An exemple of strict inequality for reverse inequality Minkowski for space $L^p$, $0 < p <1$ Let  be $0<p<1$. Suppose that we know that 
$$ \bigg(\int (u + v)^p\bigg)^{1/p} \geq \bigg(\int (u)^p\bigg)^{1/p} +\bigg(\int (v)^p\bigg)^{1/p}$$
for all $u,v \geq 0$ in $L^p$. I need find an exemple of functions how $u$ and $v$ such that the inequality above is strict.

Standard answer: Take any $u=v\in L^p$ (ie. almost everywhere). Equality is obvious. A bit more generally, you can take $u=cv$ a.e. where $c$ is a constant.
You can get here, as I mentioned in my comment, by observing that the inequality ultimately comes from the concavity of $\log$, and that inequality is an equality exactly when the two points on the graph that you're looking at are the same.
