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I am having trouble proving this modified mean-value inequality.

Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$

Prove that there exists constants $r_0,C>0$ depending only on $c$ so that

$$u(0)\le \frac{C}{r^n}\int_{B(r)}u\,\mathrm{dVol},$$

for all $r\le r_0$.

This was mentioned in passing in a paper I was reading (without any reference or any indication on how to prove it).

Any help would be much appreciated!

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    $\begingroup$ If c<0 there is nothing to prove because u is subharmonic. If $c>0$ consider $v(x,t)=u(x)e^{\sqrt{c}t}$ in $R^{n+1}$. This is subharmonic in variables $(x,t)$. Apply mean value inequality for a ball of radius $r$ and use some simple estimates, for example $\{x_{1}^{2}+...+x_{n}^{2}+t^{2}\leq r\}\subset \{x_{1}^{2}+...+x_{n}^{2}\leq r^{2}\}\cap \{ |t|\leq r\}$ $\endgroup$
    – Paata Ivanishvili
    Commented Jan 21, 2021 at 21:19
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    $\begingroup$ @ Paata Ivanishvili Despite its simplicity, I think that your comment deserves an answer. $\endgroup$
    – Giorgio Metafune
    Commented Jan 22, 2021 at 14:52

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