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Sharp asymptotics and compactness for local low energy solutions of critical elliptic systems in potential form

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Abstract

Let (M, g) be a smooth compact Riemannian n-manifold, n ≥ 3. Let also p ≥ 1 be an integer, and \(M_p^s(\mathbb {R})\) be the vector space of symmetrical p × p real matrix. We consider critical elliptic systems of equations which we write in condensed form as

$$\Delta_g^p\mathcal {U} + A(x)\mathcal {U} = \mathcal {U}^{2*-1},$$

where \(A: M \to M_p^s(\mathbb {R})\) , \(\mathcal {U}: M \to \mathbb {R}^p\) is a p-map, \(\Delta_g^p\) is the Laplace–Beltrami operator acting on p-maps, and 2* is the critical Sobolev exponent. We fully answer the question of getting sharp asymptotics for local minimal type solutions of such systems. As an application, we prove compactness of minimal type solutions and prove that the result is sharp by constructing explicit examples where blow-up occurs when the compactness assumptions are not fulfilled.

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Authors and Affiliations

  1. Département de Mathématiques, UMPA, Ecole normale supérieure de Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Olivier Druet

  2. Département de Mathématiques, Université de Cergy-Pontoise, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France

    Emmanuel Hebey

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  1. Olivier Druet
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Druet, O., Hebey, E. Sharp asymptotics and compactness for local low energy solutions of critical elliptic systems in potential form. Calc. Var. 31, 205–230 (2008). https://doi.org/10.1007/s00526-007-0111-z

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