A VARIATIONAL ANALYSIS OF EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS

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1 A VARIATIONAL ANALYSIS OF EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS ON COPACT RIEANNIAN ANIFOLDS EANUEL HEBEY, FRANK PACARD, AND DANIEL POLLACK Abstract. We establis new existence and non-existence results for positive solutions of te Einstein scalar field Licnerowicz equation on compact manifolds. Tis equation arises from te Hamiltonian constraint equation for te Einstein scalar field system in general relativity. Our analysis introduces variational tecniques, in te form of te mountain pass lemma, to te analysis of te Hamiltonian constraint equation, wic as been previously studied by oter metods. 1. Introduction One of te foundations in te matematical analysis of te Einstein field equations of general relativity is te rigorous formulation of te Caucy problem. Te basic local existence result of Coquet-Bruat [10], and te important extension of tis due to Coquet-Bruat and Geroc [5], allows one to approac te study of globally yperbolic spacetimes via te analysis of initial data sets. Te Gauss and Codazzi equations impose constraints on te coices of initial data in general relativity, and tese constraints are expressed by te Einstein constraint equations. Tis perspective, originally studied in te context of vacuum spacetimes, as also been successfully employed in te study of many non-vacuum models obtained by minimally coupling gravity to many of te classical matter and field sources, suc as electromagnetism via te axwell equations), Yang-ills fields, fluids, and oters [8, 11, 12]. One of te simplest non-vacuum systems is te Einstein-scalar field system wic arises in coupling gravity to a scalar field satisfying a linear or nonlinear wave equation wit respect to te Lorentz metric describing te gravitational field. Te Einstein-scalar field system, wen posed in tis generality, includes as special cases te massless or massive) Einstein-Klein-Gordon equations as well as te vacuum Einstein equations wit a positive or negative) cosmological constant. Einstein scalar field teories ave been te subject of interesting developments in recent years. Among tese are te recent attempts to use suc teories to explain te observed acceleration of te expansion of te universe [15, 16, 17, 18]. Using te conformal metod, Coquet-Bruat, Isenberg, and Pollack [6, 7] reformulated te constraint equations for te Einstien scalar field system as a determined system of nonlinear partial differential equations. Te equations are semi-decoupled in te constant mean curvature CC) setting. One of tese equations, te conformally formulated momentum constraint, is a linear elliptic equation and its solvability is easy to address. Te oter one, te conformally formulated Hamiltonian constraint, is a nonlinear elliptic equation te Einstein scalar field Licnerowicz equation) as Date: February 2,

2 2 EANUEL HEBEY, FRANK PACARD, AND DANIEL POLLACK in 1.1) below see [3] for a survey on te constraint equations, and in particular, te conformal metod). Tis nonlinear equation, wic contains bot a positive critical Sobolev nonlinearity and a negative power nonlinearity, turns out to be of great matematical interest. In tis paper we provide a variational analysis of tis equation under certain conditions on its coefficients. Te analysis of te Licnerowicz equations wic arise as te conformally formulated Hamiltonian constraint equations in bot vacuum and non-vacuum settings as, in te past, been conducted primarily by eiter te metod of sub and supersolutions i.e. a barrier metod) or by perturbation or fixed point metods. Tis approac as been sufficient to allow for a complete understanding of solvability in, for example, te case of constant mean curvature vacuum initial data on compact manifolds [11]. In [7] tis metod was applied to constant mean curvature initial data for te Einstein-scalar field system on compact manifolds. In a number of cases, te metod of sub and supersolutions was sown to be sufficient to completely analyze te solvability of te Einstein scalar field Licnerowicz equation. In oter cases, te limitations of tis metod were exposed and only partial results were obtained. We establis ere two general teorems concerning non-existence and existence respectively, of positive solutions to te Einstein scalar field Licnerowicz equation 1.1). Tese results are of interest due bot to teir application to questions of existence and non-existence of solutions of te Einstein scalar field constraint equations, as well as, more generally, te introduction of variational tecniques to te analysis of te constraint equations. We expect tat similar variational tecniques will be of use in resolving oter open questions concerning initial data for te Caucy problem in general relativity. In wat follows we let, g) be a smoot compact Riemannian manifold of dimension n 3. We let also H 1 ) be te Sobolev space of functions in L 2 ) wit one derivative in L 2 ). Te H 1 norm on H 1 ) is given by u H 1 = u 2 + u 2 ) dv g. Let 2 = 2n n 2, so tat 2 is te critical Sobolev exponent for te embedding of H 1 into Lebesgue s spaces. Let also, A, and B be smoot functions on. We consider te following Einstein scalar field Licnerowicz type equations g u + u = Bu A u 2 +1, 1.1) were g = div g is te Laplace-Beltrami operator, and u > 0. Unless oterwise stated, solutions are always required to be smoot and positive. Te relationsip between te coefficients in 1.1) and initial data for te Einstein scalar field system are as follows see [7] for more details). We first note tat te sign convention for te Laplace-Beltrami operator wic we use ere is te opposite of te one used in [7]. Te conformal initial data for te purely gravitational portion of te Einstein scalar field system consists of a background Riemannian metric g indicating a coice of conformal class for te pysical metric) togeter wit a symmetric 0, 2)-tensor σ wic is divergence-free and trace-free wit respect to g so tat σ is wat is commonly referred to as a transverse-traceless, or TT-tensor) and a scalar function τ representing te mean curvature of te Caucy surface in te spacetime development of te initial data set. Te initial data for te scalar field

3 EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS 3 consists of two functions, ψ and π on, representing respectively te initial value for te scalar field and its normalized time derivative. Wit respect to tis set of conformal initial data, te constraint equations for te Einstein scalar field system can be realized as a determined elliptic system wose unknowns consist of a positive scalar function φ and a vector field W on. As previously remarked, in te CC case wen τ is constant) tis system becomes semi-decoupled. Tis means tat te portion of it corresponding to te momentum constraint equation is a linear, elliptic, vector equation for W in wic te unknown φ does not appear. Tis equation as a unique solution wen, g) as no conformal Killing vector fields. Te solution, W, of tis conformally formulated momentum constraint equation ten appears in te one of te coefficients of te conformally formulated Hamiltonian constraint equation wic is wat we refer to as te Einstein scalar field Licnerowicz equation. A positive solution φ of te Einstein scalar field Licnerowicz equation is ten used wit te vector field W to transform te conformal initial data set g, σ, τ, ψ, π) into a pysical initial data set satisfying te Einstein scalar field constraint equations see [7]). In terms of te conformal initial data set and te vector field W satisfying te conformally formulated momentum constraint equation) te coefficients of te Einstein scalar field Licnerowicz equation 1.1) are = c n Rg) ψ 2 g ), A = cn σ + DW 2 g + π 2) and ) n 1 B = c n n τ 2 4V ψ) were c n = n 2 4n 1), Rg) is te scalar curvature, is te covariant derivative for g, V ) is te potential in te wave equation for te scalar field, and te operator D is te conformal Killing operator relative to g, defined by DW ) ab := a W b + b W a 2 n g ab m W m. Te kernel of D consists of te conformal Killing fields on, g). Note tat relative to te notation of [7], we ave = R g,ψ, B = B τ,ψ and A = A g,w,π. We assume in wat follows tat A 0 in. Tis assumption implies no pysical restrictions since we always ave tat A 0 in te original Einstein scalar field teory. One of te results of [7] is te definition of a conformal invariant, te Yamabe scalar field conformal invariant, wose sign can be used, troug a judicious coice of te background metric g, to control te sign of. We prove two type of results in tis paper. Te first one, in Section 2, establises a set of sufficient conditions to guarantee te nonexistence of positive solutions of 1.1). Te second one, in Section 3, is concerned wit te existence of positive solutions of 1.1). Our existence result corresponds to but generalizes) te case of initial data wit a positive Yamabe scalar field conformal invariant considered in [7]. ore specifically te results presented ere sould be contrasted wit te partial results indicated in te tird row of Table 2 of [7], and specifically wit Teorems 4 and 5 in of [7]. Te results presented ere apply, for example, wen considering initial data for te Einstein massive Klein Gordon system wit small relative to te mass), or zero, values of te mean curvature. Te basic variational metod employed ere is to use te mountain pass lemma [1] to solve a family of ε-approximated equations, and let ten ε 0 to obtain a solution of

4 4 EANUEL HEBEY, FRANK PACARD, AND DANIEL POLLACK 1.1). Finally, Section 4 contains a brief discussion of a class of sligtly more general equations wic arise wen considering te Einstein axwell scalar field teory. 2. Nonnexistence of smoot positive solutions Examples of nonexistence results involving pointwise conditions on, A, and B are easy to get. Let u be a smoot positive solution of 1.1), and x 0 be a point were u is minimum. Ten g ux 0 ) 0 and we get tat x 0 )ux 0 ) Bx 0 )ux 0 ) Ax 0 )ux 0 ) 2 1. Let us assume tat bot A and B are positive functions. We ave x 0 ) Bx 0 ) X + Ax 0 ) X 1 n, 2.1) were we ave set X = ux 0 ) 4 n 2. Studying te least value of te rigt and side of 2.1) considered as a function of X), we get tat 1.1) does not possess a smoot positive solution if n n + ) n ) > max n 1) n 1 A B n ) It also follows from 2.1) tat ) n 2 4n 1) A ux) ux 0 ) min + for all x. Te idea of getting suc a bound will be used again in Section 3 wen proving Teorem 3.1. We now obtain a nonexistence result involving te Lebesgue norm of te functions A, B and. Teorem 2.1. Let, g) be a smoot compact Riemannian manifold of dimension n 3. Let also, A, and B be smoot functions on wit A 0 in. If B > 0 in, and n n n 1) n 1 ) n+2 4n A n+2 3n 2 4n B 4n dvg > + ) n+2 4 B 2 n 4 dv g, 2.3) were + = max0, ), ten te Einstein scalar field Licnerowicz equation 1.1) does not possess any smoot positive solution. Proof. We assume first tat B > 0. Let u be a smoot positive solution of 1.1). Integrating 1.1) over we get tat Bu 2 1 Adv g dv g + u 2 +1 = udv g. 2.4) By Hölder s inequality, ) 4 ) n 2 udv g + ) n+2 4 B 2 n n+2 4 dv g B u 2 1 n+2 dv g. Again by using Hölder s inequality, ) 3n 2 ) n+2 A n+2 3n 2 4n B 4n dvg B u 2 1 4n 4n Adv g dv g u Collecting tese inequalities and using 2.4), we get ) 4n ) 4 X + A n+2 3n 2 n+2 4n B 4n dvg X 1 n + ) n+2 4 B 2 n n+2 4 dv g 2.5)

5 EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS 5 were we ave set X = B u 2 1 dv g ) 4 n+2 Te study of te minimal value of te function of X wic appears on te left and side of 2.5) implies tat n n ) 4n ) 4n n 1) n 1 A n+2 3n 2 n+2 4n B 4n dvg + ) n+2 4 B 2 n n+2 4 dv g Tis completes te proof of te teorem. any more restrictive nonexistence conditions can be obtained easily from 2.3). For example, replacing B by min B in te two integrals in 2.3), we get tat if n n ) n+2 4n n 1) n 1 A n+2 4n dvg > + ) n+2 4 dv g min B) n 1)n+2) 4n is fulfilled, ten 2.3) olds true and te Einstein scalar field Licnerowicz equation 1.1) does not possess any smoot positive solution. In te same spirit, note tat condition 2.2) is more restricitive tan 2.3) since, for any triple of functions satisfying 2.2) we ave raising tis to te power n+2 4n n n 3n 2 A B n+2 n 1) n 1 > + ) n B n2 n) n+2 and integrating te result over yields 2.3). In wat follows we let S = S, g), S > 0, be te Sobolev constant of, g) defined as te smallest S > 0 suc tat u 2 dv g S u 2 + u 2) ) 2 2 dv g 2.6) for all u H 1 ). Explicit upper bounds for S can be given in special geometries, like, see Ilias [13], wen te Ricci curvature of te manifold is positive. Concerning lower bounds, it is well-known tat S Kn 2, were K n is te sarp Sobolev constant in te n-dimensional Euclidean space for te Sobolev inequality u L 2 K n u L 2. By letting u = 1 in 2.6) we also get tat S V 2 /n g, were V g is te volume of wit respect to g. Using tis, we prove some nonexistence result for solutions wit bound an a priori bound on teir H 1 energy. Teorem 2.2. Let, g) be a smoot compact Riemannian manifold of dimension n 3. Let also, A, and B be smoot functions on wit A 0 in. If B is arbitrary, not necessarily positive, and A 1 2 dvg > SΛ max 2 B + max 1, max + ) SΛ 4 n 2 ) ) for some Λ > 0, were B = max0, B) and S is as in 2.6), ten te Einstein scalar field Licnerowicz equation 1.1) does not possess smoot positive solutions of energy u H 1 Λ. oreover, 2.7) is sarp in te sense tat te power p = 1 2 in te left and side of 2.7) cannot be improved, and tat te bound on te energy cannot be removed.

6 6 EANUEL HEBEY, FRANK PACARD, AND DANIEL POLLACK Proof. We prove ere tat 2.7) proibits te existence of positive solutions of 1.1). Te discussion on te sarpness of tis condition is postponed after te proof. Let u be a smoot positive solution of 1.1) suc tat u H 1 Λ, Λ > 0. Let C = max 1, max + ), were + = max0, ). Ten, u 2 + u 2) dv g C u 2 + u 2) dv g. 2.8) ultiplying 1.1) by u, and integrating over, we get by 2.8) tat Adv Bu 2 g dv g + C Λ ) u 2 By te Sobolev inequality 2.6) we can write tat Bu 2 dv g max B ) SΛ 2, 2.10) were B = max 0, B). Ten, by combining 2.9) 2.10) we get tat Adv g C Λ 2 + max u B ) SΛ ) 2 Now, Hölder inequality yields A 1 2 dvg ) 1 Adv ) g u 2 dv g. 2.12) u 2 By combining tis inequality wit 2.11), and by te Sobolev inequality 2.6), we get tat A 1 2 dvg SΛ max 2 B + C ) 1 2. SΛ 4 n 2 Tis proves te teorem. We now discuss te sarpness of 2.7) in Teorem 2.2. Te Yamabe equation on a Riemannian manifold, g) may be written as g u + n 2 Rg)u = u2 1, 2.13) 4n 1) were Rg) is te scalar curvature of g. A positive solution u > 0 of 2.13) corresponds to a conformally related metric g = u 2 2 g wit constant positive scalar curvature R g) = 4n 1) n 2. Now, any solution of 2.13) is a solution of 1.1) wen we let = n 2 4n 1) Rg), B = α, and A = 1 α)u22 for some α R. Tis provides a transformation rule for rewriting equations like 2.13) into equations like 1.1). On te unit spere S n, g), for wic Rg) = nn 1), we know see, for instance, Aubin [2]) tat tere exist families u ε ) ε of solutions of 2.13), ε > 0, suc tat u ε H 1 = Kn n + o1) for all ε > 0, and u ε L p + as ε 0 for all p > 2, were K n is te sarp Sobolev constant in te n-dimensional Euclidean space for te Sobolev inequality u L 2 K n u L 2. Letting α = 1 2, te above transformation rule 2.13) 1.1) provides a family of Einstein scalar field Licnerowicz type equations indexed by ε > 0, wit and B independent of ε, suc tat any equation in te family possesses a solution of energy less tan or equal to 2Kn n, and for wic Ap εdv g + as ε 0 for all p > 1 2. Tis proves tat te power p = 1 2 in te left and side of 2.7) cannot be improved. Tis example can be modified in

7 EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS 7 different ways wit te constructions given in Brendle [4] and in Druet and Hebey [9]. We prove next tat te bound on te energy in Teorem 2.2 cannot be removed. By Druet and Hebey [9] we know tat on te unit spere in dimension n 6, or on any quotient, g) of te unit spere in dimension n 6, tere exist families ε ) ε of smoot functions, suc tat ε nn 2) 4 in C 1 ), and families u ε ) ε of smoot positive functions suc tat, for any ε > 0, u ε solves te Yamabe type equation g0 u ε + ε u ε = u 2 1 ε, 2.14) and suc tat u ε H 1 + as ε 0. Rewriting 2.14) wit te transformation rule 2.13) 1.1), we see tat te u ε s solve 1.1) wit = ε, B = α, and A = 1 α)u 22 ε for some α R. Letting α = 1 2, we get families of Einstein scalar field Licnerowicz type equations indexed by ε > 0 suc tat any equation in te family possesses a solution, B is independent of ε, te ε s converge in te C 1 -topology to a positive constant function, and A1/2 ε dv g + as ε 0. In particular, we cannot ope to get tat tere exists C = Cn,, B), depending on te manifold and continuously on and B in te C 0 -topology, like tis is te case for te constant in 2.7) wen Λ is fixed, suc tat if A1/2 dv g C, ten te Einstein scalar field Licnerowicz type equation 1.1) does not possess a smoot positive solution. Tis proves tat te bound on te energy in Teorem 2.2 cannot be removed. In te same circle of ideas, we mention tat if B > 0 in, ten we can give anoter form to 2.7) were te constant appears as CΛ 2. In order to get tis dependancy in Λ 2 we may proceed as in te proof of Teorem 2.2, but now getting bounds from te estimate 2.9). By 2.9), since we assumed tat B > 0 in, we can write tat u 2 dv g C Λ 2 min B and Adv g C Λ ) u 2 Ten, by 2.12) as in te proof of te second part of Teorem 2.2, we get from 2.15) tat 1.1) does not possess a smoot positive solution if A 1 2 dvg > max 1, max + ) Λ ) min B) 1 2 Condition 2.16) is complementary to te condition in Teorem 2.2. For large Λ s, 2.16) is better tan 2.7) since it involves te energy Λ 2 and not Λ 2n 1)/n 2). 3. Existence of a smoot positive solution In tis section we use te mountain pass lemma [1], to get existence results tat complement te nonexistence results presented in Teorem 2.2. ore precisely, we prove tat if Adv g is sufficiently small, and A > 0 in, ten 1.1) possesses a solution. Wen A 0, 1.1) is te prescribed scalar curvature equation and we know from Kazdan and Warner [14] tat tere are situations in wic te equation does not possess a solution.

8 8 EANUEL HEBEY, FRANK PACARD, AND DANIEL POLLACK In te sequel we assume tat te function is cosen so tat g + is coercive. Tis amounts to say tat tere exists a constant K = K, g, ) > 0, suc tat u 2 dv g K u 2 + u 2) dv g for all u H 1 ). It will be convenient to define u H 1 = u 2 + u 2) ) 1 2 dv g. 3.1) We also denote by S = S, g, ) > 0, te Sobolev constant defined to be te smallest constant S > 0 suc tat u 2 dv g S u 2 + u 2) ) 2 2 dv g 3.2) for all u H 1 ). Observe tat, if > 0 in, ten g + is coercive and conversely coercivity implies tat dv g > 0, and tus tat max > 0. Also observe tat if A, B 0, A + B > 0, and if 1.1) possesses a smoot positive solution, ten g + is coercive. Indeed, in tat case, tere exists a function u > 0 suc tat g u+u > 0 everywere in, and te existence of suc an u implies te coercivity of g +. Finally, as already mentioned, wen > 0 in, ten g + is coercive and we ave te bound ) S max 1, S. min were S = S, g) > 0 is te Sobolev constant defined in 2.6). We prove ere tat te following existence result olds true. Teorem 3.1. Let, g) be a smoot compact Riemannian manifold of dimension n 3. Let, A, and B be smoot functions on for wic g + is coercive, A > 0 in, and max B > 0. Tere exists a constant C = Cn), C > 0 depending only on n, suc tat if and ϕ 2 H 1 A ϕ 2 dv g C S max B ) n 1 3.3) Bϕ 2 dv g > 0 for some smoot positive function ϕ > 0 in, were H 1 is as in 3.1) and S is as in 3.2), ten te Einstein scalar field Licnerowicz equation 1.1) possesses a smoot positive solution. Proof. Preliminary computations We define I 1) : H 1 ) R by I 1) u) = 1 u 2 + u 2) dv g Bu + ) 2 dv g, 3.4) and if we fix ε > 0 we define I 2) ε : H 1 ) R by I ε 2) u) = 1 2 Adv g, 3.5) ε + u + ) 2 2 )

9 EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS 9 were 2 = 2 2. Obviously, for any u H 1 ) we can write Φ u H 1 ) I 1) u) Ψ u H 1 ) 3.6) if te functions Φ, Ψ : [0, + ) R are defined by and Φt) = 1 2 t2 max B 2 S t 2 Ψt) = 1 2 t2 + max B 2 S t 2 for t R, were S > 0 and H 1 are as in 3.1) and 3.2). Let t 0 > 0 be given by t 0 = 1 S max B ) n ) 3.8) 3.9) so tat Φ is increasing in [0, t 0 ], and decreasing in [t 0, + ). We define θ > 0 suc tat θ 2 1 = 2n 1) and t 1 = θ t 0 for t 0 as in 3.9). It is easy to ceck tat Ψt 1 ) θ Φt 0) 1 2 Φt 0), 3.10) were Φ and Ψ are as in 3.7) and 3.8). Finally, we define te functional I ε = I 1) + I 2) ε, 3.11) were I 1) and I ε 2) are as in 3.4) and 3.5). Let ϕ C ), ϕ > 0 in, be te function in te statement of te teorem. In particular Bϕ 2 dv g > ) and, witout loss of generality, we can assume tat ϕ H 1 = 1. Now, provided te constant C in 3.3) is cosen to be C = θ 2 2 2, 4 we find tat 3.3) precisely translates into 1 A 2 dv g 1 t 1 ϕ) 2 2 Φt 0) 3.13) and by 3.6), 3.10), and 3.13) we get tat Finally, 3.12) implies tat I ε t 1 ϕ) Φt 0 ) < I ε t 0 ϕ) 3.14) lim I εt ϕ) =. +

10 10 EANUEL HEBEY, FRANK PACARD, AND DANIEL POLLACK Hence we can coose t 2 > t 0 suc tat were I ε is te functional in 3.11). I ε t 2 ϕ) < 0, 3.15) Application of te ountain Pass Lemma By 3.14) and 3.15), we can apply te mountain pass lemma [1] to te functional I ε. Let c ε = inf γ Γ max u γ I εu), 3.16) were Γ stands for te set of continuous pats joining u 1 = t 1 ϕ to u 2 = t 2 ϕ. Observe tat c ε > Φt 0 ) and, taking te pat γt) = t ϕ, for t [t 1, t 2 ], we see tat c ε is bounded uniformly as ε tends to 0. We will keep in mind, for furter use tat for all ε small enoug, were c > 0 is independent of ε. Φt 0 ) < c ε c 3.17) By te mountain pass lemma we get tat tere exists a sequence u k ) k in H 1 ) suc tat I ε u k ) c and I εu k ) ) as k +. By 3.18), = u k ϕ)dv g + u k ϕdv g Bu + 1 k ϕdv )2 g Au + k ϕdv ) 3.19) g ε + u + k )2 ) + o ϕ 2 +1 H 1 for all ϕ H 1 ), were u k ϕ) stands for te pointwise scalar product of u k and ϕ wit respect to g, and 1 uk 2 + u 2 ) k dvg Bu + k dv )2 g ) Adv g 2 ε + u + k )2 ) = c ε + o1). 2 Combining 3.19) wit ϕ = u k, and 3.20), we get tat 1 Bu + k dv n )2 g + 1 Au + k )2 dv g 2 ε + u + k )2 ) Adv ) g 2 ε + u + k )2 ) = c ε + o u k H + o1), ) and it follows from 3.21) tat for k sufficiently large, 1 ) Bu + k dv n )2 g 2c ε + o u k H. 3.22) 1 By 3.20) and 3.22) we ten get tat for k sufficiently large uk 2 + u 2 k) dvg n 2 Bu + k dv n )2 g + 4 c ε ) 2n c ε + o u k H )

11 EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS 11 In particular, by 3.22) and 3.23), uk 2 + u 2 ) k dvg 2n c ε + 1, and 4n n 2 c ε Bu + k )2 dv g 3nc ε 3.24) for k sufficiently large, were c ε is as in 3.16). By 3.24), te sequence u k ) k is bounded in H 1 ). Up to passing to a subsequence we may ten assume tat tere exists u ε H 1 ) suc tat u k u ε weakly in H 1 ), u k u ε strongly in L p ) for some p > 2, and u k u ε almost everywere in as k +. As a consequence, u + k )2 1 u + ε ) 2 1 weakly in L 2 /2 1) ), and u + k ε + u + k )2 ) q u + ε ε + u + ε ) 2 ) q strongly in L2 ) 3.25) for all q > 0, as k +. Indeed, by 3.24), te u + k 1 s are bounded in )2 L 2 /2 1) ). Since tey converge almost everywere to u + ε ) 2 1, te first equation in 3.25) follows from standard integration teory. By te Lebesgue s dominated convergence teorem we also ave tat ε + u + k )2 ) q ε + u + ε ) 2 ) q strongly in L p ) for all p 1 and all q > 0, and since u k u ε in L p ) for some p > 2, we easily get tat te second equation in 3.25) olds true. By 3.25), letting k + in 3.19), it follows tat u ε satisfies g u ε + u ε = Bu + ε ) Au + ε ε + u + ε ) 2 ) ) in te weak sense. Te weak maximum principle and 3.26) imply tat u ε 0. As a consequence, g u ε + u ε = Bu 2 1 Au ε ε ) ε + u 2 ε) 2 +1 in te weak sense. Regularity and positivity of te solution We may rewrite 3.27) as ) A g u ε + u ε + u 2 ε) 2 +1 ε = Bu 2 1 ε, and since A ε + u 2 ε) 2 +1 L ), te regularity arguments developed in Trudinger [19] apply to 3.27). It follows tat u ε L s ) for some s > 2. Since we ave tat Aε + u 2 ε) 2 +1 u ε L p ) if u ε L p ), and u ε L s ) for some s > 2, te standard bootstrap procedure, togeter wit regularity teory, gives tat u ε H 2,p ) for all p 1, were H 2,p is te Sobolev space of functions in L p wit two derivatives in L p. By te Sobolev embedding teorem we ten get tat te rigt and side in 3.27) is in C 0,α ) for α 0, 1), and by regularity teory it follows tat u ε C 2,α ) for α 0, 1). In particular, te strong maximum principle can be applied and we get tat eiter

12 12 EANUEL HEBEY, FRANK PACARD, AND DANIEL POLLACK u ε 0, or u ε > 0 in. Ten we easily get tat u ε C ) is smoot. By 3.24) and 3.25), letting k + in 3.21), we get tat 1 Adv g 2 ε + u 2 ε) 2 1)c. 3.28) 2 were c is te upper bound for c ε. If, for a sequence of ε j tending to 0, u εj were to be equal to 0, we would conclude tat 1 Adv g c 2 2 1)ε 2 j 3.29) wic is clearly impossible since we ave assumed tat A > 0. Terefore, for ε small enoug u ε 0. Ten, according to te above discussion, u ε is a smoot positive solution of 3.27). By 3.24), and standard properties of te weak limit, we also get tat uε 2 + u 2 ) ε dvg 2nc ε ) for all ε > 0 small enoug. Passing to te limit as ε tends to 0 In wat follows we let ε k ) k be a sequence of positive real numbers suc tat ε k 0 as k + and 3.29) olds true wit ε = ε k for all k, and let u k = u εk. Ten u k is a smoot positive function in suc tat g u k + u k = Bu 2 1 k + Au k ε k + u 2 k ) ) in wile, by 3.17) and 3.30), te sequence u k ) k is bounded in H 1 ). Let x k be a point were u k is minimum. Ten g u k x k ) 0 and we get wit 3.31) tat Let δ 0 > 0 be suc tat x k ) + B x k )u k x k ) 2 2 δ 22 +1) 0 max Ax k ) ε k + u k x k ) 2 ) ) ) + max 2 B )δ2 0 = min A 2 By 3.32) we obtain tat u k x k ) δ 0, and tus tat min u k δ ) wen k is sufficiently large. Since u k ) k is bounded in H 1 ) we may assume tat tere exists u H 1 ) suc tat, up to passing to a subsequence, u k u weakly in H 1 ), u k u strongly in L p ) for some p > 2, and u k u almost everywere in as k +. By 3.33), u δ 0 almost everywere in. Still by 3.33), we get wit similar arguments to tose used to prove 3.25) tat u 2 1 k u 2 1 weakly in L 2 /2 1) ), and u k ε k + u 1 2 k )2 +1 u 2 +1 strongly in L2 ). 3.34) as k +. By 3.31) and 3.34), letting k + in 3.31), we get tat u is a weak solution of te Einstein scalar field Licnerowicz equation 1.1). Rewriting 1.1) as g u + A ) u = Bu 2 1, u 2 +2

13 EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS 13 and since Au 2 2 L ), te regularity arguments developed in Trudinger [19] apply to 1.1). It follows tat u L s ) for some s > 2. Since u δ 0 almost everywere, and δ 0 > 0, te standard bootstrap procedure, togeter wit regularity teory, gives tat u is a smoot positive solution of 1.1). Tis ends te proof of te teorem. As a remark, te above proof provides an explicit expression for te dimensional constant C in 3.3). As anoter remark, it can be noted tat wen Bdv g > 0, ten we can take ϕ to be constant in 3.12). In particular, our existence result as te following Corollary. Corollary 3.1. Let, g) be a smoot compact Riemannian manifold of dimension n 3 and a smoot functions on for wic g + is coercive. Tere exists a constant C = Cn, ), C > 0, suc tat if A and B are smoot functions on, wit A > 0 in, max B > 0, and Bdv g > 0, and if we furter assume tat max B )n 1 A dv g Cn, ), 3.35) ten te Einstein scalar field Licnerowicz equation 1.1) possesses a smoot positive solution. Wen A > 0 and B > 0, we can also take ϕ = A n 2 4n result as te following Corollary. in 3.12), and our existence Corollary 3.2. Let, g) be a smoot compact Riemannian manifold of dimension n 3 and a smoot functions on for wic g + is coercive. Tere exists a constant C = Cn, ), C > 0, suc tat if A and B are smoot functions on, wit A > 0 and B > 0 in and if we furter assume tat max B )n 1 A n 2 4n 2 H 1 A 1 2 dvg Cn, ), 3.36) ten te Einstein scalar field Licnerowicz equation 1.1) possesses a smoot positive solution. Interestingly, Sobolev embedding implies tat A 1 2 dvg S A n 2 4n 2 H 1 and so, if A and B satisfy 3.36), ten max B )n 1 A 1 2 dvg ) 2 Cn, ) S wic is reminiscent of condition tat ensured te non existence of a solution wic was obtained in Teorem Einstein-axwell-scalar field teory Te metods employed in sections 2 and 3 are strong enoug to deal wit additional nonlinear negative power terms in te equation of te form Cu p for C 0 and p > 1. Suc terms arise, for example, in te Einstein-axwell-scalar field teory. Given, g) compact of dimension n 3, we let, A, B, and C be smoot functions in, and we briefly discuss in tis section equations of te form g u + u = Bu A u C u p, 4.1),

14 14 EANUEL HEBEY, FRANK PACARD, AND DANIEL POLLACK were A, C 0 and p > 1. In te case of te Einstein-axwell-scalar field teory in spatial) dimension n = 3 we ave p = 3 and C 0 represents te sum of te squares of te norms of te electric and magnetic fields on. Te approac we used to prove Teorem 2.2 deals wit inequalities resulting from te signs of te coefficients and te powers of te unknown function u and tus applies to 4.1). Let ˆp = 2 +p Ten, if we concentrate on getting nonexistence results of smoot positive solutions wit no a priori bound on te energy, te approac we used to prove Teorem 2.2 gives in particular tat 4.1) does not possess a smoot positive solution if B > 0 in, A, C 0 in, and eiter 2.3) olds true, or α + 1) α+1 α α ) 1 ˆp C 1ˆp B ˆp 1 ˆp dv g > + ) n+2 4 B 2 n 4 dv g, 4.2) were α = n 2)p + 1)/4. We also do get similar conditions to 4.2) for te nonexistence of solutions of 4.1) of energy bounded by Λ. Te metod we used to prove Teorem 3.1 applies to 4.1) as well. Assume g + is coercive, A, C 0 in, A + C > 0 in, and max B > 0. Following te proof of Teorem 3.1 we get tat tat tere exists Λ = Λn, p), Λ > 0 depending only on n and p, suc tat if and A ϕ 2 dv g Λ S max B ) n 1, Bϕ 2 dv g > 0 C ϕ p 1 dv Λ g S max B ) α 4.3) for some smoot positive function ϕ > 0 in suc tat ϕ H 1 = 1, were H 1 is as in 3.1), S is as in 3.2), and α is as in 4.2), ten 4.1) possesses a smoot positive solution. As for 3.3), te constant Λ in 4.3) can be made explicit. References [1] A. Ambrosetti and P. Rabinowitz, Dual variational metods in critical point teory and applications, J. Functional Analysis ), [2] T. Aubin, Nonlinear analysis on manifolds. onge-ampre equations. Grund. der at. Wissenscaften, 252 Springer-Verlag, New York, [3] R. Bartnik and J. Isenberg, Te Constraint Equations, in Te Einstein equations and te large scale beavior of gravitational fields P.T. Cruściel and H. Friedric, eds.), Birkäuser, Basel, 1 39, [4] S. Brendle, Blow-up penomena for te Yamabe PDE in ig dimensions, Preprint 2006). [5] Y. Coquet-Bruat and R. Geroc, Global aspects of te Caucy problem in general relativity, Comm. at. Pys ), [6] Y. Coquet-Bruat, J. Isenberg and D. Pollack Te Einstein scalar field constraints on asymptotically Euclidean manifolds, Cin. Ann. at., Vol 27, ser. B, no ), 31 52; ArXiv: gr-qc/ [7] Coquet-Bruat, Y., Isenberg, J., and Pollack, D., Te constraint equations for te Einstein scalar field system on compact manifolds, Class. Quantum Grav ) : ArXiv: gr-qc/ [8] Y. Coquet-Bruat and J. York, Te Caucy Problem, in General Relativity and Gravitation - Te Einstein Centenary, A. Held ed.), Plenum, , [9] O. Druet and E. Hebey, Blow-up examples for second order elliptic PDEs of critical Sobolev growt, Trans. Amer. at. Soc ) [10] Y. Foures-Bruat, Téorème d existence pour certains systèmes d équations aux dérivées partialles non linéaires, Acta. at ) [11] J. Isenberg, Constant mean curvature solutions of te Einstein constraint equations on closed manifolds, Class. Quantum Grav )

15 EINSTEIN SCALAR FIELD LICHNEROWICZ EQUATIONS 15 [12] J. Isenberg, D. axwell and D. Pollack, A gluing constructions for non-vacuum solutions of te Einstein constraint equations, Adv. Teor. at. Pys ), no. 1, ; ArXiv: gr-qc/ [13] S. Ilias, Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes, Ann. Inst. Fourier ) [14] J.L. Kazdan and F.W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry ) [15] A. Rendall, Accelerated cosmological expansion due to a scalar field wose potential as a p ositive lower bound, Class. Quantum Grav. 21, ); ArXive: gr-qc/ [16] A. Rendall, atematical properties of cosmological models wit accelerated expansion., Analytical and numerical approaces to matematical relativity, , Lecture Notes in Pys., 692, Springer, Berlin, 2006; ArXive: gr-qc/ [17] A. Rendall, Intermediate inflation and te slow-roll approximation, Class. Quantum Grav. 22, ); ArXive: gr-qc/ [18] V. Sani, Dark matter and dark energy, In E. Papantonopoulos ed.) Pysics of te early universe. Springer, Berlin. 2005): ArXive: astro-p/ [19] N.S. Trudinger, Remarks concerning te conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa ) Emmanuel Hebey, Université de Cergy-Pontoise, Département de atématiques, Site de Saint-artin, 2 avenue Adolpe Cauvin, Cergy-Pontoise cedex, France address: Emmanuel.Hebey@mat.u-cergy.fr Frank Pacard, Université Paris XII, Département de atématiques, 61 avenue du Général de Gaulle, Créteil cedex, France address: pacard@univ-paris12.fr Daniel Pollack, University of Wasington, Department of atematics, Box , Seattle, WA , USA address: pollack@mat.wasington.edu