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

Weiwei Ao; Juncheng Wei; Wen Yang

doi:10.3934/dcds.2017242

Article Contents

2017, Volume 37, Issue 11: 5561-5601. Doi: 10.3934/dcds.2017242

This issue Previous Article A discrete Bakry-Emery method and its application to the porous-medium equation Next Article The index bundle and multiparameter bifurcation for discrete dynamical systems

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

1.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2.
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
3.
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Received: February 28, 2017

Revised: May 31, 2017

Published: October 2017

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Abstract

We consider the fractional nonlinear Schrödinger equation

\begin{equation*}(-Δ)^su+V(x)u=u^p \mbox{ in }\mathbb{R}^N, u→0~\mathrm{as}~|x|→+∞,\end{equation*}

where $V(x)$ is a uniformly positive potential and $p>1.$ Assuming that

\begin{equation*}V(x)=V_{∞}+\frac{a}{|x|^m}+O\Big(\frac{1}{|x|^{m+σ}}\Big)~\mathrm{as}~|x|→+∞,\end{equation*}

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.

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Mathematics Subject Classification: 35J10, 35J61.

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