Infinitely many solutions to a fractional nonlinear Schr\"{o}dinger equation

Abstract

This paper considers the fractional Schr\"{o}dinger equation \begin{equation}\label{abstract} (-\Delta)^s u + V(|x|)u-u^p=0, \quad u>0, \quad u\in H^{2s}(\R^N) \end{equation} where $0<s<1$, $1<p<\frac{N+2s}{N-2s}$, $V(|x|)$ is a positive potential and $N\geq 2$. We show that if $V(|x|)$ has the following expansion: \[ V(|x|)=V_0 + \frac{a}{|x|^m} + o\left(\frac{1}{|x|^m}\right) \qquad \mbox{as} \ |x| \rightarrow +\infty, \] in which the constants are properly assumed, then (\ref{abstract}) admits infinitely many non-radial solutions, whose energy can be made arbitrarily large. This is the first result for fractional Schr\"{o}dinger equation. The $s=1$ case corresponds to the known result in Wei-Yan \cite{WY}.