Klein-Gordon-Maxwell-Proca Type Systems in the Electro-Magneto-Static Case

HEBEY Emmanueland THIZY Pierre-Damien

1Universit′edeCergy-Pontoise,D′epartementdeMath′ematiques,SitedeSaint-Martin,2 avenue Adolphe Chauvin,95302 Cergy-Pontoise cedex,France.

2Institut Camille Jordan,Universit′e Claude Bernard Lyon 1,B?atiment Braconnier,21 avenue Claude Bernard,69622 Villeurbanne Cedex,France.

Abstract.We investigate a Klein-Gordon-Maxwell-Proca type system in the context of closed 3-dimensional manifolds.We prove existence of solutions and compactness of the system both in the subcritical and in the critical case.

Key Words:Elliptic systems;Klein-Gordon-Maxwell-Proca equations;critical exponent;compactness;stability.

1 Introduction

Let(M,g)be a 3-dimensional closed manifold.We look for systems of equations with unknowns(u,v,A),where u and v are functions and A is a 1-form,which express like

where q＞0,Φ(x,v,A)=a?ω2(qv?1)2+|?S?qA|2,ω∈R,a,b,S∈C∞(M)are smooth functions with a,b＞0 in M,Δg=?divg?is the Laplace-Beltrami operatorwhenacting on functions u and v,Δg=δd+dδ is the Hodge-deRham Laplacian whenacting on1-forms A and p∈(2,6](d is the differential,δ is the codifferential,6 is the critical Sobolev exponent).Systems of equations like(1.1)are derived from the full KGMP system when we look for solutions of such systems in the form Ψ(x,t)=u(x,t)eiS(x,t)with u depending only on x and S in the splitted form S(x,t)=S(x)?ωt.The full KGMP system(see the discussion in Hebey and Truong[1]and Section 2 below)for solutions like Ψ(x,t)=u(x,t)eiS(x,t)is written as

where Δg=?divg? is the Laplace-Beltrami operator,is half the Laplacian acting on forms,and δ is the codifferential.We assume here that we are in the static case,where?tu≡0,?tA≡0 and ?t?≡0.We look for solutions with S(x,t)=S(x)?ωt.Such type of solutions were introduced in the very nice paper by Benci and Fortunato[2]for the Klein-Gordon-Maxwell equations in R3(see also D’Avenia,Mederskiand Pomponio[3]).A special choice of S in these papers gives rise to vortex solutions of the system.In the case of(1.2),for such solutions,namely for solutions like Ψ(x,t)=u(x)ei(S(x)?ωt),the full KGMP system(1.2)in the static case writes as

Thankstothe fourthequationinthis system,sinceand δΔg=0,thesecondequation in(1.3)can be omitted and replaced by the Coulomb gauge equation δA=0.Also the secondequationis automatically satis fiedin theKlein-Gordon-Maxwell settingforwhich m1=0(sinceWhen δA=0,we get thatwhere Δg=dδ+δd is the Hodge-de Rham Laplacian on forms.Letting ?=ωv,and if we replaceby a positive function a,andby a positive function b,the system(1.3)reduces to(1.1)when we forget about the Coulomb gauge equation δA=0.It’s not clear that much can be said about(1.1).The goal in this paper is to prove that despite it’s intricate structure,strong results can be proved on this system.We recall that a coercive operator like Δg+Λghas positive mass if the regular part of its Green’s function is positive on the diagonal.Our main result is thefollowing theorem.More resultsare provedin the sequel.The subscript R in the notationrefers to the fact that a,b,S in the theoremare(real-valued)functions.

Theorem 1.1.Let(M,g)be a smooth closed 3-manifold,q＞0 and a,b,be smooth functions with a＞0 and b≥0 in M.We assume that Rcg+bg＞0 in M,in the sense of bilinear forms,where Rcgis the Ricci curvature of g.Let ω∈R be such thatLet p∈(2,6].When p is critical from the viewpoint of Sobolev embeddings,namely when p=6,we also assume that

where Λg＞ 0 is a smooth positive function such that Δg+Λghas positive mass.Then(1.1)possesses a smooth nontrivial solution(u,v,A)with u＞0 and v＞0,and the setSof solutions(u,v,A)of(1.1)with u≥0 and v≥0 is compact in the C2-topology.

The above theorem includes both the subcritical case for which p＜6 and the critical case for which p=6.Condition(1.4)is required in the sole critical case.By the positive mass theorem of Schoen and Yau[4](see also Witten[5])when the scalar curvature Sgof g is positive,any function Λgsuch thatthe inequalitybeing strict at least at one point if(M,g)is conformally diffeomorphic to the unit 3-sphere,satis fies that the mass of Δg+Λgis positive.In other words,we can takein(1.4)and the existence of a smooth positive function Λgfor which Δg+Λghas positive mass is guaranteed when Sg＞0.As another remark,adding the Coulomb gauge equation δA=0 to(1.1)obviously does not change anything to the compactness part of Theorem 1.1.An extended version of the compactness part of the theorem to the notion of stability,where the coefficients and the exponent in the equations are allowed to vary,is proved in Theorem 6.1 and Theorem 8.1 below.Consequences of the stability of(1.1)with respect to its coefficients are derived in Proposition 8.1.

2 A quick origin of the problem

TheKlein-Gordon-Maxwell-Proca system(1.2)describesaninteracting fieldtheorymodel in theoretical physics.The interaction in this model is described by the minimum substitution rule

in a nonlinear Klein-Gordon Lagrangian.As for the external massive vector field it is governed by the Maxwell-Proca Lagrangian.More precisely,assuming for short that the manifold is orientable,we de fine the Lagrangian densitiesLNKGandLMPof ψ,?,and A by

Writing ψ in polar form as ψ(x,t)=u(x,t)eiS(x,t),taking the variation ofSwith respect to u,S,?,and A,we get(1.2),and if we let

then the two last equations in(1.2)give rise to the first pair of the Maxwell-Proca equations

with ∈0=μ0=1(units are chosen such that c=1)andwhile the two first equations in(2.4)give rise to the second pair of the equations.These massive Maxwell equations,as modi fied to Proca form,appear to have been first written in modern format by Schr¨odinger[6].The Proca formalism a priori breaks Gauge invariance.Gauge invariance can be restaured by the Stueckelberg trick,as pointed out by Pauli[7],and then by the Higgs mechanism.We refer to Goldhaber and Nieto[8,9],Luo,Gillies and Tu[10],and Ruegg and Ruiz-Altaba[11]for very complete references on the Proca approach.The first equation in(1.2)gives rise to the nonlinear Klein-Gordon matter equation.The second equation in(1.2)gives rise to the charge continuity equationwhich,thanks to(2.5),is equivalent to the Lorentz conditionIn the stationnary setting,as discussedin the introduction,we are back to the Coulomb gauge equation.When discussing an equation like(1.1)we essentially discuss the part of(1.3)which consists of the Maxwell-Proca equations coupled with the nonlinear Klein-Gordon matter equation,and we forget about the gauge condition in Coulomb form?·A=0(the notations?·and δ represent the same object here).

3 The auxiliary map problem 1

Lemma 3.1.Assume that Rcg+bg＞0 in M,in the sense of bilinear forms,where Rcgis the Ricci curvature of g.For any,there exists a uniquesuch that

and let

where

By Rellich-Kondrakov,the embeddingsare compact for any 1≤p＜6.By the Weitzenb¨ock formula we get that for any 1-form A,

where A?is the vector field we get from A by the musical isomorphism.Let(Aα)αbe a minimizing sequence forμ.Then,by(3.4),

is nonnegative.IfQis the quadratic form associated toB,then(same proof as for the Cauchy-Schwarz inequality),

and thus that

and contracting by A2?A1,since Rcg+bg＞0 in M and by theWeitzenb¨ockformula again,we get thatThis ends the proof of the lemma.

Now we can prove the following important a priori bound lemma.

Lemma3.2.Assumethat Rcg+bg＞0 in M,in the sense of bilinear forms.There exists aconstant C＞0,depending on q and S,such that|Au|≤C in M for allwhere Auis as in Lemma 3.1.

In particular,by(3.5),

Since Rcg+bg＞0 in M there exists ε0＞0 such that Rcg≥?(b?ε0)g in M.In particular

in M.Then,by(3.6),

in M.By the weak maximum principle it follows thatand thus that|A|≤in M.This ends the proof of the lemma.

As a remark,since b≥0 and Rcg+bg＞0,there holds with(3.4)and the Sobolev inequality for 1-forms that for any smooth 1-form A,and any

where C,C′＞0 are independent of A and ε＞0 is independent of A and u.Now we prove the following derivation lemma.

Lemma 3.3.Assume that Rcg+bg＞0 in M,in the sense of bilinear forms.Letbe the map de fined by Φ(u)=Au,where Auis as in Lemma 3.1.Then Φ is locally Lipschitz,differentiable and its differentialatis given by

Contracting(3.10)with Av?Auand integrating over M we get with(3.4)that

by H¨older’s inequalities and Lemma 3.2,where C＞0 does not depend on u and v.By Sobolev,and since we assumed that Rcg+bg＞0,we then get from(3.11)that

where C＞0 does not depend on u and v.This proves that Φ is locally Lipschitz.Now we turn our attention to the proof that Φ is differentiable.By Lemma 3.2,and with the same arguments than those used in the proof of Lemma 3.1,we get thatexists for all,whereis given by(3.9).Also we easily get from(3.4),Lemma 3.2 and the assumption Rcg+bg＞0 thatis(linear)continuous.Now we letbe given,and forwe de fine

There holds that

Contracting(3.13)with□hand integrating over M we get with H¨older’s inequalities,(3.4)and Lemma 3.2 that

where C＞0 is independent of u and h.Since Rcg+bg＞0 it follows from(3.12)and(3.14)that

After(3.11)we can also write that

by Sobolev inequality.Then we get that

where C＞0 is independent of u and v.The following lemma is an important(technical)consequence of Lemma 3.3.

Lemma 3.4.The functionde fined by

is C1inand its differential is given by

Proof.By composition,and Lemma 3.3,H is differentiable.Moreover,using Stoke’s formula,(3.1)and(3.9),we get that it’s differential is given by

4 The auxiliary map problem 2

Following a very nice idea going back to Benci-Fortunato[12],we introduce the auxiliary mapde fined by the equation

It follows from standard variational arguments that Ψ is well-de fined inas soon as b＞0(and we assume that b＞0 in this section).Noting thatsince u and Ψ(u)are init follows from(4.1)that.We further get with(4.1)thatElementary though useful properties of Ψ are as follows.

Lemma 4.1.The mapis C1and its differential DΨ(u)=Vuat u is the map de fined by

it also followsfrom themaximum principle that,and thisprovesthatfor allGivenwe compute

Multiplying(4.4)by Ψ(u+h)?Ψ(u),and integrating over M,by the coercivity of Δg+b,by the Sobolev embedding theorem,by H¨older’s inequality,and by what we proved above,we get that

Multiplying(4.5)by Ψ(u+h)?Ψ(u)?Vu(h)and integrating over M,by the coercivity of Δg+b,by the Sobolev embedding theorem,by H¨older’s inequality,and by the inequality on Ψ we proved above,we get that

Coming back to(4.4),and the procedure described after(4.4),it also holds true that

Lemma 4.2.The mapgiven by

is C1inand,for any

Proof.It follows from Lemma 4.1 that Θ is differentiable.By(4.1),

Noting that

we get that(4.8)holds true.It is easy to check that Θ is actually C1.This ends the proof of the lemma.

5 The variational analysis of the problem

Solutions(u,v,A)of(1.1)are(formally)critical points of the functional

given by

As one can check,I is a highly competitive functional.The following lemma avoids dealing with the competition between u,v and A.

Lemma 5.1.Let a,b＞0 in M and ω ∈R be such thatLet p∈(2,6].Assume that Rcg+bg＞0 in M,in the sense of bilinear forms.De fine the functional

where Auis as in Lemma 3.1,Ψ is as in Lemma 4.1 and u+=max(0,u).The functional Ipis differentiable and if u is a critical point of Ip,then(u,Ψ(u),Au)is a smooth solution of(1.1)with u,v≥0.

Proof.The functional Ipis differentiable by Lemmas 3.4 and 4.2.Moreover,

RelatedreferencesintheProcacontext,orinthecompact case,are byClapp,Ghimenti and Micheletti[14]and by d’Avenia,Pisani and Siciliano[15].

6 The subcritical case

A key point in the analysis,which plays its role even in the subcritical case,is the lowdimensionalcompensationphenomenonattached totheseequations.It holdsbothfor Auand Ψ(u).As a remark,see Thizy[50],the phenomenon is lost in high dimensions.The key lemma for the low-dimensional compensation phenomena for Auis the following.

Lemma 6.1.[Local control for Au]Let(M,g)be a smooth closed 3-manifold,q＞0 and b,S∈C∞(M)be smooth functions with b＞0 in M.We assume that Rcg+bg＞0 in M,in the sense of bilinear forms,where Rcgis the Ricci curvature of g.Then

where Auis as in Lemma 3.1.

Proof.By(3.1)and Lemma 3.2,

noting that 4＞12/5,there holds that

Now we prove the following existence lemma.

Lemma 6.2.Let(M,g)be a smooth closed 3-manifold,q＞0 and a,b,S∈C∞(M)be smooth functions with a,b＞0 in M.We assume that Rcg+bg＞0 in M,in the sense of bilinear forms,where Rcgis the Ricci curvature of g.Let ω ∈R be such thatFor any p∈(2,6),(1.1)possesses a smooth nontrivial solution(u,v,A)with u＞0 and v＞0.

Proof.Let p∈(2,6)and Ipbe the functional in(5.1).It is easily checked that Ip(0)=0 and that for u0arbitrarily given inwiththere holds thatBy Lemma 4.1,

By Lemma 6.1,for any ε＞0 there exists δ0＞0 such that

wherePdenotes the class of continuous paths joining 0 to T0u0.By the mountain pass lemma,see Ambrosettiand Rabinowitz[16],there exists(uα)αin H1Rsuch thatandinas α→+∞.Writing that Ip(uα)=cp+o(1)and that DIp(uα)·we get that

and since

for all α,where C,C′＞0 are independent of α.By(6.6)and(6.8),also by Lemmas 3.2 and 4.1,and since p＞2,the sequence(uα)αis bounded inSince p＜6 we may thenassume that there existssuch that,up to a subsequence,inand uα→upinas α→+∞.Sinceinas α→+∞,there holds that up≥0 in M,and since cp＞0,it is clear from(6.6)thatFor any.Hence,see(5.2),we have that

we get by Lemmas 3.2 and 4.1,that for i=1,2,

where C1,C2＞0 do not depend on α.Letting α→+∞ it follows from(3.16),(4.6),(6.9)and(6.10)that for any

In particular,by(6.11),upis a critical point of Ipand we can apply Lemma 5.1.This ends the proof of Lemma 6.2.

A complementary lemma to Lemmas 3.2 and 4.1 includes the case b≡0.We letbe the set of smooth nonnegative functions ?≥0 in M.Given Λ＞0 arbitrary,we let

Lemma 6.3.Let Λ＞0 arbitrary.There exists a constant CΛ＞0 such that for anyanyand anysuch that Rcg+bg＞0 in M in the sense of bilinear forms,ifandare such that

Proof.Given u and A as in(6.12),we get the boundfor A exactly as in the proof of Lemma 3.2.Same thing with the proof of the inequalityin Lemma 4.1.There we just need to have thatto avoid v=Cte.This is where the conditioncomes from.The problem does not occur with A when b≡ 0 as there are no nontrivial harmonic 1-forms when Rcg＞0.As a possible bound given by this proof we can take.Lemma 6.3 is proved.

For the sake of clearness we prove the following result in this section.

Lemma 6.4.Let(M,g)be a smooth closed 3-manifold,q＞0,a,b,S∈C∞(M)be smooth functions with a＞0 in M,b≥0 in M,ω ∈R and p∈(2,6].Let(aα)α,(bα)αand(Sα)αbe sequences of smooth functions such that aα→a and bα→b inas α→+∞,such that bα≥0 in M for α?1,such that Sα→S inas α→+∞ for some θ∈(0,1)and such that Rcg+bαg＞0 in M in the sense of bilinear forms for all α.Let also(ωα)αand(pα)αbe sequences of real numbers such that ωα→ω and pα→p in(2,6]as α→+∞.Let(uα,vα,Aα)αbe a sequence of solutions of

where

and uα＞0 in M.AssumeThen,up to passing to a subsequence,uα→u,vα→v inand Aα→A inas α→+∞,for someandwhich solve(1.1).

Proof.By the Weitzenb¨ock formula,for any 1-form A,

where Δgis the rough Laplacian and Rcg(A)is the 1-form with coordinates Rcg(A)i=gαβRiαAβ.In particular,in local coordinates,for any i=1,...,3,

As a remark on Lemma 6.4,the following result holds true.

for some C＞0 independent of α.This implies that u＞0.Passing to the limit we get that

and it follows that v ＞0.Also.The lemma follows.

At this point we prove the following stability proposition.

Theorem 6.1.Let(M,g)be a smooth closed 3-manifold,q＞0,a,b,S∈C∞(M)be smooth functions with a＞0 in M,b≥0 in M,ω∈R and p∈(2,6).Let(aα)α,(bα)αand(Sα)αbe sequences of smooth functions such that aα→a and bα→b inas α→+∞,such that bα≥0 in M for α?1,such that Sα→S inas α→+∞ for some θ∈(0,1)and such that Rcg+bαg＞0 in M in the sense of bilinear forms for all α.Let also(ωα)αand(pα)αbe sequences of real numbers such that ωα→ω and pα→p as α→+∞.Then for any sequence(uα,vα,Aα)αof solutions of

where

and uα＞0 in M,there holds that,up to a subsequence,uα→u,vα→v inand Aα→A inas α→+∞,for someandwhich solve(1.1).

Proof.By Lemma 6.3 there exists C ＞ 0 such thatfor all α,where Φα=Φα(·,vα,Aα).At this point we assume by contradiction that

as α→+∞.Let xα∈M and μα＞0 be such that

By(6.16),μα→0 as α→+∞.De fineby

and gαbyforwhere δ＞0 is small.Sinceμα→0,we get that gα→ξ inas α→+∞.Moreover,by(6.15),for all α,whereis given by

in R3,where Δξis the Euclidean Laplacian and,since 2＜ p＜6,we get a contradiction with the Liouville result of Gidas and Spruck[18].As a conclusion,(6.16)is not possible and we get thatThen the result follows from Lemma 6.4.This ends the proof of Theorem 6.1.

Now we are in position to prove Theorem 1.1 in the subcritical case.

Proof of Theorem 1.1 in the subcritical case.Existence in Theorem 1.1 when p＜6 and b＞0 follows from Lemma 6.2.The compactness part in Theorem 1.1 in the subcritical case is an easy consequence of the stability proved in Theorem 6.1.Existence in the b≥0 case follows from the existence in the b＞0 case together with the stability proved in Theorem 6.1.It suf fices to let bα=b+εαand leave fixed the other parameters,where(εα)αis a sequence of positive real numbers converging to zero.This proves Theorem 1.1 in the subcritical case.As a remark,it can be noted that by Lemma 6.5,if(uα,vα,Aα)αis a sequence of solutions of(1.1)which converge to(u,v,A)in C2,then u＞0 and v＞0 as soon as uα＞0 in M for all α.Also A6≡0 if S is not a constant.

7 Existence in the critical case

A key point here,in addition to the low-dimensional compensation phenomenon for Au,is the low-dimensional compensation phenomenon for Ψ.We look in this section,as in Lemma 6.2,for“variational” solutions of our problem.

Lemma 7.1(Local control for Ψ).Let(M,g)be a smooth closed 3-manifold,q＞0 and b∈C∞(M)be a smooth function with b＞0 in M.Then

where Ψ is as in(4.1).

Proof.The proof goes as for the proof of Lemma 6.1.By(4.1)and Lemma 4.1,there holds that

Let Λg＞0 be such that Δg+Λghas positive mass.Let G be the Green’s function of Δg+Λg.Let x0be given in M and G(x)=G(x0,x).In geodesic normal coordinates,

where ω2is the volume of the unit 2-sphere,A ＞ 0 is the mass at x0andFollowing Schoen[19],we let ρ0＞0 be a small radius and ε0＞0 to be chosensmall relative to ρ0.Let also ψ be a piecewise smooth decreasing function of|x|such that ψ(x)=1 forfor|x|≥2ρ0,andfor ρ0≤|x|≤2ρ0.We de fine uε,ε＞0,by

where dgis the Riemannian distance,and we require that

Computing as in Schoen[19]we get that

for ε?1,where K3is the sharp constant in the Euclidean Sobolev inequality for(see Aubin[20]and Talenti[21]).Also there holds that

In particular,since uε→0 a.e.as ε→0,we get from(7.5),(7.6)and the compactness of the embeddingsfor p＜6 that uε→0 infor all p＜6 as ε→0.Givenwe de fine

wherePdenotes the class of continuous paths joining 0 to u0and I6is the functional in(5.1)in the critical case p=6.The following lemma holds true.

Lemma 7.2.Let(M,g)be a smooth closed 3-manifold,q＞0 and a,b,S∈C∞(M)be smooth functions with a,b＞0 in M.We assume that Rcg+bg＞0 in M,in the sense of bilinear forms,where Rcgis the Ricci curvature of g and that(1.4)holds true.Then I6(u0)＜0 and

Proof.We let(uε)εbe as in(7.4).By Lemma 3.2,Lemma 4.1 and(7.6)there exists T0?1 such that I6(T0uε)＜0 for all 0＜ε?1.We fix such a T0?1.Since uε→0 inas ε→0,there holds from the low-dimensional compensation lemmas,Lemma 6.1 and Lemma 7.1,that

as ε→0.Then,by(1.4)and(7.9),

for all 0＜ε?1,where

for all 0＜ε?1.By(7.5),letting u0=T0uεfor ε＞0 suf ficiently small,we get that c6(u0)≤for some δ0＞0.Concerning the left hand side inequality in(7.8),we proceed as in the beginning of the proof of Lemma 6.2.By Lemma 4.1,

By Lemma 6.1,for any ε′＞0 there exists δ1＞0 such that

At this point it remains to prove that the following result holds true.

we get that

By Lemma 4.1 we have that

and that

and it follows from Lemmas 3.2 and 4.1,and from(7.17),that

Then,by(7.16)and(7.18)we get thatand thus,coming back to(7.15),we get thatUp to passing to a subsequence we may assume thatininand uα→u a.e.for someSinceinthere holds that u≥0.Let.Then.Hence,see(5.2),

As in the proof of Lemma 6.2,letting α→+∞ it follows from(7.19)that for any

In particular,by(7.20),u is a critical point of I6and we can apply Lemma 5.1.If,then u＞0,also Ψ(u)＞0,by the maximum principle and(1.1)possesses a smooth nontrivial solution(u,v,A)with u＞0 and v＞0.Then,at this point,it remains to prove that we cannot have that u≡0.We assume by contradiction that u≡0.Then we get from(7.15)and(7.17)that

By the sharp Sobolev inequality,see Hebey and Vaugon[22,23],

Combining(7.21)and(7.22)we get thata contradiction with our choice of u0by Lemma 7.2.Hence u6≡0 and this ends the proof of Lemma 7.3.

8 Stability in the critical case

Compactness and stability results for Yamabe type equations have a long history.Among possible references,we refer to Brendle[24],Brendle and Marques[25,26],Druet[27,28],Druet,Hebey and Robert[29],Esposito,Pistoia and V′etois[30],Khuri,Marques and Schoen[31],Li and Zhang[32–34],Li and Zhu[35],Marques[36]and Schoen[37–40].A reference in book form is Hebey[41].This list is far from being exhaustive.The goal in this section is to prove the following stability proposition.

Theorem 8.1.Let(M,g)be a smooth closed 3-manifold,q＞0,a,b,S∈C∞(M)be smooth functions with a＞0 and b≥0 in M.Let(aα)α,(bα)αand(Sα)αbe sequences of smooth functions such that aα→a and bα→b inas α→+∞,such that bα≥0 in M for α?1,and such that Sα→S inas α→+∞ for some θ∈(0,1).We assume that(1.4)holds true and that Rcg+bg＞0 in M in the sense of bilinear forms.Let also(ωα)αand(pα)αbe sequences of real numbers such that ωα→ω,where ω∈R is such thatand such that pα→2?in(2,2?]as α→+∞.Then for any sequence(uα,vα,Aα)αof solutions of

where

and uα＞0 in M,there holds that,up to a subsequence,uα→u,vα→v inand Aα→A inas α→+∞,for someandwhich solve(1.1).

We prove Theorem 8.1 in several steps.We let(M,g)be a smooth closed 3-manifold,q＞0,a,b,S∈C∞(M)be smooth functions with a＞0 and b≥0 in M.Let(aα)α,(bα)αand(Sα)αbe sequences of smooth functions such that aα→a and bα→b inas α→+∞,such that bα≥0 in M for α?1,and such that Sα→S inas α→+∞ for some θ∈(0,1).We assume that Rcg+bg＞0 in M in the senseof bilinear forms.We let also(ωα)αand(pα)αbe sequences of real numbers such that ωα→ω,where ω∈R,and such that pα→2?in(2,2?]as α→+∞,and we let(uα,vα,Aα)αbe a sequence of solutions of(8.1)with uα＞0 for all α.

By Lemma 6.3 there exists C＞0 such thatfor all α,whereWe assume by contradiction that

as α→+∞.We let(xα)αbe a sequence of points in M,and(ρα)αbe a sequence of positive real numbers with 0＜ρα＜ig/7 for all α,where igis the injectivity radius of(M,g).We assume that the xα’s and ρα’s satisfy

We letμαbe given by

Since the Φα’s are L∞-bounded we can apply the asymptotic analysis in Hebey and Thizy[42].Closely related arguments were first developed by Schoen[37],and then by Druet[27,28]and Li and Zhu[35]assuming C1-convergences of the potentials.We refer also to Druet and Hebey[43–45],Druet,Hebey and V′etois[46,47]and Hebey[41].Unstability in our problem(in the case A≡0 and S≡0)has been discussed in Hebey and Wei[48].Assuming(8.3),and coming back to the analysis in Hebey and Thizy[42],we can write thatμα→0 andas α→+∞,and that

It follows from(8.5)that

as α→+∞,while the de finition of rαgives that

and that

As in Druet and Hebey[43]and Hebey and Thizy[42]there holds that,after passing to a subsequence,

Lemma 8.1.There holds that,and if we assume that rα→0 as α→+∞,where rαis as in(8.7),then ρα=O(rα)and

Proof.Let ζαbe the 1-form given by

where

and ν is the unit outward normal derivative to ?Bxα(rα).We have that

for all i,j.By(8.11)and(8.15)we then get that

Similarly,

By(8.11)and(8.17),we then get that

and

Now we prove that

for α large,where C ＞0 is independent of α.By(8.17)there holds that

Using(8.5)and(8.11)we get that

By(8.5)and(8.8),we get that

and thanks to(8.22)–(8.24)we get that(8.21)holds true.Using(8.8),(8.14),(8.16),(8.18),(8.19),(8.20)and(8.21),since pα→6 and μα→0,we get that

At this point we assume that rα→0 as α→+∞.Let R≥3 be such that Rrα≤3ραfor α?1.In what follows we assume that rα→0 as α→+∞.For x∈B0(R)we set

Since rα→0 as α →+∞,we have thatinas α → +∞,where ξ is the Euclidean metric.Thanks to(8.11)we also have that

in B0(R){0}.By(8.1),(8.8),and thanks to standard elliptic theory we can write that,after passing to a subsequence,inas α→+∞,where?u satis fiesin B0(R){0}.By(8.26),in B0(R){0}.Thus we can write that

Combining(8.28)and(8.29),we get with(8.8)that

and thus thatH(0)≤0 by(8.21).At this point it remains to prove that ρα=O(rα).We proceed by contradiction and assume that

Then,(8.27)holds true in B0(R)for all R＞1.SinceHis harmonic,we then get from(8.27)that

Following Druet and Hebey[43,44],see also Druet,Hebey and V′etois[46]and Hebey[41],there exists C＞0 such that for any α there exist Nα∈N?and Nαcritical points of uα,denoted by(x1,α,x2,α,...,xNα,α),such that

for all i,j∈{1,...,Nα},and

for all x∈M and all α.We de fine dαby

If Nα=1,we setwhere igis the injectivity radius of(M,g).The second important lemma we prove in order to get Theorem 8.1 is that blow-up points are necessarily isolated in the strong sense thatas α→+∞.

Lemma 8.2.There holds thatas α→+∞,where dαis as in(8.33).

Proof.The proof is rather standard in the 3-dimensional blow-up analysis of critical elliptic equations and speci fic to dimension 3.We proceed by contradiction and assume that dα→0 as α→+∞.Then Nα≥2 for α?1,and we can assume that the xi,α’s are such thatfor all i=2,...,Nα?1.Letbe given.Forwe let

We split the proof into the study of two cases.In the first case we assume that there exist R＞0 and 1≤i≤NR,αsuch thatThen,by(8.36),for all 1≤i≤NR,αand all R＞0.Noting that the two first equations in(8.3)are satis fied byand,it follows from(8.5)that the sequenceis uniformly bounded in the balls B?xi,α(1/2).Thus,by(8.35)and elliptic theory,the sequenceis bounded inUp to a subsequence,still thanks to(8.35),we get that theconverge inas α →+∞to someu? which satis fiesin R3.There holds thatby(8.31).Moreover,u? has two critical points which are 0 and the limitas α →+∞of the?x2,α’s in(8.37).By the classi fication result of Caffarelli,Gidas,and Spruck[49],this is impossible.In particular,we are left with the second case of our study,where we assume that there exist R＞0 and 1≤i≤NR,αsuch thatas α→+∞.Then,by(8.36),as α→+∞for all 1≤i≤NR,αand all R＞0.The assumptions(8.3)are satis fied by xα=x1,αand.Let.By(8.35),

Up to passing to a subsequence,we letbe the limit ofin(8.37)and I={1,...,.We pick a compact set.The sequenceis bounded in K by(8.32).Hence by(8.38),we get that

in K,where(Fα)αis bounded in,and by the Harnack inequality

Choosing R?1 suf ficiently large,we get that X(0)＞0 and this is a contradiction with X(0)=0.This proves Lemma 8.2.

The following lemma improves the control on Φα.We de fine Φ∞to be given by

Lemma 8.3.There holds thatand,up to passing to a subsequence,in C0,θas α→+∞ for some 0＜θ＜1,where Φ∞is as in(8.41).

Proof.By Lemma 8.2,M being compact,we can assume,up to a subsequence,that Nα=N for all α,N≥1,and that xi,α→xias α→+∞ for all 1≤i≤ N.Without loss of generality,we may also assume that for any δ＞0,

as α→+∞,for all i=1,...,N.It follows that there exists δ0＞0,suf ficiently small,such that(8.3)holds true with xα=xi,αand ρα=δ0for all i=1,...,N.We fix i=1,...,N arbitrary.By Lemma 8.1,as α→+∞.Then,it follows from(8.11)that there exist r＞0 and C＞0 such that

for all x∈ Bxα(r){xα}and all α.In particular,together with(8.5)and the equation on pαin Lemma 8.1,this implies that for α large

On the other hand,(8.32)implies that(uα)αis uniformly bounded in the setHence we have thatThensince pα→6,and sincewe can write that

It remains to prove that v is constant,thatand that A≡0.Let Φ=Φ(·,v,A).Up to passing to a subsequence we can assume thatinand uα→u infor some u.By the first equation in(8.1),and according to the above,

It follows from the second equation in(8.45)that v ≡ 0 ifand that v is constant ifSince Rcg+bg＞0 in M in the sense of bilinear forms,we get with the third equation in(8.45)and the Weitzenb¨ock formula that A≡0.This ends the proof of Lemma 8.3.

Inwhatfollowswelet δ＞0be given,suf ficientlysmall,andlet η∈C∞(M×M),0≤η≤1,be such that η(x,y)=1 if dg(x,y)≤δ and η(x,y)=0 if dg(x,y)≥2δ.Forwe de fine

where ω2is the volume of the unit 2-sphere.The following lemma establishes basic estimates for the Green’s functions of Schr¨odinger’s operators as well as a positive mass propertyforsuchoperatorsthat we deducefrom themaximum principle and thepositive mass theorem of Schoen and Yau[4].It is extracted from Druet and Hebey[45]and we refer to this paper for its proof.

Lemma 8.4.Let(M,g)be a smooth compact Riemannian 3-dimensional manifold and Φ∈C∞(M)be such that Δg+Φ is coercive.The Green’s function G of Δg+Φ can be written as

for all(x,y)∈M×MD,where D is the diagonal in M×M,and R is continuous in M×M.Moreover,for any x∈M,there exists C＞0 such that

for all y∈M{x},where Rx(y)=R(x,y),and there also holds that

for all sequences(δα)αof positive real numbers converging to zero.At last,if we assume that Φ≤Λg,the inequality being strict at least at one point if(M,g)is conformally diffeomorphic to the unit 3-sphere,then R(x,x)＞0 for all x∈M.

Now we are in position to prove Theorem 8.1.Up to now,(1.4)hasn’t been used.

Proof of Theorem 8.1.We let(M,g)be a smooth closed 3-manifold,q＞0,a,b,S∈C∞(M)be smooth functions with a＞0 and b≥0 in M.Let(aα)α,(bα)αand(Sα)αbe sequences of smooth functions such that aα→a and bα→b inas α→+∞,such that bα≥0 in M for α ?1,and such that Sα→ S inas α→+∞ for some θ∈(0,1).We assume that(1.4)holds true and that Rcg+bg＞0 in M in the sense of bilinear forms.We let also(ωα)αand(pα)αbe sequences of real numbers such that ωα→ ω,where ω ∈ R is such thatand such that pα→2?in(2,2?]as α →+∞,and we letbe a sequence of solutions of(8.1)with uα＞0 for all α.By Lemma 6.4 it suf fices to prove thatWe proceed by contradiction and we assume thatas α→+∞.By(1.4)and Lemma 8.3 there holds thatin C0,θas α→+∞ for some 0＜θ＜1,where Φ∞is smooth and satis fies that

in M.Let Nαand the xi,α’s be as in(8.31)–(8.32).By Lemma 8.2 we get that(Nα)αis bounded and we can assume that Nα=N for all α,where N≥1.We let the μi,α’s be given by

and let xibe the limit of(xi,α)αfor i=1,...,N.Without loss of generality,we can assume that μi,α→ 0 as α → +∞ for all i.We reorganize the i’s such that,up to passing to a subsequence,,and we de fine μi≥0 by

By(8.11),the isolation of bubbles,and the Harnack inequality,for any δ＞0 small,there exists C＞0 such that

in Bxi,α(δ)for all i and all α.From(8.1),letting

there holds that

Combining(8.52)and(8.54)we get that,after passing to a subsequence,

in the distribution sense,where,for z∈ M,δzis the Dirac mass at z.Since Φ∞is smooth and positive,we get from Lemma 8.4 and(8.56)that

for x∈MS.Let i∈{1,...,N}be arbitrary and Xαbe the vector field given by Xα=?fα,whereWe apply the Pohozaev identity in Druet and Hebey[43],and we get that

where

and ν is the unit outward normal to Bxi,α(r).For r＞0 small enough,there holds that

for α large,and by(8.55),

while by(8.53),there also holds thatin a neighbourhood of xi.From(8.48),we have in addition that dg(xi,x)|?R(xi,x)|≤C for allIt follows that

and that

as r→0.Now we choose δ＞0 in the de finition of(8.46)such that dg(xi,xj)≥4δ for all i,j=1,...,N such thatSince G is nonnegative,it follows from our choice of δ that R(xj,xi)is nonnegative for allIn a neighbourhood of xi,we get from(8.57)that

By(8.49)and(8.63),we compute as r→0

Noting that

Still by(8.11),

forall i.By construction,R(xj,xi)≥0 foralland by Lemma8.4 and(8.50),R(xi,xi)＞0.Wehave thatμ1=1.Thecontradictionfollowsfrom(8.67).ThisendstheproofofTheorem 8.1.

Now we are in position to prove Theorem 1.1 in the critical case.

Proof of Theorem 1.1 in the critical case.Existence in Theorem 1.1 when p=6 and b＞0 follows from Lemma 7.2 and Lemma 7.3,and we get a mountain pass type solution.Existencein the b≥0 case follows from the existenceinthe b＞0 case togetherwiththe stability proved in Theorem 8.1.It suf fices to let bα=b+εαand leave fixed the other parameters,where(εα)αis a sequence of positive real numbers converging to zero.Anotherapproach can be to use existence in the subcritical case and then let the subcritical exponent tend to the critical exponent using the stability of the system in the critical case.In the same vein,the compactness part in Theorem 1.1 in the critical case is an easy consequence of the stability proved in Theorem 8.1.This proves Theorem 1.1 in the critical case.As a remark,it can be noted that by Lemma 6.5,if(uα,vα,Aα)αis a sequence of solutions of(1.1)which converge to(u,v,A)in C2,then u＞0 and v＞0 as soon as uα＞0 in M for all α.AlsoS is not a constant.

The following proposition is an immediate consequence of the existence results we discussed above and the stability proved in Theorems 6.1 and 8.1.

Proposition 8.1.Let(M,g)be a smooth closed 3-manifold,q＞0 andbe a smooth function with a＞0.Let ω ∈R be such thatLet p∈(2,6].

then(1.1)possesses a smooth nontrivial solution(uS,vS,AS)and if S→1 inthen,up to passing to a subsequence,any smooth solution(uS,vS,AS)of(1.1)satis fies that AS→0 inuS→u and vS→v inwhere(u,v)is a pair of solutions of(1.1)with A≡0.In particular,δAS→0 inas S→1 inand for S close to 1 the Coulomb gauge equation is close to be satis fied.

As a remark,v in case(ii)has to be a constant,and we need to have thatifAlsoif S is not constant.