Abstract
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.
Citation
Emmanuel Hebey. "Critical elliptic systems in potential form." Adv. Differential Equations 11 (5) 511 - 600, 2006. https://doi.org/10.57262/ade/1355867695
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