Regularity of solutions for a non linear elliptic equation

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 $$\lim_{k\to+\infty} v_k=v$$ in $C^{3}_{loc}(\mathbb{R}^4)$?

• Can you say something about context of your question? -- And why do you know that the assertion you mention is indeed true? Nov 20, 2013 at 17:52
• The reference is a paper from F. Robert: "Concentration phenomena for a fourth order equation with exponential growth: the radial case." Exactly is the proof of Lemma 5.1, equation (32). Nov 20, 2013 at 19:45
• What does radial sequence mean? How is $\Omega$ related to $B_R(0)$? Mar 30, 2021 at 14:32

By the second assumption, $e^{v_k}$ are uniformly bounded in $L^\infty$. Standard $W^{2,p}$ estimate with the third assumption, implies that $\Delta v_k$ are uniformly bounded in $W^{2,p}_{loc}$ for any $p<\infty$. Then you can use Sobolev embedding and bootstrapping.
• I don't know where to use Harnack. Maybe the bound on $v_k$ (which I forgot)? But it seems unnecessary to use Harnack inequality. By the second assumption with uniform bound on $\Delta v_k$, I think we can get a uniform bound for $\|v_k\|_{L^1}$. Nov 22, 2013 at 4:23