Electron. J. Differential Equations, Vol. 2018 (2018), No. 142, pp. 1-21.

Existence of ground states for fractional Kirchhoff equations with general potentials via Nehari-Pohozaev manifold

Jing Chen, Xianhua Tang, Sitong Chen

Abstract:
We consider the nonlinear fractional Kirchhoff equation
$$ \Big(a+b\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,dx\Big) (-\Delta)^\alpha u+V(x)u=f(u) \quad \text{in } \mathbb R^3, u\in H^{\alpha}(\mathbb R^3), $$
where a>0, $b\ge 0$, $\alpha\in(3/4, 1)$ are three constants, V(x) is differentiable and $f\in C^1(\mathbb R, \mathbb R)$. Our main results show the existence of ground state solutions of Nehari-Pohozaev type, and the existence of the least energy solutions to the above problem with general superlinear and subcritical nonlinearity. These results are proved by applying variational methods and some techniques from [27].

Submitted November 15, 2017. Published July 13, 2018.
Math Subject Classifications: 35J20, 35J65.
Key Words: Fractional Kirchhoff equation; Nehari-Pohozaev manifold; ground state solutions.

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Jing Chen
School of Mathematics and Computing Sciences
Hunan University of Science and Technology
Xiangtan, Hunan 411201, China
email: cjhnust@aliyun.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: tangxh@mail.csu.edu.cn
Sitong Chen
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: mathsitongchen@163.com

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