In this variational form, as far as I can see, you are enforcing Neumann and Robin conditions on the same boundary. That is not possible. If you could make a minimal working example, as described here.
The only thing that differs from a problem with dirichlet conditions is that after doing integration by parts of -\int_\Omega -\nabla(k \nabla \cdot u )\cdot v = \int_\Omega k \nabla u : \nabla v - \int_{\partial \Omega} k\frac{\partial u}{\partial n } v, is that you replace \frac{\partial u}{\partial n} with your condition.
For Neumann that is k\frac{\partial u}{\partial n} = g, and for Robin it is k\frac{\partial u}{\partial n} =-ru +rs in your setting.