The biharmonic operator is $\nabla^4 \phi \equiv \nabla^2 (\nabla^2 \phi)$. Are there any identities for it? I need to find $\phi$ such that
$~\\$ $\nabla^4 \phi = \frac{1}{3}\nabla^4 u^3 - u \nabla^4 u^2$,
where we know $\nabla^2 u = 0$. Alternatively, also $\psi$ such that $\nabla^2 \psi = u \nabla^4 u^2$ would be great. Any help is greatly appreciated.