Abstract: This paper deals with the Neumann problem for an elliptic
equation
$$
\left\{
\begin{array}{ll}
\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,
\ \ &x\in\Omega,\\
D_\gamma{u}+\alpha(x)u=0, &x\in\partial\Omega\backslash\{0\},
\end{array}
\right.
$$
where $\Omega $ is a bounded domain in $ R^N$ with $ C^1$
boundary, $ 0\in\partial\Omega$, $N\ge5$.
$2^{*}(s)=\frac{2(N-s)}{N-2}$ ($0\leq s\leq 2$) is the critical
Sobolev-Hardy exponent, $1 the unit outward normal to boundary $\partial\Omega$. By
variational method and the dual fountain theorem, the existence of
infinitely many solutions with negative energy is proved.

Key words: Neumann problem, Critical Sobolev-Hardy exponent, (ps)_c*condition, Dual fountain theorem