I want to prove existence -not really care for uniqueness- to the Helmholtz equation \begin{equation}\tag{1} (\Delta+\lambda)u= g \quad \text{in } \Omega, \end{equation} with Neumann boundary condition \begin{equation} \partial_\nu u = 0 \quad \text{on } \partial\Omega. \end{equation} Here, $\Omega$ would be a nice (say smooth) domain in $\mathbb{R}^n$ with $n\geq 3$, $\partial_\nu$ would denote normal outward pointing derivative and $\lambda>0$. In particular, my data $f$ is in $L^2$, and is actually compactly supported inside of $\Omega$.
I was trying to follow Folland's Introduction to Partial Differential Equations. There, the author proves existence for the Laplace equation $\Delta u =g$. He actually translates the problem to solving \begin{equation} \Delta u= 0 \quad \text{in } \Omega, \end{equation} with Neumann boundary condition \begin{equation} \partial_\nu u = f \quad \text{on } \partial\Omega, \end{equation} which can also be done for the Helmholtz equation (1). In theorem (3.40) (of the second edition), it is stated that a solution exists if and only if \begin{equation} \int_{\partial\Omega}f=0, \end{equation} which I would say translates to \begin{equation} \int_{\Omega}g=0. \end{equation} It would be really disappointing for me if this condition was also necessary for the Helmholtz equation (1). However, in Colton and Kress' Inverse Acoustic and Electromagnetic Theory, theorem 3.12 (third edition), it is plainly stated that the exterior Neumann problem has a solution, no conditions imposed. That makes me think that the Helmholtz equation, although similar to the Laplace equation, has some better behaviour with respect to existence of solutions for the Neumann problem. Is this the case? Does anybody know any good references that cover this matter?
EDIT: In Colton and Kress' Integral Equation Methods in Scattering Theory, theorem 3.20 says that the interior Neumann problem is solvable if and only if \begin{equation} \int_{\partial\Omega}fv=0, \end{equation} for every $v$ solution of the homogeneous Neumann problem (i.e. $\partial_\nu v = 0$). That is already interesting, but there may be other similar results.