"Large" conformal metrics of prescribed Q-curvature in the negative case

Abstract

Given a compact and connected four dimensional smooth Riemannian manifold $(M,g_0)$ with $k_P:={\int }_M{Q}_{g_0}d{V}_{g_0}<0$ and a smooth non-constant function $f_0$ with ${\max }_{p\in M}{f}_0(p)=0$, all of whose maximum points are non-degenerate, we assume that the Paneitz operator is nonnegative and with kernel consisting of constants. Then, we are able to prove that for sufficiently small $\lambda >0$ there are at least two distinct conformal metrics $g_{\lambda }=e^{2u_{\lambda }}{g}_0$ and $g^{\lambda }=e^{2u^{\lambda }}{g}_0$ of $Q$-curvature $Q_{g_{\lambda }}=Q_{g^{\lambda }}=f_0+\lambda$. Moreover, by means of the "monotonicity trick", we obtain crucial estimates for the "large" solutions $u^{\lambda }$ which enable us to study their "bubbling behavior" as $\lambda \downarrow 0$.

References in corpus (1)

• Concentration-compactness phenomena in the higher order Liouville's equation