Non-stability of Paneitz–Branson type equations in arbitrary dimensions

Laurent Bakri 1 Jean-Baptiste Casteras 2
2 MEPHYSTO - Quantitative methods for stochastic models in physics
Inria Lille - Nord Europe, ULB - Université Libre de Bruxelles [Bruxelles], LPP - Laboratoire Paul Painlevé - UMR 8524
Abstract : Let (M, g) be a compact riemannian manifold of dimension n ≥ 5. We consider a Paneitz-Branson type equation with general coefficients ∆ 2 g u − div g (A g du) + hu = |u| 2 * −2−ε u on M, (E) where A g is a smooth symmetric (2, 0)-tensor, h ∈ C ∞ (M), 2 * = 2n n − 4 and ε is a small positive parameter. Assuming that there exists a positive nondegenerate solution of (E) when ε = 0 and under suitable conditions, we construct solutions u ε of type (u 0 − BBl ε) to (E) which blow up at one point of the manifold when ε tends to 0 for all dimensions n ≥ 5.
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Laurent Bakri, Jean-Baptiste Casteras. Non-stability of Paneitz–Branson type equations in arbitrary dimensions. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2014, 107, pp.118-133. ⟨hal-01981188⟩

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