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Critical functions and elliptic PDE on compact Riemannian manifolds. (English) Zbl 1157.53334

Summary: We study in this work the existence of minimizing solutions to the critical-power type equation \(\Delta_gu + h.u = f.u^{(n+2)/(n-2)}\) on a compact Riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of “critical function” that was originally introduced by E. Hebey and M. Vaugon for the study of the second best constant in the Sobolev embeddings. Along the way we prove an important estimate concerning concentration phenomena when \(f\) is a non-constant function.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35J20 Variational methods for second-order elliptic equations
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