A threshold phenomenon for embeddings of {H^m_0} into Orlicz spaces

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 λk → 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 Λ1 = (2m − 1)!vol(S2m) is the total Q-curvature of S2m.