The normal trick is to set it up as an eigenvalue problem, namely to look instead at
$$\Delta u = 0, ~~\frac{\partial u}{\partial n} = \lambda b(x) u(x)~~ \forall x \in \partial D.$$ 
You first establish that this corresponds to an eigenvalue problem  on $L^2(\partial \Omega)$ for a compact operator, thanks to the compact embedding of $H^{1/2}(\partial \Omega)$ into $L^2((\partial \Omega)$. 

Then the usual machinery rolls out. There is a discrete spectrum, and there are eigenvalues. And if it so happens that $\lambda=1$ is one of those, then your problem has non trivial solutions. 

The advantage of this point of view is that the problem is well posed, there is a clear functional analytic setup, and you don't need much on $b$, some integrability (if it is bounded as you liked to consider, it is fine).