Ground state solutions for fractional Schrödinger–Choquard–Kirchhoff type equations with critical growth

ABSTRACT

In this paper, we investigate the existence of ground state solutions for fractional Schrödinger–Choquard–Kirchhoff type equations with critical growth $\left\{\begin{array}{l}\left(a+b{\int }_{{\mathbb{R}}^N}|(-\mathrm{\Delta }{)}^{\frac{s}{2}}u{|}^2\mathrm{d}x\right)(-\mathrm{\Delta }{)}^{s}u+V(x)u\\ =\lambda f(x,u)+[|x{|}^{-\mu }\ast |u{|}^{2_{\mu ,s}^{\ast }}]|u{|}^{2_{\mu ,s}^{\ast }-2}u,x\in {\mathbb{R}}^N,\\ u\in H^s\left({\mathbb{R}}^N\right),\end{array}\right.$ where a, b>0 are constants, $\lambda >0$ is a parameter, 0<s<1, $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian of order s, N>2s, $0<\mu <2s$ and $2_{\mu ,s}^{\ast }=\frac{2N-\mu }{N-2s}$. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method.

Keywords:

• Fractional Schrödinger–Choquard–Kirchhoff equation
• ground state solutions
• Nehari method
• critical growth

AMS SUBJECT CLASSIFICATIONS:

• 35R11
• 35A15
• 35B33

Acknowledgments

The authors would like to express their sincere gratitude to one anonymous referee for his/her constructive comments for improving the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (NSFC) (11701178), the Natural Science Foundation program of Jiangxi Province (20202BABL201011) and the Science and Technology Project of Education Department in Jiangxi province (GJJ190337).