Conformal metrics on \({R}^n\) with arbitrary total \(Q\)-curvature

Abstract: I will talk about the existence of solution to the \(Q\)-curvature problem

\((1)\)

\((-\Delta)\)\(^\frac n2\) u=\(Qe\)\(^{nu}\)\(\quad\text{in }\)\(\mathbb{R}^n,\)\(\quad \kappa:\)=\(\int_{\mathbb{R}^n}\)\(Qe^{nu}dx\)\(<\infty,\)

where \(Q\) is a non-negative function and \(n>2\). Geometrically, if \(u\) is a solution to \((1)\) then \(Q\) is the \(Q\)-curvature of the conformal metric \(g_u\) = \(e^{2u}\)\(|dx|^2\) \((|dx|^2\) is the Euclidean metric on \(\mathbb{R}^n)\), and \(\kappa\) is the total \(Q\)-curvature of \(g_u\).

 Under certain assumptions on \(Q\) around origin and at infinity, we prove the existence of solution to \((1) \)for every \(\kappa\) > \(0\).