Normalized solutions to the Sobolev critical Kirchhoff-type equation with non-trapping potential

Abstract

The paper is concerned about the existence of solutions with prescribed $L^2$-norm to the following Kirchhoff-type equation $-(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla u}{|}^2)\mathrm{\Delta u}+(V+\lambda )u=|u{|}^{p-2}u+\mu |u{|}^{q-2}u\mathrm{in}{\mathbb{R}}^3,$where $a,b>0,2<q<14/3<p⩽6$ or $14/3<q<p⩽6$, $\mu >0$. Noting that 14/3 is the mass critical exponent, a Pohozaev constraint method is adopted in two cases. In the mass mixed critical case, i.e., $2<q<10/3,14/3<p⩽6$, we get a normalized solution to above equation with small enough μ by Ekeland's variational principle. In the mass supercritical case, i.e., $14/3<q<p⩽6$, we obtain a positive ground state normalized solution, and energy comparison argument is used in the Sobolev critical case.

Keywords:

• Kirchhoff-type equation
• non-trapping potential
• Sobolev critical
• normalized solutions

Mathematics subject classification:

• 35J60

Acknowledgments

The author would like to thank the reviewers for careful reading and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Grant Nos. 12101376, 12271313, 12071266, 12026217 and 12026218) and Fundamental Research Program of Shanxi Province (20210302124528,202203021211309).