Existence and nonexistence of extremals for critical Adams inequalities in [formula omitted] and Trudinger-Moser inequalities in [formula omitted

Abstract

Though much progress has been made with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W1,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 formS(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⁎≥(32π2)2B22 and B2≥124π2, 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.