Advances in Differential Equations

Critical elliptic systems in potential form

Emmanuel Hebey

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Let $(M,g)$ be a smooth compact Riemannian $n$-manifold, $n \ge 3$. Let also $p \ge 1$ be an integer, and $M_p^s({\mathbb R})$ be the vector space of symmetrical $p\times p$ real matrices. For $A: M \to M_p^s({\mathbb R})$ smooth, $A = (A_{ij})$, we consider vector-valued equations, or systems, like $$\Delta_g^p{\mathcal U} + A(x){\mathcal U} = \frac{1}{2^\star}D_{{\mathcal U}}\vert{\mathcal U}\vert^{2^\star}, $$ where ${\mathcal U}: M \to {\mathbb R}^p$ is a $p$-map, $\Delta_g^p$ is the Laplace-Beltrami operator acting on $p$-maps, and $2^\star$ is critical from the Sobolev viewpoint. We investigate various questions for this equation, like the existence of minimizing solutions, the existence of high energy solutions, blow-up theory, and compactness. We provide the complete $H_1^2$-theory for blow-up, sharp pointwise estimates, and prove compactness when the equations are not trivially coupled and of geometric type.

Article information

Adv. Differential Equations, Volume 11, Number 5 (2006), 511-600.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 35J55 35J60: Nonlinear elliptic equations


Hebey, Emmanuel. Critical elliptic systems in potential form. Adv. Differential Equations 11 (2006), no. 5, 511--600.

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