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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

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Abstract. We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.

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Following standard notations, · p in the above expression stands for the L p -norm (with respect to the Riemannian measure dv g ). As is well known and easy to see, for all u ∈ C ∞ (M), ( g u) 2 ≤ n|∇ 2 u| 2 .Conversely, by the Bochner-Lichnerowicz-Weitzenböck formula we get that DJADLI, HEBEY, AND LEDOUX

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