Ground states of nonlinear Schrödinger equations with fractional Laplacians

Zupei Shen; Zhiqing Han; Qinqin Zhang

doi:10.3934/dcdss.2019136

Article Contents

2019, Volume 12, Issue 7: 2115-2125. Doi: 10.3934/dcdss.2019136

This issue Previous Article Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction Next Article On periodic solutions in the Whitney's inverted pendulum problem

Ground states of nonlinear Schrödinger equations with fractional Laplacians

1.
Center for Applied Mathematics, Guangzhou University, Guangzhou 510405, China
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Qinqin Zhang
* Corresponding author: Qinqin Zhang

Received: November 30, 2017

Revised: March 31, 2018

Published: June 2019

This work was supported by National Natural Science Foundation (11701114, 11471085) and the program for Changjiang scholars and Innovative Recearch Team in univesity (Grant No.IRT1226)

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Abstract

Inspired by Schaftingen [15], we develop a symmetric variational principle for the field equation involving a fractional Laplacians

$ \begin{equation*} \left\{ \begin{aligned} (-\Delta)^\alpha u+u& = f(u), x\in\mathbb{R}^N,\\ u(x)&\geq 0. \end{aligned} \right. \end{equation*} $

As an application, we prove the existence of symmetric ground states in the fractional Sobolev space $ H^\alpha (\mathbb{R}^N) $. These results improve some known ones in the literature. An example is also given to illustrate our results.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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