Some non-stability results for geometric Paneitz–Branson type equations

Laurent Bakri; Jean-Baptiste Casteras

doi:10.1088/0951-7715/28/9/3337

Nonlinearity

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Some non-stability results for geometric Paneitz–Branson type equations

Laurent Bakri^1,3 and Jean-Baptiste Casteras^2,4

Published 20 August 2015 • © 2015 IOP Publishing Ltd & London Mathematical Society
Nonlinearity, Volume 28, Number 9 Citation Laurent Bakri and Jean-Baptiste Casteras 2015 Nonlinearity 28 3337 DOI 10.1088/0951-7715/28/9/3337

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laurent.bakri@gmail.com

Jean-Baptiste.Casteras@univ-brest.fr

Author affiliations

¹ Departamento de Matemática, Universidad Técnica Federico Santa María, 1680 Av. españa Valparaíso, Chile

² UFRGS, Instituto de Matemática, Av. Bento Goncalves 9500, 91540-000 Porto Alegre-RS, Brasil

³ The first author was supported by PROYECTO BASAL PFB03 CMM, Universidad de Chile, Santiago (Chile)

⁴ The second author was supported by the CNPq (Brazil) project 501559/2012-4

Dates

Received 31 March 2015
Accepted 21 July 2015
Published 20 August 2015

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Abstract

Let (M, g) be a compact riemannian manifold of dimension $n\geqslant 5$ . We consider two Paneitz–Branson type equations with general coefficients

$\begin{eqnarray}&&\Delta _{g}^{2}u-\text{di}{{\text{v}}_{g}}\left({{A}_{g}}\text{d}u\right)+hu=\,|u{{|}^{{{2}^{*}}-2-\varepsilon}}u\,\text{on}\,M,\end{eqnarray} \tag{ E.1 }$

and

$\begin{eqnarray}&&\Delta _{g}^{2}u-\text{di}{{\text{v}}_{g}}\left(\left({{A}_{g}}+\varepsilon {{B}_{g}}\right)\text{d}u\right)+hu=|u{{|}^{{{2}^{*}}-2}}u\,\text{on}\,M,\end{eqnarray} \tag{ E.2 }$

where A_g and B_g are smooth symmetric (2, 0)-tensors, $h\in {{C}^{\infty}}(M)$ $h\in {{C}^{\infty}}(M)$ , ${{2}^{*}}=\frac{2n}{n-4}$ ${{2}^{*}}=\frac{2n}{n-4}$ and ε is a small positive parameter. Under suitable assumptions, we construct solutions ${{u}_{\varepsilon}}$ ${{u}_{\varepsilon}}$ to (E.1) and (E.2) which blow up at one point of the manifold when ε tends to 0. In particular, we extend the result of Deng and Pistoia 2011 (to the case where A_g is the one defined in the Paneitz operator) and the result of Pistoia and Vaira (2013 Int. Math. Res. Not. 2013 3133–58) (to the case n = 8 and (M, g) locally conformally flat).

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