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Sharp asymptotics and compactness for local low energy solutions of critical elliptic systems in potential form
Authors:Olivier Druet  Emmanuel Hebey
Institution:(1) Département de Mathématiques, UMPA, Ecole normale supérieure de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France;(2) Département de Mathématiques, Université de Cergy-Pontoise, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France
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.
Keywords:Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000)  Primary: 58E30  Secondary: 58J05
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