  Given a compact and connected four dimensional smooth Riemannian manifold
$(M,g_0)$ with $k_P := \int_M Q_{g_0} dV_{g_0} <0$ and a smooth non-constant
function $f_0$ with $\max_{p\in M}f_0(p)=0$, all of whose maximum points are
non-degenerate, we assume that the Paneitz operator is nonnegative and with
kernel consisting of constants. Then, we are able to prove that for
sufficiently small $\lambda>0$ there are at least two distinct conformal
metrics $g_\lambda=e^{2u_\lambda}g_0$ and $g^\lambda=e^{2u^\lambda}g_0$ of
$Q$-curvature $Q_{g_\lambda}=Q_{g^\lambda}=f_0+\lambda$. Moreover, by means of
the "monotonicity trick", we obtain crucial estimates for the "large" solutions
$u^\lambda$ which enable us to study their "bubbling behavior" as $\lambda
\downarrow 0$.

Analysis of PDEs