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Let $n\geq 3$, and $u$ satisfies $$ -\Delta u=K(x)u^{\frac{n+2}{n-2}} \quad x\in B_1\setminus \{0\}, $$ where

  • $|K(x)|\leq A$ in $B_1\setminus \{0\}$, and
  • $u\geq 0$ in $B_1\setminus \{0\}$.

Can we use the Brezis-Kato theorem to show $u\in L^{\infty}(B_{0.9})$?

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  • $\begingroup$ usually in the Brezis-Kato theorem you need a linear bound for the RHS $-\Delta u=g(x,u(x))$ i.e. $$|g(x,u(x))|\leq \alpha(x)(1+|u(x)|).$$ But in this article they also cover a generalization with power exponent arxiv.org/pdf/2202.11408.pdf "Brezis–Kato Type Regularity Results for Higher Order Elliptic Operators." $\endgroup$
    – Thomas Kojar
    Commented Jul 25, 2023 at 15:08
  • $\begingroup$ Thanks a lot for your help. I still have a question that if 0 is removable singularity so it's bounded near 0 and have $L^{\infty}$ bound near 0. sorry for late comment. $\endgroup$
    – Davidi Cone
    Commented Aug 6, 2023 at 15:07

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