*I know classical strong maximum principles for supersolutions of Laplacian operator, which says:*

Suppose $ u \in C^2(\Omega)\cap C(\overline{\Omega}) $ satisfies $ -\Delta u \geq 0 $ in $ \Omega$.

If $ u \geq 0 $ on $ \partial \Omega $ then $ u \geq 0 $ in $ \Omega $. In fact, either $u > 0$ in $\Omega$ or $ u \equiv 0 $ in $\Omega$.

I am reading an article, there author use this version of strong maximum principle:

Assume $ u \in L_{\mathrm{loc}}^1(\Omega) $ is a very weak supersolutions of $ - \Delta u =0 $ in the sense of distributions, means for every non-negative $\phi \in C_0^{\infty}(\Omega) $ we have $$ \int_{\Omega} -u \, \Delta \phi \, \mathrm{d}x \geq 0 $$

Also assume $ u \geq 0 $ on $ \partial \Omega $ and $ u \not\equiv 0 $. Then

$$ u > 0 \quad \mathrm{in} \,\, \Omega $$

There is no reference for it, and I could not find any proof for it. Does anyone know a proof for it, or a reference for it's proof.

Thanks.