w)∈[Zλ,Y]to(Qλ)thecorrespondingoperatorTu,λhasspectralradiusr(Tu,λ)<1.

Proof.Weprovethelemmabyacontradictionargument.Assumethatit×doesnothold.Thenthereexistasequence{λn}∞n=1withλ<λn→∞andsolutions(un,wn):=(uλn,wλn)∈[Zλn,Y]to(Qλ)withλ=λnsuchthatrn≥1,withrndenotingthespectralradiusofTn:=Tun,λn.Let( n,ψn)∈Xbethepositiveeigenfunctionpertainingtorn.Wenormalizetheeigenfunctionsuchthatmax n=1.Thiscanbedonesince n=0

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM201

impliesthatψn=0,Itholdsthat 1 rnλn n=1 rnλ n ψn= n=acontradictionbecause( n,ψn)isaneigenfunction.(f (un)+ω β) n+βψn rnω n(f (un)+M) n+(ω α)ψn rnωψn

ψn=0

1/kin ,in ,on .Becauseoperatornorms Tn L(X)areuniformlyboundeditfollowsfrom k rn= Tn k→∞L(X)≤ Tn L(X)

thatthesequence{rn}∞n=1isbounded.Bygoingovertoasubsequence,still∞denotedby{rn}n=1,wecanassumethatrn→r¯≥1.Withθn:=β( n ψn)onehasthat 1 θ rnλnin ,n=δ n γθn+(1 rn)ωθnθn=0on .

1 Thisshowsthat n≥ψnandhence rnλ n n≤f(un) ingesti-1 1/2+ρσ0mate(15)inLemma3.4wehaveforallx∈ withdist(x,Γ)>b 0λnthatf (un(x))≤0andconsequently 1 1/2+ n≤0in(36)x∈ ;dist(x,Γ)>b0λnρσ0.

1¯nwithdist(¯xn,Γ)≤b ρ+Hence nattainsitsmaximuminapointxσ0.0λnLetx¯Γ,n∈Γbesuchthat|x¯n x¯Γ,n|=dist(¯xΓ,n,Γ).Bygoingovertoasubsequencewecanassumethatx¯Γ,n→x¯∈Γ.Byablow-upargumentaroundx¯ ,similartotheargumentin[4],one constructsU,W,Φ,Ψ∈C2RN+∩C+suchthat(U,W)satis es

inRN +, U=f(U) βU+βW

W=f(U)+MU αWinRN+, U=W=0on RN+, 1/2

and(Φ,Ψ)satis es ¯ Φ=(f (U)+ω β)Φ+βΨ r¯ωΦ r

¯ωΨ r¯ Ψ=(f (U)+M)Φ+(ω α)Ψ r Φ=Ψ=0inRN+,inRN+,

on RN+.

Thenormalizationmaxφn=1leadstosupΦ=1.Furthermore,usingtheuniformestimate(14)itfollowsthat

(37)x1→∞+, ρlim(U,W)(x1,x )=(ρ+δ/γδ/γ)uniformlyinx ∈RN 1.

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

202C.REINECKEANDG.SWEERS

Hence,byProposition3.13,(U,W)(x1,x )=(u,w)κ¯,ν¯(x1).Then(Φ,Ψ)isaboundedpositivesolutionto(30)withr¯≥1.ByProposition3.14,(Φ,Ψ)≡(0,0),incontradictionwithsupΦ=1. Weshallusethislemmatoprovethattherecanbeatmostonesolutionto(Qλ)intheorderinterval[Zλ,Y].Firstwede netheoperatorHλ:X→X

Hλ:=Kω,λ(F+ωI),

whereF:X→Xisde nedby f(u) βu+βwu.=Ff(u)+Mu αww

WeshallshowthatHλhasatmostone xedpointin[Zλ,Y].InordertousetheLeray-Schauderdegreewehavetoconsiderthe xedpointproblem

Hλ(u,w)=(u,w)

inanappropriatespace. ¯∩C2( )thecorrespond-Letµbetheprincipaleigenvalueande∈C1 ingeigenfunctiontotheproblem e=µein ,e=0onΓ.

Wenormalizeesuchthatmaxe=1.FollowingAmann[1]wede ne ¯:=u∈C ¯; t>0suchthat|u|≤te,Ce (38)

equipped withthenorm u e=inf{t>0; te≤u≤te}.Itholdsthat¯isaBanachspace,infactaBanachlattice,withclosedunitballCe ¯; e≤u≤e.LetXe=Ce ¯×Ce ¯.Orderintervalsinu∈C Xe willbedenotedby[·,·]e.Letj1,j2betheembeddingsofXeinXand 1 ¯×C1 ¯inXerespectivelyandde neHe:Xe→XebyC00λ 1 1 λ +ω00e 1 1 (F+ωI) j1.:=j2 Hλ0 λ +ω0 1 ¯intoWerecallthat λ 1 +ω0wasde nedasanoperatorfromC 1 ¯.Wenotethat(u,w)isa xedpointofHeifandonlyifj1(u,w)isC0λehasaunique xeda xedpointofHλ.Henceitsu cestoshowthatHλpointin[Zλ,Y]∩Xe.Italsoholdsfor(ui,wi)∈Xwith(u1,w1)<(u2,w2)that

(39)Hλ(u1,w1)<Hλ(u2,w2).

InfactHλ(u2,w2) Hλ(u1,w1)isanelementoftheinteriorofthepositiveconeofXe,orequivalently,thereexistst>0suchthat

(40)Hλ(u2,w2) Hλ(u1,w1)≥(te,te).

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a tr

ONAFITZHUGH-NAGUMOTYPEELLIPTICSYSTEM203

SinceneitherZλnorYare xedpointsofHλwe nd,using(39),thatany xedpoint(u,w)∈[Zλ,Y]satis es

:=HλZλ<(u,w)<HλY=:Y .Zλ

,Y ],theFrom(40)weevenhavethestrongerresultthat(u,w)∈int[Zλe ,Y ]withrespecttothe · -topology.Theuniquenessofainteriorof[Zλee xedpointofHλin[Zλ,Y]forλ≥λ thenfollowsfromthenextlemma.

einLemma4.2.Foreveryλ>λ thereexistsaunique xedpointofHλ ,Y ].int[Zλe

Proof.Wehaveforeveryλ>λ thatthereexistsatleastonesolutionto(Qλ)intheorderinterval[Zλ,Y]which,asweobserved,isa xedpointofeand(u,w)∈int[Z ,Y ].Toshowthatthisistheonlysolution,weHλλeshalluseadegreeargument. ,Y ],withλ>λ ,isa xedpointofHe.TheSuppose(u,w)∈int[ZλλeeeeoperatorHλisdi erentiableandTu,λ:=dHλ(u,w)∈L(Xe)givenby 1 1 λ +ω00e 1 1 (Mu+ωI) j1,Tu,λ=j2 0 λ +ω0

withMuasde nedin(34).FromLemma4.1wehavethatthespectrale)<1.IndeedµisaneigenvalueofTeifandonlyifµisanradiusr(Tu,λu,λe)>0.ButTeisaeigenvalueofTu,λ.Sincer(Tu,λ)>0itholdsthatr(Tu,λu,λeepositivecompactoperatorandhencer(Tu,λ)isaneigenvalueofTu,λtowhiche)=r(Tapositiveeigenfunctionpertains.Thisimpliesthatr(Tu,λu,λ)<1.eandconsequentlytheindexofInparticular1isnotaneigenvalueofTu,λthe xedpoint(u,w)iswellde nedwith

index(u,w)=1,

see[17,p.66].Usingthehomotopyinvarianceofthedegreeandthefact ,Y ]isconvex,itfollowsthatthatint[Zλe e degreeI Hu,λ,int[Zλ,Y ]e,0=1.

,Y ]bearbitraryandde nethehomotopyIndeed,letz¯∈int[Zλe

eGt=(1 t)(I z¯)+t(I Hu,λ).

ezandhencez∈¯+tHu,λItholdsthatGtz=0ifandonlyifz=(1 t)z ,Y ].SinceGhasnozerosontheboundaryint[Z ,Y ]wehaveint[Zλtλeethat

,Y ]e,0)=degree(G0,int[Zλ,Y ]e,0)=1.degree(G1,int[Zλ

ecanhaveatmostBytheadditivitypropertyofthedegreeweseethatHλ ,Y ].one xedpointinint[Zλ e

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