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 $C^{3}_{loc}(\mathbb{R}^4)$?