Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

Xudong Shang; Jihui Zhang

doi:10.3934/cpaa.2018107

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2018, Volume 17, Issue 6: 2239-2259. Doi: 10.3934/cpaa.2018107

This issue Previous Article A Cameron-Storvick theorem for the analytic Feynman integral associated with Gaussian paths on a Wiener space and applications Next Article The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities

Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

Xudong Shang^1, , and
Jihui Zhang^2,

1.
School of Mathematics, Nanjing Normal University Taizhou College, Taizhou 225300, China
2.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

* Corresponding author
* Corresponding author

Received: March 31, 2017

Revised: August 31, 2017

Published: June 2018

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Abstract

In this paper we consider the multiplicity and concentration behavior of positive solutions for the following fractional nonlinear Schrödinger equation

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

where $\varepsilon$ is a positive parameter, $(-Δ)^{s}$ is the fractional Laplacian, $s ∈ (0,1)$ and $N> 2s$. Suppose that the potential $V(x) ∈\mathcal{C}(\mathbb{R}^{N})$ satisfies $\text{inf}_{\mathbb{R}^{N}} V(x)>0$, and there exist $k$ points $x^{j} ∈ \mathbb{R}^{N}$ such that for each $j = 1,···,k$, $V(x^{j})$ are strict global minimum. When $f$ is subcritical, we prove that the problem has at least $k$ positive solutions for $\varepsilon>0$ small. Moreover, we establish the concentration property of the solutions as $\varepsilon$ tends to zero.

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Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35B38.

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