I have to prove uniqueness of solutions to the following problem:
Since it is a uniqueness problem, I took $\textbf{u}$ and $\textbf{v}$ both to be solutions. I replaced them in both the equation and boundary condition and substracted one from the other. So taking $\textbf{w}=\textbf{u}-\textbf{v}$, I have
$$-\nabla^2\textbf{w}-\kappa\nabla\nabla\cdot\textbf{w}=0$$ $$\textbf{w}=\textbf{0}$$
Then I multipled the equation through by $\textbf{w}$ and took the integral to apply the Green's Identity from which I'm finally at
$$\int_\mathbb{R^3}|\nabla\textbf{w}|^2dx=\kappa\int_\mathbb{R^3}\textbf{w}\nabla\nabla\cdot\textbf{w}dx$$
I think I'm supposed to show somehow that $\textbf{w}=\textbf{0}$ everywhere on the domain and not only on the boundary. But I'm stuck.
First note that you have an error in the sets over which you're integrating. You multiply by $w$ and integrate over $\Omega$ to get $$ \int_{\Omega} | \nabla w|^2 = \kappa \int_{\Omega} w \cdot \nabla (\nabla \cdot w). $$ Here's a hint for how to conclude. On the right side you can integrate by parts again: $$ \int_{\Omega} w\cdot \nabla f = -\int_{\Omega} (\nabla \cdot w) f + \int_{\partial \Omega} f w\cdot n. $$