I want to show $$\begin{cases} \Delta(\Delta u) - \nabla\cdot (\Delta u \cdot \nabla u)=0\\ \int \Delta u < \infty\\ \Delta u \ge0 \end{cases}$$ in $\mathbb{R}^2$ only has a solution such that $\Delta u =0$.
How would I approach this? I want to treat $\Delta u$ as a variable say $v$, but the thing is I don't know how to deal with $\nabla u$. Or maybe there is another way?
Can anyone give me some idea? Thanks a lot!