DSpace at VNU: Existence Results for the Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds in the Null Case

DSpace at VNU: Existence Results for the Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds in...

Commun Math Phys Digital Object Identifier (DOI) 10.1007/s00220-014-2133-7 Communications in Mathematical Physics Existence Results for the Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds in the Null Case ´ Anh Ngô1,2 , Xingwang Xu3 Quôc Laboratoire de Mathématiques et de Physique Thộorique (LMPT), CNRS-UMR 7350, Universitộ Franỗois Rabelais de Tours, Parc de Grandmont, 37200 Tours, France E-mail: quoc-anh.ngo@lmpt.univ-tours.fr Department of Mathematics, College of Science, Vietnam National University, Hanoi, Vietnam E-mail: bookworm_vn@yahoo.com Department of Mathematics, National University of Singapore, Block S17 (SOC1), 10 Lower Kent Ridge Road, Singapore 119076, Singapore E-mail: matxuxw@nus.edu.sg Received: October 2013 / Accepted: 15 April 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract: This is the second in our series of papers concerning positive solutions of the Einstein-scalar field Lichnerowicz equations Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n 3, f and a are two smooth functions on M with M f dvg < 0, sup M f > 0, and M a dvg > In this article, we prove two results involving the following equation arising from the Hamiltonian constraint equation for the Einstein-scalar field equation in general relativity gu = f u2 −1 + a u +1 , where g = − divg (∇g ) and = 2n/(n − 2) First, we prove that if either sup M f and M a dvg or sup M a is sufficiently small, the equation admits one positive smooth solution Second, we show that the equation always admits one and only one positive smooth solution provided sup M f We should emphasize that we allow a to vanish somewhere Along with these two results, existence and non-existence for related equations are also considered Contents Introduction Preliminary 2.1 Notations and basic properties for λ f and λ f,η,q 2.2 Basic properties for positive solutions 2.3 A necessary condition for f 2.4 The non-existence of smooth positive solutions of suitable small energy The Analysis of the Energy Functionals When sup M f > 3.1 Functional setting 3.2 Asymptotic behavior of μεk,q Proof of Theorem 1.1 Q A Ngô, X Xu 4.1 The case inf M a > 4.2 The case inf M a = Proof of Theorems 1.2 and 1.3 5.1 Proof of Theorems 1.2 5.2 Proof of Theorem 1.3 Some Remarks 6.1 Construction of functions satisfying (1.9) 6.2 A relation between sup M f and M a dvg 6.3 A stability result for small |h| 6.4 Proof of Lemma 2.1 References Introduction This is the second in our series of papers concerning positive solutions of the Einsteinscalar field Lichnerowicz equations (ELEs for short) on compact Riemannian manifolds Roughly speaking, given a smooth compact Riemannian manifold (M, g) without the boundary of dimension n 3, the ELEs can be written as the following simple partial differential equation gu + hu = f u −1 + a u +1 , u > 0, (1.1) where g = −divg (∇g ) is the Laplace-Beltrami operator, = 2n/(n − 2) is the critical Sobolev exponent, and h, f, a ( 0) are smooth functions in M In the literature, Eq (1.1) is motivated by the Hamiltonian constraint equations naturally arising when solving the Cauchy problem in general relativity through the conformal method Due to the nature of their origin, Eq (1.1) has recently received much considerable attention from the mathematical analysis point of view, for example [7,11,16–20] For the sake of clarity and in order to make the paper self-contained, let us briefly recall how the conformal method can be used when the Cauchy problem in general relativity is studied and how we come up with (1.1) Mathematically, for a given initial data set (M, g, K ) consisting of an n-dimensional Riemannian manifold (M, g) and a symmetric (0, 2)-tensor K , the initial value problem asks for a Cauchy development of (M, g, K ), denoted by (V, g), which is a Lorentzian manifold of dimension n + Here the spacetime metric g is required to satisfy the following Einstein equation Ric g − Scal g g = T, where Ric g and Scal g are the Ricci tensor and the scalar curvature of the spacetime metric g Also, the symmetric (0, 2)-tensor T appearing in the Einstein equation is the energy-momentum tensor which is supposed to present the density of all the energies, momenta and stresses of the sources, see [4, Chapter III] In order for (V, g) to be a Cauchy development of (M, g, K ), it is required that (M, g, K ) must embed isometrically to (V, g) as a slice with the second fundamental form K ; and the metric g becomes the pullback of the spacetime metric g by the embedding It turns out that the initial data (g, K ) cannot be arbitrary, they must satisfy some conditions In view of the Gauss and Codazzi equations, those conditions can be rewritten in a form consisting of two equations known as the Hamiltonian and momentum constraints defined on (M, g) as shown below Lichnerowicz Equations on Closed Manifolds in the Null Case Scalg −|K |2g + (tr g K )2 = 2ρ, divg K − d(tr g )K = J, (1.2) where all quantities of (1.2) involving a metric are computed with respect to g and Scalg is the scalar curvature of g Also in (1.2), ρ is a scalar field on M representing the energy density and J is a vector field on M representing the momentum density of the nongravitational fields; they are related to the energy-momentum tensor T as follows ρ = T (n, n), J = −T (n, ·), where n is the unit timelike normal to the slice M × {0}, see [4,5] and [6, Section 5] It follows from a simple dimension counting argument that the constraint equations in (1.2) form an under-determined system; thus they are in general hard to solve However, it was remarked in [4] that the conformal method can be effectively applied in the constant mean curvature setting, that is to look for (g, K ) of the following form g = u −2 g, τ K i j = u −2 g i j + u −2 (σ + LW )i j , n (1.3) where g is fixed, u is a positive (smooth) function, and W is a 1-form Note that the operator L appearing in (1.3) is the conformal Killing operator acting on W defined in local coordinates by LWi j = ∇i W j + ∇ j Wi − k (∇ Wk )gi j , n where ∇ and ∇ are the Levi-Civita connections associated to the metrics g and g respectively Here by τ = g i j K i j , we mean the mean curvature of M as a slide of V The choice for σ is somehow arbitrary; however, it is related to the York splitting The novelty of using the decomposition (1.3) is that the system (1.2) is easily transformed to a new determined system of partial differential equations of variable (u, W ) given as follows ⎧ 4(n − 1) n−1 ⎪ ⎪ τ − 2ρ u −1 + |σ + LW |2g u −2 −1 , (1.4a) ⎪ g u + Scal g u = − ⎨ n−2 n ⎪ 2(n+2) n−1 ⎪ ⎪ ⎩ u dτ + u n−2 J (1.4b) divg (LW ) = n When τ is constant, Eq (1.4b) then only involves W and generically implies W ≡ (for example, if M admits no conformal Killing vector field) Therefore, one is left with solving Eq (1.4a) In the vacuum case, e.g T ≡ and hence ρ ≡ and J ≡ as well, we know exactly which sets of data lead to solutions and which not, see [12] However, in the non-vacuum case, it should be pointed out that there are several cases for which either partial result or no result was achieved when solving (1.4a), especially when gravity is coupled to scalar field sources To see this more precisely, we assume the presence of a real scalar field ψ in the space time (V, g) with a potential U being a function of ψ The energy-momentum tensor T of a real scalar field is then given by Ti j = ∇i ψ ∇ j ψ − g i j ∇k ψ ∇k ψ −g i j U (ψ) Q A Ngô, X Xu A direct computation then leads us to 2ρ = π + |∇ψ|2g + 2U (ψ), (1.5) J = −π ∇ψ, where π is the normalized time derivative of ψ restricted to M and ψ is the restriction of ψ to M, see [4,5] for details As already shown in [5], to avoid introducing any new variable, the only way to decompose the scalar field (ψ, π ) is the following ψ = ψ, (1.6) π = u −2 π Combining (1.4), (1.5), and (1.6), we obtain from (1.4a) the following equation 4(n − 1) n−2 =− gu + (Scalg −|∇ψ|2g )u n−1 τ − 2U (ψ) u n −1 + (|σ + LW |2g + π )u −2 −1 (1.7) In view of (1.1), if we set n−2 Scalg −|∇ψ|2 , 4(n − 1) n−2 |σ + LW |2 + π , a= 4(n − 1) n−2 n−1 τ − 2U (ψ) , f =− 4(n − 1) n h= then we easily verify that (1.1) is nothing but (1.7) Based on a division recently obtained in [5], one can see that when solving (1.1) there are two cases corresponding to either h < or h ≡ with sign-changing f , for which no result was achieved This is basically due to the fact that the method of sub- and super-solutions does not work, thus forcing us to develop a new approach In the preceding paper [17], we have already proved that, in the case h < 0, a suitable balance between coefficients h, f , a of (1.1) is enough to guarantee the existence of one positive smooth solution In addition, it was found that under some further conditions we may or we may not have the uniqueness property for solutions of (1.1) This paper is a continuation of the paper [17] To be precise, in the present paper, we continue our study of the non-existence and the existence of positive smooth solutions to (1.1) when h = which was also left as an open question in the classification of [5], that is, we are interested in the following simple partial differential equation gu = f u2 −1 + au −2 −1 , u > (1.8) We assume hereafter that f and a are smooth functions on M with a The latter assumption implies no physical restrictions since we always have a in the original Einstein-scalar field theory Besides, in order to avoid studying the same equation arising from the prescribing scalar curvature problem in the null case, see [8], it is natural to assume M a dvg > Thanks to the conformally covariance property of (1.2), we can freely choose a background metric g such that the manifold M has unit volume Lichnerowicz Equations on Closed Manifolds in the Null Case In the first part of the present paper, we mainly consider the case sup M f > and M f dvg < (this is also a necessary condition if a ≡ 0) Before stating the result, let us denote by f ± the positive and negative parts of f , i.e., we define f − = min{ f, 0} and f + = max{ f, 0} Using these notations, we are able to show that if sup M f + and M a dvg are bounded from above by constants depending on n and f − , then (1.8) possesses at least one smooth positive solution Following is the statement: Theorem 1.1 Let (M, g) be a smooth compact Riemannian manifold without the boundary of dimension n Assume that f and a are smooth functions on M such that M a dvg > 0, M f dvg < 0, and sup M f > Then there exist two positive constants C1 and C2 to be specified such that if sup f + < C1 (1.9) M and a dvg < C2 (1.10) M hold, then (1.8) possesses at least one smooth positive solution To be precise, the constants C1 and C2 appearing in Theorem 1.1 are given in (4.1)– (4.2) below The question of whether we can find an explicit formula for C1 and C2 turns out to be difficult, even for the prescribed scalar curvature equation, for interested readers, we refer to [2] Combining with [17, Theorem 1.1], it turns out that existence result for the cases h = and h < are in a similar fashion However, as already seen in the case h > where we are able to keep either (1.9) or (1.10) and drop the other condition, the requirement for the case h = cannot be as strong as that in the case h < Surprisingly enough, in the next result of the present paper, we would like to emphasize that if we replace the estimate for L -norm of a in (1.10) by a suitable estimate for L ∞ -norm of a, then the condition (1.9) can be dropped The proof we provide here is based on the method of sub- and super-solutions, see [13,15] Theorem 1.2 Let (M, g) be a smooth compact Riemannian manifold without the boundary of dimension n Assume that f and a are smooth functions on M such that M a dvg > 0, M f dvg < 0, and sup M f > Then there exists a positive constant C depending only on f and n such that if sup a < C3 , (1.11) M then (1.8) possesses at least one smooth positive solution Again, the constant C3 appearing in the theorem above which is given in (5.1) below is less explicit Concerning (1.1), using previous results for the negative case in [17] and for the positive case in [10,18] together with Theorems 1.1–1.2 above, one can obtain in the case when f changes sign a picture of the interaction between the coefficients of (1.1) when h varies from −λ f to +∞ in order for (1.1) to get solutions, see Table for details In the last part of the present paper, we focus our attention to the case sup M f We shall prove the following result Q A Ngô, X Xu Theorem 1.3 Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n Let f and a be smooth functions on M with a in M, M a dvg > 0, and f Then (1.8) possesses a unique positive solution Concerning Theorem 1.3, it is worth noticing that it generalizes the same result obtained in [5] where the method of sub- and super-solutions was used In this paper, we provide a variational approach to prove this result As can be seen in both theorems, we allow a to have zeros in M In order to achieve this goal, we make use of the sub- and super-solution method as suggested in [9] While the existence of a super-solution is quite easy to see, a sub-solution is rather hard to construct We believe that our construction of a sub-solution could be useful elsewhere Before closing this section, let us briefly mention the organization of the paper and highlight some techniques used In Sect 2, we discuss the quantities λ f and λ f,η,q concerning the positive part f + of f as well as basic properties of positive solutions of (1.8) Also in this section, we prove that the condition M f dvg < is necessary In Sect 3, a careful analysis of the energy functional under the case sup M f > is presented Having these preparations, we spend Sect to prove Theorems 1.1 while Theorems 1.2 and 1.3 will be proved in Sect In Sect 6, we provide a procedure to construct a function f which satisfies (1.9) In addition, we also comment on the relation between sup M f and M a dvg Preliminary 2.1 Notations and basic properties for λ f and λ f,η,q Let M be a compact Riemannian manifold without boundary and H p (M) be the usual Sobolev space It is well known that there exist two constants K1 and A1 such that, for all u ∈ H (M), the following Sobolev inequality u 2L K1 ∇u 2L + A1 u 2L (2.1) holds For simplicity, we denote by the average of and , that is, = (2n − 2)/(n − 2) Following [21], we define the following number ⎧ ⎪ |∇u|2 dvg ⎨ inf M , if A = ∅, (2.2) λ f = u∈A M |u|2 dvg ⎪ ⎩+∞, if A = ∅, where A = u ∈ H (M) : u | f − |u dvg = 0, u ≡ 0, M Intuitively, functions in A can be thought of as functions that vanish on the support of f − As can be seen from [17], the number λ f plays an important role when solving Eq (1.1) in the negative case, namely h < 0, see also [21] We recall from [17, Proposition 2.5] that in the negative case, it is necessary to have λ f > |h| in order for (1.1) to have positive solutions In addition, the strict inequality λ f > |h| is important for arguments used in [17] Then, it is naturally to expect that the following condition λ f > should hold in the null case Up to this point, it is not clear whether or not the strict inequality λ f > actually holds in the null case This is because the method used in [17, Proposition 2.5] does not Lichnerowicz Equations on Closed Manifolds in the Null Case work for the case h = However, in view of [21, Lemme 1], the number λ f , if finite, coincides with the first positive eigenvalue λ1 of the associated Dirichlet problem over the region M1 = {x ∈ M : f (x) 0} Hence, we obtain the following result Lemma 2.1 There holds λ f > Surprisingly, although we cannot directly adopt the method used in [17] to prove Lemma 2.1 above, a small change to Eq (1.1) simply by adding the term −u/n to the left hand side of (1.1) leads to a proof for Lemma 2.1 For the sake of completeness, we provide this proof in Subsection 6.4 at the end of the paper In the following, we approximate λ f by using λ f,η,q as proposed in [21] For each η ∈ (0, 1) fixed, we let A (η, q) be a subset of H (M) defined as A (η, q) = u ∈ H (M) : u Lq | f − ||u|q dvg = η = 1, M | f − | dvg M We then define the following number λ f,η,q = inf u∈A (η,q) |∇u|2 dvg M |u| dvg M (2.3) With q ∈ (2 , ) and η > 0, the set A (η, q) is not empty Consequently, λ f,η,q is welldefined and finite It was proved in [17] that if η > is small then λ f,η,q is achieved by some positive function v ∈ A (η, q) We now mention one useful property of λ f,η,q whose proof can be found in [17,21] For the sake of clarity, we divide the statement of that property into the following two lemmas Lemma 2.2 Suppose λ f < +∞ For each δ > fixed, there exists η0 > such that for all η < η0 , there exists qη ∈ (2 , ) so that λ f,η,q λ f − δ for every q ∈ (qη , ) Lemma 2.3 Suppose λ f = +∞ There exists η0 > such that for all η < η0 , there exists qη ∈ (2 , ) so that λ f,η,q for every q ∈ (qη , ) Having the number λ f , we then introduce the following quantity ⎧ ⎨ K1 + 2A1 /λ f −1 , if λ f < +∞, λ= ⎩ (K + A )−1 , if λ f = +∞ 1 In view of Lemmas 2.2 and 2.3, there exist two numbers η0 ∈ (0, 1) and qη0 ∈ (2 , ) so that the following estimate λ f,η0 ,q λ f /2, if λ f < +∞, 1, if λ f = +∞, (2.4) holds for every q ∈ (qη0 , ) In addition, since λ f,η,q is monotone decreasing in η, see [17,21], we may assume in the case λ f = +∞ that η0 < λn 2n n−2 1/(n−2) 1/(2−n) | f − | dvg a dvg M M (1−n)/(n−2) ,2 Q A Ngô, X Xu since we may take η0 as small as we wish It is important to note that, in the case sup M f > 0, the number η0 depends only on the negative part f − of f Unless otherwise stated, from now on, we fix such an η0 and we only consider q ∈ [qη0 , ) Finally, we introduce the following numbers k1,q = η0 2q η0 η0 λq − M | f | dvg λq − M | f | dvg q/(q−2) q/(q−2) = k2,q (2.5) From the choice of η0 , one can see that k1,q < k2,q for any q ∈ [2 , ) One can easily bound k1,q from below and k2,q from above, to be precise, there exists two positive k For numbers k < and k > independent of q and ε such that k k1,q < k2,q example, since /(2 − 2) = n − one can choose k= η0 and k= 2n n−2 η0 λ − M | f | dvg n−1 max η0 n−1 ,1 (2.6) n−1 λ − | M f | dvg ,1 (2.7) 2.2 Basic properties for positive solutions As already used in [17] for the case h < 0, the original idea of our approach was based on a mini-max method in a paper by Rauzy [21] However, we find that in the case considered in [21], the assumption of the negative Yamabe invariant is important; in fact, his approach does not work for the case of the null Yamabe invariant in the prescribing scalar curvature problem Moreover, the standard sub- and super-solutions method also does not work either since f changes sign As a first step to tackle (1.8), we look for positive smooth solutions of the following subcritical equation au q−2 u+ , (2.8) g u = f |u| (u + ε)q/2+1 which does include (1.8) as a particular case We spend this subsection studying several properties of positive solutions of (2.8) First, we derive a lower bound for a positive C solution u of (2.8) which is independent of q and ε Lemma 2.4 Let u be a positive C solution of (2.8) Then there holds u M inf M a − inf M f 1/(22 ) ,1 (2.9) for any q ∈ [2 , ) provided ε< inf M a − inf M f 1/2 ,1 (2.10) Lichnerowicz Equations on Closed Manifolds in the Null Case Proof Let us assume that u achieves its minimum value at x0 For simplicity, let us denote u(x0 ), f (x0 ), and a(x0 ) by u , f , and a0 respectively Notice that u > since u is a positive solution We then have g u|x0 and a0 u f (u )q−1 + (2.11) 1+q/2 ((u ) + ε) Consequently, we get that f < Using (2.11) we can see that − f (u )q−2 ((u )2 + ε)q/2+1 a0 − f ((u )2 + ε)q which implies that a0 − f0 (u )2 + ε 1/q inf M a − inf M f 1/q Thus, one can conclude that u satisfies (2.9) for any q ∈ [2 , ) and any ε verifying the condition (2.10) The proof is complete As can be seen from the proof above, although our lower bound is independent of q and ε, it depends on inf M a A recent attempt due to Premoselli suggests that, in the case h ≡ 0, we could have an uniformly positive lower bound for a sequence of solutions {u q }q of (2.8) as q regardless of inf M a For the interested reader, we refer to [20, Proposition 3.1] We now quote the following regularity whose proof can be mimicked from a similar result proved in [17] Lemma 2.5 Assume that u ∈ H (M) is an almost everywhere non-negative weak solution of Eq (2.8) We assume further that inf M a > Then (a) If ε > 0, then u ∈ C ∞ (M) In particular, u in M (b) If ε = and u −1 ∈ L p (M) for all p 1, then u ∈ C ∞ (M) It is worth mentioning that there is an extra assumption in Lemma 2.5 above compared to [17, Lemma 2.2] To be precise, we require inf M a > in Lemma 2.5 and it seems that this assumption is just a technical assumption The reason is that we need to make sure that any C -solution of Eq (2.8) stays away from zero; and in view of Lemma 2.4, such a conclusion is guaranteed provided inf M a > 2.3 A necessary condition for f The purpose of this subsection is to derive a necessary condition for the function f so that (2.8) admits a positive smooth solution Proposition 2.6 The necessary condition for (2.8) to have positive smooth solutions is that M f dvg < In particular, the necessary condition for (1.8) to have a positive smooth solution is that M f dvg < Proof We assume that u > is a solution of (2.8) By multiplying both sides of (2.8) by u 1−q , and integrating over M, one gets ( g u)u 1−q dvg = au 2−q f dvg + (u + ε)1+q/2 By the divergence theorem and the fact that q > 2, one obtains M M au 2−q f dvg + M Thus, M M (u + ε)1+q/2 f dvg < as claimed dvg M u −q |∇u|2 dvg < dvg = (1 − q) M Q A Ngô, X Xu 2.4 The non-existence of smooth positive solutions of suitable small energy Inspired by [10, Section 2] and [17, Subsection 2.5], this subsection is devoted to proving some non-existence results for smooth positive solution of (1.8) with finite energy In order to state the result, let us assume that u is a smooth positive solution of (1.8) and α ∈ (0, 1), β are constants The restriction on α is given as follows 22 + Observe that, by the Hölder inequality, the following estimates hold 0 in order to guarantee the existence of one solution Second, by using a simple sub- and super-solutions argument, we prove that (1.8) still admits one positive smooth solution even that inf M a = 4.1 The case inf M a > In this subsection, we obtain the existence of one solution of (1.8) under the assumption inf M a > For the sake of clarity, we divide the proof into several claims Claim There holds με(k1,q ),q < min{μεk ε ,q , μ(k2,q ),q } for all q ∈ (qη0 , ) and for all ε ∈ (0, k ) satisfying (2.10) Lichnerowicz Equations on Closed Manifolds in the Null Case Proof of Claim This is a consequence of Lemma 3.6 To see this, we have to derive (3.10) for suitable q sufficiently close to To so, we first fix C1 = and C2 = η0 | f − | dvg (4.1) M 2n λn 4(η0 )n−2 n − n−2 1−n | f − | dvg (4.2) M and suppose that sup M f and M a dvg verify (1.9)–(1.10) for such constants C1 and C2 Clearly, the assumption (1.10) is equivalent to a dvg < M (2 )n−2 λη0 η0 λ − | M f | dvg n−1 Observe that limq (q + 2)/(q − 2) = n − and limq q 4/(q−2) = (2 )n−2 Hence, we can choose qη0 ∈ [2 , ) sufficiently close to in such a way that (2.4) and a dvg < M λη0 4q η0 q+2 q−2 λq − M | f | dvg , (4.3) hold for any q ∈ [qη0 , ) This settles Claim It is important to note that qη0 is independent of q and ε Thus, from now on, we only consider q ∈ [qη0 , ) Claim Equation (2.8) with ε replaced by has a positive solution, say u 1,q , that is, u 1,q solves the following subcritical equation g u 1,q = f (u 1,q )q−1 + a (u 1,q )q+1 (4.4) Proof of Claim We define μqε = inf Fqε (u) u∈Dq where Dq = u ∈ H (M) : k u q Lq k2,q Clearly, the set Dq can be rewritten as Dq = Bk,q k∈[k ,k2,q ] Thanks to k1,q ∈ (k , k2,q ), we can bound μqε from the above as follows μqε με(k1,q ),q − k f dvg + M 2k a dvg M Keep in mind that the curve k → μεk,q is continuous in [k , k2,q ] by Proposition 3.4, we conclude that μqε is also bounded from below Q A Ngô, X Xu Since Fqε is differentiable and lower semi-continuous, as a consequence of the Ekeland Variational Principle, there exists a minimizing sequence for μqε in Dq , see [22, Corollary 3.5] By standard arguments, it is easy to show that any minimizing sequence for μqε in Dq is bounded in H (M) Therefore, a similar argument to that we have used before shows that μqε is achieved by some positive function u ε1,q ∈ Dq Notice that one can claim u ε1,q ∈ Dq since q < Obviously, u ε1,q is a weak solution of (2.8) By applying Lemma 2.5(a) to (2.8), we conclude that u ε1,q ∈ C ∞ (M) Since u ε1,q L q > (k )1/2 , we know u ε1,q ≡ Lemma 2.4 and the Minimum Principle imply u ε1,q > Next, in order to send ε 0, we need a uniformly boundedness of u ε1,q in H (M) Using the Hölder inequality and the fact that u ε1,q L u ε1,q L q , it is not hard to ε prove that u 1,q H is bounded from above with the bound independent of q and ε In what follows, we let {ε j } j be a sequence of positive real numbers such that ε j as εj j → ∞ For each j, let u 1,q be a smooth positive function in M such that εj au 1,q εj ε j q−1 + (4.5) g u 1,q = f (u 1,q ) 1+q/2 εj ((u 1,q ) + εj) in M Being bounded, there exists u 1,q ∈ H (M) such that up to subsequences ε j • u 1,q • • εj u 1,q εj u 1,q u 1,q in H (M); → u 1,q strongly in L (M); and → u 1,q almost everywhere in M Using Lemma 2.4, the Lebesgue Dominated Convergence Theorem can be applied to conclude that M (u 1,q )− p dvg is finite for all p Now sending j → ∞ in (4.5), we get that u 1,q is a weak solution of (4.4) Again, by applying Lemma 2.5(b) to (4.4), we conclude that u 1,q ∈ C ∞ (M) Using the strong convergence in L p (M) and the fact that εj u 1,q (k )1/q , one can see that u 1,q ≡ Therefore, u 1,q > by using Lemma 2.4 Lq and the Strong Minimum Principle Keep in mind that we still have u 1,q L q (k2,q )1/q This settles Claim Claim Equation (1.8) has at least one positive solution Proof of Claim Let us denote by μk1 ,q the energy of u 1,q found in Claim We now estimate the H -norm of the sequence {u 1,q }q Since k1 ∈ [k , k2,q ], we obtain ∇u 1,q 2 L2 μk1 ,q + q k − f dvg + f (u 1,q )q dvg M M a dvg + M k sup f M imply that the sequence {u 1,q }q remains This and the fact that bounded in H (M) Thus, up to subsequences, there exists u ∈ H (M) such that, as q , u 1,q 2L (k)2/2 2k u in H (M); • u 1,q • u 1,q → u strongly in L (M); and • u 1,q → u almost everywhere in M Lichnerowicz Equations on Closed Manifolds in the Null Case Recall that u 1,q solves (4.4) in the weak sense, that is, the following ∇u 1,q , ∇v g dvg − M f (u 1,q )q−1 v dvg − M M a v dvg = (u 1,q )q+1 (4.6) holds for any v ∈ H (M) Observe that ∇u 1,q − ∇u , ∇v g dvg → 0, M (4.7) u 1,q − u v dvg → M as q While the latter immediately follows from the strong convergence in L (M), the former can be proved easily since ∇u 1,q ∇u weakly in L (M) In addition, thanks to inf M a > and Lemma 2.4, a strictly positive lower bound for u 1,q helps us to conclude that av av dvg → dvg (4.8) q+1 +1 (u ) (u 1,q 1) M M as q So far, we can pass to the limit every terms on the left hand side of (4.6) except the term involving f By the Hölder inequality, one obtains ⎛ ⎞1−1/2 q−1 (u 1,q )q−1 L2 /(2 −1) ⎝ −1 (u 1,q )2 dvg ⎠ M = u 1,q q−1 L2 (4.9) Making use of the Sobolev inequality (2.1) and (4.9), we can prove the boundedness of (u 1,q )q−1 in L /(2 −1) (M) In addition, since u 1,q → u almost everywhere, (u 1,q )q−1 → (u )2 −1 almost everywhere According to [1, Theorem 3.45], we con(u )2 /(2 −1) weakly in L /(2 −1) (M) Therefore, by definition clude that (u 1,q )q−1 of weak convergence and the smoothness of f , one has f (u 1,q )q−1 v dvg → f (u )2 M −1 v dvg (4.10) M as q Combining (4.7), (4.8), and (4.10), one can see that u is a weak solution to (1.8) Using Lemma 2.5(b) we conclude that u ∈ C ∞ (M) and u > in M 4.2 The case inf M a = Under this context, making use of the method of sub- and super-solutions is the key argument, see [9] for a similar approach when h > We let ε0 > sufficiently small and then fix it so that the following inequality a dvg +ε0 < M 2n λn 4(η0 )n−2 n − n−2 | f − | dvg 1−n M still holds Since the manifold M has unit volume, we can conclude from the preceding inequality that the function a +ε0 verifies all assumptions in the previous subsection, thus showing that there exists a positive smooth function u solving the following equation gu = f u2 −1 + a + ε0 u2 +1 Q A Ngô, X Xu Obviously, u is a super-solution to (1.8), that is −1 f u2 gu + a u +1 Our aim is to find a sub-solution to (1.8) Indeed, since a dvg f − dvg = 0, − | dv | f g M M a+ M there exists a function u ∈ H (M) solving g u0 a dvg f− − | dv | f g M =a+ M (4.11) Since the right hand side of (4.11) is of class L p (M) for any p < +∞, the CaldéronZygmund inequality tells us that the solution u is of class W 2, p (M) for any p < +∞ Thanks to the Sobolev Embedding theorem [1, 2.10], we can conclude that u ∈ C 0,α (M) for some α ∈ (0, 1) In particular, the solution u is continuous Therefore, by adding a sufficiently large constant C to the function u if necessary, we can always assume that M u > We now find the sub-solution u of the form εu for small ε > to be determined To this end, we first write gu = εa + a dvg − M | f | dvg M f − (max M u )−(2 Since max M u < +∞, for any < ε a εa Besides, since f − ε ε2 +1 u 02 +1 (4.12) +1)/(2 +2) , we know that (4.13) and > 2, the following inequality ε a dvg − M | f | dvg M f− ε2 −1 −1 − u0 f holds provided ε a dvg − | dv | f g M 1/(2 −2) M (max u )(1−2 M )/(2 −2) (4.14) In particular, the following ε a dvg − M | f | dvg M f− ε2 −1 −1 u0 f (4.15) holds so long as (4.14) holds Combining (4.12), (4.13), and (4.15), we conclude that for small ε a ε2 −1 u 02 −1 f + +1 +1 gu ε u0 In other words, we have shown that Lichnerowicz Equations on Closed Manifolds in the Null Case −1 f u2 gu + a u +1 Finally, since u has a strictly positive lower bound, we can choose ε > sufficiently small such that u u Using the sub- and super-solutions method, see [15, Lemma 2.6], we can conclude the existence of a positive solution u to (1.8) By a regularity result developed in [15], we know that u is smooth Proof of Theorems 1.2 and 1.3 In this section, we provide proofs for Theorems 1.2 and 1.3 To prove Theorem 1.2, we employ the method of sub- and super-solutions While the existence of a sub-solution for (1.8) can be seen from the last part of the proof of Theorem 1.1, the existence of a super-solution for (1.8) needs some special treatment 5.1 Proof of Theorems 1.2 We first construct a positive super-solution u for (1.8) By using the change of variable u = exp(v), we get that − gu + f u2 −1 + au −2 −2 = ev (− gv + |∇v|2 ) + f e(2 −1)v + ae−(2 +1)v Hence, it suffices to find v satisfying − gv + |∇v|2 + f e(2 In order to this, thanks to M −2)v + ae−(2 +2)v f dvg < 0, we can pick b > small enough such that sup f e(2 −2)bϕ − −1 M f dvg M and b|∇ϕ|2 < − f dvg , M where ϕ is a positive smooth solution of the following equation gϕ = f − f dvg M We now find the function v of the form v = bϕ + log b −2 Indeed, by calculations, we have + |∇v|2 + f e(2 −2)v + ae−(2 +2)v log b = − g bϕ + −2 b log b (2 −2) bϕ+ 2log−2 −(2 + ∇ bϕ + + fe + ae −2 − gv b +2) bϕ+ 2log−2 Q A Ngô, X Xu = −b gϕ + b2 |∇ϕ|2 + b f e(2 −2)bϕ f dvg + b2 |∇ϕ|2 + b f (e(2 =b + ae−(2 −2)bϕ +2)bϕ 1−n b − 1) + ae−(2 +2)bϕ 1−n b M f dvg − b M b b f dvg + b sup f e(2 M f dvg + ae−(2 −2)bϕ − + ae−(2 +2)bϕ 1−n b M +2)bϕ 1−n b M Therefore, if we set C3 = − bn (2 e f dvg , +2)bϕ (5.1) M and assume that (1.11) holds with this constant, we then get that − gv + |∇v|2 + f e(2 −2)v + ae−(2 +2)v b f dvg < 0, M which concludes the existence of a super-solution u We now turn to the existence of a sub-solution Before doing so, we can easily check that u = exp bϕ + (log b)/(2 − 2) > exp (log b)/(2 − 2) = b(n−2)/4 Since u has a strictly positive lower bound and thanks to the second stage of the proof of Theorem 1.1 (see the last part of the previous section), we can easily construct a sup-solution u with u < u It is important to note that the existence of a sub-solution depends heavily on the conditions a and a ≡ 0; and here is the only place we make use of that fact in the proof The proof of the theorem is now complete Remark 5.1 Having this theorem in hand, one can observe that our problem (1.8) possesses the same phenomena of the Brezis–Nirenberg problem [3] Although we not know, under the conditions sup M f > and M f dvg < 0, whether the prescribing scalar curvature equations in the null case, see [8], gu = f u2 −1 , u > 0, always admit one positive smooth solution or not, but by adding a term with a negative exponent, that is, gu = f u2 −1 + λu −2 −1 , u > 0, λ > 0, the perturbed equation always has at least one positive solution provided the constant λ is small enough Lichnerowicz Equations on Closed Manifolds in the Null Case μεk,q k k1,q k2,q Fig The asymptotic behavior of μεk,q when sup M f k 5.2 Proof of Theorem 1.3 To conclude the existence part of Theorem 1.3, one can use the method of sub- and super-solutions.1 Here we note that the method used in [17, Theorem 1.2] still works in this context, thus providing another approach Due to the limit of the length of the present paper and the fact that (1.8) takes a simpler form than that of [17], we only indicate the main argument and leave proofs for the reader Indeed, the key ingredient is to study the asymptotic behavior of μεk,q When 0, one can obtain a behavior as shown in Fig To see this, we observe sup M f that, for large k, we obtain the following result Proposition 5.2 Suppose sup M f 0, then μεk,q → +∞ as k → +∞ for any ε > and any q ∈ [2 , ) but all are fixed While for small k, despite the fact that we are under the case sup M f 0, we can still go through Lemma 3.3 to conclude that, for small ε, μεk,q → +∞ as k To obtain the same result as in Lemma 3.6, it is necessary to change ki,q , i = 1, However, we note in our context that sup M f + = Therefore, we can still use ki,q but need to choose η0 small enough in such a way that the condition (3.10) is fulfilled Hence, we obtain the following lemma Lemma 5.3 There holds με(k1,q ),q < min{μεk ε ,q , μ(k2,q ),q } for any ε ∈ (0, k ) and any q ∈ (qη0 , ) By using C= M a dvg 1/(22 ) − M f dvg one can verify that f C −1 + M a dvg = C +1 We then select v as a smooth positive solution of the equation g v = f C −1 + aC −2 −1 The monotonicity of the map t → f t −1 + at −2 −1 guarantees that v ± ε will be a super- and sub-solutions for some small ε > Q A Ngô, X Xu With the same argument as that used before, we are now in a position to conclude the existence of at least one positive smooth solution of (1.8) To conclude the uniqueness part, we as follows Suppose that there exists two positive smooth solution u and u of (1.8) By setting w(x) = u (x) − u (x) with x ∈ M, we arrive at w gw = f (u 12 −1 −1 − u 22 )(u − u ) + a(u 1−2 −1 − u 2−2 −1 )(u − u ) Integrating both sides over M gives |∇w|2 dvg M = M f (u 12 −1 − u 22 −1 )(u − u ) dvg + M a(u 1−2 −1 − u 2−2 −1 )(u − u ) dvg Thanks to f and a with a ≡ 0, the only possibility for which the preceding inequality holds is that w vanishes in M, thus proving the uniqueness of positive smooth solution of (1.8) Some Remarks 6.1 Construction of functions satisfying (1.9) In this section, we provide functions f such that the condition (1.9) is fulfilled, i.e, given f − , we construct f + as small as we want For the sake of clarity, we summarize our construction in Fig For this purpose, we take a smooth function f with sup M f > and M f dvg < The idea is to lower sup M f but still keep the negative part f − of f For each number η > 0, let us denote η = {x ∈ M : f (x) > η} By the Morse–Sard theorem, there exist two numbers ξ and η with η0 0 We then take φ : M → [0, 1] to be a (smooth) cut-off function such that φ(x) = if x ∈ M\ if x ∈ η 0, 1, ξ, Having such a cut-off function φ, we construct the function f (x) = f (x)e−tφ(x) where t > is a parameter to be determined later Obviously, f |{M\ ξ } ≡ f |{M\ ξ } In particular, there holds f |{ f 0} ≡ f |{ f 0} From the choice of η, for any x ∈ ξ \ η , there holds f (x) < f (x) η η0 | f − | dvg M Lichnerowicz Equations on Closed Manifolds in the Null Case f η η0 M |f − | η0 M |f − | ξ M Fig Construction of functions satisfying (1.9) For x ∈ η, since f (x) = f (x)e−t , one can choose t sufficiently large such that f (x) η0 | f − | dvg M Notice that, this choice of t is independent of x, for example, t = log + (sup f ) η0 M | f − | dvg −1 M It is now clear to see that the function f satisfies all conditions in Theorem 1.1 6.2 A relation between sup M f and M a dvg Throughout this subsection, we always assume sup M f > We spend this subsection to point out a connection between sup M f and M a dvg To be precise, we conclude that if we lower sup M f but still keep f − , then we may find a better upper bound for M a dvg We note that although in the statement of Theorem 1.1, the right hand side of (1.10) only depends on the negative part f − , there is no contradiction to what we are going to discuss here because (1.10) is just a sufficient condition for the solvability of (1.8) More than that, this connection explains why in the case sup M f 0, we require no condition on M a dvg rather than its positivity In order to see this, let us first introduce a scaling constant τ > Using (4.1)–(4.2), we can rewrite (1.9) and (1.10) as follows sup f < M η0 | f − | dvg (6.1) M and a dvg < M 2n λn 4(η0 )n−2 n − n−2 | f − | dvg 1−n (6.2) M We now assume that sup M f satisfies the following inequality sup f M η0 2τ | f − | dvg M (6.3) Q A Ngô, X Xu Since τ > 1, it is clear that (6.3) is stronger than (6.1) Our aim is to show that we can find a better upper bound for M a dvg rather than that in (6.2) Indeed, let us introduce a variant of ki,q , i = 1, 2, as follows k 1,q = η0 2τ q η0 τ λq − M | f | dvg q/(q−2) and k 2,q = τ λq − M | f | dvg η0 q/(q−2) Clearly, k1,q < k 1,q < k 2,q and k 2,q > k2,q Notice that η0 and λ remain unchanged since f − is being kept Following the proof of Proposition 3.5, it follows from (6.3) that Fqε (u) λ (k 2,q )2/q for any u ∈ Bk 2,q ,q Therefore, in view of Lemmas 3.3 and 3.6, it suffices to compare με and με Indeed, a simple calculation shows that (k 1,q ),q (k 2,q ),q με(k 1,q ),q < με(k 2,q ),q provided a dvg < M By sending q λη0 4τ q η0 τ λq − M | f | dvg q+2 q−2 , one arrives at a dvg < M n−2 2n τ n−2 λn 4(η0 )n−2 n − Obviously, (6.4) is better than (6.2) since n | f − | dvg 1−n (6.4) M and τ > 6.3 A stability result for small |h| It would be interesting to note that the result in Theorem 1.2 still holds for (1.1) if we require h sufficiently close to 0, thus giving us a stability result for solutions of (1.1) for certain sets of data We formulate this observation through the following theorem Theorem 6.1 Let (M, g) be a smooth compact Riemannian manifold without the boundary of dimension n Assume that h, f , and a are smooth functions on M such that M a dvg > and sup M f > We assume further that M f dvg < in the case h Then there exist two positive constants C4 and C5 such that if |h| < C4 and sup a < C5 M hold, then (1.1) possesses at least one smooth positive solution Lichnerowicz Equations on Closed Manifolds in the Null Case Table Interaction between the coefficients of (1.1) for any h h f a −λ f < h < sup f + < C( f − ) a < C( f − ) h=0 sup f + < C( f − ) a < C( f − ) h=0 h>0 sup a < C( f ) a < C( f ) h>0 sup f + < C( f − , a) h We will not provide any proof for this result but will only indicate necessary steps for a proof First, as can be seen from Table 1, when h > 0, it is obvious to see the existence of such a constant C5 since our requirement for sup M a is stronger than the requirement for M a dvg Second, when h < 0, we can follow the proof of Theorem 1.2 Indeed, it suffices to find v in such a way that − gv + |∇v|2 + f e(2 −2)v + ae−(2 +2)v −h holds Still using the same form for v, we arrive at − gv + |∇v|2 + f e(2 −2)v + ae−(2 +2)v b −h f dvg + ae−(2 +2)bϕ 1−n b − h M Therefore, in this case, we can choose C4 = − b f dvg M and C5 = − bn e(2 +2)bϕ f dvg M or smaller if necessary to conclude the proof From this stability result, knowing that |h| < λ f is necessary if h < 0, one can ask what the optimal value for C4 is Naturally, there would be C4 λ f While seeking for an answer, we found that the constant C4 does depend on f as the constant C4 can be made arbitrarily close to To prove this, we borrow an idea essentially due to Kazdan and Warner [14, Theorem 10.1] Lemma 6.2 Suppose that h < is constant If a solution to (1.1) exists, then the unique solution ϕ of − g ϕ + (2 − 2)hϕ = (2 − 2) f (6.5) must be positive Proof Assume that u is a positive solution of (1.1) Using the substitution v = u −2 one can easily see that − gv = (−2 + 2)(2 + 1)u −2 |∇u|2 − (−2 + 2)u −2 |∇u|2 gu = (2 − 2)(2 − 1)v + (2 − 2)v u u and |∇v|2 = (2 − 2)2 (u −2 +1 )2 |∇u|2 |∇u|2 = (2 − 2)2 v 2 u +1 gu +2 , Q A Ngô, X Xu In particular, there holds − gv = − |∇v|2 gu + (2 − 2)v −2 v u Using the equation for u, we further arrive at − gv = − |∇v|2 − (2 − 2)hv + (2 − 2) f + (2 − 2)au −22 −2 v (6.6) Let ϕ be the unique solution of (6.5) and let w = ϕ − v Then − gw + (2 − 2)hw = − − |∇v|2 − (2 − 2)au −22 −2 v A standard argument shows that the minimum of w must be non-negative Therefore, ϕ v > as claimed Now, for h < given, we show that there is a smooth function f with M f dvg < for which (1.1) does not have any solution Indeed, let ψ ∈ C ∞ (M) satisfy M ψ dvg = but ψ ≡ We choose a constant α > so small in such a way that ψ + α still changes sign We then set f =− g ψ/(2 − 2) + h(ψ + α) Clearly, ψ + α solves (6.5) since − g ψ + α + (2 − 2)h ψ + α = (2 − 2) f Clearly, M f dvg = hα < However, by Lemma 6.2, (1.1) does not admit any solution for this f since the unique solution of (6.5) turns out to be ϕ = ψ +α, which now changes sign From the argument above, it is not clear whether λ− g ψ/(2 −2)+h(ψ+α) decreases or not since the set A for this new f is less understood However, it is worth noticing that the number λ f tends to if the set {x ∈ M : f (x) = 0} tends to M In addition, we suspect that the constant C4 should be λ f , unfortunately, all our attempts in this regard up to now have failed Finally, we note that the construction of (1.1) with no solution above suggests that the constant C5 appearing in Theorem 6.1 also depends on f since our construction is valid for any function a 6.4 Proof of Lemma 2.1 In this subsection, we prove Lemma 2.1 The idea is to make use of Theorem 1.2 and [17, Proposition 2.5] Indeed, it follows from Theorem 1.2 that the following equation gu = f u2 −1 + 2εu −2 −1 has a positive continuous solution, say u, for some positive small constant ε We now fix such ε and u Since the manifold M is compact, we can choose n > in such a way that − u > −εu −2 n −1 Lichnerowicz Equations on Closed Manifolds in the Null Case Combining this inequality and the equation satisfied by u, we arrive at gu − u > f u2 n −1 + εu −2 −1 Hence, we have shown that u is a super-solution to the equation gu − u = f u2 n −1 + εu −2 −1 (6.7) Now we can use the same technique that we have used several times to construct a sub-solution, say u The method of sub- and super-solutions now tells us that Eq (6.7) admits a positive solution Thus, we are now able to apply [17, Proposition 2.5] which tells us that λ f > 1/n is necessary Acknowledgements Ngô and Xu want to thank Professor Frank Pacard for his interest and enlightening discussion on the problem and related questions, especially during his short visit to Singapore in January 2013 Ngô also wants to thank Professors Emmanuel Humbert and Romain Gicquaud for their interest in the problem Part of the paper was done when he was under the support of the Région Centre through the convention no 00078780 during the period 2012–2014 Xu acknowledges the support under the NUS Research Grant R-146-000-169-112 Last but not least, Ngô and Xu are very grateful to the two anonymous referees for careful reading of the manuscript and for valuable comments and suggestions which substantially improved the exposition of the article References Aubin, T.: Some nonlinear problems in Riemannian geometry In: Springer monographs in mathematics Springer, New York Aubin, T., Cotsiolis, A.: Problème de la courbure scalaire prescrite sur les variétés riemanniennes complètes J Math Pures Appl 81, 999–1009 (2002) Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents Comm Pure Appl Math 36, 437–477 (1983) Choquet-Bruhat, Y.: General relativity and the Einstein equations In: Oxford 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