*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$.