Published Paper

Inserted: 7 dec 2022
Last Updated: 7 dec 2022

Journal: Communications in Contemporary Mathematics
Year: 2022
Doi: 10.1142/S0219199722500535

Abstract:

We consider the following prescribed $Q$-curvature problem \[\Delta^2 u=(1-
x
^p)e^{4u}, \quad\text{on}\,\,\mathbb{R}^4\] requiring also $\Lambda:=\int_{\mathbb{R}^4}(1-
x
^p)e^{4u}dx<\infty$. We show that for every polynomial $P$ of degree 2 such that $\lim\limits_{
x
\to+\infty}P=-\infty$, and for every $\Lambda\in(0,\Lambda_\mathrm{sph})$, there exists at least one solution which assume the form $u=w+P$, where $w$ behaves logarithmically at infinity. Conversely, we prove that all solutions have the form $v+P$, where \[v(x)=\frac{1}{8\pi^2}\int\limits_{\mathbb{R}^4}\log\left(\frac{
y
}{
x-y
}\right)(1-
y
^p)e^{4u}dy\] and $P$ is a polynomial of degree at most 2 bounded from above. Moreover, if $u$ is a solution to the previous problem, it has the following asymptotic behavior \[u(x)=-\frac{\Lambda}{8\pi^2}\log
x
+P+o(\log
x
),\quad\text{as}\,\,
x
\to+\infty.\] As a consequence, we give a geometric characterization of solutions in terms of the scalar curvature at infinity of the associated conformal metric $e^{2u}
dx
^2$.