Hello all,

Does the following boundary value problem admit unique solutions $q$:

$- \Delta q + \beta q = f$,  $x \in \Omega$

$ \nabla q \cdot \vec{n} = g $, $x \in \Gamma := \partial \Omega$,

where $\beta > 0$ is reasonably small? I am not clear if the pure Neumann boundary conditions make the solution non-unique; does the inhomogeneity in the volume equation take care of this problem? What are the spaces for $f$ and $g$ such that we have uniqueness?