We consider the existence of single and multi-peak solutions of the following nonlinear elliptic Neumann problem
$$\begin{aligned} \left\{ \begin{aligned} -\Delta u+\lambda ^{2} u&=Q(x)|u|^{p-2}u \qquad&\text {in} ~~~~\mathbb {R}^{N}_{+}, \\ \frac{\partial u }{\partial n}&=f(x,u) \qquad&\text {on}~~\partial \mathbb {R}^{N}_{+}, \end{aligned}\right. \end{aligned}$$
where
\(\lambda \) is a large number,
\(p\in (2,\frac{2N}{N-2})\) for
\(N\ge 3\),
f(
x,
u) is subcritical about
u and
Q is positive and has some non-degenerate critical points in
\(\mathbb {R}^{N}_{+}\). For
\(\lambda \) large, we can get solutions which have peaks near the non-degenerate critical points of
Q.