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

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 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 Pohozaev constraint method based on the splitting properties of the Brezis-Lieb lemma, and make some subtle energy estimates. By decomposing Pohozaev 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.