suppose $\overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is whether (edited) $$ u(r,w) \leq C\overline u(r),$$ where $C$ is independent of $u$. If this inequality is true, How can I prove it?
This question is from Aviles' article (see inequality(2.3)),Local Behavior of Solutions of Some Elliptic Equations, Commun.Math.Phys.108,177-192(1987).
Added information (from comments):
In this article, $u$ is a non-negative solution of \begin{equation} Δu+u^{\frac{n}{n−2}}=0 \mbox{ in } B_1\setminus{0}. \tag{1} \end{equation} So $u$ is superharmonic, $\Delta u \leq 0$. Based on Aviles' Lemma 1. Any non-negative solution of (1) satisfies \begin{equation} (−\ln r)^{\frac{n−2}{2}}r^{n−2}\bar{u}(r)≤\left(\frac{n-2}{\sqrt{2}}\right)^{n−2},\mbox{ for all } 0<r<r_0 \end{equation} for some $1>r_0>0$. In the following step, the author sets $t=−\ln|x|=−\ln r$, and $ϕ(t,w)=|x|^{n−2}u(x)$.
Obviously $r^{n−2}\bar{u}(r)≤Ct^{\frac{2−n}{2}}$, but Alives writes directly, $$ ϕ(t,w)≤Ct^{\frac{2−n}{2}}, $$ So I guess $u(r,w)≤C\bar{u}(r)$ is true...but I don't see why.
