Existence and nonexistence of extremals for critical Adams inequalities in R⁴ and Trudinger-Moser inequalities in R²

Though much progress has been made with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W^1,n(Rⁿ) and higher order Adams inequalities on finite domain Ω⊂Rⁿ, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space Rⁿ still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R⁴. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Pólya-Szegö type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31]), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R⁴ of the form S(α)=sup‖u‖_H² =1⁡∫R⁴(exp⁡(32π²|u|²)−1−α|u|²)dx, where α∈(−∞,32π²). We establish the existence of the threshold α^⁎, where α^⁎≥[Formula presented] and B₂≥[Formula presented], such that S(α) is attained if 32π²−α<α^⁎, and is not attained if 32π²−α>α^⁎. This phenomenon has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R². Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.