Sharp Critical and Subcritical Trace Trudinger–Moser and Adams Inequalities on the Upper Half-Spaces

In this paper, we establish the sharp critical and subcritical trace Trudinger–Moser and Adams inequalities on the half-spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic extension and scaling invariance. These trace Trudinger–Moser (Theorems 1.1 and 1.2) and trace Adams inequalities (Theorems 1.4, 1.5, 1.10 and 1.11) can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders on half-spaces. Furthermore, as an application, we show the existence of the least energy solutions for a class of bi-harmonic equations with nonlinear Neumann boundary condition associated with the trace Adams inequalities (Theorem 1.13). It is interesting to note that there are two types of trace Trudinger–Moser and trace Adams inequalities: critical and subcritical trace inequalities under different constraints. Moreover, trace Trudinger–Moser and trace Adams inequalities of exact growth also hold on half-spaces (Theorems 1.6, 1.8 and 1.12).