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