reference for poisson equation on unbounded domain

Question

Let $\Omega\subset\mathbb{R}^3$ be some unbounded exterior domain, with $C^{\infty}$ smooth boundary. Consider the Poisson equation with Neumann boundary condition \begin{align} &\Delta u = f & \text{ in }\Omega\\ &\partial_{\nu}u = g &\text{ on }\partial\Omega\\ &u\to u_{\infty} &\text{ as }|\mathbf{x}|\to\infty \end{align} where $u_{\infty}$ is some constant. There is a lot of literature on the well posedness in the space of $C^{2}(\Omega)\cap C^1(\bar{\Omega})$ when $f$ and $g$ are $C^0$.
I wonder if there is any reference that considers higher regularity? I suppose we should have well-posedness in the space $C^{\infty}(\bar{\Omega})$, if $f,g$ are both $C^{\infty}$. Does anyone know any reference to the infitely smooth case?