Advances in Differential Equations

Critical functions and elliptic PDE on compact Riemannian manifolds

Stephane Collion

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We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle _{\textbf{g}}u+h.u=f .u^{\frac{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 phenomenas when $f$ is a non-constant function.

Article information

Adv. Differential Equations, Volume 12, Number 1 (2007), 55-120.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]
Secondary: 35B33: Critical exponents 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]


Collion, Stephane. Critical functions and elliptic PDE on compact Riemannian manifolds. Adv. Differential Equations 12 (2007), no. 1, 55--120.

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