Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ satisfy the inhomogeneous Neumann boundary condition $\frac{\partial u}{\partial y}(x, 0) = -u(x, 0)f(x)$, where $f(x)$ is smooth and bounded for $|x| \leq 1$. I am looking for a reference stating that $u$ is bounded around $(0, 0)$. Also, what is the biggest set around $(0, 0)$ that $u$ is necessarily bounded on? Thanks!

The present form of the question is very confusing. First of all, the boundary values are defined only for the flat part of the boundary. Secondly, even on the flat part, the condition given is not a Neumann condition due to the presence of $u(x,0)$ on the right hand side. If $f(x)$ is a constant function, say $f(x)= c$ for every $-2 \leq x \leq 2,$ then it is the homogeneous Robin boundary condition $\frac{\partial u}{\partial y} + c u =0$ if $c \neq 0$ and a homogeneous Neumann condition if $c=0.$ Hope it helps.