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Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior

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  • We study the solutions $u\in C^\infty(\mathbb{R}^{2m})$ of the problem \begin{equation}\label{P0} (-\Delta)^mu=\bar Qe^{2mu}, \text{ where }\bar Q=\pm (2m-1)!, \quad V :=\int_{\mathbb{R}^{2m}}e^{2mu}dx <\infty,(1) \end{equation} particularly when $m>1$. Problem (1) corresponds to finding conformal metrics $g_u:=e^{2u}|dx|^2$ on $\mathbb{R}^{2m}$ with constant $Q$-curvature $\bar Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $|x|\to \infty$ can be simultaneously prescribed, under certain restrictions. When $\bar Q= (2m-1)!$ we need to assume $V < vol(S^{2m})$, but surprisingly for $\bar Q=-(2m-1)!$ the volume $V$ can be chosen arbitrarily.
    Mathematics Subject Classification: Primary: 35J30, 35J60, 53A30.


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