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The concentration of solutions to a fractional Schrödinger equation

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Abstract

In this paper, we study the following fractional Schrödinger equation

$$\left\{\begin{array}{ll}({-} \mathit{\Delta} )^su +V(x)u=a|u|^{\frac{4s}{N}} u,& \quad x\in \mathbb{R}^N, \\ u > 0,& \quad u\in H^s(\mathbb{R}^N),\end{array}\right.$$

where \({N \geq 1, s \in (0,1)}\), \({a \in \mathbb{R}}\) and V(x) is a measurable function. We prove the existence or nonexistence of ground states for the above equation under \({L^{2}}\)-constraint and some assumptions on \({V(x)}\) and a. Besides, we also analyze the behavior of ground states as a tends to some fixed constant.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, 430079, Wuhan, People’s Republic of China

    Qihan He

  2. College of Mathematics and Information Science, Jiangxi Normal University, 330022, Nanchang, Jiangxi, People’s Republic of China

    Wei Long

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  1. Qihan He
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  2. Wei Long
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Correspondence to Wei Long.

Additional information

The authors sincerely thank the anonymous reviewer’s professional comments and suggestions, which improves greatly the quality of this manuscript. The authors sincerely thank Professor Shuangjie Peng for helpful suggestions and discussions.

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He, Q., Long, W. The concentration of solutions to a fractional Schrödinger equation. Z. Angew. Math. Phys. 67, 9 (2016). https://doi.org/10.1007/s00033-015-0607-x

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  • DOI: https://doi.org/10.1007/s00033-015-0607-x

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