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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 apositive function, $1<p <\frac{N+2}{N-2}$. We show that if $V(r)$ has thefollowing 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 manynon-radial positive} solutions, whose energy can be made arbitrarily large.

Infinitely many positive solutions for the nonlinear Shcrodinger equations in $R^N$