EXISTENCE AND UNIQUENESS OF CONSTRAINED MINIMIZERS FOR FRACTIONAL KIRCHHOFF TYPE PROBLEMS IN HIGH DIMENSIONS

Shulin Zhang; Hua Jin

doi:10.11948/20230311

2024 Volume 14 Issue 3

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Shulin Zhang, Hua Jin. EXISTENCE AND UNIQUENESS OF CONSTRAINED MINIMIZERS FOR FRACTIONAL KIRCHHOFF TYPE PROBLEMS IN HIGH DIMENSIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1598-1612. doi: 10.11948/20230311

Citation:

Shulin Zhang, Hua Jin. EXISTENCE AND UNIQUENESS OF CONSTRAINED MINIMIZERS FOR FRACTIONAL KIRCHHOFF TYPE PROBLEMS IN HIGH DIMENSIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1598-1612. doi: 10.11948/20230311

EXISTENCE AND UNIQUENESS OF CONSTRAINED MINIMIZERS FOR FRACTIONAL KIRCHHOFF TYPE PROBLEMS IN HIGH DIMENSIONS

Shulin Zhang^1,2, ,,
Hua Jin^1,

1.
School of Mathematics, China University of Mining and Technology, 221116 Xuzhou, China
2.
School of Mathematics, Xuzhou Vocational Technology Academy of Finance and Economics, 221116 Xuzhou, China

Author Bio: Email: huajin@cumt.edu.cn (H. Jin)
Corresponding author: Email: zhangshulin0228@126.com(S. Zhang)
Fund Project: The authors were supported by the Fundamental Research Funds for the Central Universities (2019XKQYMS90)

Abstract

Abstract

In this paper, we investigate the existence and uniqueness of solutions with prescribed $L^{2}$-norm for a class of fractional Kirchhoff type problems. Firstly, we prove the existence of global constraint minimizers for the exponent $2 < p<2 + \frac{4\theta s}{N}$. Secondly, we obtain the existence of solutions with prescribed $L^{2}$-norm for the exponent $2 + \frac{4\theta s}{N}\leq p< 2^{*}_{s}$ by mountain pass theorem. Furthermore, all these solutions are unique up to translations and our methods only rely on scaling transformations and simply energy estimates. We point out that these obtained results extend the previous results for $0<s<1$ and $\theta=2$ or $s=1$ and $\theta=2$ in low dimensions. To the best of our knowledge, with respect to the $L^{2}$-subcritical or $L^{2}$-critical constrained variational problem for fractional Kirchhoff type problems, the critical exponent $p=2 + \frac{4\theta s}{N}$ is properly established for the first time.
- Fractional Kirchhoff type problems /
- $L^{2}$-normalized critical point /
- existence and uniqueness /
- mountain pass theorem
MSC: 35J20, 35J60

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References

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