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Positive solutions for a fourth order equation invariant under isometries


Author: Frédéric Robert
Journal: Proc. Amer. Math. Soc. 131 (2003), 1423-1431
MSC (2000): Primary 35J35, 58J99
DOI: https://doi.org/10.1090/S0002-9939-02-06676-5
Published electronically: September 5, 2002
MathSciNet review: 1949872
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Abstract: Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 5$. We consider the problem

\begin{displaymath}(\star) \quad\quad\qquad\qquad\qquad\quad\Delta_g^2 u+\alpha\... ...=f u^{\frac{n+4}{n-4}}, \ \quad \qquad\qquad\qquad\qquad\quad \end{displaymath}

where $\Delta_g=-div_g(\nabla)$, $\alpha, a\in \mathbb{R} $, $u,f\in C^{\infty}(M)$. We require $u$ to be positive and invariant under isometries. We prove existence results for $(\star)$ on arbitrary compact manifolds. This includes the case of the geometric Paneitz-Branson operator on the sphere.


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Additional Information

Frédéric Robert
Affiliation: Département de Mathématiques-Site Saint-Martin, Université de Cergy-Pontoise, 2, Avenue Adolphe Chauvin, F 95302 Cergy-Pontoise Cedex, France
Email: Frederic.Robert@math.u-cergy.fr

DOI: https://doi.org/10.1090/S0002-9939-02-06676-5
Received by editor(s): December 12, 2000
Received by editor(s) in revised form: December 7, 2001
Published electronically: September 5, 2002
Communicated by: Bennett Chow
Article copyright: © Copyright 2002 American Mathematical Society

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