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

Abstract

We consider the fractional nonlinear Schrödinger equation\begin{document}\begin{equation*}(-Δ)^su+V(x)u=u^p \mbox{ in }\mathbb{R}^N, u→0~\mathrm{as}~|x|→+∞,\end{equation*}\end{document}where $V(x)$ is a uniformly positive potential and $p>1.$ Assuming that\begin{document}\begin{equation*}V(x)=V_{∞}+\frac{a}{|x|^m}+O\Big(\frac{1}{|x|^{m+σ}}\Big)~\mathrm{as}~|x|→+∞,\end{equation*}\end{document}and $p,m,σ,s$ satisfy certain conditions, we prove the existence of infinitely many positive solutions for $N=2$. For $s=1$, this corresponds to the multiplicity result given by Del Pino, Wei, and Yao [24] for the classical nonlinear Schrödinger equation.

Locations

• Discrete and Continuous Dynamical Systems - View - PDF