I am trying to figure out if the following problem is well posed:
$$\cases{\Delta^2u - \nabla \cdot (k\nabla u) + u\lambda = f\\u= 0,\quad \vec x \in \partial\Omega\\ \Delta u=0,\quad \vec x \in \partial\Omega\\}$$
With $$k(\vec x)>0\qquad \lambda>0$$
I can assume that a solution exists and should now decide whether the given boundary conditions are sufficient for the solution to be unique. I cannot figure out whether this is true or not. I tried finding a counter example i.e. two functions that solve the problem that are not identical, but failed in doing so.
Is there a simple way to "see" the amount of BCs needed in a higher order problem?
The standard way to test for uniqueness is to multiply the differential equation (with homogeneous right-hand side) with the solution itself and then integrate by parts. If you have the "right" number of BC prescribed, you sometimes obtain an identity that leads to uniqueness.
In your case: Assume we have two solutions $v$ and $w$. Then the difference $u:=v-w$ is a solution to $$\cases{\Delta^2u - \nabla \cdot (k\nabla u) + u\lambda = 0\\u= 0,\quad \vec x \in \partial\Omega\\ \Delta u=0,\quad \vec x \in \partial\Omega.\\}$$ Multiplying the first equation with $u$ and integrating over $\Omega$, we get $$ \int_\Omega \Delta^2 u\, u\ dx - \int_\Omega \big[ \nabla\cdot (k\nabla u) \big]\, u\, dx + \int_\Omega \lambda |u|^2\, dx = 0. $$ Taking into account the boundary conditions, we can integrate by parts to obtain $$ \int_\Omega |\Delta u|^2 dx + \int_\Omega k\,|\nabla u|^2 dx + \int_\Omega \lambda |u|^2\, dx = 0. $$ Since $k>0$ and $\lambda>0$, it follows that $u=0$. Consequently $v=w$, that is, we have uniqueness.