Say, we're given smooth functions $f_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying

- $\Delta f_n\ge 0$ (subharmonic)
- $f_n\ge 0$
- $\int_\Omega f_n=I>0$ for all $n\in\mathbb{N}$
- ${f_n}_{|\partial\Omega}=n$

Then, say $B\subset\subset \Omega$. Can we conclude that $\int_B f_n\to 0$?

When I visualize these functions, I suspect this might be true, but I can't come up with a proof nor a counterexample. Any help would be appreciated.