I am having a hard time understanding the passage at the end of page 478 and the beginning of page 479 from Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part II : $N\geq 4$ by Rey & Wei.
In short, their argument is as follows: let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ ($N\geq 2$); $\mu>0$ and suppose that $u\in C^2(\overline{\Omega})$ satisfies $$ \begin{cases} -\Delta u+\mu u=0 &\text{in}~\Omega;\\ \partial_\nu u=g &\text{on}~\partial\Omega. \end{cases} $$ They then proceed to say that standard elliptic theory implies a bound on $\|u\|_\infty$ in function of $g$. Unfortunately, I have been unable to identify which result would let one deduce such a conclusion.