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

Abstract

Let (M, g) be a compact riemannian manifold of dimension . We consider two Paneitz–Branson type equations with general coefficientsE.1Δg2u−divg(Agdu)+hu= |u|2∗−2−εu on M, ?>andE.2Δg2u−divg((Ag+εBg)du)+hu=|u|2∗−2u on M, ?>where Ag and Bg are smooth symmetric (2, 0)-tensors, , and ε is a small positive parameter. Under suitable assumptions, we construct solutions 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 Ag 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).