Diff. Geom, Math. Phys., PDE Seminar: Ali Hyder

I will talk about the existence of solution to the Q-curvature problem

\begin{align}\label{1}

(-\Delta)^\frac n2 u=Qe^{nu}\quad\text{in }\mathbb{R}^n,\quad

\kappa:=\int_{\mathbb{R}^n}Qe^{nu}dx<\infty,

\end{align}

where Q is a non-negative function and n>2.

Geometrically, if u is a solution to \eqref{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 \eqref{1} for every \kappa > 0.