Normalized solutions to the fractional Kirchhoff equations with a perturbation

Abstract

In this paper, we study the following fractional Kirchhoff equation $(a+b{\int }_{{\mathbb{R}}^3}|(-\mathrm{\Delta }{)}^{\frac{s}{2}}u{|}^2\mathrm{d}x)(-\mathrm{\Delta }{)}^{s}u=\lambda u+\mu |u{|}^{q-2}u+|u{|}^{p-2}u\text{in}{\mathbb{R}}^3,$with a prescribed mass ${\int }_{{\mathbb{R}}^3}|u{|}^2\mathrm{d}x=c^2,$where $s\in (\frac{3}{4},1)$, a, b, c>0, $2<q<p<2_s^{\ast }=\frac{6}{3-2s}$, $\mu >0$ and $\lambda \in \mathbb{R}$ as a Lagrange multiplier. By decomposing Pohozaev set and constructing fiber map, the existence and properties of normalized ground states are established.

Keywords:

• Normalized solutions
• fractional Kirchhoff equations
• Lagrange multiplier
• decreasing rearrangement

2010:

• 35Q55
• 35J20

Disclosure statement

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

Additional information

Funding

L. Liu is supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2021zzts0048). H.B. Chen issupported by the National Natural Science Foundation of China (No. 12071486).