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

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  • $\begingroup$ Can you say something about context of your question? -- And why do you know that the assertion you mention is indeed true? $\endgroup$
    – Stefan Kohl ♦
    Commented Nov 20, 2013 at 17:52
  • $\begingroup$ 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). $\endgroup$
    – gin111
    Commented Nov 20, 2013 at 19:45
  • $\begingroup$ What does radial sequence mean? How is $\Omega$ related to $B_R(0)$? $\endgroup$
    – Deane Yang
    Commented Mar 30, 2021 at 14:32

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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.

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  • $\begingroup$ ok, this is a standard argument but where do I use Harnack? $\endgroup$
    – gin111
    Commented Nov 21, 2013 at 10:04
  • $\begingroup$ 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 $\|v_k\|_{L^1}$. $\endgroup$
    – Kelei Wang
    Commented Nov 22, 2013 at 4:23
  • $\begingroup$ Harnack is for the convergence $\endgroup$
    – username
    Commented Mar 30, 2021 at 9:13

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