  In this paper, we aim to investigate the following class of singularly
perturbed elliptic problem $$ \left\{
  \begin{array}{ll}
  \displaystyle -\varepsilon^2\triangle {u}+|x|^\eta u =|x|^\eta f(u)&
\mbox{in}\,\, A,
  u=0 & \mbox{on}\,\, \partial A,
  \end{array} \right. $$ where $\varepsilon>0$, $\eta\in\mathbb{R}$,
$A=\{x\in\R^{2N}:\,\,0<a<|x|<b\}$, $N\ge2$ and $f$ is a nonlinearity of $C^1$
class with supercritical growth. By a reduction argument, we show that there
exists a nodal solution $u_\e$ with exactly two positive and two negative
peaks, which concentrate on two different orthogonal spheres of dimension $N-1$
as $\e\rightarrow0$. In particular, we establish different concentration
phenomena of four peaks when the parameter $\eta>2$, $\eta=2$ and $\eta<2$.
