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Regularity of solutions for a non linear elliptic equation

Discussion in 'Mathematics' started by gin111, Aug 2, 2020 at 2:25 AM.

  1. gin111

    gin111 Guest

    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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