My question is related to the following.
Prove the uniqueness of poisson equation with robin boundary condition
I was thinking about the use of $a$ being positive. So I tried to find an example that fails the uniqueness when $a$ is a negative constant. The easiest one might be something working on one dimentional, let us say $\Omega$ is a bounded open interval. For simplicity I took $(0,1)$ and tried some $u_1, u_2$ linear/quadratic different to each other but no success. Can somebody give an idea of how to find such example?
Consider $\Omega = (0,\ell)$ for simplicity. Then $\Delta u = u'' =0$ requires that $u(x) = cx + d$ for $c,d \in \mathbb{R}$. Plugging into the boundary conditions $\partial_n u -a u =0$ for $a >0$ yields $$ c - a(c\ell+d) =0 \text{ and } -c-ad =0. $$ This is equivalent to the linear system $$ \begin{pmatrix} 1-a\ell & -a \\ -1 & -a \end{pmatrix} \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} 0 \\0 \end{pmatrix}. $$ We have a nontrivial solution if and only if the determinant of the matrix vanishes, which is equivalent to $$ (1-a\ell)(-a) -a =0 \Leftrightarrow a(a\ell-2) =0 $$ and so we have a degenerate matrix if and only if $a = 2/\ell$.
As long as $a$ and $\ell$ are related in this way we will not have unique solutions to the problem $$ \begin{cases} u''(x) = h(x) & \text{for }x \in (0,\ell) \\ \partial_n u - au =g & \text{for }x=0,\ell. \end{cases} $$