Li Ma

Xingwang Xu

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 347 (2009) 805–808Differential Geometry/Mathematical PhysicsUniform bound and a non-existence result for Lichnerowiczequation in the whole n-space ✩Li Ma a, Xingwang Xu ba Department of Mathematical Sciences, Tsinghua University, Beijing 100084, Chinab Mathematics Department, The National University of Singapore, 10, Kent Ridge Crescent, Singapore 119260Received 30 January 2009; accepted after revision 9 April 2009Available online 9 May 2009Presented by Étienne GhysAbstractIn this Note, we give a uniform bound and a non-existence result for positive solutions to the Lichnerowicz equation in Rn. Inparticular, we show that positive smooth solutions to:u + f (u) = 0, u > 0, in Rnwheref (u) = u−p−1 − up−1,are uniformly bounded. To cite this article: L. Ma, X. Xu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméUne estimation uniforme et un résultat de non-existence pour l’équation de Lichnerowicz sur n-espace. Dans cette Note,nous donnons une estimation uniforme et un résultat de non-existence pour les solutions positives de l’équation de Lichnerowiczsur Rn. En particulier, nous montrons que les solutions positives régulières de :u + f (u) = 0, u > 0, dans Rnoùf (u) = u−p−1 − up−1,sont bornées. Pour citer cet article : L. Ma, X. Xu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.✩ The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20060003002.E-mail addresses: lma@math.tsinghua.edu.cn (L. Ma), matxuxw@math.nus.edu.sg (X. Xu).1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.04.017806 L. Ma, X. Xu / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 805–8081. IntroductionIn the Einstein-scalar field theory one has the Lichnerowicz equation on a Riemannian manifold (M,γ ) of dimen-sion n  3 (see [2,3,6]). The aim of this paper is to give some results for positive solutions to this equation in thewhole Euclidean space.Given a smooth symmetric 2-tensor σ , a smooth vector field W , and a triple data (π, τ,ϕ) of smooth functionson M . Setcn = n − 24(n − 1) , p =2nn − 2 ,and letRγ,ϕ = cn(R(γ ) − |∇ϕ|2γ), Aγ,W,π = cn(|σ + DW |2γ + π2)andBτ,ϕ = cn(n − 1nτ 2 − V (ϕ))where V : R → R is a given smooth function and R(γ ) is the scalar curvature function of γ . Then the Lichnerowiczequation for the Einstein-scalar conformal data (γ, σ,π, τ,ϕ) with the given vector field W isγ u − Rγ,ϕu + Aγ,W,πu−p−1 − Bτ,ϕup−1 = 0, u > 0, (1)where γ is the Laplacian operator of γ . We use the convention that γ u = u′′ on the real line R. Note that Aγ,W,π 0. This equation is closely related to the Yamabe problem and the prescribing scalar curvature problems (see [1,7,8]).We shall consider a special case when (M,γ ) = Rn is the standard Euclidean space with radial symmetry data(σ,π, τ,ϕ). In this case, we can simply rewrite the equation in the following formu + R(x)u + A(x)u−p−1 + B(x)up−1 = 0, u > 0, on Rn (2)where R(x)  0, A(x)  0, and B(x) are given smooth functions of x ∈ Rn.Theorem 1. Suppose that A := A(x)  0, B := B(x)  0, and R(x)  0. Let β = p+12p . Assume that+∞∫0dr(r1−n∫Br (0)A1−βBβ dx)= +∞. (3)Then there exists no positive solution to (2).Note that β = 3n−24n , so the condition (3) can be written as+∞∫0dr(r1−n∫Br (0)A(x)n+24n B(x)3n−24n dx)= +∞.As a particular example, we note that when A1−βBβ  C > 0 for some positive constant C > 0, there exists nopositive solution to (2).This result may be extended to other case (see Theorem 3 in next section).We also have the following uniform bound for any positive solution to (2).Proposition 2. Assume that R(x) = 0 and A(x) = 1 is a positive constant and B(x) = −B is a negative constantin (2). Then any positive solution to (2) is uniformly bounded.In a recent paper, O. Druet and E. Hebey [4] have proved a very interesting result which says that for Lichnerowiczequation on a compact Riemannian manifold, the stability holds true when the dimension n is such that n  5 andfails to hold in general when n  6.L. Ma, X. Xu / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 805–808 8072. Non-existence resultsIn this section we prove Theorem 1.Recall our assumption that B(x)  0 and R(r)  0. We remark that for each fixed x ∈ Rn,A(x)X−p−1 + B(x)Xp−1is a convex function in X.Proof of Theorem 1. Let ū := ū(r) be the average of u(x) on the sphere Sn−1r (0) of radius r .Note that taking this average operation and using Jensen’s inequality to Eq. (2) we have−ū′′ − n − 1rū′  R(x)u + A(x)u−p−1 + B(x)up−1. (4)Using the Holder inequality to the right side of (4), we haveA(x)u−p−1 + B(x)up−1  A1−βBβwhereβ = p + 12p.Then we have−(rn−1ū′)′  rn−1(R(x)u + A1−βBβ ),which implies that−rn−1ū′ ∫Br (0)A1−βBβ dx +∫Br(0)Ruafter an integration. Dividing both side by rn−1 and integrating this inequality over [0, r0] , we haveū(0) − ū(r0) r0∫0dr(r1−n∫Br(0)A1−βBβ dx)+r0∫0r1−n∫Br(0)Ru.Sending r0 → ∞ we haveū(0) ∞∫0dr(r1−nr∫0τn−1A1−βBβ dτ),which is impossible by our assumption that+∞∫0dr(r1−n∫Br (0)A1−βBβ dx)= +∞.Then the proof of Theorem 1 is complete.We remark that from our proof above, we use the interaction between A and B . If we use the interaction betweenR and A, we can have the following result by the same argument.Theorem 3. Suppose that A := A(x)  0, B(x)  0, and R := R(x)  0. Let β = p+12p . Assume that+∞∫0dr(r1−n∫Br (0)A(x)n−24(n−1) R(x)3n−24(n−1) dx)= +∞.Then there exists no positive solution to (2).808 L. Ma, X. Xu / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 805–8083. Proof of Proposition 2In this section, we assume that R(r) = 0 and A(r) = 1 is a positive constant and B(r) = −B is a negative constantin (2). Then we may reduce (2) into the following form:u + f (u) = 0, u > 0, on Rn (5)wheref (u) = u−p−1 − Bup−1.Denote by BR any ball of radius R > 0 in Rn.We shall use a trick used in [5]. We look for a positive radial super-solution v = v(r) to (5) in the ball BR with thepositive infinity boundary condition. This is equivalent to finding v = v(r) > 0 such that{v + f (v)  0, in BR,v = +∞, on ∂BR.Note thatf ′ = −(p + 1)u−p − B(p − 1)up−2 < 0for u > 0. Then the comparison lemma is true for (5) in the ball BR . Hence, we haveu(x)  v(r), in BR.From this we know that u is uniformly bounded in Rn.Let v(r) = (R2 − r2)−α for large α > 1 and small R  1. By direct computation, we know that v is the rightsuper-solution v = v(r) to (5) in the ball BR with positive infinity boundary condition. Henceu(x)  2αR−2α, in BR/2.This proves our Proposition 2.It is clear that our argument can be generalized to treat positive solutions to the following equation:u + A(x)u−p−1 − Bup−1 = 0, in Rn,where A(x) is a smooth uniformly bounded function in Rn. It is an open question if the Liouville type theorem is truefor positive solutions to (5).AcknowledgementsThe authors would like to thank Prof. F. Pacard for sending us Ref. [4].References[1] Th. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer, New York, 1998.[2] Y. Choquet-Bruhat, J. Isenberg, D. Pollack, The Einstein-scalar field constraints on asymptotically Euclidean manifolds, Chinese Ann. Math.Ser. B 27 (1) (2006) 31–52.[3] Y. Choquet-Bruhat, J. Isenberg, D. Pollack, The constraint equations for the Einstein-scalar field system on compact manifolds, Class. QuantumGrav. 24 (2007) 809–828.[4] O. Druet, E. Hebey, Stability and instability for Einstein-scalar field Lichnerowicz equations on a compact Riemannian manifolds, 2008,preprint.[5] Y. Du, L. Ma, Logistic type equations on RN by a squeezing method involving boundary blow-up solutions, J. London Math. Soc. 64 (2001)107–124, MR 2002d:35089.[6] E. Hebey, F. Pacard, D. Pollack, A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds,Comm. Math. Phys. 278 (1) (2008) 117–132.[7] J. Lee, Th. Parker, The Yamabe problem, Bull. Am. Soc. 17 (1) (1987) 37–91.[8] R. Schoen, A report on some recent progress on nonlinear problems in differential geometry, Surveys in Differential Geometry, 1991, pp.201–241.

Uniform bound and a non-existence result for Lichnerowicz equation in the whole <i>n</i>-space

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Uniform bound and a non-existence result for Lichnerowicz equation in the whole <i>n</i>-space

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