Normal conformal metrics with prescribed $Q$-Curvature in ${\mathbb{R}}^{2n}$

Abstract

We consider the $Q$-curvature equation \begin{equation}\label{0.1} (-Δ)^n u = K(x)e^{2nu}\quad\text{in} ~\mathbb{R}^{2n} \ (n \geq 2) \end{equation} where $K$ is a given non constant continuous function. Under mild growth control on $K$, we get a necessary condition on the total curvature ${\Lambda }_u$ for any normal conformal metric $g_u=e^{2u}|dx{|}^2$ satisfying $Q_{g_u}=K$ in ${\mathbb{R}}^{2n}$, or equivalently, solutions to equation with logarithmic growth at infinity. Inversely, when $K$ is nonpositive satisfying polynomial growth control, we show the existence of normal conformal metrics with quasi optimal range of total curvature and precise asymptotic behavior at infinity. If furthermore $K$ is radial symmetric, we establish the same existence result without any growth assumption on $K$.

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