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

Druet, Olivier; Hebey, Emmanuel
February 2008
Calculus of Variations & Partial Differential Equations;Feb2008, Vol. 31 Issue 2, p205
Academic Journal
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 aswhere $$A: M \to M_p^s(\mathbb {R})$$ , $$\mathcal {U}: M \to \mathbb {R}^p$$ 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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