Non-stability of Paneitz–Branson type equations in arbitrary dimensions
Résumé
Let $(M, g)$ be a compact riemannian manifold of dimension $n\geq 5$. We consider a Paneitz-Branson type equation with general coefficients $(E)\Delta_{g}^{2} u − div_g (A_g du) + hu = |u| \frac{2 * −2−\epsilon} u$ on $M$, (E) where $A_g$ is a smooth symmetric (2, 0)-tensor, $h\in C^\infty (M), 2 * = \frac{2n}{n − 4}$ and $\epsilon$ is a small positive parameter. Assuming that there exists a positive nondegenerate solution of (E) when $\epsilon = 0$ and under suitable conditions, we construct solutions $u_\epsilon$ of type $(u_0 − BBl_\epsilon)$ to (E) which blow up at one point of the manifold when $\epsilon$ tends to 0 for all dimensions $n\geq 5$.
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