I have seen a lemma which I do not have any reference or hint for it.

Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution to \begin{cases} \Delta^2 u = 0 \hspace{.9cm} \mathrm{in} \hspace{.1cm} Ω \\ u =-\Delta u= 0 \hspace{.9cm} \mathrm{on} \hspace{.1cm} ∂Ω \end{cases} then for any $B_R(x_0) ⊂⊂ Ω $, there exists a positive constant $C = C(θ, ρ, q,R)$ , $0 <q< \dfrac{N}{ N−2}$ , $0 <θ<ρ< 1$ , such that

$$||u||_{L^q(B_{ρR}(x_0))} \leq C \hspace{.2cm} ess \inf_{B_{θR}(x_0)}u$$

I need a reference or hint, or any help to prove it.


  • 1
    $\begingroup$ The normal approach for Navier BC would be to take $v:= - \Delta u$ and see what $v$ satisfies. (Note in this case I think the only solution is $u=0$)... $\endgroup$ – Math604 Oct 11 '15 at 16:32
  • $\begingroup$ Thanks, I found the version of laplacian operator case in Lecture notes of courant institute ( QING HAN and FANGHUA LIN ). and with this change of variable, I achieve the result. For solutions, $u=0$ is the only one. but in the question I mean supersolution. $\endgroup$ – Hheepp Oct 11 '15 at 19:07
  • $\begingroup$ oh sorry...didn't see "supersolution" $\endgroup$ – Math604 Oct 11 '15 at 21:14

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