First, which of the following, concerning the solution of the Neumann problem for the Laplace equation, $$\Delta u=0~ \text{on}~~\Omega, \frac{\partial u}{\partial n}=f(x,y),\text{on}~\partial\Omega$$ on a smooth bounded domain $\Omega$ is true ?
$1$. Solution is unique.
$2$. Solution is unique up to an additive constant .
$3$. Solution is unique up to multiplicative constant.
$4$. No conclusion can be drawn about uniqueness.
I know that the result is if solution exists for Neumann problem then solution is unique up to additive constant. So can I say that option $D$ is correct option ?
Secondly I know only necessary condition for existence of solution of the Neumann problem $$\Delta u=0~ \text{on}~~\Omega, \frac{\partial u}{\partial n}=f(x,y),\text{on}~\partial\Omega$$ which is $\int_{\partial\Omega}fds=0$ for bounded domain . I want to know sufficient condition for existence of solution for this Neumann problem . I searched in my local books but didn’t get any exact answer for sufficient condition. There is also necessary and sufficient condition for the Poisson's equation to admit a solution$?$ Please help. Thank you.