Let $v_k$ be a radial sequence of function that satisfies in $\Omega\subset\mathbb{R}^4$


 - $(-\Delta)^2 v_k=e^{v_k}$
 - $v_k(x)\leq v_k(0)=0$
 - $\left\Vert (-\Delta)v_k\right\Vert_{L^1(B_R(0))}=O(1)\qquad R>0$
 - $\left\Vert (-\Delta)v_k \right\Vert_{C^1(B_{R/2}(0))}=O(1).$

How can I prove that from those assumptions and Harnack's inequality and Elliptic theory follows that there exists $v\in C^{3}(\mathbb{R}^{4})$ such that
\begin{equation}
\lim_{k\to+\infty} v_k=v
\end{equation} 
in $C^{3}_{loc}(\mathbb{R}^4)$?