  We consider the following nonlinear problem in $\R^N$ $$\label{eq} - \Delta u
+V(|y|)u=u^{p},\quad u>0 {in} \R^N, u \in H^1(\R^N) $$ where $V(r)$ is a
positive function, $1<p <\frac{N+2}{N-2}$. We show that if $V(r)$ has the
following expansion: There are constants $a>0$, $m>1$, $\theta>0$, and $V_0>0$,
such that \[ V(r)= V_0+\frac a {r^m} +O\bigl(\frac1{r^{m+\theta}}\bigr),\quad
\text{as $r\to +\infty$,} \] then \eqref{eq} has {\bf infinitely many
non-radial positive} solutions, whose energy can be made arbitrarily large.
