Electron. J. Differential Equations, Vol. 2018 (2018), No. 82, pp. 1-21.

Fractional minimization problem on the Nehari manifold

Mei Yu, Meina Zhang, Xia Zhang

In the framework of fractional Sobolev space, using Nehari manifold and concentration compactness principle, we study a minimization problem in the whole space involving the fractional Laplacian. Firstly, we give a Lions type lemma in fractional Sobolev space, which is crucial in the proof of our main result. Then, by showing a relative compactness of minimizing sequence, we obtain the existence of minimizer for the above-mentioned fractional minimization problem. Furthermore, we also point out that the minimizer is actually a ground state solution for the associated fractional Schrodinger equation

Submitted September 23, 2017. Published March 26, 2018.
Math Subject Classifications: 35A15, 35J60, 46E35.
Key Words: Minimization problem; fractional Schrodinger equation; ground state; Nehari manifold; concentration compactness principle.

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Mei Yu
Department of Applied Mathematics
Northwestern Polytechnical University
Xi'an, Shannxi, 710129, China
email: yumei@nwpu.edu.cn
Meina Zhang
College of Science
Harbin Engineering University
Harbin 150001, China
email: meina_zhang1@163.com
Xia Zhang
Department of Mathematics
Harbin Institute of Technology, China
email: zhangxia@hit.edu.cn

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