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.
Categories: math.AP