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Normalized Ground States for the Mass-Energy Doubly Critical Kirchhoff Equations

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Abstract

In this paper, we study the normalized solutions for the nonlinear critical Kirchhoff equations with combined nonlinearities in \(\mathbb{R}^{4}\). In particular, in the case of \(N=4\), there is a new mass-energy doubly critical phenomenon for Kirchhoff equation with combined nonlinearities that the mass critical exponent \(2+\frac{8}{N}\) is equal to the energy critical exponent \(\frac{2N}{N-2}\), which remains unsolved in the existing literature. To deal with the special difficulties created by the nonlocal term and doubly critical term, we develop a perturbed Pohožaev constraint method based on the splitting properties of the Brézis-Lieb lemma, and make some subtle energy estimates. By decomposing Pohožaev manifold and constructing fiber map, we prove the existence of a positive normalized ground state. Moreover, we also explore the asymptotic behavior of the obtained normalized solutions. These conclusions extend some known ones in previous papers.

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Funding

This work is supported by National Natural Science Foundation of China (No. 12071486).

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  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, P.R. China

    Lingzheng Kong & Haibo Chen

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  1. Lingzheng Kong
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Kong, L., Chen, H. Normalized Ground States for the Mass-Energy Doubly Critical Kirchhoff Equations. Acta Appl Math 186, 5 (2023). https://doi.org/10.1007/s10440-023-00584-4

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