A Variational Analysis of Einstein–Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds

Hebey, Emmanuel; Pacard, Frank; Pollack, Daniel

doi:10.1007/s00220-007-0377-1

Cited by 52 publications

(97 citation statements)

References 18 publications

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“…The Positive case. It is known that θ 2 is finite when f > 0, see Theorem 2.1 in HebeyPacard-Pollack [12]. To conclude the proof of Theorem 1.1 we now show, when f > 0, that for every θ < θ 2 equation (EL θ ) has at least two distinct solutions and that for θ = θ 2 there is exactly one solution.…”

Section: Multiplicity Of Solutions Of (El)mentioning

confidence: 51%

“…When f ≤ 0 the equation is fully understood, see Isenberg [15] or Choquet-Bruhat, Isenberg and Pollack [7]. Partial existence results are known when max M f > 0, see Hebey, Pacard and Pollack [12] and Ngô and Xu [20]. The main result of the paper is as follows.…”

Section: Statement Of the Resultsmentioning

confidence: 96%

“…We compute the degree of (EL θ ) mimicking a computation carried out by Schoen [26] for the Yamabe equation. Since f > 0 both the existence of a solution of (EL θ ) when θ ≪ 1 and the non-existence when θ ≫ 1 were proven in Hebey-Pacard-Pollack [12]. Let for any positive M…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…We have S h,2 * = S h , where S h is as in (3.2). Following Hebey-Pacard-Pollack [12], to get rid of the negative exponent we consider for any ε > 0 the perturbed functional on H 1 (M ):…”

Section: A General Mountain-pass Situationmentioning

confidence: 99%

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Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

Premoselli

2014

Calc. Var.

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Abstract. We prove the stability of the Einstein-scalar field Lichnerowicz equation under subcritical perturbations of the critical nonlinearity in dimensions n = 3, 4, 5. As a consequence, we obtain the existence of a second solution to the equation in several cases. In particular, in the positive case, including the CMC positive cosmological constant case, we show that each time a solution exists, the equation produces a second solution with the exception of one critical value for which the solution is unique. Statement of the results.Let (M, g) be a smooth closed Riemannian manifold of dimension n ≥ 3. We are interested in the Einstein-scalar field Lichnerowicz equation in M :(EL) △ g u + hu = f u 2 * −1 + a u 2 * +1 , where h, f and a are smooth functions on M ,n−2 is the critical Sobolev exponent for the Sobolev space H 1 , and we assume that △ g + h is coercive, a ≥ 0, a ≡ 0, and max M f > 0. By closed, following standard terminology, we mean compact without boundary. Equation (EL) arises in the mathematical analysis of general relativity when solving the Einstein equations in a scalar-field setting, when the gravity is coupled to a scalar-field ψ. Special important cases include the massive Klein-Gordon setting or the case of a positive cosmological constant Λ. Given a closed manifold (M, g) of dimension n ≥ 3 endowed with two smooth functions π and ψ and a (2, 0)-symmetric tensor field K, the Cauchy problem in general relativity consists in finding a Lorentzian manifold (M × R,g) together with a smooth functionψ on M × R such thatψ |M = ψ and ∂ nψ|M = π, where ∂ n denotes the normalized time derivative, such that K is the second fundamental form of the embedding M ⊂ M × R, and such that (M × R,g) satisfies the Einstein equations:where Ric(g) is the Ricci tensor ofg, R(g) is its scalar curvature and T is the stress-energy tensor-field. This tensor field depends ong, onψ and on some potential V , itself related toψ by some wave equation. As shown first by for the vacuum case, see also Choquet-Bruhat-Isenberg-Pollack [6], a necessary and sufficient condition for the existence of such ag on M × R is that the following system of equations in M is satisfied:where R(g) is the scalar curvature of g and ∇ refers to the Levi-Civita connection of g. By specifying some of the unknown initial data (g, K, ψ, π) and solving the system for the remaining data, the conformal method initiated by Lichnerowicz [17] allows to turn (1.1) into a system of 1 2 BRUNO PREMOSELLI elliptic partial differential equations of critical Sobolev growth, called the conformal constraint system of equations. For a survey reference on the constraint equations see and for further informations on the conformal method see Choquet-Bruhat, Isenberg and Pollack [7]. Essentially, the set of free data consists of (ψ, τ, π, U ), where ψ, τ, π are smooth functions in M and U is a smooth symmetric traceless and divergence-free (2, 0)-tensor in M . Given (ψ, τ, π, U ) an initial free data set, the conformal constraint system of equations, whose unknown...

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Section: Multiplicity Of Solutions Of (El)mentioning

confidence: 51%

Section: Statement Of the Resultsmentioning

confidence: 96%

Section: Statement Of the Resultsmentioning

confidence: 99%

“…We have S h,2 * = S h , where S h is as in (3.2). Following Hebey-Pacard-Pollack [12], to get rid of the negative exponent we consider for any ε > 0 the perturbed functional on H 1 (M ):…”

Section: A General Mountain-pass Situationmentioning

confidence: 99%

See 2 more Smart Citations

Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

Premoselli

2014

Calc. Var.

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show abstract

“…The spirit of the proof of this theorem is different from [18]. The point being that we want to obtain a stable solution φ 0 , meaning that φ 0 is a stable local minimum for the functional I defined in (4.2), while [18] uses the mountain pass lemma.…”

Section: The Lichnerowicz Equationmentioning

confidence: 99%