Multiple solutions for a fractional nonlinear Schrödinger equation with local potential

Wulong Liu; Guowei Dai

doi:10.3934/cpaa.2017104

Article Contents

2017, Volume 16, Issue 6: 2105-2123. Doi: 10.3934/cpaa.2017104

This issue Previous Article Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations Next Article Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels

Multiple solutions for a fractional nonlinear Schrödinger equation with local potential

Wulong Liu^1, , and
Guowei Dai^2,

1.
School of Science, Jiangxi University of Science and Technology, Ganzhou, Jiangxi 341000, China
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

* Corresponding author
* Corresponding author

Received: November 30, 2016

Revised: March 31, 2017

Published: September 2017

The first author is partially supported by the Youth Science Foundation of Jiangxi Provincial Department of Education (GJJ14460), the NSFC Grant(61364015), the Foundation of the Jiangxi University of Science and Technology (NSFJ2015-G25). The second author is supported by NNSF of China (No. 11261052,11401477) and the Fundamental Research Funds for the Central Universities (No. DUT15RC(3)018, DUT17LK05)

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Abstract

Using penalization techniques and the Ljusternik-Schnirelmann theory, we establish the multiplicity and concentration of solutions for the following fractional Schrödinger equation

$\left\{ \begin{align} &{{\varepsilon }^{2\alpha }}{{\left( -\Delta \right)}^{a}}u+V\left( x \right)u=f\left( u \right),\ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{a}}\left( {{\mathbb{R}}^{N}} \right),u>0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ \end{align} \right.$

where $0<α<1$ , $N>2α$ , $\varepsilon>0$ is a small parameter, $V$ satisfies the local condition, and $f$ is superlinear and subcritical nonlinearity. We show that this equation has at least $\text{cat}_{M_{δ}}(M)$ single spike solutions.

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Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35S15.

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