  In this paper, we consider the existence and multiplicity of solutions for
the critical Neumann problem \begin{equation}\label{1.1ab}
  \left\{
  \begin{aligned}
  -\Delta {u}-\frac{1}{2}(x \cdot{\nabla u})&= \lambda{|u|^{{2}^{*}-2}u}+{\mu
{|u|^{p-2}u}}& \ \ \mbox{in} \ \ \ {{\mathbb{R}}^{N}_{+}},
  \frac{{\partial u}}{{\partial n}}&=\sqrt{\lambda}|u|^{{2}_{*}-2}u \ &
\mbox{on}\ {{\partial {{\mathbb{R}}^{N}_{+}}}},
  \end{aligned}
  \right. \end{equation} where $ \mathbb{R}^{N}_{+}=\{(x{'}, x_{N}): x{'}\in
{\mathbb{R}}^{N-1}, x_{N}>0\}$, $N\geq3$, $\lambda>0$, $\mu\in \mathbb{R}$, $2<
p <{2}^{*}$, $n$ is the outward normal vector at the boundary ${{\partial
{{\mathbb{R}}^{N}_{+}}}}$, $2^{*}=\frac{2N}{N-2}$ is the usual critical
exponent for the Sobolev embedding
$D^{1,2}({\mathbb{R}}^{N}_{+})\hookrightarrow
{L^{{2}^{*}}}({\mathbb{R}}^{N}_{+})$ and ${2}_{*}=\frac{2(N-1)}{N-2}$ is the
critical exponent for the Sobolev trace embedding
$D^{1,2}({\mathbb{R}}^{N}_{+})\hookrightarrow {L^{{2}_{*}}}(\partial
\mathbb{R}^{N}_{+})$. By establishing an improved Pohozaev identity, we show
that the problem has no nontrivial solution if $\mu \le 0$; By applying the
Mountain Pass Theorem without $(PS)$ condition and the delicate estimates for
Mountain Pass level, we obtain the existence of a positive solution for all
$\lambda>0$ and the different values of the parameters $p$ and ${\mu}>0$.
Particularly, for $\lambda >0$, $N\ge 4$, $2<p<2^*$, we prove that the problem
has a positive solution if and only if $\mu >0$. Moreover, the existence of
multiple solutions for the problem is also obtained by dual variational
principle for all $\mu>0$ and suitable $\lambda$.
