Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds

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Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds

T. T. Truong, E. Hebey

To cite this version:

T. T. Truong, E. Hebey. Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed mani- folds. Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2011, pp.2350_110-6983.

�10.1515/crelle.2011.130�. �hal-00671790�

STATIC KLEIN-GORDON-MAXWELL-PROCA SYSTEMS IN 4-DIMENSIONAL CLOSED MANIFOLDS

EMMANUEL HEBEY AND TRONG TUONG TRUONG

Abstract. We prove existence and uniform bounds for critical static Klein- Gordon-Maxwell-Proca systems in the case of 4-dimensional closed Riemann- ian manifolds.

Static Klein-Gordon-Maxwell-Proca systems are massive versions of the electro- static Klein-Gordon-Maxwell Systems. The vector field in these systems inherits a mass and is governed by the Proca action which generalizes that of Maxwell.

Klein-Gordon-Maxwell systems are intended to provide a dualistic model for the description of the interaction between a charged relativistic matter scalar field and the electromagnetic field that it generates. The electromagnetic field is both gener- ated by and drives the particle field. In the electrostatic form of the Klein-Gordon- Maxwell systems, looking for standing wavesue^iωt, the matter field is characterized by the property that u, together with a gauge potential v, solve the electrostatic Klein-Gordon-Maxwell systems (0.3) withm1= 0. In the case of a closed manifold we discuss here the two equations in (0.3) are independent one of another when m1 = 0 and the system reduces to the sole Schr¨odinger equation. The Proca for- malism, form1>0, leads to a deeper phenomenon and is more appropriate to the closed case. The particle in this model interacts via the minimum coupling rule

∂_t→∂_t+iqϕ and ∇ → ∇ −iqA (0.1) with an external massive vector field (ϕ, A) which is governed by the Maxwell-Proca Lagrangian. The Proca action is a gauge-fixed version of the Stueckelberg action in the Higgs mechanism (see Goldhaber and Nieto [25], and Ruegg and Ruiz-Altaba [42]). In the Proca formalism, developped under the influence of de Broglie, the photon inherits a nonzero mass. This issue is of considerable importance and inten- sively studied in modern physics (see for instance Adelberger, Dvali and Gruzinov [1], Byrne [12], Goldhaber and Nieto [24, 25], Luo and Tu [37], Luo, Gillies and Tu [36] and the references in these papers). Whenn= 3, the KGMP equations consist in the nonlinear Klein-Gordon matter equation, the charge continuity equation and the massive modified Maxwell equations in SI units, which are hereafter explicitly written down:

∇.E =ρ/ε0−µ²ϕ ,

∇ ×H =µ0

J +ε0

∂E

∂t

−µ²A ,

∇ ×E+∂H

∂t = 0 and∇.H= 0.

(0.2)

Date: October 29, 2010.

1

These massive Maxwell equations, as modified to Proca form, appear to have been first written in modern format by Schr¨odinger [45]. The Proca formalism a priori breaks Gauge invariance. Gauge invariance can be restaured by the Stueckelberg trick, as pointed out by Pauli [40], and then by the Higgs mechanism. We refer to Goldhaber and Nieto [25], Luo, Gillies and Tu [36], and Ruegg and Ruiz-Altaba [42] for very complete references on the Proca approach.

In what follows we let (M, g) be a smooth compact 3,4-dimensional Riemannian manifold. We let also 2^? = _n−2²ⁿ be the critical Sobolev exponent, where n is the dimension ofM. Given real numbersq >0,m0, m1>0,ω∈(−m0, m0), and p∈(2,2^?], the derivation of the Klein-Gordon-Maxwell-Proca system we investigate in this paper is written as

(∆_gu+m²₀u=u^p−1+ω²(qv−1)²u

∆gv+ m²₁+q²u²

v=qu², (0.3)

where ∆_g = −div_g∇ is the Laplace-Beltrami operator. The system (0.3) corre- sponds to looking for standing wavesue^iωtfor the full KGMP system in the static case where the massive vector field (ϕ, A) depends on the sole spatial variable. The system is energy critical when n= 3 andp= 6 and when n= 4 andp= 4. It is subcritical otherwise, namely when n= 3 and p∈(2,6) or n= 4 and p∈ (2,4).

In the above model,m1is a coupling constant which makes that the two equations in (0.3) are trully coupled (m1is the Proca mass in the Maxwell-Proca formalism) whilem0 is the mass of the particle,qis the charge of the particle,uis the ampli- tude in the writing of the particle,ω is its temporal frequency (referred to as the phase in the sequel), andv is the electric potential.

Let S_g stand for the scalar curvature of g, andS_p(ω) be the set consisting of the positive smooth solutionsU = (u, v) of (0.3) with phaseω and nonlinear term u^p−1. Namely,

Sp(ω) =n

(u, v) smooth s.t.u >0, v >0, and (u, v) solve (0.3)o

. (0.4)

Givenω∈[0, m0), we let

K₀(ω) = (−m₀,−ω][

[ω, m₀). (0.5)

Whenω= 0, K0(0) = (−m0, m0) is the full admissible phase range. Forθ∈(0,1), andU = (u, v), we letkU k_C2,θ =kuk_C2,θ+kvk_C2,θ. The following result was proved in Druet and Hebey [19].

Theorem 0.1 (The 3-dimensional case - Druet and Hebey [19]). Let (M, g) be a smooth compact 3-dimensional Riemannian manifoldm0, m1>0,ω ∈(−m0, m0), andp∈(2,6]. When p= 6assume

m²₀< ω²+1

8S_g(x) (0.6)

for allx∈M. Then (0.3)possesses a smooth positive solution. Moreover, for any p∈(2,6), and anyθ∈(0,1), there exists C >0such that for any ω⁰∈K0(0), and anyU ∈ Sp(ω⁰),kU k_C2,θ ≤C, whereSp(ω⁰)is as in (0.4)andK0(0)is as in (0.5).

Assuming again (0.6), there also holds that for any θ∈(0,1),kU k_C2,θ ≤C for all U ∈ S6(ω⁰)and allω⁰∈K₀(ω), where C >0 does not depend onω⁰ andU.

This result exhibits phase compensation in the 3-dimensional case. We aim in this paper in proving that a similar phenomenon holds true when n = 4. In this dimension the second equation in (0.3) becomes critical and this leads to serious difficulties. We prove below the existence of smooth positive solutions and the existence of uniform bounds for (0.3) in the subcritical cases p ∈ (2,4) without any conditions, and in the critical case p = 4 assuming that the mass potential, balanced by the phase, is smaller than the geometric threshold potential of the conformal Laplacian. In doing so we prove that phase compensation still holds true for our systems whenn= 4. Our result, in the subcritical case, is as follows.

Theorem 0.2 (The subcritical 4-dimensional case). Let (M, g) be a smooth com- pact 4-dimensional Riemannian manifold, q > 0, m0, m1 > 0, ω ∈ (−m0, m0), andp∈(2,4). Then (0.3)possesses a smooth positive solution. Moreover, for any θ ∈(0,1), there exists C >0 such that for any ω⁰ ∈K0(0), and anyU ∈ Sp(ω⁰), kU k_C2,θ ≤C, whereSp(ω⁰)is as in (0.4)andK0(0) is as in (0.5).

In the critical case we prove the following result. The geometry of the ambiant inhomogeneous space, through the scalar curvature of g, comes to play a role as in the 3-dimensional case. However, the result now turns out to be local in its existence part.

Theorem 0.3 (The critical 4-dimensional case). Let (M, g)be a smooth compact 4-dimensional Riemannian manifold, q > 0, m0, m1 > 0, ω ∈ (−m0, m0), and p= 4. Assume

m²₀< ω²+1

6Sg(x) (0.7)

for some x∈M. Then (0.3) possesses a smooth positive solution. Assuming that (0.7)holds true for all x∈M there also holds that for anyθ∈(0,1),kU k_C2,θ ≤C for allU ∈ S4(ω⁰)and allω⁰ ∈K0(ω), whereC >0 does not depend on ω⁰ and U, S4(ω⁰)is as in (0.4), andK0(ω)is as in (0.5).

There are two consequences to Theorem 0.3. We list them in points (i)-(ii) below.

In point (i) we illustrate the phase compensation effect associated with (0.3). There we always get existence and a priori bounds for all phasesω which are close tom0. Point (ii) concerns the full range of phases when we assumem0is not too large.

(i) Phase compensation in the critical case - Assume p= 4 andSg >0 in M. Then there exists ε > 0 such that for any m0−ε <|ω| < m0, (0.3) possesses a smooth positive solution. Moreover, for anyθ∈(0,1), there existsC >0such that kU k_C2,θ ≤C for all U ∈ S4(ω)and all ωa−ε <|ω|< ωa.

(iii) Full phase range in the critical case -Assume p= 4and m²₀ < ¹₆Sg inM. For any ω ∈(−m0, m0),(0.3)possesses a smooth positive solution. Moreover, for any θ∈(0,1), there exists C >0 such that kU k_C2,θ ≤C for allU ∈ S4(ω) and all ω∈(−ωa, ωa).

As an immediate consequence of theC^2,θ-bounds in the above results we obtain phase stability for standing waves of the Klein-Gordon-Maxwell-Proca equations in electrostatic form. Standing waves for the Klein-Gordon-Maxwell-Proca equations in electrostatic form are written as S =ue^iωt and they are coupled with a gauge potentialv, where (u, v) solves (0.3). Roughly speaking, phase stability means that for any arbitrary sequence of standing wavesu_αe^iωαt, with gauge potentialsv_α, the convergence of the phases ω_α in R implies the convergence of the amplitudes u_α

and of the gauge potentials vα in the C²-topology. Phase stability prevents the existence of arbitrarily large amplitude standing waves.

High dimensional KGM systems in Coulomb gauge have been recently investi- gated by Rodnianski and Tao [41] and with special emphasis in (1 + 4)-dimensions by Klainerman and Tataru [28] and Selberg [46]. Electrostatic KGM systems in the three dimensional case have been investigated by several authors. Possible ref- erences on the physics side are by Benci and Fortunato [5], Long [34], Long and Stuart [35]. Blowing-up solutions to the electrostatic Schr¨odinger-Maxwell system, a cousin of the electrostaic KGM type systems that we consider here, have been constructed in D’Aprile and Wei [2, 3].

We briefly discuss in Section 1 the physics relevance of (0.3). We prove our theorem in Sections 2 to 4. The existence part in the theorem is proved in Section 2. The C^2,θ-bound in the subcritical case is established in Section 3. The more delicate C^2,θ-bound in the critical case is established in Sections 4 . The phase compensation phenomenon in the theorem holds true thanks to the 4-dimensional log effectµ²=o(µ²logµ) asµ→0.

1. The physics origin of the system

The Klein-Gordon-Maxwell-Proca system discussed in this work describes an interacting field theory model in theoretical physics. Most electromagnetic phe- nomena are described by conventional electrodynamics, which is a theory of the coupling of electromagnetic fields to matter fields. Of prime importance for par- ticle physics is fermion electrodynamics in which matter is represented by spinor fields. However one may have also boson electrodynamics in which matter is de- scribed by integer spin or bosonic fields. The simplest one is of course the complex scalar field, describing spinless particles having electric charges ±q. It gives rise to scalar electrodynamics, which describes in the non-relativistic limit the super- conductivity of metals at very low temperatures. In the more general context of particle physics, a complex scalar field ψ may serve to describe scalar mesons in nuclear matter interacting via a massive vector boson field (ϕ, A).

The interaction in this model is described by the minimum substitution rule (0.1) 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 define the Lagrangian densitiesLN KGandLM P

ofψ,ϕ, andA by LN KG(ψ, ϕ, A) = 1

2

(∂

∂t+iqϕ)ψ

2

−1

2|(∇ −iqA)ψ|²+m²₀

2 |ψ|²−1 p|ψ|^p,

LM P(ϕ, A) = 1 2

∂A

∂t +∇ϕ

2

−1

2|∇ ×A|²+m²₁

2 |ϕ|²−m²₁ 2 |A|²,

(1.1)

where∇×=?d,?is the Hodge dual,ψrepresents the matter complex scalar field, m₀ its mass,qits charge, (ϕ, A) the electromagnetic vector field, andm₁its mass.

It can be noted that k(ϕ, A)k²_L = |ϕ|²− |A|² is the square of the Lorentz norm of (ϕ, A) with respect to the Lorentz metric diag(1,−1, . . . ,−1). The total action functional forψ,φ, and Ais then given by

S(ψ, ϕ, A) = Z Z

(LN KG+LM P)dvgdt . (1.2)

Writingψ in polar form asψ(x, t) =u(x, t)e^iS(x,t), taking the variation ofS with respect tou,S,ϕ, andA, we get four equations which are written as















∂²u

∂t² + ∆_gu+m²₀u=u^p−1+

∂S

∂t +qϕ²

− |∇S−qA|² u

∂

∂t

∂S

∂t +qϕ u²

− ∇. (∇S−qA)u²

= 0

−∇. ^∂A_∂t +∇ϕ

+m²₁ϕ+q ^∂S_∂t +qϕ u²= 0

∆_gA+_∂t^{∂ ∂A}_∂t +∇ϕ

+m²₁A=q(∇S−qA)u²,

(1.3)

where ∆g=−divg∇is the Laplace-Beltrami operator, ∆g=δdis half the Laplacian acting on forms, andδ is the codifferential. We refer to this system as a nonlinear Klein-Gordon-Maxwell-Proca system. Whenn= 3, ∆gA=∇ ×(∇ ×A) and if we let

E=− ∂A

∂t +∇ϕ

, H =∇ ×A , ρ=−

∂S

∂t +qϕ

qu², andj = (∇S−qA)qu²,

(1.4)

then the two last equations in (1.3) give rise to the first pair of the Maxwell- Proca equations (0.2) with 0 = µ0 = 1 (units are chosen such that c = 1) and µ² = m²₁, while the two first equations in (1.4) give rise to the second pair of the equations. The first equation in (1.3) gives rise to the nonlinear Klein-Gordon matter equation. The second equation in (1.3) gives rise to the charge continuity equation ^∂ρ_∂t+∇.j= 0 which, thanks to (0.2), is equivalent to the Lorentz condition

∇.A+^∂ϕ_∂t = 0.

We assume in what follows thatu(x, t) =u(x) does not depend ont,S(x, t) =ωt does not depend onx, andϕ(x, t) =ϕ(x),A(x, t) =A(x) do not depend ont. In other words, we look for standing waves solutions of (1.3) and assume that we are in the static case of the system where (ϕ, A) depends on the sole spatial variable.

By the fourth equation in (1.3) we then get that

∆_gA+ (q²u²+m²₁)A= 0. This clearly implies that, and is equivalent to,A≡0 since

Z

(∆gA, A) = Z

|dA|².

As a remark, assuming thatA ≡0, the Lorentz condition for the external Proca field (ϕ, A) would makeϕdependent on the sole spatial variables. As for the second equation in (1.3) it reduces to^∂_∂t^2S2 = 0. It is automatically satisfied whenS(t) =ωt, and we are thus left with the first and third equations in (1.3). LettingS =−ωt, andϕ=ωv, these equations are rewritten as

(∆gu+m²₀u=u^p−1+ (qϕ−ω)²u

∆_gϕ+m²₁ϕ+q(qϕ−ω)u²= 0. (1.5) In particular, letting ϕ = ωv, in (1.5), we recover our original system (0.3). In other words, our original system (0.3) corresponds to looking for standing waves solutions of the Klein-Gordon-Maxwell-Proca system (1.3) in static form.

2. Existence Theory

We prove the existence part in Theorems 0.2 and 0.3. Formally, solutions of (0.3) are critical points of the functionalS defined by

S(u, v) = 1 2 Z

M

|∇u|²dvg−ω² 2

Z

M

|∇v|²dvg+m²₀ 2

Z

M

u²dvg

−ω²m²₁ 2

Z

M

v²dvg−1 p

Z

M

u^pdvg−ω² 2

Z

M

u²(1−qv)²dvg.

(2.1)

The functionalS is strongly indefinite because of the competition between uand v. Following a very nice idea going back to Benci-Fortunato [5], we introduce the auxiliary functional Φ given by

∆gΦ(u) + (m²₁+q²u²)Φ(u) =qu², (2.2) and then consider thatuin (0.3) can be seen as a critical point of

Ip(u) = 1 2

Z

M

|∇u|²dvg+m²₀ 2

Z

M

u²dvg−1 p

Z

M

(u⁺)^pdvg

−ω² 2

Z

M

(1−qΦ(u))u²dvg ,

(2.3)

whereu⁺= max(u,0). Let Ψ :H¹(M)→Rbe defined by Ψ(u) = 1

2 Z

M

(1−qΦ(u))u²dvg. (2.4) The following lemma establishes the existence and differentiability of Φ, as well as theC¹-smoothness of Ψ. Equation (2.2) is critical whenn= 4 because of the term u²Φ(u).

Lemma 2.1. Let (M, g)be a smooth compact Riemannian 4-manifold and q >0.

There existsΦ :H¹(M)→H¹(M)such that (2.2)holds true and0≤Φ(u)≤ ¹_q for all u∈H¹(M). Moreover,Φ is locally Lipschitz and differentiable. Its differential DΦ(u) =V_u atuis given by

∆_gV_u(ϕ) + (m²₁+q²u²)V_u(ϕ) = 2qu(1−qΦ(u))ϕ (2.5) for all ϕ ∈ H¹(M). The functional Ψ : H¹(M) → R defined in (2.4) is C¹ in H¹(M)and

DΨ(u).(ϕ) = Z

M

(1−qΦ(u))²uϕdvg (2.6) for allu, ϕ∈H¹(M).

Proof of Lemma 2.1. We briefly sketch the proof. Letu∈H¹ and H_u : H¹ →R be defined by

Hu(ϕ) = Z

M

|∇ϕ|²dvg+ Z

M

(m²₁+q²u²)ϕ²dvg.

The functional is well defined sinceH¹⊂L⁴. Letting Φ(0) = 0 we can assume that u6≡0. Let

µ= inf

u∈H¹,Ru²ϕ=1

H_u(ϕ).

By standard minimization arguments there existsϕ∈H¹(M) such thatR

Mu²ϕdv_g= 1 andH_u(ϕ) =µ. In particular,µ >0. Letting Φ(u) =_µ^qϕwe get that Φ(u) solves

(2.2) in H¹. It is easily seen that Φ(u) is unique. By the maximum principle, Φ(u)≥0. Noting that

∆_g 1

q −Φ(u)

+ (m²₁+q²)u² 1

q−Φ(u)

≥0

it also follows from the maximum principle that Φ(u) ≤ ¹_q. Now we let u, v ∈ H¹(M). We have that

∆_g(Φ(v)−Φ(u)) + (m²₁+q²)u²(Φ(v)−Φ(u)) =q(v²−u²) (1−qΦ(v)) . Multiplying the equation by Φ(v)−Φ(u), integrating overM, and by the Sobolev emedding theorem, we get that

kΦ(v)−Φ(u)kH¹ ≤C(kukH¹+kvkH¹)kv−ukH¹ . (2.7) In particular, Φ is locally Lipschitz continuous. We can prove the existence ofV_u(ϕ) in (2.5) as when proving the existence of Φ(u). Writing the equation satisfied by Φ(u+ϕ)−Φ(u)−V_u(ϕ), multiplying the equation by Φ(u+ϕ)−Φ(u)−V_u(ϕ) and integrating overM, we get that

kΦ(u+ϕ)−Φ(u)−Vu(ϕ)k_H1 ≤Ckϕk_H1(kϕk_H1+kuk_H1kΦ(u+ϕ)−Φ(u)k_H1) Then the differentiability of Φ follows from the continuity of Φ. In particular, Ψ is differentiable. By (2.2),

Ψ(u) = 1 2 Z

M

|∇Φ(u)|²+m²₁Φ(u)²

dvg+1 2

Z

M

(1−qΦ(u))²u²dvg, and we also have that ^∂H_∂Φ(u,Φ(u)) = 0, whereH(u,Φ) =¹₂H_u(Φ)−qR

Mu²Φdv_g. Noting that

Ψ(u) =H(u,Φ(u)) +1 2

Z

M

u²dv_g,

we get that (2.6) holds true. The continuity of DΨ can be proved directly from (2.6) and the continuity of Φ. This ends the proof of the lemma.

Now we prove the subcritical existence of Theorem 0.2. We proceed by applying the mountain pass lemma to the functionalIp in (2.3).

Proof of existence in Theorem 0.2. By Lemma 2.1, I_p is C¹ in H¹. Let u₀ ∈ H¹ such that u⁺₀ 6≡0. There holds Ip(0) = 0 and Ip(tu0) → −∞ as t → +∞ since p >2. Since 0≤Φ(u)≤ ¹_q for allu, we also have that

Ip(u) ≥ 1 2

Z

M

|∇u|²dvg+ (m²₀−ω²) Z

M

u²dvg

−1 p

Z

M

|u|^pdvg

≥ C1kuk²_H1−C2kuk^p_H1

for all u∈ H¹, whereC₁, C₂ >0 do not depend on u. In particular, there exist δ, C >0 such thatI_p(u)≥C for allu∈H¹such thatkuk_H1 =δ. LetT₀=T₀(u₀), T01, be such thatIp(T0u0)<0, andcp=cp(u0) be given by

cp= inf

P∈Pmax

u∈PIp(u), (2.8)

whereP is the class of continuous paths joining 0 toT₀u₀. According to the above we can apply the mountain pass lemma and we get the existence of a sequence

(uα)α in H¹ such thatIp(uα)→cp and DIp(uα)→0 as α→+∞. Writing that Ip(uα) =cp+o(1) and thatDIp(uα).(uα) =o(kuαkH¹), we get by Lemma 2.1 that

1 2

Z

M

|∇uα|²+m²₀u²_α dv_g

= 1 p

Z

M

(u⁺_α)^pdv_g+c_p+ω² 2

Z

M

(1−qΦ(u_α))u²_αdv_g+o(1) 1

2 Z

M

|∇u_α|²+m²₀u²_α dv_g

= 1 2

Z

M

(u⁺_α)^pdvg+ω² 2

Z

M

(1−qΦ(uα))²u²_αdvg+o(kuαk_H1)

(2.9)

for all α. Writing thatDIp(uα).(u⁻_α) =o(ku⁻_αk_H1) we get that u⁻_α → 0 inH¹ as α→+∞. By (2.9) we then get that (uα)α is bounded inH¹. In particular, there existsup∈H¹(M) such that, up to passing to a subsequence,

(i)uα* upweakly inH¹, (ii)u_α→u_p inL^p,

anduα→up a.e. asα→+∞. Substracting one equation to another in (2.9), let- tingα→+∞, and sincecp6= 0, we get thatup6≡0. Writing the equation satisfied by Φ(uα)−Φ(up), multiplying the equation by Φ(uα)−Φ(up) and integrating over M, we get that

Φ(uα)→Φ(up) inH¹ (2.10)

as α → +∞. Now we let ϕ ∈ H¹. There holds DIp(uα).(ϕ) = o(1). Hence, by Lemma 2.1,

Z

M

∇uα∇ϕdvg+m²₀ Z

M

uαϕdvg

= Z

M

(u⁺_α)^p−1ϕdvg+ω² Z

M

(1−qΦ(uα))²uαϕdvg+o(1).

(2.11)

Lettingα→+∞in (2.11) we then get by (2.10) that

∆_gu_p+m²₀u_p= (u⁺_p)^p−1+ω²(1−qΦ(u_p))²u_p

in H¹. Multiplying the equation by u⁻_p and integrating over M, it follows that u⁻_p ≡0. In particular,u_p≥0,u_p6≡0, and

∆gup+m²₀up=u^p−1_p +ω²(1−qΦ(up))²up (2.12) in H¹. By regularity results we get from (2.12) that up ∈ H^2,s for all s. Then, by regularity results, Φ(up)∈H^2,s for alls. By the Sobolev embedding theorem, regularity theory, and the maximum principle, it follows thatup,Φ(up) ∈C²(M) and that up,Φ(up) > 0 in M. Letting u = up and v = Φ(up), this proves the

existence part in Theorem 0.2.

An additional information we obtain is thatup realizescp. Indeed, sinceup≥0, u⁻_α →0 inH¹, and Φ(uα)→Φ(up) inH¹, there holds that

Z

M

(u⁺_α)^pdvg→ Z

M

u^p_pdvg, and Z

M

(1−qΦ(uα))²u²_αdvg→ Z

M

(1−qΦ(up))²u²_pdvg

asα→+∞. The second equation in (2.9) together with (2.12) then give that Z

M

|∇uα|²dvg→ Z

M

|∇up|²dvg.

It follows thatuα→up in H¹as α→+∞. By the first equation in (2.9) we then get thatIp(up) =cp. In other words,cp is realized byup. Now, givenx0∈M and ρ0>0 small, sufficiently small, we defineuεby

(uε(x) = _ε2+r^{ε 2} −_ε2+ρ^ε 2 0

ifr≤ρ0,

uε(x) = 0 ifr≥ρ0, (2.13)

wherer=dg(x0, x). Then, see Aubin [4], for anyλ∈R, Jλ(uε) = 1

K₄²

1 +C 1

6Sg(x0)−λ

ε²lnε+o(ε²lnε)

, (2.14) where

J_λ(u_ε) = R

M |∇uε|²+λu²_ε dv_g R

Mu⁴_εdvg

1/2 , andC >0 is independent ofα. Also there holds

Z

M

u⁴_εdv_g= Z

R³

1 1 +|x|²

⁴

dx+o(1), Z

M

|∇uε|²dvg= 8 Z

M

u⁴_εdvg+o(1).

(2.15)

In what follows we prove the existence part of Theorem 0.3.

Proof of existence in Theorem 0.3. As a preliminary remark, by standard argu- ments such as developed in Aubin [4] and Br´ezis and Nirenberg [11], we just need to prove that we can choseu₀∈H¹,u⁺₀ 6≡0, such that

δ₀≤c_p≤ 1

4K₄⁴ −δ₀ (2.16)

for allp∈(4−ε,4) and someε, δ0 >0, where cp =cp(u0) is as in (2.8). Now we assume that (0.7) holds true for somex∈M, in particular for x∈M where Sg is maximum. We let x0 ∈ M be such that Sg is maximum atx0, and (tε)ε be any family of positive real numbers such that thetε’s are bounded. The first estimate we prove is that

Z

M

Φ(tεuε)u²_εdvg=O ε²

, (2.17)

where theuε’s are as in (2.13). By definition,

∆gΦ(tεuε) + (m²₁+q²)t²_εu²_εΦ(tεuε) =qt²_εu²_ε. (2.18) Multiplying (2.18) by Φ(t_εu_ε) and integrating overM we get by H¨older’s inequalities that

kΦ(tεuε)k²_H1 = qt²_ε Z

M

u²_εΦ(tεuε)dvg

≤ C Z

M

u^8/3_ε dv_g ^3/4

kΦ(t_εu_ε)k_L4

and it follows from the Sobolev inequality that kΦ(t_εu_ε)k_H1 ≤C

Z

M

u^8/3_ε dv_g ^3/4

. (2.19)

Then, by (2.19), Z

M

Φ(tεuε)u²_εdvg ≤C Z

M

u^8/3_ε dvg

3/4

kΦ(tεuε)k_L4

≤C Z

M

u^8/3_ε dvg

3/2

.

(2.20)

There holds, Z

M

u^8/3_ε dvg≤ω3

Z ρ₀ 0

ε ε²+r²

8/3

r³dr

=ω3ε^4/3 Z ρ₀/ε

0

1 1 +r²

8/3

r³dr

=O(ε^4/3).

(2.21)

By (2.20) and (2.21), this proves (2.17). Let (εα)α be a sequence of positive real numbers such thatεα→0 asα→+∞,uα=uε_α, andF4be the functional defined inH¹ by

F4(u) =1 2

Z

M

|∇u|²dvg+1

2(m²₀−ω²) Z

M

u²dvg−1 4

Z

M

|u|⁴dvg. (2.22) By (2.15), there exists T0 1 such thatI4(T0uα)<0 for allα 1. There also holds that

0≤t≤Tmax0

I4(tuα) ≤ max

0≤t≤T0

F4(tuα) +CT₀² max

0≤t≤T0

Z

M

Φ(tuα)u²_αdvg

≤ 1

4Jλ(uα)²+CT₀² max

0≤t≤T0

Z

M

Φ(tuα)u²_αdvg

for allα, where λ=m²₀−ω². By (2.14) and (2.17) we thus get that max

0≤t≤T0

I₄(tu_α)≤ 1 K₄⁴

1 +C

1

6S_g(x₀)−m²₀+ω²

ε²_αlnε_α+o(ε²_αlnε_α)

,

whereC >0 is independent ofα. By assumption theε²_αlnεαcoefficient is positive.

Letu0=uα, whereα1 is sufficiently large such that

0≤t≤Tmax0

I4(tuα)≤ 1 4K₄⁴ −δ0

for someδ₀>0. Sinceu₀ is now fixed, we can write that

0≤t≤Tmax0

Ip(tu0)≤(1 +δε) max

0≤t≤T0

I4(tu0) (2.23)

for allp∈(4−ε,4), whereδε>0 is such thatδε→0 asε→0. Noting that Ip(u) ≥ 1

2 Z

M

|∇u|²+ (m²₀−ω²)u²

dvg−1 p

Z

M

|u|^pdvg,

≥ C₁kuk²_H1−C₂kuk^p_H1

whereC₁, C₂>0 are independent ofu, there holds that there existδ₁, δ₂>0 such thatδ₁, δ₂ are as small as we want, andI_p(u)≥δ₂ for allusuch that kukH¹ =δ₁.

As a conclusion, there exist δ0 >0 and ε > 0 such that (2.16) holds true for all p∈(4−ε,4). This ends the proof of the existence part in Theorem 0.3.

3. A priori bounds in the subcritical case

We prove the uniform bounds in the subcritical case of Theorem 0.2. In what followsp∈(2,4).

Proof of the uniform bounds in Theorem 0.2. Let (ωα)αbe a sequence in (−m0, m0) such that ωα → ω as α→ +∞ for some ω ∈ [−m0, m0]. Also let p∈ (2,4) and

(uα, vα)

α be a sequence of smooth positive solutions of (0.3) with phases ωα. Then,

(∆guα+m²₀uα=u^p−1_α +ω²_α(qvα−1)²uα

∆_gv_α+ m²₁+q²u²_α

v_α=qu²_α (3.1)

for all α. By the second equation in (3.1), 0 ≤ v_α ≤ ¹_q for all α. Assume by contradiction that

max

M uα→+∞ (3.2)

asα→+∞. Letxα∈M andµα>0 be given by uα(xα) = max

M uα=µ^−2/(p−2)_α . By (3.2),µα→0 asα→+∞. Define ˜uα by

˜

uα(x) =µ

2 p−2

α uα exp_xα(µαx) andg_αbyg_α(x) = exp^?_x

αg

(µ_αx) for x∈B₀(δµ⁻¹_α ), whereδ >0 is small. Since µ_α→0, we get thatg_α→ξinC_loc² (R³) asα→+∞. Moreover, by (3.1),

∆g_α˜uα+µ²_αm²₀u˜α= ˜u^p−1_α +ω²_αµ²_α(qˆvα−1)²u˜α, (3.3) where ˆv_αis given by ˆv_α(x) =v_α exp_x

α(µ_αx)

. We have ˜u_α(0) = 1 and 0≤u˜_α≤1.

By (3.3) and standard elliptic theory arguments, we can write that, after passing to a subsequence, ˜uα→uin C_loc^1,θ(R⁴) as α→+∞, whereuis such thatu(0) = 1 and 0≤u≤1. Then

∆u=u^p−1

in R⁴, where ∆ is the Euclidean Laplacian. It follows that u is actually smooth and positive, and, since 2< p <4, we get a contradiction with the Liouville result of Gidas and Spruck [23]. As a conclusion, (3.2) is not possible and there exists C >0 such that

uα+vα≤C (3.4)

inM for allα. Coming back to (3.1) it follows that the sequences (u_α)_αand (v_α)_α are actually bounded inH^2,sfor alls. Pushing one step further the regularity argu- ment they turn out to be bounded inH^3,sfor alls, and by the Sobolev embedding theorem we get that they are also bounded inC^2,θ, 0< θ <1. This ends the proof of the uniform bounds in Theorem 0.2 whenp∈(2,4).

If we assume that ωα → ω as α → +∞ for some ω ∈ (−m0, m0), p ∈ (2,4], and u_α → u and v_α → v in C² as α → +∞, then u > 0, v > 0, and u, v are smooth solutions of (0.3). Indeed, givenε >0 sufficiently small, sincem²₀−ω²>0,

∆g+ (m²₀−ω²−ε) is coercive. There holds that 0≤vα≤ ¹_q for allα. In particular, by (3.1) and the Sobolev inequality, for anyα1 sufficiently large,

Z

M

|∇uα|²+ m²₀−ω²−ε u²_α

dvg

≤ Z

M

|∇uα|²dvg+m²₀ Z

M

u²_αdvg−ω_α² Z

M

(qv_α²−1)²u²_αdvg

= Z

M

u^p_αdv_g≤C Z

M

|∇uα|²+ m²₀−ω²−ε u²_α

dv_g p/2

for someC >0 independent of α. This implies u >0 and then v >0. Obviously the positivity ofuandv does not hold anymore if we allowω²=m²₀. Let (ε_α)_α be a sequence of positive real numbers such that ε_α →0 asα→+∞. Letu_α =ε_α and

vα= qε²_α m²₁+q²ε²_α .

Thenu_α→0 andv_α→0 inC² as α→+∞, and we do have that (uα, v_α) solves (3.1) , where

ω²_α= 1

(qv_α−1)² m²₀−ε^p−2_α .

In this caseω_α² →m²₀asα→+∞and the construction provides a counter example to the above statement about the positivity ofuandv.

4. A priori bounds in the critical case

In what follows we let (M, g) be a smooth compact 4-dimensional Riemannian manifold,m0, m1>0, and (ωα)αbe a sequence in (−m0, m0) such thatωα→ω as α→+∞for someω∈[−m0, m0]. Also we let (uα, vα)

αbe a sequence of smooth positive solutions of (0.3) with phasesω_α andp= 4. Namely,

(∆_gu_α+m²₀u_α=u³_α+ω²_α(qv_α−1)²u_α

∆gvα+ m²₁+q²u²_α

vα=qu²_α (4.1)

for all α. By the second equation in (4.1), 0≤v_α ≤ ¹_q for all α. In particular, if we let

hα=m²₀−ω²_α(qvα−1)² , (4.2) thenkhαkL^∞ ≤C for allα, whereC >0 is independent ofα. Assume by contra- diction that

max

M u_α→+∞ (4.3)

asα→+∞. In what follows we let (xα)αbe a sequence of points inM, and (ρα)α

be a sequence of positive real numbers, 0 < ρα < ig/7 for all α, where ig is the injectivity radius of (M, g). We assume that thexα’s andρα’s satisfy







∇u_α(x_α) = 0 for allα,

dg(xα, x)uα(x)≤Cfor allx∈Bx_α(7ρα) and allα , lim_α→+∞ρ_αsup_B

xα(6ρ_α)u_α(x) = +∞.

(4.4) We letµα be given by

µ_α=u_α(x_α)⁻¹. (4.5)

Since the hα’s in (4.2) are L^∞-bounded we can apply the asymptotic analysis in Druet and Hebey [17] and Druet, Hebey and V´etois [21]. In particular, we get that

ρ_α

µα →+∞as α→+∞and that µ_αu_α exp_x

α(µ_αx)

→

1 + |x|² 8

⁻¹

(4.6) in C_loc¹ (R⁴) as α→ +∞, where µα is as in (4.5). As a consequence, µα → 0 as α→+∞. Now we defineϕα: (0, ρα)7→R⁺ by

ϕα(r) = 1

|∂B_xα(r)|_g Z

∂B_xα(r)

uαdσg, (4.7)

where |∂Bx_α(r)|_g is the volume of the sphere of center xα and radius r for the induced metric. Let Λ = 4√

2. We definerα∈[Λµα, ρα] by rα= sup

r∈[Λµα, ρα] s.t. (sϕα(s))⁰ ≤0 in [Λµα, r] . (4.8) It follows from (4.6) that

rα

µα

→+∞ (4.9)

asα→+∞, while the definition ofr_αgives that

rϕα(r) is non-increasing in [Λµα, rα] (4.10) and that

(rϕα(r))⁰(rα) = 0 ifrα< ρα. (4.11) LetBα be defined inM by

B_α(x) = µα

µ²_α+^dg(xα₈^,x)2

, (4.12)

whereµ_αis as in (4.5). The following sharp estimates, see Druet, Hebey and Robert [20] and Druet, Hebey and V´etois [21], hold true.

Lemma 4.1. Let(M, g)be a smooth compact Riemannian4-dimensional manifold, and (u_α, v_α)

α be a sequence of smooth positive solutions of (4.1)such that (4.3) holds true. Let(xα)α and(ρα)α be such that (4.4)hold true, and letR≥6be such that Rrα ≤6ρα for all α 1. There exists C > 0 such that, after passing to a subsequence,

uα(x) +dg(xα, x)|∇uα(x)| ≤Cµαdg(xα, x)⁻² (4.13) for allx∈B_xα(^R₂r_α)\ {xα} and allα, whereµ_α is as in (4.5), and where r_α is as in (4.8). In addition, there also exist C >0 and(ε_α)_α such that

|u_α−B_α| ≤Cµ_α r⁻²_α +S_α

+ε_αB_α (4.14)

in Bx_α(2rα)\{xα} for all α, where εα→0 asα→+∞and Sα(x) =dg(xα, x)⁻¹ forx∈M\{xα}.

Lemma 4.1 provide a sharp control on theuα’s, but we need more to conclude.

We prove that the following fundamental asymptotic estimate holds true. Lemma 4.2 is the key estimate we need to prove the a priori bounds in the critical case discussed in this section.

Lemma 4.2. Let(M, g)be a smooth compact Riemannian4-dimensional manifold and (uα, vα)

α be a sequence of smooth positive solutions of (4.1)such that (4.3) holds true. Let(x_α)_αand(ρ_α)_αbe such that(4.4)holds true. Assume(0.7). There holds thatr_α→0 asα→+∞, wherer_α is as in (4.8). Moreoverρ_α=O(r_α)and

r_α²µ⁻¹_α u_α exp_x

α(r_αx)

→ 8

|x|²+H(x) (4.15)

in C_loc² (B₀(2)\{0}) as α → +∞, where µ_α is as in (4.5), and H is a harmonic function inB₀(2) which satisfies thatH(0)≤0.

Proof of Lemma 4.2. LetR ≥6 be such that Rr_α ≤6ρ_α for α1. We assume first thatr_α→0 asα→+∞. Forx∈B₀(3) we define

˜

uα(x) = r²_αµ⁻¹_α uα exp_xα(rαx) ,

˜

gα(x) = exp^?_xαg

(rαx) , and

˜hα(x) = hα exp_xα(rαx) ,

wherehαis as in (4.2). Sincerα→0 asα→+∞, we have that ˜gα→ξinC_loc² (Rⁿ) asα→+∞, whereξis the Euclidean metric. Thanks to Lemma 4.1,

|˜uα(x)| ≤C|x|⁻² (4.16)

inB₀(^R₂)\{0}. By (4.1),

∆g˜_αu˜α+r_α²˜hαu˜α= µα

r_α ²

˜

u³_α (4.17)

in B0(^R₂). Thanks to (4.9) and by standard elliptic theory, we then deduce that, after passing to a subsequence,

˜

uα→u˜ (4.18)

in C_loc² B₀(^R₂)\{0}

as α→+∞, whereW satisfies ∆˜u= 0 in B₀(^R₂)\{0} and ∆ is the Euclidean Laplace Beltrami operator. Moreover, thanks to (4.16), we know that

|˜u(x)| ≤C|x|⁻² (4.19)

inB0(^R₂)\{0}. Thus we can write that

˜

u(x) = Λ

|x|² +H(x) (4.20)

where Λ≥0 and H satisfies ∆H= 0 in B0(^R₂). In order to see that Λ = 8, it is sufficient to integrate (4.17) inB0(1) to get that

− Z

∂B0(1)

∂νu˜αdσ˜g_α= µα

r_α ²Z

B0(1)

˜

u³_αdv˜g_α−r_α² Z

B0(1)

˜hαu˜αdv˜g_α. (4.21) By (4.16),

Z

B0(1)

˜

uαdv˜g_α≤C (4.22)

and by changingxinto ^µ_r^α

αx, we can write that Z

B₀(1)

˜

u³_αdvg_α =r_α²µ⁻²_α Z

B₀(^rα

µα)

ˆ u³_αdvˆg_α,

where ˆuα(x) = µαuα exp_xα(µαx)

and ˆgα(x) = exp^?_xαg

(µαx). By (4.6) and Lemma 4.1, we then get that

α→+∞lim µα

rα

²Z

B0(1)

˜

u³_αdv˜g_α= 16ω3. (4.23) Noting that by (4.18) and (4.20),

α→+∞lim Z

∂B₀(1)

∂νu˜αdσg˜_α =−2ω3Λ, (4.24) we get that Λ = 8 thanks to (4.22)–(4.24) by passing into the limit in (4.21) as α→+∞. At this point we claim that there exists β∈(0,1] andC >0 such that

vα≤Cu^β_α inM (4.25)

for allα. Letxα∈M be a point where ^vα

u^β_α is maximum. Then,

∆gvα(xα)

v_α(x_α) ≥∆gu^β_α(xα) u^βα(xα) and it follows from (4.1) that

quα(xα)²

v_α(x_α) −m²₁−q²u_α(x_α)²

≥ −β(β−1)|∇uα(x_α)|²

uα(xα)² +βu_α(x_α)²−βm²₀+βω_α²(qv_α(x_α)−1)² .

(4.26)

Choosing β ∈ (0,1] such that m²₁−βm²₀ > 0, since 0 < vα ≤ ¹_q, we get that u^β_α(xα)≥Cvα(xα) for someC >0 independent ofα. This proves (4.25). In what follows we letXα be the 1-form given by

X_α(x) =

1− 1

18Rc^]_g(x).(∇f_α(x),∇f_α(x))

∇f_α(x), (4.27) where fα(x) = ¹₂dg(xα, x)², Rcg is the Ricci curvature of g, and ] is the musical isomorphism. We apply the Pohozaev identity in Druet-Hebey [18] with the vector field Xα to uα in Bx_α(rα). We separate the regular part Aα=m²₀−ω²_α from the singular part inhα. Then,hα=Aα+O(vα) and we get that

Z

B_xα(r_α)

A_αu_αX_α(∇u_α)dv_g+1 8

Z

B_xα(r_α)

(∆_gdiv_gX_α)u²_αdv_g +1

4 Z

B_xα(rα)

(div_gX_α)A_αu²_αdv_g

=Q1,α+Q2,α+Q3,α+O Z

B_xα(rα)

vαu²_αdvg

!

+O Z

B_xα(r_α)

vαuα|Xα(∇uα)|dvg

! ,

(4.28)