  We consider the standing-wave problem for a nonlinear Schr\"{o}dinger
equation, corresponding to the semilinear elliptic problem \begin{equation*}
-\Delta u+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where
$V(x)$ is a uniformly positive potential and $p>1$. Assuming that
\begin{equation*}
V(x)=V_\infty+\frac{a}{|x|^m}+O\Big(\frac{1}{|x|^{m+\sigma}}\Big),\ \text{as}\
|x|\rightarrow+\infty, %\tag{$V2$} \end{equation*} for instance if $p>2$, $m>2$
and $\sigma>1$ we prove the existence of infinitely many positive solutions. If
$V(x)$ is radially symmetric, this result was proved in \cite{WY-10}. The proof
without symmetries is much more difficult, and for that we develop a new {\em
intermediate Lyapunov-Schmidt reduction method}, which is a compromise between
the finite and infinite dimensional versions of it.
