Given an open bounded domain
\({\Omega\subset\mathbb {R}^{2m}}\) with smooth boundary, we consider a sequence
\({(u_k)_{k\in\mathbb{N}}}\) of positive smooth solutions to
$\left\{\begin{array}{ll} (-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2} \quad\quad\quad\quad\quad {\rm in}\,\Omega\\ u_k=\partial_\nu u_k=\cdots =\partial_\nu^{m-1} u_k=0 \quad {\rm on }\, \partial \Omega, \end{array}\right.$
where λ
k → 0
+. Assuming that the sequence is bounded in
\({H^m_0(\Omega)}\) , we study its blow-up behavior. We show that if the sequence is not precompact, then
$\liminf_{k\to\infty}\|u_k\|^2_{H^m_0}:=\liminf_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx\geq \Lambda_1,$
where Λ
1 = (2
m ? 1)!vol(
S 2m ) is the total
Q-curvature of
S 2m .