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

• Published in: Acta applicandae mathematicae 2023-08, Vol.186 (1), p.5, Article 5
• Main Authors: Kong, Lingzheng, Chen, Haibo
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Asymptotic properties; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Energy; Ground state; Lagrange multiplier; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinearity; Partial Differential Equations; Probability Theory and Stochastic Processes
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-023-00584-4

In this paper, we study the normalized solutions for the nonlinear critical Kirchhoff equations with combined nonlinearities in 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 + 8 N is equal to the energy critical exponent 2 N 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.