Infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

Abstract

We consider the standing-wave problem for a nonlinear Schrödinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -Δu+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+σ}}\Big),\ \text{as}\ |x|\rightarrow+\infty, %\tag{$V2$} \end{equation*} for instance if $p>2$, $m>2$ and $σ>1$ 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.

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