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

Let (M,g)(M,g) be a compact riemannian manifold of dimension n≥5n≥5. We consider a Paneitz–Branson type equation with general coefficients equation(E)Δg2u−divg(Agdu)+hu=|u|2∗−2−εuon  M, where AgAg is a smooth symmetric (2,0)(2,0)-tensor, h∈C∞(M)h∈C∞(M), 2∗=2nn−4 and εε is a small positive parameter. Assuming that there exists a positive nondegenerate solution of (E) when ε=0ε=0 and under suitable conditions, we construct solutions uεuε of type (u0−BBlε)(u0−BBlε) to (E) which blow up at one point of the manifold when εε tends to 00 for all dimensions n≥5n≥5.