Electron. J. Differential Equations, Vol. 2024 (2024), No. 10, pp. 1-14.

Ground state solutions for fractional Kirchhoff type equations with critical growth

Kexue Li

We study the nonlinear fractional Kirchhoff problem $$\displaylines{ \Big(a+b\int_{\mathbb{R}^3}|(-\Delta)^{s/2}u|^2dx\Big) (-\Delta)^su+u=f(x,u)+|u|^{2_s^{\ast}-2}u \quad \text{in }\mathbb{R}^3,\\ u\in H^s(\mathbb{R}^3), }$$ where \(a,b>0\) are constants, \(s(3/4,1)\), \(2_s^{\ast}=6/(3-2s)\), \((-\Delta)^s\) is the fractional Laplacian. Under some relaxed assumptions on \(f\), we prove the existence of ground state solutions.

Submitted February 13, 2023. Published January 29, 2024.
Math Subject Classifications: 35R11, 35B50, 34A08.
Key Words: Ground state solution; fractional Kirchhoff equation; critical exponent.
DOI: 10.58997/ejde.2023.10

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Kexue Li
School of Mathematics and Statistics
Xi'an Jiaotong University
Xi'an, 710049, China
email: kxli@mail.xjtu.edu.cn

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