let u be the eigenfunction of $-\Delta$ with eigenvalue $\lambda$ on domain $D$. Suppose $u < c_0r$ in $B_r(0)$, show that for every $y \in B_(r/2)$, $r<r_0$,$$u(y) \leq C\frac{1}{r^n}\int_{B_r/4}|u| + Cc_0r^3\lambda$$.
My professor hinted that to apply mean value property to $|u|^2 + A|x|^2$ for some $A$.
My attempt:
Direct calculation shows that for every $y$, $|u|^2 + A|x-y|^2$ is subharmonic for $A$ large. To apply the mean value theorem, we have$$|u|^2 \leq \frac{C}{r^n}\int_{B_r/4}|u(x)|^2 + A|x-y|^2 dx. $$We can integrate the second part out and get $r^2$, but I don't get $r^3$. I wonder if any one can help me proceed from here?Thank you!