If I understand things correctly, $u$ is vanishing on some non-empty  open subset. Moreover, you have pointwisely on $\mathbb R^3$ the differential inequality
$$
\vert \Delta u\vert\le C\vert u\vert^5.
$$
Also, you know that the function $u$ is in $\dot H^1(\mathbb R^3)$, which is a subset of $L^6(\mathbb R^3)$. This implies that 
$$
\vert u\vert^5=\vert u\vert\underbrace{\vert u\vert^4}_{L^{3/2}},
$$
implying that you have a differential inequality
$
\vert \Delta u\vert\le V\vert u\vert, \ V \in L^{d/2}
$
in $\mathbb R^d, d\ge 3$.
Then the strong unique continuation result due to Jerison \& Kenig [MR0794370] entails that $u\equiv 0$. Note that $d/2$ is critical and that $L^{d/2}_{\text{loc}}$ is enough.